Phase Transformations in Materials

Gernot Kostorz (Editor)

Phase Transformations in Materials

Phase Transformations in Materials. Edited by Gernot KostorzCopyright © 2001 WILEY-VCH Verlag GmbH, WeinheimISBN: 3-527-30256-5

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Gernot Kostorz (Ed.)

Phase Transformationsin Materials

Weinheim · New York · Chichester · Brisbane · Singapore · Toronto

Editor:Prof. Gernot KostorzETH ZürichInstitut für Angewandte PhysikCH-8093 ZürichSwitzerland

This book was carefully produced. Nevertheless, authors, editor and publisher do not warrant the informa-tion contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illus-trations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No. applied for

British Library Cataloguing-in-Publication Data:A catalogue record for this book is available from the British Library.

Die Deutsche Bibliothek – Cataloguing-in-Publication Data:A catalogue record for this book is available from Die Deutsche BibliothekISBN 3-527-30256-5

© WILEY-VCH Verlag GmbH, D-69469 Weinheim (Federal Republic of Germany), 2001

All rights reserved (including those of translation in other languages). No part of this book may be reproducedin any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into machine lan-guage without written permission from the publishers. Registered names, trademarks, etc. used in this book, evenwhen not specifically marked as such, are not to be considered unprotected by law.

Printed in the Federal Republic of GermanyPrinted on acid-free paper.

Indexing: Borkowski & Borkowski, Schauernheim

Composition, Printing and Bookbinding: Konrad Triltsch, Print und digitale Medien GmbH, 97199 Ochsenfurt-Hohestadt

To the memory of Peter Haasen(1927–1993)


Preface

In 1991, the late Peter Haasen, who had set out to edit a comprehensive treatment of “Ma-terials Science and Technology” together with Robert W. Cahn and Edward J. Kramer, avery successful series of up-to-date coverages of a broad range of materials topics, wrotethe following in a Preface to Volume 5.

“Herewith we proudly present the first volume of this Series, the aim of which is to pro-vide a comprehensive treatment of materials science and technology. The term ‘materials’encompasses metals, ceramics, electronic and magnetic materials, polymers and compos-ites. In many cases these materials have been developed independently within different dis-ciplines but are now finding uses in similar technologies. Moreover, similarities found amongthe principles underlying these various disciplines have led to the discovery of commonphenomena and mechanisms. One of these common features, phase transformations, con-stitutes the topic of this volume and rates among the fundamental phenomena central to theSeries. A phase transformation often delivers a material into its technologically useful formand microstructure. For example, the major application of metals and alloys as mechanical-ly strong materials relies on their multi-phase microstructure, most commonly generated byone or more phase transformations.”

Peter Haasen, who passed away in 1993, did not live to see the overwhelming success ofthis volume. A revised edition was planned as early as 1996 and finally, work on the indi-vidual chapters began in 1998. Almost all of the original authors agreed to update their ear-lier work, in many cases they arranged for the participation of younger colleagues who hadmade major contributions to the field. It is thus with similar pride the present editor sub-mits the second edition of “Phase Transformations” to the scholarly public. It was with somehesitation that he assumed the task of editor, as he did not like giving the impression of pla-giarizing a successful work. All the credit for the idea, the original chapter definitions andthe selection of the initial authors remains with Peter Haasen. He would certainly have likedto work on a new edition himself and, assuming that he would have judged it timely to ac-complish it about ten years after the first edition, this editor tried to help reaching this goalin the original spirit.

Thus, all the chapters kept their original titles. The contents have been thoroughly re-edited and updated and reflect the progress in the field up to about the middle of the year2000. As before, the book starts with the foundations of phase transformations (“Thermo-dynamics and Phase Diagrams” by A. D. Pelton). The sequence of the following Chaptershas been slightly modified. As most of the volume concerns the solid state, Chapter 2 (byH. Müller-Krumbhaar, W. Kurz and E. Brener) is devoted to solidification, a subject of greatbasic and technological relevance. Chapter 3 by G. E. Murch covers the most importantideas and methods of diffusion kinetics in solids, an indispensible ingredient to many phasetransformations. Statistical theories of phase transformations are presented by K. Binder inChapter 4, featuring phenomenological concepts and computational methods. Diffusion con-trolled homogeneous phase transformations are treated in Chapter 5 (“Homogeneous Sec-


ond Phase Precipitation” by R. Wagner, R. Kampmann and P. W. Voorhees) and Chapter 6(“Spinodal Decomposition” by K. Binder and P. Fratzl), looking at the very complex kinet-ical aspects of the formation of new phases from the points of view of metastability and in-stability in an initially homogeneous system. Heterogeneous phase transformations are treat-ed in Chapter 7 (“Transformations Involving Interfacial Diffusion” by G. R. Purdy and Y.Bréchet) while Chapter 8 by G. Inden deals with atomic ordering, mostly involving substi-tutional alloys and intermetallic phases. Though much progress has been made in elucidat-ing equilibrium ordered states, kinetical aspects are still widely unexplored in this field. Fi-nally, the numerous aspects of diffusionless transformations in the solid state are taken upby L. Delaey (Chapter 9), and a completely new Chapter on the effects of pressure on phasetransformations has been provided by M. Kunz (Chapter 10).

In working on this new edition, the editor had great pleasure interacting with the authors,those of the first edition as well as those who joined for the new edition. He is grateful toall of them for their friendly and competent co-operation. Thanks are due to the publisherfor expedient support and for the preparation of the subject index.

It is hoped that this book will be useful as a source of reference to active researchers andadvanced students; more up-to-date and more detailed than encyclopedic articles, but notas complete and extensive as any monographs. Phase transformations are among the mostcomplex and most versatile phenomena in solid state physics and materials science – andhave considerable impact on production and processing technology. The present book shouldencourage the reader to enter and more deeply appreciate this challenging field.

Gernot Kostorz, ZürichApril 2001

Preface VII

Prof. Kurt BinderUniversität MainzInstitut für PhysikStaudinger Weg 7D-55099 MainzGermanyChapters 4 and 6

Prof. Yves J. M. BréchetLaboratoires de Thermodynamiquesat Physico-Chimique Métallurgiques BP75Domaine Universitaire de GrenobleF-38402 Saint Martin d’HeresFranceChapter 7

Dr. Efim BrenerInstitut für FestkörperforschungForschungszentrum JülichD-52425 JülichGermanyChapter 2

Prof. Luc DelaeyKatholieke Universiteit LeuvenDept. Metaalkunde en ToegepasteMateriaalkundeDecroylaan 2B-3030 Heverlee-LeuvenBelgiumChapter 9

Prof. Peter FratzlInstitut für MetallphysikMontan-Universität LeobenJahnstraße 12A-8700 LeobenAustriaChapter 6

Prof. Gerhard IndenMax-Planck-Institut für Eisenhütten-forschung GmbHMax-Planck-Str. 1D-40237 DüsseldorfGermanyChapter 8

Dr. Reinhard KampmannGKSS-Forschungszentrum GeesthachtGmbHInstitut für WerkstoffforschungPostfach 1160D-21494 GeesthachtGermanyChapter 5

Prof. Martin KunzETH ZürichLabor für KristallographieSonneggstr. 5CH-8092 ZürichSwitzerlandnow at:Naturhistorisches Museum BaselAugustinerstr. 2CH-4053 BaselSwitzerlandChapter 10

Prof. Wilfried KurzÉcole Polytechnique de LausanneDMX-G, EcublensCH-1015 LausanneSwitzerlandChapter 2

List of Contributors


List of Contributors IX

Prof. Gary R. PurdyDept. of Materials Science and EngineeringMcMaster University1280 Main StreetHamilton, Ontario L8S 4L7CanadaChapter 7

Prof. Peter W. VoorheesDept. of Materials Science and EngineeringNorthwestern University2225 N. Campus DriveEvanston, IL 60208-3108USAChapter 5

Prof. Richard WagnerForschungszentrum JülichD-52425 JülichGermanyChapter 5

Prof. Heiner Müller-KrumbhaarInstitut für FestkörperforschungForschungszentrum JülichD-52425 JülichGermanyChapter 2

Prof. Graeme E. MurchUniversity of NewcastleDept. of Chemical and MaterialsEngineeringRomkin DriveNewcastle, NSW 2308AustraliaChapter 3

Prof. Arthur D. PeltonÉcole Polytechnique de MontréalCentre de Recherche en CalculThermochimiqueCP 6079Succursale AMontréal, Québec H3C 3A7CanadaChapter 1

Contents

1 Thermodynamics and Phase Diagrams of Materials . . . . . . . . . . . . 1A. D. Pelton

2 Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81H. Müller-Krumbhaar, W. Kurz, E. Brener

3 Diffusion in Crystalline Solids . . . . . . . . . . . . . . . . . . . . . . . 171G. E. Murch

4 Statistical Theories of Phase Transitions . . . . . . . . . . . . . . . . . . 239K. Binder

5 Homogeneous Second Phase Precipitation . . . . . . . . . . . . . . . . . 309R. Wagner, R. Kampmann, P. W. Voorhees

6 Spinodal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 409K. Binder, P. Fratzl

7 Transformations Involving Interfacial Diffusion . . . . . . . . . . . . . . . 481G. R. Purdy, Y. J. M. Bréchet

8 Atomic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519G. Inden

9 Diffusionless Transformations . . . . . . . . . . . . . . . . . . . . . . . . 583L. Delaey

10 High Pressure Phase Transformations . . . . . . . . . . . . . . . . . . . . 655M. Kunz

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697


1 Thermodynamics and Phase Diagrams of Materials

Arthur D. Pelton

Centre de Recherche en Calcul Thermochimique, École Polytechnique, Montréal, Québec, Canada

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Gibbs Energy and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Predominance Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Calculation of Predominance Diagrams . . . . . . . . . . . . . . . . . . . 71.3.2 Ellingham Diagrams as Predominance Diagrams . . . . . . . . . . . . . . 81.3.3 Discussion of Predominance Diagrams . . . . . . . . . . . . . . . . . . . 91.4 Thermodynamics of Solutions . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Gibbs Energy of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Tangent Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.4 Gibbs-Duhem Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.5 Relative Partial Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.6 Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.7 Ideal Raoultian Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.8 Excess Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.9 Activity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.10 Multicomponent Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5.1 Systems with Complete Solid and Liquid Miscibility . . . . . . . . . . . . 141.5.2 Thermodynamic Origin of Phase Diagrams . . . . . . . . . . . . . . . . . 161.5.3 Pressure-Composition Phase Diagrams . . . . . . . . . . . . . . . . . . . 191.5.4 Minima and Maxima in Two-Phase Regions . . . . . . . . . . . . . . . . . 201.5.5 Miscibility Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5.6 Simple Eutectic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.7 Regular Solution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.5.8 Thermodynamic Origin of Simple Phase Diagrams Illustrated by Regular

Solution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5.9 Immiscibility – Monotectics . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.10 Intermediate Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.11 Limited Mutual Solubility – Ideal Henrian Solutions . . . . . . . . . . . . 291.5.12 Geometry of Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . 31


1.6 Application of Thermodynamics to Phase Diagram Analysis . . . . . . 341.6.1 Thermodynamic/Phase Diagram Optimization . . . . . . . . . . . . . . . . 341.6.2 Polynomial Representation of Excess Properties . . . . . . . . . . . . . . . 341.6.3 Least-Squares Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 351.6.4 Calculation of Metastable Phase Boundaries . . . . . . . . . . . . . . . . . 391.7 Ternary and Multicomponent Phase Diagrams . . . . . . . . . . . . . . 391.7.1 The Ternary Composition Triangle . . . . . . . . . . . . . . . . . . . . . . 391.7.2 Ternary Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.7.3 Polythermal Projections of Liquidus Surfaces . . . . . . . . . . . . . . . . 411.7.4 Ternary Isothermal Sections . . . . . . . . . . . . . . . . . . . . . . . . . 431.7.4.1 Topology of Ternary Isothermal Sections . . . . . . . . . . . . . . . . . . 451.7.5 Ternary Isopleths (Constant Composition Sections) . . . . . . . . . . . . . 461.7.5.1 Quasi-Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 471.7.6 Multicomponent Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . 471.7.7 Nomenclature for Invariant Reactions . . . . . . . . . . . . . . . . . . . . 491.7.8 Reciprocal Ternary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . 491.8 Phase Diagrams with Potentials as Axes . . . . . . . . . . . . . . . . . . 511.9 General Phase Diagram Geometry . . . . . . . . . . . . . . . . . . . . . 561.9.1 General Geometrical Rules for All True Phase Diagram Sections . . . . . . 561.9.1.1 Zero Phase Fraction Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 581.9.2 Choice of Axes and Constants of True Phase Diagrams . . . . . . . . . . . 581.9.2.1 Tie-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.9.2.2 Corresponding Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . 601.9.2.3 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 601.9.2.4 Other Sets of Conjugate Pairs . . . . . . . . . . . . . . . . . . . . . . . . 611.10 Solution Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.10.1 Sublattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.10.1.1 All Sublattices Except One Occupied by Only One Species . . . . . . . . . 621.10.1.2 Ionic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.10.1.3 Interstitial Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.10.1.4 Ceramic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.10.1.5 The Compound Energy Formalism . . . . . . . . . . . . . . . . . . . . . . 651.10.1.6 Non-Stoichiometric Compounds . . . . . . . . . . . . . . . . . . . . . . . 651.10.2 Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661.10.3 Calculation of Limiting Slopes of Phase Boundaries . . . . . . . . . . . . 661.10.4 Short-Range Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.10.5 Long-Range Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711.11 Calculation of Ternary Phase Diagrams From Binary Data . . . . . . . 721.12 Minimization of Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . 741.12.1 Phase Diagram Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 761.13 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761.13.1 Phase Diagram Compilations . . . . . . . . . . . . . . . . . . . . . . . . . 761.13.2 Thermodynamic Compilations . . . . . . . . . . . . . . . . . . . . . . . . 771.13.3 General Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2 1 Thermodynamics and Phase Diagrams of Materials

List of Symbols and Abbreviations 3

List of Symbols and Abbreviations

Symbol Designation

ai activity of component iC number of componentscp molar heat capacityE electrical potential of a galvanic cellF degrees of freedom/varianceG Gibbs energy in Jg molar Gibbs energy in J/molgi partial molar Gibbs energy of iGi

0 standard Gibbs energy of igi

0 standard molar Gibbs energy of iDgi relative partial Gibbs energy igE excess molar Gibbs energygi

E excess partial Gibbs energy of iDG Gibbs energy changeDG0 standard Gibbs energy changeDgm molar Gibbs energy of mixingDgf

0 standard molar Gibbs energy of fusionDgv

0 standard molar Gibbs energy of vaporizationH enthalpy in Jh molar enthalpy in J/molhi partial enthalpy of iHi

0 standard enthalpy of ihi

0 standard molar enthalpy of iDhi relative partial enthalpy of ihE excess molar enthalpyhi

E excess partial enthalpy of iDH enthalpy changeDH 0 standard enthalpy changeDhm molar enthalpy of mixingDhf

0 standard molar enthalpy of fusionDhv

0 standard molar enthalpy of vaporizationK equilibrium constantkB Boltzmann constantn number of molesni number of moles of constituent iNi number of particles of iN 0 Avogadro’s numberpi partial pressure of iP total pressureP number of phasesqi general extensive variable

R gas constantS entropy in J/Ks molar entropy in J/mol Ksi partial entropy of iSi

0 standard entropy of isi

0 standard molar entropy of iDsi

0 relative partial entropy of is E excess molar entropysi

E excess partial entropy of iDS entropy changeDS0 standard entropy changeDsm molar entropy of mixingDsf

0 standard molar entropy of fusionDsv

0 standard molar entropy of vaporizationT temperatureTf temperature of fusionTc critical temperatureTE eutectic temperatureU internal energyvi molar volume of ivi

0 standard molar volume of iXi mole fraction of iZ coordination number

g i activity coefficient of ie bond energyh empirical entropy parameterm i chemical potential of in number of moles of “foreign” particles contributed by a mole of solutex molar metal ratios vibrational bond entropyfi generalized thermodynamic potentialw empirical enthalpy parameter

b.c.c. body-centered cubicf.c.c. face-centered cubich.c.p. hexagonal close-packed


1.2 Gibbs Energy and Equilibrium 5

1.1 Introduction

An understanding of thermodynamicsand phase diagrams is fundamental and es-sential to the study of materials science. Aknowledge of the equilibrium state under agiven set of conditions is the starting pointin the description of any phenomenon orprocess.

The theme of this chapter is the relation-ship between phase diagrams and thermo-dynamics. A phase diagram is a graphicalrepresentation of the values of thermody-namic variables when equilibrium is estab-lished among the phases of a system. Mate-rials scientists are used to thinking of phasediagrams as plots of temperature versus com-position. However, many other variablessuch as total pressure and partial pressuresmay be plotted on phase diagrams. In Sec.1.3, for example, predominance diagramswill be discussed, and in Sec. 1.8 chemicalpotential–composition phase diagrams willbe presented. General rules regarding phasediagram geometry are given in Sec. 1.9.

In recent years, a quantitative couplingof thermodynamics and phase diagramshas become possible. With the use of com-puters, simultaneous optimizations of ther-modynamic and phase equilibrium data canbe applied to the critical evaluation of bi-nary and ternary systems as shown in Sec.1.6. This approach often enables good esti-mations to be made of the thermodynamicproperties and phase diagrams of multi-component systems as discussed in Sec.1.11. These estimates are based on structu-ral models of solutions. Various modelssuch as the regular solution model, the sub-lattice model, and models for interstitialsolutions, polymeric solutions, solutions ofdefects, ordered solutions, etc. are dis-cussed in Secs. 1.5 and 1.10.

The equilibrium diagram is always cal-culated by minimization of the Gibbs en-

ergy. General computer programs are avail-able for the minimization of the Gibbs en-ergy in systems of any number of phases,components and species as outlined in Sec.1.12. When coupled to extensive databasesof the thermodynamic properties of com-pounds and multicomponent solutions,these provide a powerful tool in the studyof materials science.

1.1.1 Notation

Extensive thermodynamic properties arerepresented by upper case symbols. For ex-ample, G = Gibbs energy in J. Molar prop-erties are represented by lower case sym-bols. For example, g = G/n = molar Gibbsenergy in J/mol where n is the total numberof moles in the system.

1.2 Gibbs Energy and Equilibrium

1.2.1 Gibbs Energy

The Gibbs energy of a system is definedin terms of its enthalpy, H, entropy, S, andtemperature, T:

G = H – T S (1-1)

A system at constant temperature and pres-sure will approach an equilibrium state thatminimizes G.

As an example, consider the question ofwhether silica fibers in an aluminum ma-trix at 500 °C will react to form mullite,Al6Si2O13

If the reaction proceeds with the formationof dn moles of mullite then, from the stoi-chiometry of the reaction, dnSi = (9/2) dn,dnAl = – 6 dn, and dnSiO2

= –13/2 dn. Sincethe four substances are essentially immis-cible at 500 °C, we need consider only the

132

692

SiO Al = Si Al Si O (1-2)2 6 2 13+ +

standard molar Gibbs energies, Gi0. The

Gibbs energy of the system then varies as:

where DG0 is the standard Gibbs energychange of reaction, Eq. (1-2), at 500 °C.

Since DG0 < 0, the formation of mulliteentails a decrease in G. Hence, the reactionwill proceed spontaneously so as to mini-mize G.

1.2.2 Chemical Equilibrium

The partial molar Gibbs energy of anideal gas is given by:

gi = gi0 + RT ln pi (1-4)

where gi0 is the standard molar Gibbs en-

ergy (at 1 bar), pi is the partial pressure inbar, and R is the gas constant. The secondterm in Eq. (1-4) is entropic. As a gas ex-pands at constant T, its entropy increases.

Consider a gaseous mixture of H2, S2

and H2S with partial pressures pH2, pS2

andpH2S. The gases can react according to

2 H2 + S2 = 2 H2S (1-5)

If the reaction, Eq. (1-5), proceeds to theright with the formation of 2 dn moles ofH2S, then the Gibbs energy of the systemvaries as:

dG/dn = 2 gH2S – 2 gH2– gS2

= (2 g0H2S – 2 g0

H2– g0

S2)

+ RT (2 ln pH2S – 2 ln pH2– ln pS2

)

= DG0 + RT ln (p2H2S p–2

H2pS2

–1)

= DG (1-6)

DG, which is the Gibbs energy change ofthe reaction, Eq. (1-5), is thus a function ofthe partial pressures. If DG < 0, then the re-action will proceed to the right so as tominimize G. In a closed system, as the re-

d d =

= = kJ (1-3)

Al Si O Si0

SiO

Al

6 2 13 2G n

G

/ g g g

g

0 0

0 0

92

132

6 830

+ −

− −D

action continues with the production ofH2S, pH2S will increase while pH2

and pS2

will decrease. As a result, DG will becomeprogressively less negative. Eventually anequilibrium state will be reached whendG/dn = DG = 0.

For the equilibrium state, therefore:

DG0 = – RT ln K (1-7)= – RT ln (p2

H2S p–2H2

pS2

–1)equilibrium

where K, the “equilibrium constant” of thereaction, is the one unique value of the ra-tio (p2

H2S p–2H2

pS2

–1) for which the system willbe in equilibrium at the temperature T.

If the initial partial pressures are suchthat DG > 0, then the reaction, Eq. (1-5),will proceed to the left in order to minimizeG until the equilibrium condition of Eq. (1-7) is attained.

As a further example, we consider thepossible precipitation of graphite from agaseous mixture of CO and CO2. The reac-tion is:

2 CO = C + CO2 (1-8)

Proceeding as above, we can write:

dG/dn = gC + gCO2– 2 gCO

= (g0C + g0

CO2– 2 g0

CO) + RT ln (pCO2/p2

CO)

= DG0 + RT ln (pCO2/p2

CO) (1-9)

= DG = – RT ln K + RT ln (pCO2/p2

CO)

If (pCO2/p2

CO) is less than the equilibriumconstant K, then precipitation of graphitewill occur in order to decrease G.

Real situations are, of course, generallymore complex. To treat the deposition ofsolid Si from a vapour of SiI4, for example,we must consider the formation of gaseousI2, I and SiI2 so that three reaction equa-tions must be written:

SiI4(g) = Si (sol) + 2 I2(g) (1-10)

SiI4(g) = SiI2(g) + I2(g) (1-11)

I2(g) = 2 I (g) (1-12)


1.3 Predominance Diagrams 7

The equilibrium state, however, is still thatwhich minimizes the total Gibbs energy ofthe system. This is equivalent to satisfyingsimultaneously the equilibrium constantsof the reactions, Eqs. (1-10) to (1-12), aswill be shown in Section 1.12 where thisexample is discussed further.

1.3 Predominance Diagrams

1.3.1 Calculation of Predominance Diagrams

Predominance diagrams are a particu-larly simple type of phase diagram whichhave many applications in the fields of hotcorrosion, chemical vapor deposition, etc.Furthermore, their construction clearly il-lustrates the principles of Gibbs energyminimization and the Gibbs Phase Rule.

A predominance diagram for the Cu–S–O system at 1000 K is shown in Fig. 1-1. The axes are the logarithms of the partial pressures of SO2 and O2 in the gas phase. The diagram is divided into areas or domains of stability of the various

solid compounds of Cu, S and O. For ex-ample, at point Z, where pSO2

= 10–2 andpO2

= 10–7 bar, the stable phase is Cu2O.The conditions for coexistence of two andthree solid phases are indicated respectivelyby the lines and triple points on the diagram.

For example, along the univariant line (phase boundary) separating the Cu2O andCuSO4 domains the equilibrium constantK = p–2

SO2pO2

–3/2 of the following reaction issatisfied:

Cu2O + 2 SO2 + –32

O2 = 2 CuSO4 (1-13)

Hence, along this line:(1-14)

log K = – 2 log pSO2– –3

2log pO2

= constant

This boundary is thus a straight line with aslope of (– 3/2)/2 = – 3/4.

In constructing predominance diagrams,we define a “base element”, in this case Cu,which must be present in all the condensedphases. Let us further assume that there isno mutual solubility among the condensedphases.

Following the procedure of Bale et al.(1986), we formulate a reaction for the for-

Figure 1-1. Predomi-nance diagram. log pSO2

versus log pO2(bar) at

1000 K for the Cu–S–Osystem (Bale et al., 1986).

mation of each solid phase, always fromone mole of the base element Cu, and in-volving the gaseous species whose pres-sures are used as the axes (SO2 and O2 inthis example):

Cu + –12

O2 = CuO

DG = DG0 + RT ln pO2

–1/2 (1-15)

Cu + –14

O2 = –12

Cu2O

DG = DG0 + RT ln pO2

–1/4 (1-16)

Cu + SO2 = CuS + O2

DG = DG0 + RT ln (pO2p–1

SO2) (1-17)

Cu + SO2 + O2 = CuSO4

DG = DG0 + RT ln (p–1SO2

p–1O2

) (1-18)

and similarly for the formation of Cu2S,Cu2SO4 and Cu2SO5.

The values of DG0 are obtained from ta-bles of thermodynamic properties. For anygiven values of pSO2

and pO2, DG for each

formation reaction can then be calculated.The stable compound is simply the onewith the most negative DG. If all the DGvalues are positive, then pure Cu is thestable compound.

By reformulating Eqs. (1-15) to (1-18) interms of, for example, S2 and O2 ratherthan SO2 and O2, a predominance diagramwith ln pS2

and ln pO2as axes can be con-

structed. Logarithms of ratios or productsof partial pressures can also be used asaxes.

1.3.2 Ellingham Diagramsas Predominance Diagrams

Rather than keeping the temperatureconstant, we can use it as an axis. Figure 1-2 shows a diagram for the Fe–S–O sys-tem in which RT ln pO2

is plotted versus Tat constant pSO2

= 1 bar. The diagram is ofthe same topological type as Fig. 1-1.

A similar phase diagram of RT ln pO2

versus T for the Cu–O system is shown in

Fig. 1-3. For the formation reaction:

4 Cu + O2 = 2 Cu2O (1-19)

we can write:

DG0 = – RT ln K = RT ln (pO2)equilibrium

= DH0 – T DS0 (1-20)

The diagonal line in Fig. 1-3 is thus a plotof the standard Gibbs energy of formationof Cu2O versus T. The temperatures indi-cated by the symbol M and M are the melt-ing points of Cu and Cu2O respectively.This line is thus simply a line taken fromthe well-known Ellingham Diagram orDG0 vs. T diagram for the formation of oxides. However, by drawing vertical linesat the melting points of Cu and Cu2O asshown in Fig. 1-3, we convert the plot to atrue phase diagram. Stability domains forCu(sol), Cu(l), Cu2O(sol), and Cu2O(l) are shown as functions of T and of imposedpO2

. The lines and triple points indicate


Figure 1-2. Predominance diagram. RT ln pO2ver-

sus T at pSO2= 1.0 bar for the Fe–S–O system.

1.4 Thermodynamics of Solutions 9

conditions of two- and three-phase equilib-rium.

1.3.3 Discussion of PredominanceDiagrams

In this section discussion is limited tothe assumption that there is no mutual sol-ubility among the condensed phases. Thecalculation of predominance phase dia-grams in which mutual solubility is takeninto account is treated in Sec. 1.9, wherethe general geometrical rules governingpredominance diagrams and their relation-ship to other types of phase diagrams arediscussed.

We frequently encounter predominancediagrams with domains for solid, liquid,and even gaseous compounds which havebeen calculated as if the compounds wereimmiscible, even though they may actuallybe partially or even totally miscible. Theboundary lines are then no longer phaseboundaries, but are lines separating regionsin which one species “predominates”. Thewell known E– pH or Pourbaix diagrams ofaqueous chemistry are examples of suchpredominance diagrams.

Predominance diagrams may also be con-structed when there are two or more baseelements, as discussed by Bale (1990).

Predominance diagrams have foundmany applications in the fields of hot cor-rosion, roasting of ores, chemical vapordeposition, etc. A partial bibliography ontheir construction and applications includesYokokowa (1999), Bale (1990), Bale et al.(1986), Kellogg and Basu (1960), Ingra-ham and Kellogg (1963), Pehlke (1973),Garrels and Christ (1965), Ingraham andKerby (1967), Pilgrim and Ingraham(1967), Gulbransen and Jansson (1970),Pelton and Thompson (1975), Shatynski(1977), Stringer and Whittle (1975), Spen-cer and Barin (1979), Chu and Rahmel(1979), and Harshe and Venkatachalam(1984).

1.4 Thermodynamics of Solutions

1.4.1 Gibbs Energy of Mixing

Liquid gold and copper are completelymiscible at all compositions. The Gibbs en-ergy of one mole of liquid solution, gl, at1400 K is drawn in Fig. 1-4 as a function ofcomposition expressed as mole fraction,XCu, of copper. Note that XAu = 1 – XCu. Thecurve of gl varies between the standard mo-lar Gibbs energies of pure liquid Au andCu, g0

Au and g0Cu.

Figure 1-3. Predominance diagram(also known as a Gibbs energy-tem-perature diagram or Ellingham dia-gram) for the Cu–O system. PointsM and M represent the meltingpoints of the metal and oxide re-spectively.

The function Dglm shown on Fig. 1-4 is

called the molar Gibbs energy of mixing ofthe liquid solution. It is defined as:

Dglm = gl – (XAu g0

Au + XCu g0Cu) (1-21)

It can be seen that Dglm is the Gibbs energy

change associated with the isothermal mix-ing of XAu moles of pure liquid Au and XCu

moles of pure liquid Cu to form one moleof solution:

XAu Au(l) + XCu Cu(l)= 1 mole liquid solution (1-22)

Note that for the solution to be stable it isnecessary that Dgl

m be negative.

1.4.2 Chemical Potential

The partial molar Gibbs energy of com-ponent i, gi , also known as the chemicalpotential, mi , is defined as:

gi = mi = (∂G/∂ni)T, P, nj(1-23)

where G is the Gibbs energy of the solu-tion, ni is the number of moles of compo-

nent i, and the derivative is taken with allnj ( j9i) constant.

In the example of the Au–Cu binary liq-uid solution, gCu = (∂G l/∂nCu)T, P, nAu

, whereG l = (nCu + nAu) gl. That is, gCu, which hasunits of J/mol, is the rate of change of theGibbs energy of a solution as Cu is added.It can be seen that gCu is an intensive prop-erty of the solution which depends uponthe composition and temperature but notupon the total amount of solution. That is,adding dnCu moles of copper to a solutionof given composition will (in the limit asdnCu Æ 0) result in a change in Gibbs en-ergy, dG, which is independent of the totalmass of the solution.

The reason that this property is called achemical potential is illustrated by the fol-lowing thought experiment. Imagine twosystems, I and II, at the same temperatureand separated by a membrane that permitsonly the passage of copper. The chemicalpotentials of copper in systems I and II are gI

Cu = ∂ GI/∂nICu and gII

Cu = ∂ GII/∂nIICu.

Copper is transferred across the membrane,with dnI = – dnII. The change in the totalGibbs energy accompanying this transfer is then: (1-24)dG = d (GI + GII) = – (gI

Cu – gIICu) dnII

Cu

If gICu > gII

Cu, then d (GI + GII) is negativewhen dnII

Cu is positive. That is, the totalGibbs energy will be decreased by a trans-fer of Cu from system I to system II.Hence, Cu will be transferred spontane-ously from a system of higher gCu to a sys-tem of lower gCu. Therefore gCu is calledthe chemical potential of copper.

An important principle of phase equilib-rium can now be stated. When two or morephases are in equilibrium, the chemical po-tential of any component is the same in allphases.


Figure 1-4. Molar Gibbs energy, g l, of liquid Au–Cu alloys at constant temperature (1400 K) illustrat-ing the tangent construction.


1.4.3 Tangent Construction

An important construction is illustratedin Fig. 1-4. If a tangent is drawn to thecurve of gl at a certain composition(XCu = 0.6 in Fig. 1-4), then the interceptsof this tangent on the axes at XAu = 1 andXCu = 1 are equal to gAu and gCu respec-tively at this composition.

To prove this, we first consider that theGibbs energy of the solution at constant Tand P is a function of nAu and nCu. Hence:

Eq. (1-25) can be integrated as follows:

where the integration is performed at con-stant composition so that the intensiveproperties gAu and gCu are constant. Thisintegration can be thought of as describinga process in which a pre-mixed solution ofconstant composition is added to the sys-tem, which initially contains no material.

Dividing Eqs. (1-26) and (1-25) by(nAu + nCu) we obtain expressions for themolar Gibbs energy and its derivative:

gl = XAu gAu + XCu gCu (1-27)

and

dgl = gAu dXAu + gCu dXCu (1-28)

Since dXAu = – dXCu, it can be seen thatEqs. (1-27) and (1-28) are equivalent to thetangent construction shown in Fig. 1-4.

These equations may also be rearrangedto give the following useful expression fora binary system:

gi = g + (1 – Xi) dg/dXi (1-29)

0

l

0Au Au

0Cu Cu

lAu Au Cu Cu

lAu Cu

d = d d

= (1-2 )

G n n

G n n

G n n

∫ ∫ ∫+

+

g g

g g 6

d = d d

= d d (1-2 )

ll

AuAu

l

CuCu

Au Au Cu Cu

GGn

nGn

n

n n

T P,∂∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝⎜

⎞⎠⎟

+g g 5

1.4.4 Gibbs–Duhem Equation

Differentiation of Eq. (1-27) yields:

dgl = (XAu dgAu + XCu dgCu)

+ (gAu dXAu + gCu dXCu) (1-30)

Comparison with Eq. (1-28) then gives theGibbs–Duhem equation at constant T and P:

XAu dgAu + XCu dgCu = 0 (1-31)

1.4.5 Relative Partial Properties

The difference between the partial Gibbsenergy gi of a component in solution andthe partial Gibbs energy gi

0 of the samecomponent in a standard state is called therelative partial Gibbs energy (or relativechemical potential ), Dgi . It is most usual tochoose as standard state the pure compo-nent in the same phase at the same temper-ature. The activity ai of the component rel-ative to the chosen standard state is thendefined in terms of Dgi by the followingequation, as illustrated in Fig. 1-4.

Dgi = gi – gi0 = mi – m i

0 = RT ln ai (1-32)

Note that gi and mi are equivalent symbols,as are gi

0 and m i0, see Eq. (1-23).

From Fig. 1-4, it can be seen that:

Dgm = XAu DgAu + XCu DgCu

= RT (XAu ln aAu + XCu ln aCu) (1-33)

The Gibbs energy of mixing can be di-vided into enthalpy and entropy terms, ascan the relative partial Gibbs energies:

Dgm = Dhm – T Dsm (1-34)

Dgi = Dhi – T Dsi (1-35)

Hence, the enthalpy and entropy of mixingmay be expressed as:

Dhm = XAu DhAu + XCu DhCu (1-36)

Dsm = XAu DsAu + XCu DsCu (1-37)

and tangent constructions similar to that ofFig. 1-4 can be used to relate the relativepartial enthalpies and entropies Dhi andDsi to the integral molar enthalpy of mix-ing Dhm and integral molar entropy of mix-ing Dsm respectively.

1.4.6 Activity

The activity of a component in a solutionwas defined by Eq. (1-32).

Since ai varies monotonically with gi itfollows that when two or more phases arein equilibrium the activity of any compo-nent is the same in all phases, provided thatthe activity in every phase is expressedwith respect to the same standard state.

The use of activities in calculations ofchemical equilibrium conditions is illus-trated by the following example. A liquidsolution of Au and Cu at 1400 K withXCu = 0.6 is exposed to an atmosphere inwhich the oxygen partial pressure ispO2

= 10–4 bar. Will Cu2O be formed? Thereaction is:

2 Cu(l) + –12

O2(g) = Cu2O(sol) (1-38)

where the Cu(l) is in solution. If the reac-tion proceeds with the formation of dnmoles of Cu2O, then 2 dn moles of Cu areconsumed, and the Gibbs energy of theAu–Cu solution changes by

– 2 (dG l/dnCu) dn

The total Gibbs energy then varies as:

dG/dn = gCu2O – –12gO2

– 2 (dG l/dnCu)

= gCu2O – –12gO2

– 2 gCu

= (g0Cu2O – –1

2g0

O2– 2 g0

Cu)

– –12

RT ln pO2– 2 RT ln aCu

= DG0 + RT ln (pO2

–1/2 a–2Cu)

= DG (1-39)

For the reaction, Eq. (1-38), at 1400 K,DG0 = – 68.35 kJ (Barin et al., 1977). Theactivity of Cu in the liquid alloy atXCu = 0.6 is aCu = 0.43 (Hultgren et al.,1973). Substitution into Eq. (1-39) withpO2

= 10–4 bar gives:

dG/dn = DG = – 50.84 kJ

Hence under these conditions the reactionentails a decrease in the total Gibbs energyand so the copper will be oxidized.

1.4.7 Ideal Raoultian Solutions

An ideal solution or Raoultian solutionis usually defined as one in which the ac-tivity of a component is equal to its molefraction:

aiideal = Xi (1-40)

(With a judicious choice of standard state,this definition can also encompass idealHenrian solutions, as discussed in Sec.1.5.11.)

However, this Raoultian definition ofideality is generally only useful for simplesubstitutional solutions. There are moreuseful definitions for other types of solu-tions such as interstitial solutions, ionic so-lutions, solutions of defects, polymer solu-tions, etc. That is, the most convenient def-inition of ideality depends upon the solu-tion model. This subject will be discussedin Sec. 1.10. In the present section, Eq. (1-40) for an ideal substitutional solutionwill be developed with the Au–Cu solutionas example.

In the ideal substitutional solution modelit is assumed that Au and Cu atoms arenearly alike, with nearly identical radii andelectronic structures. This being the case,there will be no change in bonding energyor volume upon mixing, so that the en-thalpy of mixing is zero:

Dhmideal = 0 (1-41)



Furthermore, and for the same reason, theAu and Cu atoms will be randomly distrib-uted over the lattice sites. (In the case of aliquid solution we can think of the “latticesites” as the instantaneous atomic positions.)

For a random distribution of NAu

gold atoms and NCu copper atoms over(NAu + NCu) sites, Boltzmann’s equationcan be used to calculate the configurationalentropy of the solution. This is the entropyassociated with the spatial distribution ofthe particles:

(1-42)Sconfig = kB ln (NAu + NCu) ! /NAu ! NCu !

where kB is Boltzmann’s constant. The con-figurational entropies of pure Au and Cuare zero. Hence the configurational entropyof mixing, DSconfig, will be equal to Sconfig.Furthermore, because of the assumed closesimilarity of Au and Cu, there will be nonon-configurational contribution to the en-tropy of mixing. Hence, the entropy ofmixing will be equal to Sconfig. Applying Stirling’s approximation, whichstates that ln N ! = [(N ln N) – N ] if N islarge, yields:

For one mole of solution, (NAu + NCu) = N0,where N0 = Avogadro’s number. We alsonote that (kB N0) is equal to the ideal gasconstant R. Hence:

(1-44)DSm

ideal = – R (XAu ln XAu + XCu ln XCu)

Therefore, since the ideal enthalpy of mix-ing is zero:

(1-45)Dgm

ideal = R T (XAu ln XAu + XCu ln XCu)

By comparing Eqs. (1-33) and (1-45) weobtain:

Dgiideal = R T ln ai

ideal = R T ln Xi (1-46)

DS S k N N

NN

N NN

NN N

mideal config

B Au Cu

AuAu

Au CuCu

Cu

Au Cu

= = (1- )− +

×+

++

⎛⎝⎜

⎞⎠⎟

( )

ln ln

43

Hence Eq. (1-40) has been demonstratedfor an ideal substitutional solution.

1.4.8 Excess Properties

In reality, Au and Cu atoms are not iden-tical, and so Au–Cu solutions are not per-fectly ideal. The difference between a solu-tion property and its value in an ideal solu-tion is called an excess property. The ex-cess Gibbs energy, for example, is definedas:

gE = Dgm – Dgmideal (1-47)

Since the ideal enthalpy of mixing is zero,the excess enthalpy is equal to the enthalpyof mixing:

hE = Dhm – Dhmideal = Dhm (1-48)

Hence:

gE = hE – T sE

= Dhm – T sE (1-49)

Excess partial properties are defined simi-larly:

giE = Dgi – Dgi

ideal

= R T ln ai – R T ln Xi (1-50)

siE = Dsi – Dsi

ideal = Dsi + R ln Xi (1-51)

Also:

giE = hi

E – T siE

= Dhi – T siE (1-52)

Equations analogous to Eqs. (1-33), (1-36) and (1-37) relate the integral andpartial excess properties. For example, inAu–Cu solutions:

gE = XAu gEAu + XCu gE

Cu (1-53)

sE = XAu sEAu + XCu sE

Cu (1-54)

Tangent constructions similar to that ofFig. 1-4 can thus also be employed for ex-cess properties, and an equation analogous

to Eq. (1-29) can be written:

giE = gE + (1 – Xi) dgE/dXi (1-55)

The Gibbs–Duhem equation, Eq. (1-31),also applies to excess properties:

XAu dgEAu + XCu dgE

Cu = 0 (1-56)

In Au–Cu alloys, gE is negative. That is,Dgm is more negative than Dgm

ideal and sothe solution is thermodynamically morestable than an ideal solution. We say thatAu–Cu solutions exhibit negative devia-tions from ideality. If gE > 0, then the solu-tion is less stable than an ideal solution andis said to exhibit positive deviations.

1.4.9 Activity Coefficient

The activity coefficient of a componentin a solution is defined as:

gi = ai /Xi (1-57)

From Eq. (1-50):

giE = R T ln gi (1-58)

In an ideal solution gi =1 and giE = 0 for

all components. If gi <1, then giE < 0 and by

Eq. (1-50), Dgi < Dgiideal. That is, the com-

ponent i is more stable in the solution thanit would be in an ideal solution of the samecomposition. If gi >1, then gi

E > 0 and thedriving force for the component to enterinto solution is less than in the case of anideal solution.

1.4.10 Multicomponent Solutions

The equations of this section were de-rived with a binary solution as an example.However, the equations apply equally tosystems of any number of components. Forinstance, in a solution of componentsA–B–C–D …, Eq. (1-33) becomes:

Dgm = XA DgA + XB DgB + XC DgC

+ XD DgD + … (1-59)

1.5 Binary Phase Diagrams

1.5.1 Systems with Complete Solidand Liquid Miscibility

The temperature–composition (T – X )phase diagram of the CaO–MnO system isshown in Fig. 1-5 (Schenck et al., 1964;Wu, 1990). The abscissa is the composi-


Figure 1-5. Phase dia-gram of the CaO–MnOsystem at P =1 bar (after Schenck et al.,1964, and Wu, 1990).

1.5 Binary Phase Diagrams 15

tion, expressed as mole fraction of MnO,XMnO. Note that XMnO = 1 – XCaO. Phase di-agrams are also often drawn with the com-position axis expressed as weight percent.

At all compositions and temperatures inthe area above the line labelled liquidus, asingle-phase liquid solution will be ob-served, while at all compositions and tem-peratures below the line labelled solidus,there will be a single-phase solid solution.A sample at equilibrium at a temperatureand overall composition between these twocurves will consist of a mixture of solidand liquid phases, the compositions ofwhich are given by the liquidus and soliduscompositions at that temperature. For ex-ample, a sample of overall compositionXMnO = 0.60 at T = 2200°C (at point R inFig. 1-5) will consist, at equilibrium, of amixture of liquid of composition XMnO =0.70 (point Q) and solid of compositionXMnO = 0.35 (point P).

The line PQ is called a tie-line or co-node. As the overall composition is variedat 2200 °C between points P and Q, thecompositions of the solid and liquid phasesremain fixed at P and Q, and only the rela-tive proportions of the two phases change.From a simple mass balance, we can derivethe lever rule for binary systems: (moles ofliquid) / (moles of solid) = PR /RQ. Hence,at 2200 °C a sample with overall composi-tion XMnO = 0.60 consists of liquid and solidphases in the molar ratio (0.60 – 0.35) /(0.70–0.60) = 2.5. Were the compositionaxis expressed as weight percent, then thelever rule would give the weight ratio ofthe two phases.

Suppose that a liquid CaO–MnO solu-tion with composition XMnO = 0.60 iscooled very slowly from an initial tempera-ture of about 2500 °C. When the tempera-ture has decreased to the liquidus tempera-ture 2270 °C (point B), the first solid appears, with a composition at point A

(XMnO = 0.28). As the temperature is de-creased further, solid continues to precipi-tate with the compositions of the twophases at any temperature being given bythe liquidus and solidus compositions atthat temperature and with their relativeproportions being given by the lever rule.Solidification is complete at 2030 °C, thelast liquid to solidify having compositionXMnO = 0.60 (point C).

The process just described is known asequilibrium cooling. At any temperatureduring equilibrium cooling the solid phasehas a uniform (homogeneous) composition.In the preceding example, the compositionof the solid phase during cooling variesalong the line APC. Hence, in order for thesolid grains to have a uniform compositionat any temperature, diffusion of CaO fromthe center to the surface of the growinggrains must occur. Since solid-state dif-fusion is a relatively slow process, equi-librium cooling conditions are only ap-proached if the temperature is decreasedvery slowly. If a sample of compositionXMnO = 0.60 is cooled very rapidly from theliquid, concentration gradients will be ob-served in the solid grains, with the concen-tration of MnO increasing towards the sur-face from a minimum of XMnO = 0.28 (pointA) at the center. Furthermore, in this casesolidification will not be complete at2030 °C since at 2030°C the average con-centration of MnO in the solid particleswill be less than XMnO = 0.60. These con-siderations are discussed more fully inChapter 2 of this volume (Müller-Krumb-haar et al., 2001).

At XMnO = 0 and XMnO = 1 in Fig. 1-5 theliquidus and solidus curves meet at theequilibrium melting points, or tempera-tures of fusion of CaO and MnO, which areT 0

f (CaO) = 2572 °C, T 0f (MnO) = 1842 °C.

The phase diagram is influenced by thetotal pressure, P. Unless otherwise stated,

T – X diagrams are usually presented forP = const. = 1 bar. For equilibria involvingonly solid and liquid phases, the phaseboundaries are typically shifted only by theorder of a few hundredths of a degree perbar change in P. Hence, the effect of pres-sure upon the phase diagram is generallynegligible unless the pressure is of the or-der of hundreds of bars. On the other hand,if gaseous phases are involved then the ef-fect of pressure is very important. The ef-fect of pressure will be discussed in Sec.1.5.3.

1.5.2 Thermodynamic Originof Phase Diagrams

In this section we first consider the ther-modynamic origin of simple “lens-shaped”phase diagrams in binary systems withcomplete liquid and solid miscibility.

An example of such a diagram was givenin Fig. 1-5. Another example is the Ge–Siphase diagram in the lowest panel of Fig.1-6 (Hansen, 1958). In the upper three pan-els of Fig. 1-6, the molar Gibbs energies ofthe solid and liquid phases, gs and gl, atthree temperatures are shown to scale. Asillustrated in the top panel, gs varies withcomposition between the standard molarGibbs energies of pure solid Ge and of puresolid Si, gGe

0(s) and gSi0(s), while gl varies

between the standard molar Gibbs energiesof the pure liquid components gGe

0(l) andgSi

0(l).The difference between gGe

0(l) and gSi0(s) is

equal to the standard molar Gibbs energyof fusion (melting) of pure Si, Dg0

f (Si) =(gSi

0(l) – gSi0(s)). Similarly, for Ge, Dg0

f (Ge) =(gGe

0(l) – gGe0(s)). The Gibbs energy of fusion of

a pure component may be written as:

Dgf0 = Dhf

0 – T Dsf0 (1-60)

where Dhf0 and Dsf

0 are the standard molarenthalpy and entropy of fusion.

Since, to a first approximation, Dhf0 and

Dsf0 are independent of T, Dgf

0 is approxi-mately a linear function of T. If T > Tf

0,then Dgf

0 is negative. If T < Tf0, then Dgf

0 ispositive. Hence, as seen in Fig. 1-6, as Tdecreases, the gs curve descends relative togl. At 1500 °C, gl < gs at all compositions.Therefore, by the principle that a systemalways seeks the state of minimum Gibbsenergy at constant T and P, the liquid phaseis stable at all compositions at 1500 °C.

At 1300 °C, the curves of gs and gl cross.The line P1 Q1, which is the common tan-gent to the two curves, divides the compo-sition range into three sections. For compo-sitions between pure Ge and P1, a single-phase liquid is the state of minimum Gibbsenergy. For compositions between Q1 andpure Si, a single-phase solid solution is thestable state. Between P1 and Q1, a totalGibbs energy lying on the tangent lineP1 Q1 may be realized if the system adoptsa state consisting of two phases with com-positions at P1 and Q1 and with relativeproportions given by the lever rule. Sincethe tangent line P1 Q1 lies below both gs andgl, this two-phase state is more stable thaneither phase alone. Furthermore, no otherline joining any point on gl to any point ongs lies below the line P1 Q1. Hence, this linerepresents the true equilibrium state of thesystem, and the compositions P1 and Q1 arethe liquidus and solidus compositions at1300 °C.

As T is decreased to 1100 °C, the pointsof common tangency are displaced tohigher concentrations of Ge. For T <937°C,gs < gl at all compositions.

It was shown in Fig. 1-4 that if a tangentis drawn to a Gibbs energy curve, then theintercept of this tangent on the axis at Xi =1is equal to the partial Gibbs energy orchemical potential gi of component i. Thecommon tangent construction of Fig. 1-6thus ensures that the chemical potentials of



Ge and Si are equal in the solid and liquidphases at equilibrium. That is:

glGe = gs

Ge (1-61)

glSi = gs

Si (1-62)

This equality of chemical potentials wasshown in Sec. 1.4.2 to be the criterion forphase equilibrium. That is, the commontangent construction simultaneously mini-mizes the total Gibbs energy and ensuresthe equality of the chemical potentials,thereby showing that these are equivalentcriteria for equilibrium between phases.

If we rearrange Eq. (1-61), subtractingthe Gibbs energy of fusion of pure Ge,Dg0

f (Ge) = (gGe0(l) – gGe

0(s)), from each side, weget:

(glGe – gGe

0(l)) – (gsGe – gGe

0(s))

= – (gGe0(l) – gGe

0(s)) (1-63)

Using Eq. (1-32), we can write Eq. (1-63)as:

DglGe – Dgs

Ge = – Dg0f (Ge) (1-64)

or

RT ln alGe – RT ln as

Ge = – Dg0f (Ge) (1-65)

Figure 1-6. Ge–Si phase diagram atP =1 bar (after Hansen, 1958) andGibbs energy composition curves atthree temperatures, illustrating the com-mon tangent construction (reprintedfrom Pelton, 1983).

where alGe is the activity of Ge (with re-

spect to pure liquid Ge as standard state) inthe liquid solution on the liquidus, and as

Ge

is the activity of Ge (with respect to puresolid Ge as standard state) in the solid solu-tion on the solidus. Starting with Eq. (1-62), we can derive a similar expression forthe other component:

RT ln alSi – RT ln as

Si = – Dg0f (Si) (1-66)

Eqs. (1-65) and (1-66) are equivalent to thecommon tangent construction.

It should be noted that absolute values ofGibbs energies cannot be defined. Hence,the relative positions of gGe

0(l) and gSi0(l) in

Fig. 1-6 are completely arbitrary. However,this is immaterial for the preceding discus-sion, since displacing both gSi

0(l) and gSi0(s) by

the same arbitrary amount relative to gGe0(l)

and gGe0(s) will not alter the compositions of

the points of common tangency.It should also be noted that in the present

discussion of equilibrium phase diagramswe are assuming that the physical dimen-sions of the single-phase regions in thesystem are sufficiently large that surface(interfacial) energy contributions to theGibbs energy can be neglected. For veryfine grain sizes in the sub-micron range,however, surface energy effects can notice-ably influence the phase boundaries.

The shape of the two-phase (solid + liq-uid) “lens” on the phase diagram is deter-mined by the Gibbs energies of fusion,Dgf

0, of the components and by the mixingterms, Dgs and Dgl. In order to observehow the shape is influenced by varyingDgf

0, let us consider a hypothetical systemA–B in which Dgs and Dgl are ideal Raoul-tian (Eq. (1-45)). Let T 0

f (A) = 800 K andT 0

f (B) = 1200 K. Furthermore, assume thatthe entropies of fusion of A and B are equaland temperature-independent. The enthalp-ies of fusion are then given from Eq. (1-60)by the expression Dhf

0 = Tf0Dsf

0 since

Dgf0 = 0 when T = Tf

0. Calculated phase dia-grams for Dsf

0 = 3, 10 and 30 J/mol K areshown in Fig. 1-7. A value of Dsf

0 ≈ 10 istypical of most metals. However, when thecomponents are ionic compounds such asionic oxides, halides, etc., Dsf

0 can be sig-nificantly larger since there are several ions per formula unit. Hence, two-phase “lenses” in binary ionic salt or oxide phasediagrams tend to be “fatter” than those encountered in alloy systems. If we areconsidering vapor– liquid equilibria ratherthan solid– liquid equilibria, then the shape is determined by the entropy of vaporization, Dsv

0. Since Dsv0 is usually an

order of magnitude larger than Dsf0, two-

phase (liquid + vapor) lenses tend to be


Figure 1-7. Phase diagram of a system A–B withideal solid and liquid solutions. Melting points of Aand B are 800 and 1200 K, respectively. Diagramsare calculated for entropies of fusion ∆S 0

f(A)= ∆S 0f(B) =

3, 10 and 30 J/mol K.


very wide. For equilibria between two solidsolutions of different crystal structure, theshape is determined by the entropy ofsolid–solid transformation, which is usu-ally smaller than the entropy of fusion byapproximately an order of magnitude.Therefore two-phase (solid + solid) lensestend to be very narrow.

1.5.3 Pressure–Composition PhaseDiagrams

Let us consider liquid–vapor equilib-rium with complete miscibility, using as anexample the Zn–Mg system. Curves of gv

and gl can be drawn at any given T and P,as in the upper panel of Fig. 1-8, and the

common tangent construction then givesthe equilibrium vapor and liquid composi-tions. The phase diagram depends upon theGibbs energies of vaporization of the com-ponents Dgv(Zn) and Dgv(Mg) as shown inFig. 1-8.

To generate the isothermal pressure–composition (P– X ) phase diagram in thelower panel of Fig. 1-8 we require theGibbs energies of vaporization as functionsof P. Assuming monatomic ideal vaporsand assuming that pressure has negligibleeffect upon the Gibbs energy of the liquid,we can write:

Dgv(i) = Dg0v(i) + RT ln P (1-67)

where Dgv(i) is the standard Gibbs energyof vaporization (when P =1 bar), which isgiven by:

Dg0v(i) = Dh0

v(i) – T Ds0v(i) (1-68)

For example, the enthalpy of vaporizationof Zn is Dh0

v(Zn) = 115 300 J/mol at its nor-mal boiling point of 1180 K (Barin et al.,1977). Assuming that Dh0

v is independentof T, we calculate from Eq. (1-68) thatDs0

v(Zn) = 115 300/1180 = 97.71 J/mol K.From Eq. (1-67), Dgv(Zn) at any T and P isthus given gy:

(1-69)Dgv(Zn) = (115 300 – 97.71 T ) + RT ln P

A similar expression can be derived for theother component Mg.

At constant temperature, then, the curveof gv in Fig. 1-8 descends relative to gl asthe pressure is lowered, and the P– X phasediagram is generated by the common tan-gent construction. The diagram at 1250 Kin Fig. 1-8 was calculated under the as-sumption of ideal liquid and vapor mixing(gE(l) = 0, gE(v) = 0).

P– X phase diagrams involving liquid–solid or solid–solid equilibria can be cal-culated in a similar fashion through the fol-

Figure 1-8. Pressure–composition phase diagramof the Zn–Mg system at 1250 K calculated for idealvapor and liquid solutions. Upper panel illustratescommon tangent construction at a constant pressure.

lowing general equation, which gives theeffect of pressure upon the Gibbs energychange for the transformation of one moleof pure component i from an a-phase to ab-phase:

where Dg0aÆb is the standard (P =1 bar)

Gibbs energy of transformation, and vib and

via are the molar volumes.

1.5.4 Minima and Maximain Two-Phase Regions

As discussed in Sec. 1.4.8, the Gibbs en-ergy of mixing Dgm may be expressed asthe sum of an ideal term Dgm

ideal and an ex-cess term gE. As has just been shown inSec. 1.5.2, if Dg s

m and Dglm for the solid

and liquid phases are both ideal, then a“lens-shaped” two-phase region always re-sults. However in most systems even ap-proximately ideal behavior is the exceptionrather than the rule.

Curves of gs and gl for a hypotheticalsystem A–B are shown schematically inFig. 1-9 at a constant temperature (belowthe melting points of pure A and B) such

D Da b a bb ag g v v→ → + −∫= d (1- )

=

0

1

70P

P

i i P( )

that the solid state is the stable state forboth pure components. However, in thissystem gE(l) < gE(s), so that gs presents aflatter curve than does gl and there exists acentral composition region in which gl < gs.Hence, there are two common tangentlines, P1 Q1 and P2 Q2. Such a situationgives rise to a phase diagram with a mini-mum in the two-phase region, as observedin the Na2CO3–K2CO3 system (Dessu-reault et al., 1990) shown in Fig. 1-10. At acomposition and temperature correspond-ing to the minimum point, liquid and solidof the same composition exist in equilib-rium.

A two-phase region with a minimumpoint as in Fig. 1-10 may be thought of as atwo-phase “lens” which has been “pusheddown” by virtue of the fact that the liquid isrelatively more stable than the solid. Ther-modynamically, this relative stability is ex-pressed as gE(l) < gE(s).

Conversely, if gE(l) > gE(s) to a sufficientextent, then a two-phase region with amaximum will result. Such maxima in (liq-uid + solid) or (solid + solid) two-phase re-gions are nearly always associated with theexistence of an intermediate phase, as willbe discussed in Sec. 1.5.10.


Figure 1-9. Isothermal Gibbs energy-compositioncurves for solid and liquid phases in a system A–B inwhich g E(l) >g E(s). A phase diagram of the type ofFig. 1-10 results.

Figure 1-10. Phase diagram of the K2CO3–Na2CO3

system at P =1 bar (Dessureault et al., 1990).


1.5.5 Miscibility Gaps

If gE > 0, then the solution is thermody-namically less stable than an ideal solution.This can result from a large difference insize of the component atoms, ions or mole-cules, which will lead to a (positive) latticestrain energy, or from differences in elec-tronic structure, or from other factors.

In the Au–Ni system, gE is positive inthe solid phase. In the top panel of Fig. 1-11,gE(s) is plotted at 1200 K (Hultgren et al.,1973) and the ideal Gibbs energy of mixing, Dgm

ideal, is also plotted at 1200 K.The sum of these two terms is the Gibbsenergy of mixing of the solid solution,Dgm

s , which is plotted at 1200 K as well as at other temperatures in the central panelof Fig. 1-11. Now, from Eq. (1-45), Dgm

ideal

is always negative and varies directly with T, whereas gE varies less rapidly withtemperature. As a result, the sum Dgm

s =Dgm

ideal + gE becomes less negative as T de-creases. However, the limiting slopes to theDgm

ideal curve at XAu =1 and XNi =1 are bothinfinite, whereas the limiting slopes of gE

are always finite (Henry’s Law). Hence,Dgs

m will always be negative as XAu Æ1and XNi Æ1 no matter how low the temper-ature. As a result, below a certain tempera-ture the curve of Dgs

m will exhibit two neg-ative “humps”. Common tangent linesP1 Q1, P2 Q2, P3 Q3 to the two humps at dif-ferent temperatures define the ends of tie-lines of a two-phase solid–solid miscibilitygap in the Au–Ni phase diagram, which isshown in the lower panel in Fig. 1-11(Hultgren et al., 1973). The peak of the gapoccurs at the critical or consolute tempera-ture and composition, Tc and Xc.

When gE(s) is positive for the solid phasein a system it is usually also the case thatgE(l) < gE(s) since the unfavorable factors(such as a difference in atomic dimensions)which are causing gE(s) to be positive will

have less of an effect upon gE(l) in the liq-uid phase owing to the greater flexibility ofthe liquid structure to accommodate differ-ent atomic sizes, valencies, etc. Hence, asolid–solid miscibility gap is often asso-ciated with a minimum in the two-phase (solid + liquid) region, as is the case in theAu–Ni system.

Figure 1-11. Phase diagram (after Hultgren et al.,1973) and Gibbs energy–composition curves of solidsolutions for the Au–Ni system at P =1 bar. Letters“s” indicate spinodal points (Reprinted from Pelton,1983).

Below the critical temperature the curveof Dgs

m exhibits two inflection points, indi-cated by the letter “s” in Fig. 1-11. Theseare known as the spinodal points. On thephase diagram their locus traces out thespinodal curve (Fig. 1-11). The spinodalcurve is not part of the equilibrium phasediagram, but it is important in the kineticsof phase separation, as discussed in Chap-ter 6 (Binder and Fratzl, 2001).

1.5.6 Simple Eutectic Systems

The more positive gE is in a system, thehigher is Tc and the wider is the miscibility

gap at any temperature. Suppose that gE(s)

is so positve that Tc is higher than the min-imum in the (solid + liquid) region. The re-sult will be a phase diagram such as that ofthe MgO–CaO system shown in Fig. 1-12(Doman et al., 1963; Wu, 1990).

The lower panel of Fig. 1-12 shows theGibbs energy curves at 2450 °C. The twocommon tangents define two two-phase re-gions. As the temperature is decreased be-low 2450 °C, the gs curve descends relativeto gl and the two points of tangency P1

and P2 approach each other until, at T =2374 °C, P1 and P2 become coincident atthe composition E. That is, at T = 2374 °C


Figure 1-12. Phase diagramat P =1 bar (after Doman etal., 1963, and Wu, 1990) andGibbs energy–compositioncurves at 2450°C for theMgO–CaO system. SolidMgO and CaO have thesame crystal structure.


there is just one common tangent line con-tacting the two portions of the gs curve atcompositions A and B and contacting the gl

curve at E. This temperature is known asthe eutectic temperature, TE, and the com-position E is the eutectic composition. Fortemperatures below TE, gl lies completelyabove the common tangent to the two por-tions of the gs curve and so for T < TE asolid–solid miscibility gap is observed.The phase boundaries of this two-phase re-gion are called the solvus lines. The wordeutectic is from the Greek for “to meltwell” since the system has its lowest melt-ing point at the eutectic composition E.

This description of the thermodynamicorigin of simple eutectic phase diagrams isstrictly correct only if the pure solid com-ponents A and B have the same crystalstructure. Otherwise, a curve for gs whichis continuous at all compositions cannot bedrawn.

Suppose a liquid MgO–CaO solution ofcomposition XCaO = 0.52 (composition P1)is cooled from the liquid state very slowlyunder equilibrium conditions. At 2450 °Cthe first solid appears with composition Q1.As T decreases further, solidification con-tinues with the liquid composition follow-ing the liquidus curve from P1 to E and thecomposition of the solid phase followingthe solidus curve from Q1 to A. The rela-tive proportions of the two phases at any Tare given by the lever rule. At a tempera-ture just above TE, two phases are ob-served: a solid of composition A and a liq-uid of composition E. At a temperature justbelow TE, two solids with compositions Aand B are observed. Therefore, at TE, dur-ing cooling, the following binary eutecticreaction occurs:

liquid Æ solid1 + solid2 (1-71)

Under equilibrium conditions the tempera-ture will remain constant at T = TE until all

the liquid has solidified, and during the re-action the compositions of the three phaseswill remain fixed at A, B and E. For thisreason the eutectic reaction is called an in-variant reaction. More details on eutecticsolidification may be found in Chapter 2(Müller-Krumbhaar et al., 2001).

1.5.7 Regular Solution Theory

Many years ago Van Laar (1908) showedthat the thermodynamic origin of a greatmany of the observed features of binaryphase diagrams can be illustrated at leastqualitatively by simple regular solutiontheory. A simple regular solution is one forwhich:

gE = XA XB (w – h T ) (1-72)

where w and h are parameters independentof temperature and composition. Substitut-ing Eq. (1-72) into Eq. (1-29) yields, forthe partial properties:

(1-73)gE

A = XB2 (w – h T ) , gE

B = XA2 (w – h T )

Several liquid and solid solutions con-form approximately to regular solution be-havior, particularly if gE is small. Examplesmay be found for alloys, molecular solu-tions, and ionic solutions such as moltensalts and oxides, among others. (The verylow values of gE observed for gaseous solu-tions generally conform very closely to Eq.(1-72).)

To understand why this should be so, weonly need a very simple model. Supposethat the atoms or molecules of the compo-nents A and B mix substitutionally. If theatomic (or molecular) sizes and electronicstructures of A and B are similar, then thedistribution will be nearly random, and theconfigurational entropy will be nearlyideal. That is:

gE ≈ Dhm – T S E(non-config) (1-74)

More will be said on this point in Sec.1.10.5.

We now assume that the bond energieseAA, eBB and eAB of nearest-neighbor pairsare independent of temperature and com-position and that the average nearest-neighbor coordination number, Z, is alsoconstant. Finally, we assume that the en-thalpy of mixing results mainly from thechange in the total energy of nearest-neigh-bor pair bonds.

In one mole of solution there are (N 0

Z /2) neareast-neighbor pair bonds, whereN 0 is Avogadro’s number. Since the distri-bution is assumed random, the probabilitythat a given bond is an A–A bond is equalto X 2

A. The probabilities of B–B and A–Bbonds are, respectively, X 2

B and 2 XA XB.The molar enthalpy of mixing is then equalto the sum of the energies of the nearest-neighbor bonds in one mole of solution,minus the energy of the A–A bonds in XA

moles of pure A and the energy of the B–Bbonds in XB moles of pure B:

Dhm = (N 0 Z /2)¥ (X 2

A eAA + X 2B eBB + 2 XAB eAB)

– (N 0 Z /2) (XA eAA) – (N 0 Z /2) (XB eBB)

= (N 0 Z ) [eAB – (eAA + eBB)/2] XA XB

= w XA XB (1-75)

We now define sAB, sAA and sBB as thevibrational entropies of nearest-neighborpair bonds. Following an identical argu-ment to that just presented for the bond energies we obtain:

sE(non-config) (1-76)= (N 0 Z ) [sAB – (sAA + sBB)/2] = h XA XB

Eq. (1-72) has thus been derived. If A–Bbonds are stronger than A–A and B–Bbonds, then (eAB – hABT ) < [(eAA– hAA T )/2+ (eBB – hBB T ) /2]. Hence, (w – h T ) < 0and gE < 0. That is, the solution is renderedmore stable. If the A–B bonds are rela-

tively weak, then the solution is renderedless stable, (w – h T ) > 0 and gE > 0.

Simple non-polar molecular solutionsand ionic solutions such as molten salts of-ten exhibit approximately regular behavior.The assumption of additivity of the energyof pair bonds is probably reasonably realis-tic for van der Waals or coulombic forces.For alloys, the concept of a pair bond is, atbest, vague, and metallic solutions tend toexhibit larger deviations from regular be-havior.

In several solutions it is found that |h T | < |w| in Eq. (1-72). That is, gE ≈ Dhm

= w XA XB, and to a first approximation gE is independent of T. This is more oftenthe case in non-metallic solutions than inmetallic solutions.

1.5.8 Thermodynamic Originof Simple Phase Diagrams Illustratedby Regular Solution Theory

Figure 1-13 shows several phase dia-grams, calculated for a hypothetical systemA–B containing a solid and a liquid phasewith melting points of T 0

f (A) = 800 K andT 0

f (B) = 1200 K and with entropies of fusionof both A and B set to 10 J/mol K, which isa typical value for metals. The solid andliquid phases are both regular with temper-ature-independent excess Gibbs energies

gE(s) = w s XA XB and gE(l) = w l XA XB

The parameters w s and wl have been variedsystematically to generate the various pan-els of Fig. 1-13.

In panel (n) both phases are ideal. Panels(l) to (r) exhibit minima or maxima de-pending upon the sign and magnitude of(gE(l) – gE(s)), as has been discussed in Sec.1.5.4. In panel (h) the liquid is ideal butpositive deviations in the solid give rise toa solid–solid miscibility gap as discussedin Sec. 1.5.6. On passing from panel (h) to



panel (c), an increase in gE(s) results in awidening of the miscibility gap so that thesolubility of A in solid B and of B in solidA decreases. Panels (a) to (c) illustrate thatnegative deviations in the liquid cause arelative stabilization of the liquid with re-

sultant lowering of the eutectic tempera-ture.

Eutectic phase diagrams are often drawnwith the maximum solid solubility occur-ring at the eutectic temperature (as in Fig.1-12). However, panel (d) of Fig. 1-13, in

Figure 1-13. Topological changes in the phase diagram for a system A–B with regular solid and liquid phases,brought about by systematic changes in the regular solution parameters ω s and ω l. Melting points of pure A andB are 800 K and 1200 K. Entropies of fusion of both A and B are 10.0 J/mol K (Pelton and Thompson, 1975).The dashed curve in panel (d) is the metastable liquid miscibility gap (Reprinted from Pelton, 1983).

which the maximum solubility of A in theB-rich solid solution occurs at approxi-mately T = 950 K, illustrates that this neednot be the case even for simple regular so-lutions.

1.5.9 Immiscibility – Monotectics

In Fig. 1-13(e), positive deviations inthe liquid have given rise to a liquid– liquidmiscibility gap. The CaO–SiO2 system(Wu, 1990), shown in Fig. 1-14, exhibitssuch a feature. Suppose that a liquid ofcomposition XSiO2

= 0.8 is cooled slowlyfrom high temperatures. At T = 1815 °C themiscibility gap boundary is crossed and asecond liquid layer appears with a compo-sition of XSiO2

= 0.97. As the temperature islowered further, the composition of eachliquid phase follows its respective phaseboundary until, at 1692 °C, the SiO2-richliquid has a composition of XSiO2

= 0.99 (point B), and in the CaO-rich liquidXSiO2

= 0.74 (point A). At any temperature,the relative amounts of the two phases aregiven by the lever rule.

At 1692 °C the following invariant bi-nary monotectic reaction occurs upon cool-ing:

Liquid B Æ Liquid A + SiO2 (solid) (1-77)

The temperature remains constant at1692 °C and the compositions of the phasesremain constant until all of liquid B is con-sumed. Cooling then continues with pre-cipitation of solid SiO2 with the equilib-rium liquid composition following the liq-uidus from point A to the eutectic E.

Returning to Fig. 1-13, we see in panel(d) that the positive deviations in the liquidin this case are not large enough to produceimmiscibility, but they do result in a flat-tening of the liquidus, which indicates a“tendency to immiscibility”. If the nuclea-tion of the solid phases can be suppressed

by sufficiently rapid cooling, then a meta-stable liquid– liquid miscibility gap is ob-served as shown in Fig. 1-13(d). For exam-ple, in the Na2O–SiO2 system the flattened(or “S-shaped”) SiO2 liquidus heralds theexistence of a metastable miscibility gap ofimportance in glass technology.

1.5.10 Intermediate Phases

The phase diagram of the Ag–Mgsystem (Hultgren et al., 1973) is shown inFig. 1-15(d). An intermetallic phase, b¢, isseen centered approximately about thecomposition XMg = 0.5. The Gibbs energycurve at 1050 K for such an intermetallicphase has the form shown schematically inFig. 1-15(a). The curve gb¢ rises quite rap-idly on either side of its minimum, whichoccurs near XMg = 0.5. As a result, the b¢phase appears on the phase diagram onlyover a limited composition range. Thisform of the curve gb¢ results from the factthat when XAg ≈ XMg a particularly stablecrystal structure exists in which Ag and Mgatoms preferentially occupy different sites.The two common tangents P1 Q1 and P2 Q2

give rise to a maximum in the two-phase(b¢ + liquid) region of the phase diagram.(Although the maximum is observed verynear XMg = 0.5, there is no thermodynamicreason for the maximum to occur exactly atthis composition.)

Another intermetallic phase, the e phase,is also observed in the Ag–Mg system,Fig. 1-15. The phase is associated with aperitectic invariant ABC at 744 K. TheGibbs energy curves are shown schemati-cally at the peritectic temperature in Fig. 1-15(c). One common tangent line can bedrawn to gl, gb¢ and ge.

Suppose that a liquid alloy of composi-tion XMg = 0.7 is cooled very slowly fromthe liquid state. At a temperature just above744 K a liquid phase of composition C and



Figure 1-14. CaO–SiO2 phase diagram at P = 1 bar (after Wu, 1990) and Gibbs energy curves at 1500 °C illus-trating Gibbs energies of fusion and formation of the stoichiometric compound CaSiO3.

a b¢ phase of composition A are observedat equilibrium. At a temperature just below744 K the two phases at equilibrium are b¢of composition A and e of composition B.The following invariant binary peritecticreaction thus occurs upon cooling:

Liquid + b¢ (solid) Æ e (solid) (1-78)

This reaction occurs isothermally at 744 Kwith all three phases at fixed compositions(at points A, B and C). For an alloy withoverall composition between points A andB the reaction proceeds until all the liquidhas been consumed. In the case of an alloywith overall composition between B and C,the b¢ phase will be the first to be com-pletely consumed.

Peritectic reactions occur upon coolingwith formation of the product solid (e inthis example) on the surface of the reactantsolid (b¢), thereby forming a coating whichcan prevent further contact between the re-actant solid and liquid. Further reactionmay thus be greatly retarded so that equi-librium conditions can only be achieved byextremely slow cooling.

The Gibbs energy curve for the e phase,ge, in Fig. 1-15(c) rises more rapidly on ei-ther side of its minimum than does theGibbs energy gb¢ for the b¢ phase in Fig. 1-15(a). As a result, the width of the single-phase region over which the e phase exists(sometimes called its range of stoichiome-try or homogeneity range) is narrower thanfor the b¢ phase.

In the upper panel of Fig. 1-14 for theCaO–SiO2 system, Gibbs energy curves at1500 °C for the liquid and CaSiO3 phasesare shown schematically. g0.5 (CaSiO3) risesextremely rapidly on either side of its min-imum. (We write g0.5 (CaSiO3) for 0.5 molesof the compound in order to normalize to abasis of one mole of components CaO andSiO2.) As a result, the points of tangencyQ1 and Q2 of the common tangents P1 Q1

and P2 Q2 nearly (but not exactly) coincide.Hence, the range of stoichiometry of theCaSiO3 phase is very narrow (but neverzero). The two-phase regions labelled (CaSiO3 + liquid) in Fig. 1-14 are the twosides of a two-phase region that passesthrough a maximum at 1540 °C just as the(b¢ + liquid) region passes through a maxi-


Figure 1-15. Ag–Mg phase diagram at P = 1 bar (af-ter Hultgren at al., 1973) and Gibbs energy curves atthree temperatures.


mum in Fig. 1-15(d). Because the CaSiO3

single-phase region is so narrow, we referto CaSiO3 as a stoichiometric compound.Any deviation in composition from thestoichiometric 1 :1 ratio of CaO to SiO2

results in a very large increase in Gibbs energy.

The e phase in Fig. 1-15 is based on thestiochiometry AgMg3. The Gibbs energycurve, Fig. 1-15(c), rises extremely rapidlyon the Ag side of the minimum, but some-what less steeply on the Mg side. As a re-sult, Ag is virtually insoluble in AgMg3,while Mg is sparingly soluble. Such aphase with a narrow range of homogeneityis often called a non-stoichiometric com-pound. At low temperatures the b¢ phaseexhibits a relatively narrow range of stoi-chiometry about the 1 :1 AgMg composi-tion and can properly be called a com-pound. However, at higher temperatures itis debatable whether a phase with such awide range of composition should be calleda “compound”.

From Fig. 1-14 it can be seen that if stoi-chiometric CaSiO3 is heated it will meltisothermally at 1540 °C to form a liquid ofthe same composition. Such a compound iscalled congruently melting or simply a con-gruent compound. The compound Ca2SiO4

in Fig. 1-14 is congruently melting. The b¢phase in Fig. 1-15 is also congruently melt-ing at the composition of the liquidus/sol-idus maximum.

It should be noted with regard to the con-gruent melting of CaSiO3 in Fig. 1-14 thatthe limiting slopes dT /dX of both branchesof the liquidus at the congruent meltingpoint (1540 °C) are zero since we are reallydealing with a maximum in a two-phase re-gion.

The AgMg3 (e) compound in Fig. 1-15 issaid to melt incongruently. If solid AgMg3

is heated it will melt isothermally at 744 Kby the reverse of the peritectic reaction,

Eq. (1-78), to form a liquid of compositionC and another solid phase, b¢, of composi-tion A.

Another example of an incongruentcompound is Ca3Si2O7 in Fig. 1-14, whichmelts incongruently (or peritectically) toform liquid and Ca2SiO4 at the peritectictemperature of 1469 °C.

An incongruent compound is always as-sociated with a peritectic. However, theconverse is not necessarily true. A peritec-tic is not always associated with an inter-mediate phase. See, for example, Fig. 1-13(i).

For purposes of phase diagram calcula-tions involving stoichiometric compoundssuch as CaSiO3, we may, to a good approx-imation, consider the Gibbs energy curve,g0.5(CaSiO3) , to have zero width. All that is then required is the value of g0.5 (CaSiO3)

at the minimum. This value is usually expressed in terms of the Gibbs energy of fusion of the compound, Dg0

f (0.5 CaSiO3)

or the Gibbs energy of formationDg0

form(0.5 CaSiO3) of the compound from the pure solid components CaO and SiO2

according to the reaction: 0.5 CaO(sol) +0.5 SiO2(sol) = 0.5 CaSiO3(sol). Both thesequantities are interpreted graphically inFig. 1-14.

1.5.11 Limited Mutual Solubility –Ideal Henrian Solutions

In Sec. 1.5.6, the region of two solids inthe MgO–CaO phase diagram of Fig. 1-12was described as a miscibility gap. That is,only one continuous gs curve was assumed.If, somehow, the appearance of the liquidphase could be suppressed, then the twosolvus lines in Fig. 1-12, when projectedupwards, would meet at a critical pointabove which one continuous solid solutionwould exist at all compositions.

Such a description is justifiable only ifthe pure solid components have the samecrystal structure, as is the case for MgOand CaO. However, consider the Ag–Mgsystem, Fig. 1-15, in which the terminal(Ag) solid solution is face-centered-cubicand the terminal (Mg) solid solution is hex-agonal-close-packed. In this case, one con-tinuous curve for gs cannot be drawn. Eachsolid phase must have its own separateGibbs energy curve, as shown schemati-cally in Fig. 1-15(b) for the h.c.p. (Mg)phase at 800 K. In this figure, gMg

0(h.c.p.) andgAg

0(f.c.c.) are the standard molar Gibbs ener-gies of pure h.c.p. Mg and pure f.c.c. Ag,while gAg

0(h.c.p.-Mg) is the standard molarGibbs energy of pure (hypothetical) h.c.p.Ag in the h.c.p. (Mg) phase.

Since the solubility of Ag in the h.c.p.(Mg) phase is limited we can, to a good ap-proximation, describe it as a Henrian idealsolution. That is, when a solution is suffi-ciently dilute in one component, we can ap-proximate gE

solute = RT ln gsolute by its valuein an infinitely dilute solution. That is, ifXsolute is small we set gsolute = g 0

solute whereg 0

solute is the Henrian activity coefficient atXsolute = 0. Thus, for sufficiently dilute solu-tions we assume that gsolute is independentof composition. Physically, this means thatin a very dilute solution there is negligibleinteraction among solute particles becausethey are so far apart. Hence, each addi-tional solute particle added to the solutionproduces the same contribution to the ex-cess Gibbs energy of the solution and so gE-solute = dGE/dnsolute = constant.

From the Gibbs–Duhem equation, Eq.(1-56), if dgE

solute = 0, then dgEsolvent = 0.

Hence, in a Henrian solution gsolute is alsoconstant and equal to its value in an infi-nitely dilute solution. That is, gsolute =1 andthe solvent behaves ideally. In summarythen, for dilute solutions (Xsolvent ≈1)Henry’s Law applies:

gsolvent ≈ 1

gsolute ≈ g 0solute = constant (1-79)

(Care must be exercised for solutions otherthan simple substitutional solutions. Henry’sLaw applies only if the ideal activity is definedcorrectly, as will be discussed in Sec. 1.10).

Treating, then, the h.c.p. (Mg) phase in the Ag–Mg system (Fig. 1-15(b)) as aHenrian solution we write:

gh.c.p. = (XAg gAg0(f.c.c.) + XMg gMg

0(h.c.p.))

+ RT (XAg ln aAg + XMg ln aMg)

= (XAg gAg0(f.c.c.) + XMg gMg

0(h.c.p.)) (1-80)

+ RT (XAg ln (g 0Ag XAg) + XMg ln XMg)

where aAg and g 0Ag are the activity and ac-

tivity coefficient of silver with respect topure f.c.c. silver as standard state. Let usnow combine terms as follows:

gh.c.p. = [XAg (gAg0(f.c.c.) + RT ln g 0

Ag)

+ XMg gMg0(h.c.p.)] (1-81)

+ RT (XAg ln XAg + XMg ln XMg)

Since g 0Ag is independent of composition,

let us define:

gAg0(h.c.p.-Mg) = (gAg

0(f.c.c.) + RT ln g 0Ag) (1-82)

From Eqs. (1-81) and (1-82) it can be seenthat, relative to gMg

0(h.c.p.) and to the hypothet-ical standard state gAg

0(h.c.p.-Mg) defined inthis way, the h.c.p. solution is ideal. Eqs.(1-81) and (1-82) are illustrated in Fig. 1-15(b). It can be seen that as g 0

Ag becomeslarger, the point of tangency N moves to higher Mg concentrations. That is, as(gAg

0(h.c.p.-Mg) – gAg0(f.c.c.)) becomes more posi-

tive, the solubility of Ag in h.c.p. (Mg) de-creases.

It must be stressed that gAg0(h.c.p.-Mg) as de-

fined by Eq. (1-82) is solvent-dependent.That is, gAg

0(h.c.p.-Mg) is not the same as, say,gAg

0(h.c.p.-Cd) for Ag in dilute h.c.p. (Cd) solidsolutions.



Henrian activity coefficients can usuallybe expressed as functions of temperature:

RT lngi0 = a – b T (1-83)

where a and b are constants. If data are lim-ited, it can further be assumed that b ≈ 0 sothat RT lngi

0 ≈ constant.

1.5.12 Geometry of Binary Phase Diagrams

The geometry of all types of phase dia-grams of any number of components isgoverned by the Gibbs Phase Rule.

Consider a system with C components inwhich P phases are in equilibrium. Thesystem is described by the temperature, thetotal pressure and the composition of eachphase. In a C-component system, (C –1) in-dependent mole fractions are required todescribe the composition of each phase(because S Xi =1). Hence, the total numberof variables required to describe the systemis [P (C –1) + 2]. However, as shown in Sec.1.4.2, the chemical potential of any compo-nent is the same in all phases (a, b, g, …)since the phases are in equilibrium. That is:

gia (T, P, X1

a, X2a, X3

a, …)

= gib (T, P, X1

b, X2b, X3

b, …)

= gig(T, P, X1

g, X2g, X3

g, …) = … (1-84)

where gia (T, P, X1

a, X2a, X3

a, …) is a func-tion of temperature, of total pressure, andof the mole fractions X1

a, X2a, X3

a, … in the a phase; and similarly for the otherphases. Thus there are C (P –1) indepen-dent equations in Eq. (1-84) relating thevariables.

Let F be the differences between thenumber of variables and the number ofequations relating them:

F = P (C – 1) + 2 – C (P – 1)F = C – P + 2 (1-85)

This is the Gibbs Phase Rule. F is calledthe number of degrees of freedom or vari-ance of the system and is the number of pa-rameters which can and must be specifiedin order to completely specify the state ofthe system.

Binary temperature–composition phasediagrams are plotted at a fixed pressure,usually 1 bar. This then eliminates one de-gree of freedom. In a binary system, C = 2.Hence, for binary isobaric T – X diagramsthe phase rule reduces to:

F = 3 – P (1-86)

Binary T – X diagrams contain single-phase areas and two-phase areas. In the sin-gle-phase areas, F = 3 – 1 = 2. That is, tem-perature and composition can be specifiedindependently. These regions are thuscalled bivariant. In two-phase regions,F = 3 – 2 = 1. If, say, T is specified, then thecompositions of both phases are deter-mined by the ends of the tie-lines. Two-phase regions are thus termed univariant.Note that the overall composition can bevaried within a two-phase region at con-stant T, but the overall composition is not aparameter in the sense of the phase rule.Rather, it is the compositions of the indi-vidual phases at equilibrium that are theparameters to be considered in counting thenumber of degrees of freedom.

When three phases are at equilibrium ina binary system at constant pressure,F = 3 – 3 = 0. Hence, the compositions ofall three phases, as well as T, are fixed.There are two general types of three-phaseinvariants in binary phase diagrams. Theseare the eutectic-type and peritectic-typeinvariants as illustrated in Fig. 1-16. Letthe three phases concerned be called a, band g, with b as the central phase as shownin Fig. 1-16. The phases a, b and g can besolid, liquid or gaseous. At the eutectic-type invariant, the following invariant re-

action occurs isothermally as the system iscooled:

b Æ a + g (1-87)

whereas at the peritectic-type invariant theinvariant reaction upon cooling is:

a + g Æ b (1-88)

Some examples of eutectic-type invari-ants are: (i) eutectics (Fig. 1-12) in whicha = solid1, b = liquid, g = solid2; the eutecticreaction is l Æ s1 + s2; (ii) monotectics (Fig.1-14) in which a = liquid1, b = liquid2, g = -solid; the monotectic reaction is l2 Æ l1 + s;(iii) eutectoids in which a =solid1, b = solid2, g = solid3; the eutectoidreaction is s2 Æ s1 + s3; (iv) catatectics inwhich a = liquid, b = solid1, g = solid2; thecatatectic reaction is s1Æ l + s2.

Some examples of peritectic-type invari-ants are: (i) peritectics (Fig. 1-15) in whicha = liquid, b = solid1, g = solid2. The peri-

tectic reaction is l + s2 Æ s1; (ii) syntectics(Fig. 1-13(k)) in which a = liquid1, b =solid, g = liquid2. The syntectic reaction isl1 + l2 Æ s; (iii) peritectoids in which a =solid1, b = solid2, g = solid3. The peritec-toid reaction is s1 + s3 Æ s2.

An important rule of construction whichapplies to invariants in binary phase dia-grams is illustrated in Fig. 1-16. This ex-tension rule states that at an invariant theextension of a boundary of a two-phase re-gion must pass into the adjacent two-phaseregion and not into a single-phase region.Examples of both correct and incorrectconstructions are given in Fig. 1-16. Tounderstand why the “incorrect extensions”shown are not right consider that the (a + g)phase boundary line indicates the composi-tion of the g-phase in equilibrium with thea-phase, as determined by the commontangent to the Gibbs energy curves. Sincethere is no reason for the Gibbs energycurves or their derivatives to change dis-continuously at the invariant temperature,the extension of the (a + g) phase boundaryalso represents the stable phase boundaryunder equilibrium conditions. Hence, forthis line to extend into a region labeled assingle-phase g is incorrect.

Two-phase regions in binary phase dia-grams can terminate: (i) on the pure com-ponent axes (at XA = 1 or XB = 1) at a trans-formation point of pure A or B; (ii) at acritical point of a miscibility gap; (iii) at aninvariant. Two-phase regions can also ex-hibit maxima or minima. In this case, bothphase boundaries must pass through theirmaximum or minimum at the same point asshown in Fig. 1-16.

All the geometrical units of constructionof binary phase diagrams have now beendiscussed. The phase diagram of a binaryalloy system will usually exhibit several ofthese units. As an example, the Fe–Mophase diagram (Kubaschewski, 1982) is


Figure 1-16. Some geometrical units of binary phasediagrams, illustrating rules of construction.


shown in Fig. 1-17. The invariants in thissystem are peritectics at 1540, 1488 and1450 °C; eutectoids at 1235 and 1200 °C;peritectoids at 1370 and 950 °C. The two-phase (liquid + g) region passes through aminimum at XMo = 0.2.

Between 910 °C and 1390 °C is a two-phase (a + g) g-loop. Pure Fe adopts thef.c.c. g structure between these two temper-atures but exists as the b.c.c. a phase athigher and lower temperatures. Mo, how-ever, is more soluble in the b.c.c. than in the f.c.c. structure. That is, gMo

0(b.c.c.-Fe)

< gMo0(f.c.c.-Fe) as discussed in Sec. 1.5.11.

Therefore, small additions of Mo stabilizethe b.c.c. structure.

In the CaO–SiO2 phase diagram, Fig. 1-14, we observe eutectics at 1439, 1466

and 2051°C; a monotectic at 1692 °C; anda peritectic at 1469 °C. The compoundCa3SiO5 dissociates upon heating to CaOand Ca2SiO4 by a peritectoid reaction at1789 °C and dissociates upon cooling toCaO and Ca2SiO4 by a eutectoid reaction at1250 °C. Maxima are observed at 2130 and1540 °C. At 1470 °C there is an invariantassociated with the tridymite Æ cristobalitetransition of SiO2. This is either a peritec-tic or a catatectic depending upon the rela-tive solubility of CaO in tridymite and cris-tobalite. However, these solubilities arevery small and unknown.

Figure 1-17. Fe–Mophase diagram at P = 1 bar(Kubaschewski, 1982).

1.6 Application of Thermodynamics to Phase Diagram Analysis

1.6.1 Thermodynamic/Phase DiagramOptimization

In recent years the development of solution models, numerical methods andcomputer software has permitted a quanti-tative application of thermodynamics tophase diagram analysis. For a great manysystems it is now possible to perform a simultaneous critical evaluation of avail-able phase diagram measurements and ofavailable thermodynamic data (calorimet-ric data, measurements of activities, etc.)with a view to obtaining optimized equa-tions for the Gibbs energies of each phasewhich best represent all the data. Theseequations are consistent with thermody-namic principles and with theories of solu-tion behavior.

The phase diagram can be calculatedfrom these thermodynamic equations, andso one set of self-consistent equations de-scribes all the thermodynamic propertiesand the phase diagram. This technique ofanalysis greatly reduces the amount of ex-perimental data needed to fully character-ize a system. All data can be tested forinternal consistency. The data can be inter-polated and extrapolated more accuratelyand metastable phase boundaries can becalculated. All the thermodynamic proper-ties and the phase diagram can be repre-sented and stored by means of a small setof coefficients.

Finally, and most importantly, it is oftenpossible to estimate the thermodynamicproperties and phase diagrams of ternaryand higher-order systems from the assessedparameters for their binary sub-systems, aswill be discussed in Sec. 1.11. The analysisof binary systems is thus the first and most

important step in the development of data-bases for multicomponent systems.

1.6.2 Polynomial Representationof Excess Properties

Empirical equations are required to ex-press the excess thermodynamic propertiesof the solution phases as functions of com-position and temperature. For many simplebinary substitutional solutions, a good rep-resentation is obtained by expanding theexcess enthalpy and entropy as polynomi-als in the mole fractions XA and XB of thecomponents:

hE = XA XB [h0 + h1 (XB – XA) (1-89)

+ h2 (XB – XA)2 + h3 (XB – XA)3 + …]

sE = XA XB [s0 + s1 (XB – XA) (1-90)

+ s2 (XB – XA)2 + s3 (XB – XA)3 + …]

where the hi and si are empirical coeffi-cients. As many coefficients are used as are required to represent the data in a given system. For most systems it is a goodapproximation to assume that the coeffi-cients hi and si are independent of tempera-ture.

If the series are truncated after the firstterm, then:

gE = hE – T sE = XA XB (h0 – T s0) (1-91)

This is the equation for a regular solutiondiscussed in Sec. 1.5.7. Hence, the polyno-mial representation can be considered to bean extension of regular solution theory.When the expansions are written in termsof the composition variable (XB – XA), as inEqs. (1-89) and (1-90), they are said to bein Redlich–Kister form. Other equivalentpolynomial expansions such as orthogonalLegendre series have been discussed byPelton and Bale (1986).

Differentiation of Eqs. (1-89) and (1-90)and substitution into Eq. (1-55) yields the


1.6 Application of Thermodynamics to Phase Diagram Analysis 35

following expansions for the partial excessenthalpies and entropies:

hAE = XB

2 Si=0

hi [(XB – XA)i

– 2 i XA (XB – XA)i –1] (1-92)

hBE = XA

2 Si=0

hi [(XB – XA)i

+ 2 i XB (XB – XA)i –1] (1-93)

sAE = XB

2 Si=0

si [(XB – XA)i

– 2 i XA (XB – XA)i –1] (1-94)

sBE = XA

2 Si=0

si [(XB – XA)i

+ 2 i XB (XB – XA)i –1] (1-95)

Partial excess Gibbs energies, giE, are

then given by Eq. (1-52).Eqs. (1-89) and (1-90), being based upon

regular solution theory, give an adequaterepresentation for most simple substitu-tional solutions in which deviations fromideal behavior are not too great. In othercases, more sophisticated models are re-quired, as discussed in Sec. 1.10.

1.6.3 Least-Squares Optimization

Eqs. (1-89), (1-90) and (1-92) to (1-95)are linear in terms of the coefficients.Through the use of these equations, all integral and partial excess properties (gE,hE, sE, gi

E, hiE, si

E) can be expressed by linear equations in terms of the one set ofcoefficients hi , si. It is thus possible toinclude all available experimental data fora binary phase in one simultaneous linearleast-squares optimization. Details havebeen discussed by Bale and Pelton (1983),Lukas et al. (1977) and Dörner et al.(1980).

The technique of coupled thermody-namic/phase diagram analysis is best illus-trated by examples.

The phase diagram of the LiF–NaFsystem is shown in Fig. 1-18. Data pointsmeasured by Holm (1965) are shown onthe diagram. The Gibbs energy of fusion of

each pure component at temperature T isgiven by:

where Dh0f (Tf) is the enthalpy of fusion at

the melting point Tf , and c lp and cs

p are theheat capacities of the pure liquid and solid.The following values are taken from Barinet al. (1977):

Dg0f (LiF) = 14.518 + 128.435 T

+ 8.709 ¥ 10–3 T 2 – 21.494 T ln T

– 2.65 ¥ 105 T –1 J/mol (1-97)

Dg0f (NaF) = 10.847 + 156.584 T

+ 4.950 ¥ 10–3 T 2 – 23.978 T ln T

– 1.07 ¥ 105 T –1 J/mol (1-98)

Thermodynamic properties along the liq-uidus and solidus are related by equationslike Eqs. (1-64) and (1-65). Taking theideal activities to be equal to the mole frac-tions:

R T ln Xil – R T ln Xi

s + giE(l) – gi

E(s)

= – Dg0f (i) (1-99)

D Dgf0

f f

pl

ps

=

d (1- )

f

f

h T T

c c T T

T

T

T

( ) ( / )

( ) ( / )

0 1

1 1 96

−

+ − −∫

Figure 1-18. LiF–NaF phase diagram at P = 1 barcalculated from optimized thermodynamic parame-ters (Sangster and Pelton, 1987). Points are experi-mental from Holm (1965). Dashed line is theoreticallimiting liquidus slope for negligible solid solubility.

where i = LiF or NaF. Along the LiF-richliquidus, the liquid is in equilibrium withessentially pure solid LiF. Hence, X s

LiF = 1and gLif

E(s) = 0. Eq. (1-99) then reduces to:

R T ln X lLiF + gLif

E(l) = – Dg0f (LiF) (1-100)

From experimental values of X lLiF on the

liquidus and with Eq. (1-97) for Dg0f (LiF) ,

values of gLifE(l) at the measured liquidus

points can be calculated from Eq. (1-100).Along the NaF-rich solidus the solid so-

lution is sufficiently concentrated in NaFthat Henrian behavior (Sec. 1.5.11) can beassumed. That is, for the solvent, gNaF

E(s) = 0.Hence, Eq. (1-99) becomes:

R T ln X (l)NaF – R T ln X (s)

NaF + gNaFE(l)

= – Dg0f (NaF) (1-101)

Thus, from the experimental liquidus andsolidus compositions and with the Gibbsenergy of fusion from Eq. (1-98), values ofgNaF

E(l) can be calculated at the measured liq-uidus points from Eq. (1-101).

Finally, enthalpies of mixing, hE, in theliquid have been measured by calorimetryby Hong and Kleppa (1976).

Combining all these data in a least-squares optimization, the following expres-sions for the liquid were obtained by Sang-ster and Pelton (1987):

hE(l) = XLiF XNaF (1-102)

¥ [– 7381 + 184 (XNaF – XLiF)] J/mol

sE(l) = XLiF XNaF (1-103)

¥ [– 2.169 – 0.562 (XNaF – XLiF)] J/mol

Eqs. (1-102) and (1-103) then permit allother integral and partial properties of theliquid to be calculated.

For the NaF-rich Henrian solid solution,the solubility of LiF has been measured byHolm (1965) at the eutectic temperaturewhere the NaF-rich solid solution is inequilibrium with pure solid LiF. That is,

aLiF =1 with respect to pure solid LiF asstandard state. In the Henrian solution atsaturation,

aLiF = g0LiF XLiF = g0

LiF (1 – 0.915) = 1

Hence, the Henrian activity coefficient in the NaF-rich solid solution at 649 °C is g0

LiF = 11.76. Since no solubilities havebeen measured at other temperatures, weassume that:

R T ln g0LiF = R (922) ln (11.76) (1-104)

= 18 900 J/mol = constant

Using the notation of Eq. (1-82):

gLiF0(s, NaF) = gLiF

0(s) + 18 900 J/mol (1-105)

where gLiF0(s) is the standard Gibbs energy of

solid LiF, and gLiF0(s, NaF) is the hypothetical

standard Gibbs energy of LiF dissolved insolid NaF.

The phase diagram drawn in Fig. 1-18was calculated from Eqs. (1-97) to (1-104).Complete details of the analysis of theLiF–NaF system are given by Sangster andPelton (1987).

As a second example of thermodynamic/phase diagram optimization, consider theCd–Na system. The phase diagram, withpoints measured by several authors (Math-ewson, 1906; Kurnakow and Kusnetzow,1907; Weeks and Davies, 1964) is shown inFig. 1-19.

From electromotive force measurementson alloy concentration cells, several au-thors have measured the activity coeffi-cient of Na in liquid alloys. The data are shown in Fig. 1-20 at 400 °C. From the temperature dependence of gE

Na =RT lngNa, the partial enthalpy of Na in theliquid was obtained via Eq. (1-52). The re-sults are shown in Fig. 1-21. Also, hE of theliquid has been measured by Kleinstuber(1961) by direct calorimetry. These ther-modynamic data for gE

Na, hENa and hE were


1.6 Application of Thermodynamics to Phase Diagram Analysis 37

Figure 1-19. Cd–Na phase diagram at P = 1 bar calculated from optimized thermodynamic parameters (Re-printed from Pelton, 1988a). Kurnakow and Kusnetzow (1907), Mathewson (1906), × Weeks and Davies(1964).

Figure 1-20. Sodium ac-tivity coefficient in liquidCd–Na alloys at 400°C.Line is calculated fromoptimized thermodynamicparameters (Reprintedfrom Pelton, 1988a). Hauffe (1940), Lantratov and

Mikhailova (1971), Maiorova et al. (1976), Alabyshev and

Morachevskii (1957), Bartlett et al. (1970).

optimized simultaneously (Pelton, 1988a)to obtain the following expressions for hE

and sE of the liquid:

hE(l) = XCd XNa [– 12 508 + 20 316 (1-106)¥ (XNa – XCd) – 8714 (XNa – XCd)2] J/mol

sE(l) = XCd XNa [– 15.452 + 15.186 (1-107)¥ (XNa – XCd) – 10.062 (XNa – XCd)2

– 1.122 (XNa – XCd)3] J/mol K

Eq. (1-106) reproduces the calorimetricdata within 200 J/mol–1. Eqs. (1-52), (1-58), (1-93) and (1-95) can be used to calcu-late hE

Na and gNa. The calculated curves arecompared to the measured points in Figs.1-20 and 1-21.

For the two compounds, Gibbs energiesof fusion were calculated (Pelton, 1988a)so as to best reproduce the measured phasediagram:

Dg0f (1/13 Cd11Na2) = 6816 – 10.724 T J/g-atom

(1-108)

Dg0f (1/3 Cd2Na) = 8368 – 12.737 T J/g-atom

(1-109)

The optimized enthalpies of fusion of 6816and 8368 J/g-atom agree within error lim-its with the values of 6987 and 7878 J/g-atom measured by Roos (1916). (See Fig.1-14 for an illustration of the relationbetween the Gibbs energy of fusion of acompound and the phase diagram.)

The phase diagram shown in Fig. 1-19was calculated from Eqs. (1-106) to (1-109) along with the Gibbs energies of fu-sion of Cd and Na taken from the literature(Chase, 1983). Complete details of theanalysis of the Cd–Na system are given byPelton (1988a).

It can thus be seen that one simple set ofequations can simultaneously and self-con-sistently describe all the thermodynamicproperties and the phase diagram of a bi-nary system.

The exact optimization procedure willvary from system to system dependingupon the type and accuracy of the avail-able data, the number of phases present, theextent of solid solubility, etc. A large num-ber of optimizations have been publishedin the Calphad Journal (Pergamon) since1977.


Figure 1-21. Partial excess enthalpy of sodium inliquid Cd–Na alloys. Line is calculated from opti-mized thermodynamic parameters (Reprinted fromPelton, 1988a). Lantratov and Mikhailova (1971), Maiorova et al. (1976), Bartlett et al. (1970).

1.7 Ternary and Multicomponent Phase Diagrams 39

1.6.4 Calculation of Metastable PhaseBoundaries

In the Cd–Na system just discussed, theliquid exhibits positive deviations fromideal mixing. That is, gE(l) > 0. This fact isreflected in the very flat liquidus in Fig. 1-19 as was discussed in Sec. 1.5.9.

By simply not including any solid phasesin the calculation, the metastable liquidmiscibility gap as well as the spinodalcurve (Sec. 1.5.5) can be calculated asshown in Fig. 1-19. These curves are im-portant in the formation of metallic glassesby rapid quenching.

Other metastable phase boundaries suchas the extension of a liquidus curve below aeutectic can also be calculated thermody-namically by simply excluding one or morephases during the computations.

1.7 Ternary and MulticomponentPhase Diagrams

This section provides an introduction toternary phase diagrams. For a more de-tailed treatment, see Prince (1966); Ricci(1964); Findlay (1951); or West (1965).

1.7.1 The Ternary Composition Triangle

In a ternary system with componentsA–B–C, the sum of the mole fractions isunity, (XA + XB + XC) = 1. Hence, there aretwo independent composition variables. Arepresentation of composition, symmetri-cal with respect to all three components,may be obtained with the equilateral “com-position triangle” as shown in Fig. 1-22 forthe Bi–Sn–Cd system. Compositions atthe corners of the triangle correspond to thepure components. Along the edges of thetriangle compositions corresponding to thethree binary subsystems Bi–Sn, Sn–Cd

and Cd–Bi are found. Lines of constantmole fraction XBi are parallel to the Sn–Cdedge, while lines of constant XSn and XCd

are parallel to the Cd–Bi and Bi–Sn edgesrespectively. For example, at point a in Fig.1-22, XBi = 0.05, XSn = 0.45 and XCd = 0.50.

Similar equilateral composition trianglescan be drawn with coordinates in terms ofwt.% of the three components.

1.7.2 Ternary Space Model

A ternary temperature–composition “phase diagram” at constant total pressuremay be plotted as a three-dimensional “space model” within a right triangularprism with the equilateral composition tri-angle as base and temperature as verticalaxis. Such a space model for a simple eu-tectic ternary system A–B–C is illustratedin Fig. 1-23. On the three vertical faces ofthe prism we find the phase diagrams of thethree binary subsystems, A–B, B–C andC–A which, in this example, are all simpleeutectic binary systems. The binary eutec-tic points are e1, e2 and e3. Within the prism we see three liquidus surfaces de-scending from the melting points of pureA, B and C. Compositions on these sur-faces correspond to compositions of liquidin equilibrium with A-, B- and C-rich solidphases.

In a ternary system at constant pressure,the Gibbs phase rule, Eq. (1-85), becomes:

F = 4 – P (1-110)

When the liquid and one solid phase are inequilibrium P = 2. Hence F = 2 and thesystem is bivariant. A ternary liquidus isthus a two-dimensional surface. We maychoose two variables, say T and one composition coordinate of the liquid, butthen the other liquid composition coordi-nate and the composition of the solid arefixed.


Figure 1-22. Projection of the liquidus surface of the Bi–Sn–Cd system onto the ternary composition triangle(after Bray et al., 1961–1962). Small arrows show the crystallization path of an alloy of overall composition atpoint a. (Reprinted from Pelton, 1996.)

Figure 1-23. Perspective view of ternary spacemodel of a simple eutectic ternary system. e1, e2, e3

are the binary eutectics and E is the ternary eutectic.The base of the prism is the equilateral compositiontriangle. (Reprinted from Pelton, 1983.)


The A- and B-liquidus surfaces in Fig. 1-23 intersect along the line e1 E. Liquidswith compositions along this line are there-fore in equilibrium with A-rich and B-richsolid phases simultaneously. That is, P = 3and so F –1. Such “valleys” are thus calledunivariant lines. The three univariant linesmeet at the ternary eutectic point E atwhich P = 4 and F = 0. This is an invariantpoint since the temperature and the compo-sitions of all four phases in equilibrium arefixed.

1.7.3 Polythermal Projections of Liquidus Surfaces

A two-dimensional representation of theternary liquidus surface may be obtained asan orthogonal projection upon the basecomposition triangle. Such a polythermalprojection of the liquidus of the Bi–Sn–Cdsystem (Bray et al., 1961–62) is shown inFig. 1-22. This is a simple eutectic ternarysystem with a space model like that shownin Fig. 1-23. The constant temperaturelines on Fig. 1-22 are called liquidus iso-therms. The univariant valleys are shownas heavier lines. By convention, the largearrows indicate the directions of decreas-ing temperature along these lines.

Let us consider the sequence of eventsoccurring during the equilibrium coolingfrom the liquid of an alloy of overall com-position a in Fig. 1-22. Point a lies withinthe field of primary crystallization of Cd.That is, it lies within the composition re-gion in Fig. 1-22 in which Cd-rich solidwill be the first solid to precipitate uponcooling. As the liquid alloy is cooled, theCd-liquidus surface is reached at T ≈ 465 K(slightly below the 473 K isotherm). Asolid Cd-rich phase begins to precipitate atthis temperature. Now, in this particularsystem, Bi and Sn are nearly insoluble insolid Cd, so that the solid phase is virtually

pure Cd. (Note that this fact cannot be de-duced from Fig. 1-22 alone.) Therefore, assolidification proceeds, the liquid becomesdepleted in Cd, but the ratio XSn /XBi in theliquid remains constant. Hence, the compo-sition path followed by the liquid (its crys-tallization path) is a straight line passingthrough point a and projecting to the Cd-corner of the triangle. This crystallizationpath is shown on Fig. 1-22 as the line ab.

In the general case in which a solid solu-tion rather than a pure component or stoi-chiometric compound is precipitating, thecrystallization path will not be a straightline. However, for equilibrium cooling, astraight line joining a point on the crystal-lization path at any T to the overall compo-sition point a will extend through the com-position, on the solidus surface, of the solidphase in equilibrium with the liquid at thattemperature.

When the composition of the liquid hasreached point b in Fig. 1-22 at T ≈ 435 K,the relative proportions of the solid Cd andliquid phases at equilibrium are given bythe lever rule applied to the tie-line dab: (moles of liquid)/(moles of Cd) = da /ab.Upon further cooling, the liquid composi-tion follows the univariant valley from b toE while Cd and Sn-rich solids coprecipitateas a binary eutectic mixture. When the liquidus composition attains the ternary eu-tectic composition E at T ≈ 380 K the invar-iant ternary eutectic reaction occurs:

liquid Æ s1 + s2 + s3 (1-111)

where s1, s2 and s3 are the three solidphases and where the compositions of allfour phases (as well as T ) remain fixed un-til all liquid is solidified.

In order to illustrate several of the fea-tures of polythermal projections of liquidussurfaces, a projection of the liquidus of ahypothetical system A–B–C is shown inFig. 1-24. For the sake of simplicity, iso-

therms are not shown, only the univariantlines with arrows to show the directions ofdecreasing temperature. The binary sub-systems A–B and C–A are simple eutecticsystems, while the binary subsystem B–Ccontains one congruent binary phase, e,and one incongruent binary phase, d, asshown in the insert in Fig. 1-24. The letterse and p indicate binary eutectic and peritec-tic points. The e and d phases are called bi-nary compounds since they have composi-tions within a binary subsystem. Two ter-nary compounds, h and z, with composi-tions within the ternary triangle, as indi-cated in Fig. 1-24, are also found in thissystem. All compounds, as well as puresolid A, B and C (the “a, b and g” phases),are assumed to be stoichiometric (i.e., thereis no solid solubility). The fields of pri-mary crystallization of all the solids are in-dicated in parentheses in Fig. 1-24. Thecomposition of the e phase lies within its

field, since e is a congruent compound,while the composition of the d phase liesoutside of its field since d is incongruent.Similarly for the ternary compounds, h is acongruently melting compound while z isincongruent. For the congruent compoundh, the highest temperature on the h liquidusoccurs at the composition of h.

The univariant lines meet at a number ofternary eutectics E (three arrows converg-ing), a ternary peritectic P (one arrow en-tering, two arrows leaving the point), andseveral ternary quasi-peritectics P¢ (twoarrows entering, one arrow leaving). Twosaddle points s are also shown. These arepoints of maximum T along the univariantline but of minimum T on the liquidus sur-face along a section joining the composi-tions of the two solids. For example, s1 is ata maximum along the univariant E1 P¢3, butis a minimum point on the liquidus alongthe straight line z s1 h.


Figure 1-24. Projection ofthe liquidus surface of asystem A–B–C. The binarysubsystems A–B and C–Aare simple eutectic systems.The binary phase diagramB–C is shown in the insert.All solid phases are assumedpure stoichiometric compo-nents or compounds. Smallarrows show crystallizationpaths of alloys of composi-tions at points a and b. (Re-printed from Pelton, 1983.)


Let us consider the events occurring dur-ing cooling from the liquid of an alloy ofoverall composition a in Fig. 1-24. The pri-mary crystallization product will be the ephase. Since this is a pure stoichiometricsolid the crystallization path of the liquidwill be along a straight line passingthrough a and extending to the compositionof e as shown in the figure.

Solidification of e continues until theliquid attains a composition on the univari-ant valley. Thereafter the liquid composi-tion follows the valley towards the point P¢1in co-existence with e and z. At point P¢1the invariant ternary quasi-peritectic reac-tion occurs isothermally:

liquid + e Æ d + z (1-112)

Since there are two reactants in a quasi-peritectic reaction, there are two possibleoutcomes: (i) the liquid is completely con-sumed before the e phase; in this case, so-lidification will be complete at the point P¢1;(ii) e is completely consumed before theliquid. In this case, solidification will con-tinue with decreasing T along the univari-ant line P¢1 E1 with co-precipitation of d andz until, at E, the liquid will solidify eutecti-cally (liquid Æ d + z + h). To determinewhether outcome (i) or (ii) occurs, we usethe mass balance criterion that, for three-phase equilibrium, the overall compositiona must always lie within the tie-triangleformed by the compositions of the threephases. Now, the triangle joining the com-positions of d, e and z does not contain thepoint a, but the triangle joining the compo-sitions of d, z and liquid at P¢1 does containthe point a. Hence, outcome (ii) occurs.

An alloy of overall composition b in Fig.1-24 solidifies with e as primary crystal-lization product until the liquid composi-tion contacts the univariant line. There-after, co-precipitation of e and b occurswith the liquid composition following the

univariant valley until the liquid reachesthe peritectic composition P. The invariantternary peritectic reaction then occurs iso-thermally:

liquid + e + b Æ z (1-113)

Since there are three reactants, there arethree possible outcomes: (i) the liquid isconsumed before either e or b and solidifi-cation terminates at P; (ii) e is consumedfirst, solidification then continues alongthe path PP¢3; or (iii) b is consumed firstand solidification continues along the pathPP¢1. Which outcome occurs depends onwhether the overall composition b lieswithin the tie-triangle (i) e b z; (ii) b z P, or(iii) e z P. In the example shown, outcome(i) will occur.

1.7.4 Ternary Isothermal Sections

Isothermal projections of the liquidussurface do not provide information on thecompositions of the solid phases at equilib-rium. However, this information can bepresented at any one temperature on an iso-thermal section such as that shown for theBi–Sn–Cd system at 423 K in Fig. 1-25.This phase diagram is a constant tempera-ture slice through the space model of Fig.1-23.

The liquidus lines bordering the one-phase liquid region of Fig. 1-25 are identi-cal to the 423 K isotherms of the projectionin Fig. 1-22. Point c in Fig. 1-25 is point con the univariant line in Fig. 1-22. An alloywith overall composition in the one-phaseliquid region of Fig. 1-25 at 423 K willconsist of a single liquid phase. If the over-all composition lies within one of the two-phase regions, then the compositions of thetwo phases are given by the ends of the tie-line which passes through the overall com-position. For example, a sample with over-all composition p in Fig. 1-25 will consist

of a liquid of composition q on the liquidusand a solid Bi-rich alloy of composition ron the solidus. The relative proportions ofthe two phases are given by the lever rule:(moles of liquid)/(moles of solid) = pr /pq,where pr and pq are the lengths of the tie-line segments.

In the case of solid Cd, the solid phase isnearly pure Cd, so all tie-lines of the (Cd+liquid) region converge nearly to the cornerof the triangle. In the case of Bi- and Sn-rich solids, some solid solubility is ob-served. (The actual extent of this solubility

is somewhat exaggerated in Fig. 1-25 forthe sake of clarity of presentation.) Alloyswith overall compositions rich enough inBi or Sn to lie within the single-phase (Sn)or (Bi) regions of Fig. 1-25 will consist at423 K of single-phase solid solutions. Al-loys with overall compositions at 423 K inthe two-phase (Cd + Sn) region will consistof two solid phases.

Alloys with overall compositions withinthe three-phase triangle dc f will, at 423 K,consist of three phases: solid Cd- and Sn-rich solids with compositions at d and f and


Figure 1-25. Isothermal section of the Bi–Sn–Cd system at 423 K at P = 1 bar (after Bray et al., 1961–1962).Extents of solid solubility in Bi and Sn have been exaggerated for clarity of presentation. (Reprinted from Pel-ton, 1996.)


liquid of composition c. To understand thisbetter, consider an alloy of composition ain Fig. 1-25, which is the same composi-tion as the point a in Fig. 1-22. In Sec.1.7.3 we saw that when an alloy of thiscomposition is cooled, the liquid followsthe path ab in Fig. 1-22 with primary pre-cipitation of Cd and then follows the uni-variant line with co-precipitation of Cd andSn so that at 423 K the liquid is at the com-position point c, and two solid phases are inequilibrium with the liquid.

1.7.4.1 Topology of Ternary IsothermalSections

At constant temperature the Gibbs en-ergy of each phase in a ternary system isrepresented as a function of compositionby a surface plotted in a right triangularprism with Gibbs energy as vertical axisand the composition triangle as base. Justas the compositions of phases at equilib-rium in binary systems are determined by the points of contact of a common tan-gent line to their isothermal Gibbs energycurves, so the compositions of phases atequilibrium in a ternary system are givenby the points of contact of a common tan-gent plane to their isothermal Gibbs energysurfaces. A common tangent plane cancontact two Gibbs energy surfaces at an in-finite number of pairs of points, therebygenerating an infinite number of tie-lineswithin a two-phase region on an isothermalsection. A common tangent plane to threeGibbs energy surfaces contacts each sur-face at a unique point, thereby generating athree-phase tie-triangle.

Hence, the principal topological units ofconstruction of an isothermal ternary phasediagram are three-phase (a + b + g) tie-tri-angles as in Fig. 1-26 with their accompa-nying two-phase and single-phase areas.Each corner of the tie-triangle contacts a

single-phase region, and from each edge ofthe triangle there extends a two-phase re-gion. The edge of the triangle is a limitingtie-line of the two-phase region.

For overall compositions within the tie-triangle, the compositions of the threephases at equilibrium are fixed at the cor-ners of the triangle. The relative propor-tions of the three phases are given by thelever rule of tie-triangles, which can be de-rived from mass balance considerations. Atan overall composition q in Fig. 1-26 forexample, the relative proportion of the gphase is given by projecting a straight linefrom the g corner of the triangle (point c)through the overall composition q to theopposite side of the triangle, point p. Then:(moles of g)/(total moles) = q p /cp if com-positions are expressed in mole fractions,or (weight of g)/(total weight) = q p /cp ifcompositions are in weight percent.

Isothermal ternary phase diagrams aregenerally composed of a number of thesetopological units. An example for the Al–Zn–Mg system at 25 °C is shown in Fig. 1-27 (Köster and Dullenkopf, 1936). Theb, g, d, q, h and z phases are binary inter-metallic compounds with small (~1 to 6%)

Figure 1-26. A tie-triangle in a ternary isothermalsection illustrating the lever rule and the extensionrule.

ranges of stoichiometry which can dissolvea limited amount (~1 to 6%) of the thirdcomponent. The t phase is a ternary phasewith a single-phase region existing over afairly extensive oval-shaped central com-position range. Examination of Fig. 1-27shows that it consists of the topologicalunits of Fig. 1-26.

An extension rule, a case of Schreine-makers’ Rule (Schreinemakers, 1915), seeSec. 1.7.5, for ternary tie-triangles is illus-trated in Fig. 1-26. At each corner, the extension of the boundaries of the single-phase regions, indicated by the brokenlines, must either both project into the tri-angle as at point a, or must both projectoutside the triangle as at point b. Further-more, the angle between these extensionsmust be less than 180°. For a proof, seeLipson and Wilson (1940) or Pelton(1995).

Many published phase diagrams violatethis rule. For example, it is violated in Fig.1-27 at the d-corner of the (e + d + t) tie-tri-angle.

Another important rule of construction,whose derivation is evident, is that withinany two-phase region tie-lines must nevercross one another.

1.7.5 Ternary Isopleths (Constant Composition Sections)

A vertical isopleth, or constant composi-tion section through the space model of theBi–Sn–Cd system, is shown in Fig. 1-28.The section follows the line AB in Fig. 1-22.

The phase fields on Fig. 1-28 indicatewhich phases are present when an alloywith an overall composition on the line ABis equilibrated at any temperature. For ex-ample, consider the cooling, from the liq-uid state, of an alloy of composition awhich is on the line AB (see Fig. 1-22). AtT ≈ 465 K, precipitation of the solid (Cd)phase begins at point a in Fig. 1-28. AtT ≈ 435 K (point b in Figs. 1-22 and 1-28)the solid (Sn) phase begins to appear. Fi-nally, at the eutectic temperature TE, theternary reaction occurs, leaving solid (Cd)+ (Bi) + (Sn) at lower temperatures. Theintersection of the isopleth with the univar-iant lines on Fig. 1-22 occurs at points f andg which are also indicated on Fig. 1-28.The intersection of this isopleth with theisothermal section at 423 K is shown inFig. 1-25. The points s, t, u and v of Fig. 1-25 are also shown on Fig. 1-28.

It is important to note that on an isopleththe tie-lines do not, in general, lie in theplane of the diagram. Therefore, the dia-gram provides information only on whichphases are present, not on their composi-tions. The boundary lines on an isopleth donot in general indicate the phase composi-tions, only the temperature at which aphase appears or disappears for a givenoverall composition. The lever rule cannotbe applied on an isopleth.


Figure 1-27. Ternary isothermal section of the Al–Zn–Mg system at 25°C at P =1 bar (after Köster andDullenkopf, 1936). (Reprinted from Pelton, 1983.)


Certain geometrical rules apply to iso-pleths. As a phase boundary line is crossed,one and only one phase either appears ordisappears. This Law of Adjoining PhaseRegions (Palatnik and Landau, 1964) is il-lustrated by Fig. 1-28. The only apparentexception occurs for the horizontal invari-ant line at TE. However, if we consider thisline to be a degenerate infinitely narrowfour-phase region (L + (Cd) + (Bi) + (Sn)),then the law is also obeyed here.

Three or four boundary lines meet at in-tersection points. At an intersection point,such as point f or g, Schreinemakers’ Ruleapplies. This is discussed in Sec. 1.9.

Apparent exceptions to these rules (suchas, for example, five boundaries meeting atan intersection point) can occur if the sec-tion passes exactly through a node (such as a ternary eutectic point) of the spacemodel. However, these apparent exceptionsare really only limiting cases (see Prince,1963 or 1966).

1.7.5.1 Quasi-Binary Phase Diagrams

Several of the binary phase diagrams in the preceding sections (Figs. 1-5, 1-10, 1-12, 1-14, 1-18) are actually isopleths of ternary systems. For example, Fig. 1-12is an isopleth at constant XO = nO /(nMg +nCa + nO) = 0.5 of the Mg–Ca–O system.However, all tie-lines lie within (or virtu-ally within) the plane of the diagram be-cause XO = 0.5 in every phase. Therefore,the diagram is called a quasi-binary phasediagram.

1.7.6 Multicomponent Phase Diagrams

Only an introduction to multicomponentphase diagrams will be presented here. For more detailed treatments see Palatnikand Landau (1964), Prince (1963), Prince(1966) and Hillert (1998).

For systems of four or more components,two-dimensional sections are usually plot-

Figure 1-28. Isopleth (constant composition section) of the Bi–Sn–Cd system at P =1 bar following the lineAB at XSn = 0.45 of Fig. 1-22. (Reprinted from Pelton, 1996).

ted with one or more compositional vari-ables held constant. Hence these sectionsare similar to the ternary isopleths dis-cussed in Sec. 1.7.5. In certain cases, sec-tions at constant chemical potential of oneor more components (for example, at con-stant oxygen partial pressure) can be use-ful. These are discussed in Sec. 1.8.

Two sections of the Fe–Cr–V–C system(Lee and Lee, 1992) are shown in Figs. 1-29 and 1-30. The diagram in Fig. 1-29 is aT-composition section at constant Cr and Vcontent, while Fig. 1-30 is a section at con-stant T = 850 oC and constant C content of0.3 wt.%. The interpretation and topologi-cal rules of construction of these sectionsare the same as those for ternary isopleths,as discussed in Sec. 1.7.5. In fact, the samerules apply to a two-dimensional constant-composition section for a system of anynumber of components. The phase fieldson the diagram indicate the phases present

at equilibrium for an overall compositionlying on the section. Tie-lines do not, ingeneral, lie in the plane of the diagram, sothe diagram does not provide informationon the compositions or amounts of thephases present. As a phase boundary iscrossed, one and only one phase appears ordisappears (Law of Adjoining Phase Re-gions). If temperature is an axis, as in Fig.1-29, then horizontal invariants like theline AB in Fig. 1-29 can appear. These canbe considered as degenerate infinitely nar-row phase fields of (C + 1) phases, where Cis the number of components (for isobaricdiagrams). For example in Fig. 1-29, on theline AB, five phases are present. Three orfour phase boundaries meet at intersectionpoints at which Schreinemakers’ Rule ap-plies. It is illustrated by the extrapolationsin Fig. 1-29 at points a, b and c and in Fig.1-30 at points b, c, n, i and s (see discus-sions in Sec. 1.9).


Figure 1-29. Section of the Fe–Cr–V–C system at 1.5 wt.% Cr and 0.1 wt.% V at P =1 bar (Lee and Lee,1992).


1.7.7 Nomenclature for Invariant Reactions

As discussed in Sec. 1.5.12, in a binaryisobaric temperature–composition phasediagram there are two possible types of invariant reactions: “eutectic-type” (b Æ a+ g), and “peritectic type” (a + g Æ b). Ina ternary system, there are “eutectic-type”(a Æ b + g + d), “peritectic-type” (a + b +g Æ d), and “quasiperitectic-type” (a + bÆ g + d) invariants (Sec. 1.7.3). In asystem of C components, the number oftypes of invariant reaction is equal to C. Areaction with one reactant, such as(a Æ b + g + d + e) is clearly a “eutectic-type” invariant reaction but in general thereis no standard terminology. These reactionsare conveniently described according to thenumbers of reactants and products (in thedirection which occurs upon cooling).Hence the reaction (a + b Æ g + d + e) is a2Æ 3 reaction; the reaction (a Æ b + g

+ d) is a 1Æ 3 reaction; and so on. The ter-nary peritectic-type 3Æ 1 reaction (a + b+ g Æ d) is an invariant reaction in a ter-nary system, a univariant reaction in a qua-ternary system, a bivariant reaction in aquinary system, etc.

1.7.8 Reciprocal Ternary Phase Diagrams

A reciprocal ternary salt system is oneconsisting of two cations and two anions,such as the Na+, K+/F –, Cl– system of Fig.1-31. The condition of charge neutrality(nNa+ + nK+ = nF – + nCl–) removes one degreeof freedom. The system is thus quasiter-nary and its composition can be repre-sented by two variables, usually chosen as the cationic mole fraction XK = nK /(nNa + nK) and the anionic mole fractionXCl = nCl /(nF + nCl), where ni = number ofmoles of ion i. Note that XNa = (1– XK) andXF = (1– XCl).

Figure 1-30. Section ofthe Fe – Cr – V – C systemat 850 oC and 0.3 wt.% Cat P = 1 bar (Lee and Lee,1992).

The assumption has, of course, beenmade that the condition (nNa + nK = nF + nCl)holds exactly in every phase. If there is adeviation from this exact stoichiometry,then the phase diagram is no longer quasi-

ternary but is an isopleth of the four-com-ponent Na–K–F–Cl system, and tie-linesno longer necessarily lie in the plane of thediagram.

In Fig. 1-31 the cationic and anionicfractions are plotted as axes of a square.Compositions corresponding to the fourneutral salts (KF, KCl, NaCl, NaF) arefound at the corners of the square. Edges ofthe square correspond to the binary sub-systems such as NaF–NaCl. A ternaryspace model (analogous to Fig. 1-23) canbe constructed with temperature as verticalaxis. The phase diagram of Fig. 1-31 is apolythermal projection of the liquidus sur-face upon the composition square.

In this system, three of the binary edgesare simple eutectic systems, while theNaCl–KCl binary system exhibits a sol-idus/liquidus minimum. There is a ternaryeutectic at 570 °C in Fig. 1-31(b). TheNaF–KCl diagonal contains a saddle pointat 648 °C in Fig. 1-31(b). This saddle point is a eutectic of the quasibinary systemNaF–KCl. That is, a binary phase diagramNaF–KCl could be drawn with one simpleeutectic at 648 °C. However, the NaCl–KFsystem, which forms the other diagonal, isnot a quasibinary system. If compositionslying on this diagonal are cooled at equilib-rium from the liquid, solid phases whosecompositions do not lie on this diagonalcan precipitate. Hence, a simple binaryphase diagram cannot be drawn for theNaCl–KF system.

For systems such as Ca2+, Na+/F –, SO42–

in which the ions do not all have the samecharge, composition axes are convenientlyexpressed as equivalent ionic fractions(e.g. YCa = 2 nCa /(2 nCa + nNa)), see Sec.1.9.2.1.

The concept of reciprocal systems can begeneralized beyond simple salt systemsand is closely related to the sublatticemodel (Sec. 1.10.1).


Figure 1-31. Projection of the liquidus surface ofthe Na+, K+/F –, Cl– reciprocal ternary system.a) Calculated from optimized binary thermodynamic

parameters.b) As reported by Polyakov (1940).

1.8 Phase Diagrams with Potentials as Axes 51

For further discussion and references,see Pelton (1988b) and Blander (1964).

1.8 Phase Diagrams with Potentials as Axes

So far we have considered mainly iso-baric temperature–composition phase dia-grams. However there are many otherkinds of phase diagrams of interest in ma-terials science and technology with pres-sure, chemical potentials, volume, etc. asaxes. These can be classified into geomet-rical types according to their rules of con-struction.

For instance, binary isothermal P– X di-agrams as in Fig. 1-8 are members of thesame type as binary isobaric T– X diagramsbecause they are both formed from thesame topological units of construction.Other useful phase diagrams of this samegeometrical type are isothermal chemicalpotential–composition diagrams for ter-nary systems. An example is shown in the lowest panel of Fig. 1-32 (Pelton andThompson, 1975) for the Co–Ni–Osystem at T = 1600 K (and at a constant to-tal hydrostatic pressure of 1 bar). Here thelogarithm of the equilibrium partial pres-sure of O2 is plotted versus the metal ratiox = nNi /(nCo + nNi), where ni = number ofmoles of i. There are two phases in thissystem under these conditions, a solid alloysolution stable at lower pO2

, and a solid so-lution of CoO and NiO stable at higher pO2

.For instance, point a gives pO2

for the equi-librium between pure Co and pure CoO at1600 K. Between the two single-phase re-gions is a two-phase (alloy + oxide) region.At any overall composition on the tie-linecd between points c and d, two phases willbe observed, an alloy of composition d andan oxide of composition c. The lever ruleapplies just as for binary T– X diagrams.

The usual isothermal section of the ter-nary Co–Ni–O system at 1600 K is shownin the top panel of Fig. 1-32. There are twosingle-phase regions with a two-phase re-gion between them. The single-phase areasare very narrow because oxygen is onlyvery slightly soluble in the solid alloy and CoO and NiO are very stoichiometricoxides. In the central panel of Fig. 1-32 this same diagram is shown with the com-position triangle “opened up” by puttingthe oxygen corner at infinity. This can bedone if the vertical axis becomes h = nO /(nCo + nNi) with the horizontal axis asx = nNi /(nCo + nNi). These are known asJänecke coordinates. It can be seen in Fig.1-32 that each tie-line, e f, of the isothermalsection corresponds to a tie-line cd of the

Figure 1-32. Corresponding phase diagrams for theCo – Ni – O system at 1600 K (from Pelton andThompson, 1973).

log pO2– x diagram. This underscores the

fact that every tie-line of a ternary isother-mal section corresponds to a constant chem-ical potential of each of the components.

Another example of a log pO2– x diagram

is shown for the Fe–Cr–O system at1573 K in the lower panel of Fig. 1-33(Pelton and Schmalzried, 1973). The corre-sponding ternary isothermal section inJänecke coordinates is shown in the upperpanel. Each of the invariant three-phasetie-triangles in the isothermal section corresponds to an invariant line in the

log pO2– x diagram. For example, the (spi-

nel + (Fe, Cr) O + alloy) triangle with cor-ners at points a, b and c corresponds to the“eutectic-like” or eutecular invariant withthe same phase compositions a, b and c atlog pO2

≈ –10.7. We can see that within athree-phase tie-triangle, pO2

is constant.An example of yet another kind of phase

diagram of this same geometrical type isshown in Fig. 1-34. For the quaternaryFe–Cr–O2–SO2 system at T = 1273 K andat constant pSO2

= 10–7 bar, Fig. 1-34 is aplot of log pO2

versus the molar metal ratio


Figure 1-33. Correspond-ing phase diagrams for theFe–Cr–O system at 1573 K(Pelton and Schmalzried,1973). Experimental pointsfrom Katsura and Muan(1964).


x. Since log pO2varies as –1/2 log pS2

whenpSO2

and T are constant, Fig. 1-34 is also aplot of log pS2

versus x. Plotting T versus x at constant pO2

in theFe–Cr–O system, or at constant pO2

andpSO2

in the Fe–Cr–SO2–O2 system, willalso result in phase diagrams of this same

geometrical type. Often for ceramicsystems, we encounter “binary” phase dia-grams such as that for the “CaO–Fe2O3”system in Fig. 1-35, which has been takenfrom Phillips and Muan (1958). How arewe to interpret such a diagram? How, forinstance, do we interpret the composition

Figure 1-34. Calculated phase diagram of log pO2versus molar metal ratio at T = 1273.15 K and pSO2

= 10–7 barfor the Fe–Cr–SO2–O2 system.

Figure 1-35. Phase diagram for the“CaO–Fe2O3” system in air ( pO2

=0.21 bar) from Phillips and Muan(1958) (Reprinted by permission of theAmerican Ceramic Society from Levinet al., 1964).

axis when applied to the magnetite phase?In light of the preceding discussion, it canbe seen that such diagrams are really T– xplots at constant pO2

, where x is the metalratio in any phase. The diagram will be dif-ferent at different oxygen partial pressures.If pO2

is not fixed, the diagram cannot beinterpreted.

It can be seen that the diagrams dis-cussed above are of the same geometricaltype as binary T– X diagrams because theyare all composed of the same geometricalunits of construction as in Fig. 1-16. Theirinterpretation is thus immediately clear to anyone familiar with binary T– X dia-grams. Chemical potential–composition


Figure 1-36. Pressure-temperature phase diagram ofH2O.

Figure 1-37. Corresponding phase diagrams for the Fe – O system at PTOTAL = 1 bar (after Muan and Osborn,1965).


diagrams (Figs. 1-32 to 1-34) are useful inthe study of hot corrosion, metallurgicalroasting processes, chemical vapor deposi-tion, and many aspects of materials pro-cessing.

Another important geometrical type ofphase diagram is exemplified by P – Tphase diagrams for one-component sys-tems, as shown for H2O in Fig. 1-36. Insuch diagrams (see also Chapter 10 byKunz (2001)) bivariant single-phase re-gions are indicated by areas, univarianttwo-phase regions by lines, and invariantthree-phase regions by triple points. Animportant rule of construction is the exten-sion rule, which is illustrated by the brokenlines in Fig. 1-36. At a triple point, the extension of any two-phase line must passinto the single-phase region of the thirdphase. Clearly, the predominance diagrams

of Figs. 1-1 to 1-3 are of this same geomet-rical type.

As yet another example of this geomet-rical type of diagram, a plot of RT ln pO2

versus T for the Fe–O system is shown inFig. 1-37(b). Again, one-, two- and three-phase regions are indicated by areas, linesand triple points respectively. Fig. 1-37(a)is the binary T–composition phase diagramfor the Fe–O system. The correspondencebetween Figs. 1-37(a) and 1-37(b) is evi-dent. Each two-phase line of Fig. 1-37(b)“opens up” to a two-phase region of Fig. 1-37(a). Each tie-line of a two-phase re-gion in Fig. 1-37(a) can thus be seen tocorrespond to a constant pO2

. Triple pointsin Fig. 1-37(b) become horizontal invari-ant lines in Fig. 1-37(a).

Yet another type of phase diagram isshown in Fig. 1-38. This is an isothermal

Figure 1-38. Phase diagram of log pS2versus log pO2

at 1273 K and constant molar metal ratio nCr /(nFe + nCr) =0.5 in the Fe – Cr – S2 – O2 system.

section at constant molar metal rationCr /(nFe + nCr) = 0.5 for the Fe–Cr–S2–O2

system. This diagram was calculated ther-modynamically from model parameters.The axes are the equilibrium sulfur andoxygen partial pressures. Three or fourboundary lines can meet at an intersectionpoint. Some of the boundary lines on Fig.1-38 separate a two-phase region (a + b)from another two-phase region (a + g).These lines thus represent the conditionsfor three-phase (a + b + g) equilibrium.

1.9 General Phase Diagram Geometry

Although the various phase diagramsshown in the preceding sections may ap-pear to have quite different geometries, itcan be shown that, in fact, all true phase di-agram sections obey the same set of geo-metrical rules. Although these rules do notapply directly to phase diagram projectionssuch as Figs. 1-22, 1-24 and 1-31, such di-agrams can be considered to consist of por-tions of several phase diagram sectionsprojected onto a common plane.

By “true” phase diagram we mean one inwhich each point of the diagram representsone unique equilibrium state. In the presentsection we give the general geometricalrules that apply to all true phase diagramsections, and we discuss the choices ofaxes and constants that ensure that the dia-gram is a true diagram.

We must first make some definitions. Ina system of C components we can define(C + 2) thermodynamic potentials fi . Theseare T, P, m1, m2, …, mC, where mi is thechemical potential defined in Eq. (1-23).For each potential there is a correspondingextensive variable qi related by:

fi = (∂U/∂qi)q j ( j ≠ i ) (1-114)

where U is the internal energy of thesystem. The corresponding potentials andextensive variables are listed in Table 1-1.It may also be noted that the correspondingpairs are found together in the terms of thegeneral Gibbs–Duhem equation:

S dT – V dP + S ni dmi = 0 (1-115)

1.9.1 General Geometrical Rules for All True Phase Diagram Sections

The Law of Adjoining Phase Regions ap-plies to all true sections. As a phase boun-dary line is crossed, one and only onephase either appears or disappears.

If the vertical axis is a potential (T, P,mi), then horizontal invariant lines like theeutectic line in Fig. 1-12 or the line AB inFig. 1-29 will be seen when the maximumnumber of phases permitted by the phaserule are at equilibrium. However, if theseare considered to be degenerate infinitelynarrow phase fields, then the Law of Ad-joining Phase Regions still applies. This isillustrated schematically in Fig. 1-39 wherethe three-phase eutectic line has been “opened up”. Similarly, if both axes are po-tentials, then many phase boundaries maybe degenerate infinitely narrow regions.For example, all phase boundaries on Figs.1-1 to 1-3, 1-36 and 1-37(b) are degeneratetwo-phase regions which are schematicallyshown “opened up” on Fig. 1-40.

All phase boundary lines in a true phasediagram meet at nodes where exactly fourlines converge, as in Fig. 1-41. N phases(a1, a2, …, aN) where N ≥1 are common


Table 1-1. Corresponding pairs of potentials fi andextensive variables qi .

fi : T P m1 m2 … mC

qi : S –V n1 n2 … nC

1.9 General Phase Diagram Geometry 57

to all four regions. Schreinemakers’ Rulestates that the extensions of the boundariesof the N-phase region must either both liewithin the (N +1)-phase regions as in Fig.1-41 or they must both lie within the(N + 2)-phase region. This rule is illustratedby the extrapolations in Fig. 1-29 at pointsa, b and c and in Fig. 1-30 at points b, c, n,i and s. The applicability of Schreine-makers’ Rule to systems of any number of

components was noted by Hillert (1985)and proved by Pelton (1995). In the case ofdegenerate phase regions, all nodes canstill be considered to involve exactly fourboundary lines if the degenerate boundar-ies are “opened up” as in Figs. 1-39 and 1-40.

An objection might be raised that a minimum or a maximum in a two-phase re-gion in a binary temperature–compositionphase diagram, as in Fig. 1-10 or in thelower panel of Fig. 1-16, represents an ex-

Figure 1-39. An isobaric binary T – Xphase diagram (like Fig. 1-12) with theeutectic line “opened up” to illustrate thatthis is a degenerate 3-phase region.

Figure 1-40. A potential–potential phase diagram(like Fig. 1-1 or Fig. 1-36) with the phase boundaries“opened up” to illustrate that they are degenerate 2-phase regions.

Figure 1-41. A node in a true phase diagram sec-tion.

ception to Schreinemakers’ Rule. How-ever, the extremum in such a case is not ac-tually a node where four phase boundariesconverge, but rather a point where twoboundaries touch. Such extrema in whichtwo phase boundaries touch with zeroslope may occur for a C-phase region in aphase diagram of a C-component systemwhen one axis is a potential. For example,in an isobaric temperature–compositionphase diagram of a four-component sys-tem, we may observe a maximum or a min-imum in a four-phase region separating twothree-phase regions. A similar maximumor minimum in a (C – n)-phase region,where n > 0, may also occur, but only for a degenerate or special composition path.For further discussion, see Hillert (1998).

1.9.1.1 Zero Phase Fraction Lines

All phase boundaries on true phase dia-gram sections are zero phase fraction(ZPF) lines, a very useful concept intro-duced by Gupta et al. (1986). There areZPF lines associated with each phase. Onone side of its ZPF line the phase occurs,while on the other side it does not. For ex-ample, in Fig. 1-30 the ZPF line for the aphase is the line abcde f. The ZPF line forthe g phase is g hi jk l. For the MC phase theZPF line is mnciopq. The ZPF line forM7C3 is rnbhspke t, and for M23C6 it isudjosv. These five ZPF lines yield the en-tire two-dimensional phase diagram. Phasediagram sections plotted on triangular co-ordinates as in Figs. 1-25 and 1-27 alsoconsist of ZPF lines.

In the case of phase diagrams with de-generate regions, ZPF lines for two differ-ent phases may be coincident over part of their lengths. For example, in Fig. 1-12,line CABD is the ZPF line of the liquid,while CEBF and DEAG are the ZPF linesfor the a and b phases respectively. In

Fig. 1-1, all lines are actually two coinci-dent ZPF lines.

The ZPF line concept is very useful inthe development of general algorithms forthe thermodynamic calculation of phase di-agrams as discussed in Sec. 1-12.

1.9.2 Choice of Axes and Constants of True Phase Diagrams

In a system of C components, a two-dimensional diagram is obtained by choos-ing two axis variables and holding (C –1)other variables constant. However, not allchoices of variables will result in a truephase diagram. For example on the P –Vdiagram for H2O shown schematically inFig. 1-42, at any point in the area where the(S + L) and (L + G) regions overlap thereare two possible equilibrium states of thesystem. Similarly, the diagram of carbonactivity versus XCr at constant T and P inthe Fe–Cr–C system in Fig. 1-43 (Hillert,1997) exhibits a region in which there is nounique equilibrium state.

In order to be sure that a diagram is atrue phase diagram, we must choose oneand only one variable (either fi or qi) from


Figure 1-42. Schematic P–V diagram for H2O. Thisis not a true phase diagram.


each of the (C + 2) conjugate pairs in Table1-1. (Also, at least one of these must be anextensive variable qi .) From among then (1≤ n ≤ C + 2) selected extensive vari-ables, (n –1) independent ratios are thenformed. These (n –1) ratios along with the(C + 2 – n) selected potentials are the (C +1)required variables. Two are chosen as axisvariables and the remainder are held con-stant.

As a first example, consider a binarysystem with components A–B. The conju-gate pairs are (T, S), (P, –V ), (mA, nA) and(mB, nB). Let us choose one variable fromeach pair as follows: T, P, nA, nB. From theselected extensive variables, nA and nB, weform a ratio such as nB/(nA + nB) = XB. Theresultant phase diagram variables are T, P,XB. Choosing any two as axes and holdingthe third constant will give a true phase di-agram as in Fig. 1-6 or Fig. 1-8.

As a second example, consider Fig. 1-38for the Fe–Cr–S2–O2 system. We chooseone variable from each conjugate pair asfollows: T, P, mS2

, mO2, nFe, nCr. From the

selected extensive variables we form theratio nFe /(nFe + nCr). Fig. 1-38 is a plot of mS2

versus mO2at constant T, P and

nFe /(nFe + nCr).In Fig. 1-28 the selected variables are T,

P, nBi , nSn and nCd, and ratios are formedfrom the selected extensive variables asnCd /(nCd + nBi) and nSn /(nCd + nBi + nSn)= XSn. Fig. 1-28 is a plot of T versusnCd /(nCd + nBi) at constant P and XSn.

Fig. 1-42, the P –V diagram for H2O, isnot a true phase diagram because P and Vare members of the same conjugate pair.For the diagram shown in Fig. 1-43, we canchoose one variable from each pair as fol-lows: T, P, mC, nFe, nCr . However the verti-cal axis is XCr = nCr /(nFe + nCr + nC). This ra-tio is not allowed because it contains nC

which is not on the list of chosen variables.That is, since we have chosen mC to be anaxis variable, we cannot also choose nC.Hence, Fig. 1-43 is not a true phase dia-gram. A permissible choice for the verticalaxis would be nCr /(nFe + nCr) (see Fig. 1-33). Note that many regions of Figs. 1-42and 1-43 do represent unique equilibriumstates. That is, the procedure given here is asufficient, but not necessary, condition forconstructing true phase diagrams.

To apply this procedure simply, the com-ponents of the system should be formallydefined to correspond to the desired axisvariables or constants. For example, in Fig.1-1 we wish to plot pSO2

and log pO2as

axes. Hence we define the components asCu–SO2–O2 rather than Cu–S–O.

In several of the phase diagrams in thischapter, log pi or RT ln pi has been substi-tuted for mi as axis variable or constant.From Eq. (1-32), this substitution canclearly be made if T is constant. However,even when T is an axis of the phase dia-gram as in Fig. 1-37(b), this substitution isstill permissible since mi

0 is a monotonicfunction of T. The substitution of ln ai for

Figure 1-43. Carbon activity versus mole fractionof Cr at constant T and P in the Fe–Cr–C system.This is not a true phase diagram (from Hillert, 1997).

mi results in a progressive expansion anddisplacement of the axis with increasing Tthat preserves the overall geometry of thediagram.

1.9.2.1 Tie-lines

If only potentials (T, P, mi) are held con-stant, then all tie-lines lie in the plane ofthe phase diagram section. In this case, thecompositions of the individual phases atequilibrium can be read from the phase diagram, and the lever rule applies as, forexample, in Figs. 1-6, 1-25, 1-33 or 1-34.However, if a ratio of extensive variables,such as a composition, is held constant asin the isopleths of Figs. 1-28 to 1-30, thenin general, tie-lines do not lie in the plane.

If both axes are composition variables(ratios of ni), and if only potentials are heldconstant, then it is desirable that the tie-lines (which lie in the plane) be straightlines. It can be shown (Pelton and Thomp-son, 1975) that this will only be the case ifthe denominators of the two compositionvariable ratios are the same. For example,in the central panel of Fig. 1-32, which is inJänecke coordinates, the composition vari-ables, nCo /(nCo + nNi) and nO /(nCo + nNi),have the same denominator. This same dia-gram can be plotted on triangular coordi-nates as in the upper panel of Fig. 1-32 andsuch a diagram can also be shown (Peltonand Thompson, 1975) to give straight tie-lines.

Similarly, in the quasiternary reciprocalphase diagram of Fig. 1-31 the vertical andhorizontal axes are nNa /(nNa + nK) andnCl /(nCl + nF). To preserve charge neutral-ity, (nNa + nK) = (nCl + nF), and so the tie-lines are straight. Generally, in quasiter-nary reciprocal salt phase diagrams,straight tie-lines are obtained by basing thecomposition on one equivalent of charge.For example, in the CaCl2–NaCl–CaO–

Na2O system we would choose as axes the equivalent cationic and anionic frac-tions, nNa /(nNa + 2nCa) and nCl /(nCl + 2nO),whose denominators are equal because ofcharge neutrality.

1.9.2.2 Corresponding Phase Diagrams

When only potentials are held constantand when both axes are also potentials,then the geometry exemplified by Figs. 1-1to 1-3, 1-26 and 1-37(b) results. Such dia-grams were called “type-1 phase diagrams”by Pelton and Schmalzried (1973). If onlypotentials are held constant and one axis isa potential while the other is a compositionvariable, then the geometry exemplified by Figs. 1-8, 1-12, 1-34, 1-37(a), and thelower panel of Fig. 1-33 results. Thesewere termed “type-2” diagrams. Finally, ifonly potentials are held constant and bothaxes are compositions, then a “type-3” dia-gram as in the upper panels of Figs. 1-32and 1-33 results.

If the fi axis of a phase diagram is re-placed by a composition variable that var-ies as its conjugate variable qi (ex: qi /qj ,qi /(qi + qj)), then the new diagram and theoriginal diagram are said to form a pair ofcorresponding phase diagrams. For in-stance, Figs. 1-37(a) and 1-37(b) are cor-responding type-1 and type-2 phase dia-grams, while Fig. 1-33 shows a corre-sponding pair of type-2 and type-3 dia-grams. It is useful to draw correspondingdiagrams beside each other as in Figs. 1-37or 1-33 because the information containedin the two diagrams is complementary.

1.9.2.3 Theoretical Considerations

A complete rigorous proof that the pro-cedure described in this section will alwaysgenerate a true phase diagram is beyondthe scope of this chapter. As an outline ofthe proof, we start with the generalized



stability criterion:(1-116)

(∂f1/∂q1)f2, f3, …, qN , qN+1, qN+2, …, qC+2≥ 0

This equation states that a potential fi al-ways increases as its conjugate variable qi

increases when either fj or qj from everyother conjugate pair is held constant. Forinstance, mi of a component always in-creases as that component is added to asystem (that is, as ni is increased) at con-stant T and P, when either the number ofmoles or the chemical potential of everyother component is held constant. In a bi-nary system, for example, this means thatthe equilibrium Gibbs energy envelope isalways convex, as shown in Fig. 1-6. If theenvelope were concave, then the systemwould be unstable and would separate intotwo phases, as shown in Fig. 1-11.

Consider first a phase diagram with axesf1 and f2 with f3, f4, …, fC+1 and qC+2

constant. Such a diagram is always a truephase diagram. If the potential f1 is now re-placed by q1, the diagram still remains atrue phase diagram because of Eq. (1-116).The sequence of equilibrium states that oc-curs as q1 is increased will be the same asthat which occurs as f1 is increased whenall the other variables (fi or qi) are heldconstant.

A true phase diagram is therefore ob-tained if the axis variables and constantsare chosen from the variables f1, f2, …,fN , qN +1, qN +2, …, qC+1 with qC+2 heldconstant. The extensive variables can benormalized as (qi /qC+2) or by any other in-dependent and unique set of ratios.

It should be noted that at least one exten-sive variable, qC+2, is considered to be con-stant across the entire diagram. In practice,this means that one of the extensive vari-ables must be either positive or negativeeverywhere on the diagram. For certainformal choices of components, extensivecomposition variables can have negative

values. For example, in the predominancediagram of Fig. 1-1, if the components arechosen as Cu–SO2–O2, then the com-pound Cu2S is written as Cu2(SO2)O–2;that is nO2

= –1. This is no problem in Fig.1-1, since mO2

rather than nO2was chosen

from the conjugate pair and is plotted as anaxis variable. However, suppose we wishto plot a diagram of mCu versus mSO2

at con-stant T and P in this system. In this case,the chosen variables would be T, P, mSO2

,nO2

. Since one of the selected extensivevariables must always be positive, andsince nO2

is the only selected extensive var-iable, it is necessary that nO2

be positiveeverywhere. For instance, a phase field forCu2S is not permitted. In other words, onlycompositions in the Cu–SO2–O2 subsys-tem are permitted. A different phase dia-gram would result if we plotted mCu versusmSO2

in the Cu–SO2–S2 subsystem with nS2

always positive. Cu2O would then not ap-pear, for example. That is, at a given mCu

and pSO2we could have a low pO2

and ahigh pS2

in equilibrium with, for example,Cu2S, or we could have a high pO2

and alow pS2

in equilibrium with, for example,Cu2O. Hence the diagram will not be a truediagram unless compositions are limited tothe Cu–SO2–O2 or Cu–SO2–S2 triangles.As a second example, if mSiO and mCO arechosen as variables in the SiO–CO–Osystem, then the diagram must be limited to nO > 0 (SiO–CO–O subsystem) or tonO < 0 (SiO–CO–Si–C subsystem).

1.9.2.4 Other Sets of Conjugate Pairs

The set of conjugate pairs in Table 1-1 isonly one of many such sets. For example, ifwe make the substitution (H = TS + S ni mi)in Eq. (1-115), then we obtain another formof the general Gibbs–Duhem equation:

– H d(1/T ) – (V/T ) dP + S ni d(mi /T ) = 0(1-117)

This defines another set of pairs of conju-gate potentials and extensive variables:(1/T, –H ), (P, –V/T ), (mi /T, ni). Choosingone and only one variable from each pair,we can construct a true phase diagram bythe procedure described above. However,these diagrams may be of limited practicalutility. This is discussed by Hillert (1997).

1.10 Solution Models

In Sec. 1.4.7, the thermodynamic expres-sions for simple ideal substitutional solu-tions were derived and in Secs. 1.5.7 and1.6.2, the regular solution model and poly-nomial extensions thereof were discussed.For other types of solutions such as ionicmixtures, interstitial solutions, polymericsolutions, etc., the most convenient defini-tion of ideality may be different. In thepresent section we examine some of thesesolutions. We also discuss structural order-ing and its effect on the phase diagram. Forfurther discussion, see Pelton (1997).

1.10.1 Sublattice Models

The sublattice concept has proved to bevery useful in thermodynamic modeling.Sublattice models, which were first devel-oped extensively for molten salt solutions,find application in ceramic, interstitial so-lutions, intermetallic solutions, etc.

1.10.1.1 All Sublattices Except OneOccupied by Only One Species

In the simplest limiting case, only onesublattice is occupied by more than onespecies. For example, liquid and solidMgO–CaO solutions can be modeled byassuming an anionic sublattice occupiedonly by O2– ions, while Mg2+ and Ca2+

ions mix on a cationic sublattice. In this

case the model is formally the same as thatof a simple substitutional solution, becausethe site fractions XMg and XCa of Mg2+ andCa2+ cations on the cationic sublattice arenumerically equal to the overall componentmole fractions XMgO and XCaO. Solid and liq-uid MgO–CaO solutions have been shown(Wu et al., 1993) to be well represented bysimple polynomial equations for gE.

As a second example, the intermetallic e-FeSb phase exhibits non-stoichiometry to-ward excess Fe. This phase was modeled(Pei et al., 1995) as a solution of Fe andstoichiometric FeSb by assuming two sub-lattices: an “Fe sublattice” occupied onlyby Fe atoms and an “Sb sublattice” occu-pied by both Fe and Sb atoms such that, pergram atom,

Dgm = 0.5 RT (yFe ln yFe + ySb ln ySb)+ a yFe ySb (1-118)

where ySb = (1 – yFe) = 2XSb is the site frac-tion of Sb atoms on the “Sb sites” and a isan empirical polynomial in ySb.

1.10.1.2 Ionic Solutions

Let us take as an example a solution,solid or liquid, of NaF, KF, NaCl and KClas introduced in Sec. 1.7.8. If the cationsare assumed to mix randomly on a cationicsublattice while the anions mix randomlyon an anionic sublattice, then the molarGibbs energy of the solution can be mod-eled by the following equation which con-tains an ideal mixing term for each sublat-tice:

g = (XNa XCl g0NaCl + XK XF g0

KF

+ XNa XF g0NaF + XK XCl g0

KCl) (1-119)

+ R T (XNa ln XNa + XK ln XK)

+ R T (XF ln XF + XCl ln XCl) + gE

where the factor (XNa XCl), for example, isthe probability, in a random mixture, of


1.10 Solution Models 63

finding a Na ion and a Cl ion as nearestneighbors. Differentiation of Eq. (1-119)gives the following expression for the ac-tivity of NaF:

R T ln aNaF = – XK XCl DGexchange (1-120)

+ R T ln (XNa XF) + gENaF

where DGexchange is the Gibbs energychange for the following exchange reactionamong the pure salts:

NaCl + KF = NaF + KCl; (1-121)

DGexchange = g0NaF + g0

KCl – g0NaCl – g0

KF

In this example, DGexchange < 0. The saltsNaF and KCl are thus said to form thestable pair. The first term on the right ofEq. (1-120) is positive. The members of thestable pair thus exhibit positive deviations,and in Fig. 1-31 this is reflected by the flatliquidus surfaces with widely spaced iso-therms for NaF and KCl. That is, the mix-ing of pure NaF and KCl is unfavorable be-cause it involves the formation of K+ – F –

and Na+ – Cl– nearest-neighbor pairs at theexpense of the energetically preferableNa+ – F – and K+ – Cl– pairs. If DGexchange issufficiently large, a miscibility gap will beformed, centered close to the stable diago-nal joining the stable pair.

Blander (1964) proposed the followingexpression for gE in Eq. (1-119):

gE = XNa gENaCl–NaF + XK gE

KCl–KF (1-122)

+ XF gENaF–KF + XCl gE

NaCl–KCl

– XNa XK XF XCl (DGexchange)2/ZRT

where, for example, gENaCl–NaF is the excess

Gibbs energy in the NaCl–NaF binarysystem at the same cationic fraction XNa asin the ternary, and where Z is the first coor-dination number. That is, gE contains acontribution from each binary system. Thefinal term in Eq. (1-122) is a first-ordercorrection for non-random mixing which

accounts for the fact that the number of Na+–F – and K+–Cl– nearest-neighborpairs will be higher than the number ofsuch pairs in a random mixture. This termis usually not negligible.

The phase diagram in Fig. 1-31(a) wascalculated by means of Eqs. (1-119) and(1-122) solely from optimized excessGibbs energies of the binary systems andfrom compiled data for the pure salts.Agreement with the measured diagram isvery good.

Eqs. (1-119) and (1-122) can easily bemodified for solutions in which the num-bers of sites on the two sublattices are notequal, as in MgCl2–MgF2–CaCl2–CaF2

solutions. Also, in liquid salt solutions theratio of the number of lattice sites on onesublattice to that on the other sublattice can vary with concentration, as in moltenNaCl–MgCl2–NaF–MgF2 solutions. Inthis case, it has been proposed (Saboungiand Blander, 1975) that the molar ionicfractions in all but the random mixingterms of these equations should be replacedby equivalent ionic fractions. Finally, theequations can be extended to multicompo-nent solutions. These extensions are all dis-cussed by Pelton (1988b).

For solutions such as liquid NaF–KF–NaCl–KCl for which DGexchange is not toolarge, these equations are often sufficient.For solutions with larger exchange Gibbsenergies, however, in which liquid immis-cibility appears, these equations are gener-ally unsatisfactory. It was recognized bySaboungi and Blander (1974) that this isdue to the effect of non-randomness uponthe second nearest-neighbor cation–cationand anion–anion interactions. To take account of this, Blander proposed addi-tional terms in Eq.(1-122). Dessureault andPelton (1991) modified Eqs. (1-119) and(1-122) to account more rigorously fornon-random mixing effects, with good re-

sults for several molten salt systems withmiscibility gaps. See also section 1.10.4.

1.10.1.3 Interstitial Solutions

As an example of the application of thesublattice model to interstitial solutions wewill take the f.c.c. phase of the Fe–V–Csystem. Lee and Lee (1991) have modeledthis solution using two sublattices: a metal-lic sublattice containing Fe and V atoms,and an interstitial lattice containing C atoms and vacancies, va. The numbers ofsites on each sublattice are equal. An equa-tion identical to Eq. (1-119) can be writtenfor the molar Gibbs energy:

g = (XFe Xva g0Feva + XFe XC g0

FeC

+ XV Xva g0Vva + XV XC g0

VC) (1-123)

+ R T (XFe ln XFe + XV ln XV)

+ R T (XC ln XC + Xva ln Xva) + gE

where XFe = (1– XV) and XC = (1– Xva) arethe site fractions on the two sublattices and“Feva” and “Vva” are simply pure Fe andV, i.e., g0

Feva = g0Fe. An expression for gE as

in Eq. (1-122), although without the finalnon-random mixing term, was used by Leeand Lee with optimized binary gE parame-ters. Their calculated Fe–V–C phase dia-gram is in good agreement with experimen-tal data. The sublattice model has beensimilarly applied to many interstitial solu-tions by several authors.

1.10.1.4 Ceramic Solutions

Many ceramic solutions contain two ormore cationic sublattices. As an example,consider a solution of Ti2O3 in FeTiO3 (il-menite) under reducing conditions. Thereare two cationic sublattices, the A and Bsublattices. In FeTiO3, Fe2+ ions and Ti4+

ions occupy the A and B sublattices, re-spectively. With additions of Ti2O3, Ti3+

ions occupy both sublattices. The solu-tion can be represented as (Fe2+

1–xTix3+)A

(Ti4+1–xTix

3+)B where x is the overall molefraction of Ti2O3. The ions are assumed tomix randomly on each sublattice so that:

Ds ideal = – 2R [(1 – x) ln (1 – x) + x ln x](1-124)

Deviations from ideal mixing are as-sumed to occur due to interlattice cation–cation interactions according to

(FeA2+ – TiB

4+) + (TiA3+ – TiB

3+)

= (FeA2+ – TiB

3+) + (TiA3+ – TiB

4+)

DG = a + b T (1-125)

where a and b are the adjustable parametersof the model. The probability that an A–Bpair is an (FeA

2+ – TiB3+) or a (TiA

3+ – TiB4+)

pair is equal to x (1 – x). Hence, gE =x (1– x) (a + bT ).

Similar models can be proposed forother ceramic solutions such as spinels,pseudobrookites, etc. These models canrapidly become very complex mathemati-cally as the number of possible species onthe lattices increases. For instance, inFe3O4–Co3O4 spinel solutions, Fe2+, Fe3+,Co2+ and Co3+ ions are all distributed overboth the tetrahedral and octahedral sublat-tices. Four independent equilibrium con-stants are required (Pelton et al., 1979) todescribe the cation distribution even for theideal mixing approximation. This com-plexity has been rendered much more tract-able by the “compound energy model”(Sundman and Ågren, 1981; Hillert et al.,1988). This is not actually a model, but israther a mathematical formalism permit-ting the formulation of various models interms of the Gibbs energies, g0 , of “pseu-docomponents” so that equations similar toEq. (1-119) can be used directly.



1.10.1.5 The Compound Energy Formalism

As an example, the model for the FeTiO3–Ti2O3 solution in Sec. 1.10.1.4will be reformulated. By taking all com-binations of an A-sublattice species and a B-sublattice species, we define two real components, (Fe2+)A(Ti4+)BO3 and(Ti3+)A(Ti3+)BO3, as well as two “pseudo-components”, (Fe2+)A(Ti3+)BO3

– and(Ti3+)A(Ti4+)BO3

+.Pseudocomponents, as in the present ex-

ample, may be charged. Similarly to Eq.(1-119) the molar Gibbs energy can bewritten

g = (1– x)2 g0FeTiO3

+ x2 g0Ti2O3

(1-126)

+ x (1– x) g0FeTiO3

– + x (1– x) g0Ti2O3

+ – T Ds ideal

Note that charge neutrality is maintained inEq. (1-126). The Gibbs energies of the twopseudocomponents are calculated from theequation

DG = a + b T

= g0FeTiO3

– + g0Ti2O3

+ – g0FeTiO3

– g0Ti2O3

(1-127)

where DG is the Gibbs energy change ofEq. (1-125) and is a parameter of themodel. One of g0

FeTiO3– or g0

Ti2O3+ may be as-

signed an arbitrary value. The other is thengiven by Eq. (1-127). By substitution ofEq. (1-127) into Eq. (1-126) it may beshown that this formulation is identical tothe regular solution formulation given inSec. 1.10.1.4. Note that excess terms, gE,could be added to Eq. (1-126), thereby giv-ing more flexibility to the model. In thepresent example, however, this was not re-quired.

The compound energy formalism is de-scribed and developed by Barry et al.(1992), who give many more examples. Anadvantage of formulating the sublatticemodel in terms of the compound energyformalism is that it is easily extended to

multicomponent solutions. It also providesa conceptual framework for treating manydifferent phases with different structures.This facilitates the development of com-puter software and databases because manydifferent types of solutions can be treatedas cases of one general formalism.

1.10.1.6 Non-Stoichiometric Compounds

Non-stoichiometric compounds are gen-erally treated by a sublattice model. Con-sider such a compound A1–dB1+d . The sub-lattices normally occupied by A and B at-oms will be called, respectively, the A-sub-lattice and the B-sublattice. Deviationsfrom stoichiometry (where d = 0) can occurby the formation of defects such as B atomson A sites, vacant sites, atoms occupyinginterstitial sites, etc. Generally, one type ofdefect will predominate for solutions withexcess A and another type will predomi-nate for solutions with excess B. These arecalled the majority defects.

Consider first a solution in which themajority defects are A atoms on B sites andB atoms on A sites: (A1–xBx)A(AyB1–y)B. Itfollows that d = (x – y). In the compoundenergy formalism we can write, for the mo-lar Gibbs energy,

g = (1– x) (1– y) g0AB + (1– x) y g0

AA

+ x (1– y) g0BB + x y g0

BA (1-128)

+ RT [x ln x + (1– x) ln (1– x)

+ y ln y + (1– y) ln (1– y)]

where g0AB is the molar Gibbs energy of

(hypothetical) defect-free stoichiometricAB. Now the defect concentrations at equi-librium are those that minimize g. There-fore, setting (∂g /∂x) = (∂g /∂y) = 0 at con-stant d, we obtain (1-129)

x yx y

x yRT( ) ( )

exp ( )1 1

1 1 2

− −− − − +⎛

⎝⎜⎞⎠⎟

=D Dg g

where Dg1 = (g0AA – g0

AB) and Dg2 =(g0

BB – g0AB) are the Gibbs energies of for-

mation of the majority defects and whereg0

BA has been set equal to g0AB. At a given

composition d = (x – y), and for given val-ues of the parameters Dg1 and Dg2, Eq. (1-129) can be solved to give x and y,which can then be substituted into Eq. (1-128) to give g. The more positive areDg1 and Dg2, the more steeply g rises on ei-ther side of its minimum, and the narroweris the range of stoichiometry of the com-pound.

Consider another model in which themajority defects are vacancies on the B-sublattice and B atoms on interstitial sites.We now have three sublattices with occu-pancies (A)A(B1–y vay)B(Bx va1–x)I where“I” indicates the interstitial sublattice. TheA-sublattice is occupied exclusively by Aatoms. A vacancy is indicated by va. Stoi-chiometric defect-free AB is representedby (A) (B) (va) and (x – y) = 2d /(1 – d).Per mole of A1–dB1+d , the Gibbs energy is:

g = (1– d) [(1– x) (1– y) g0ABva

+ (1– x) y g0Avava + x (1– y) g0

ABB + x y g0AvaB

+ RT [x ln x + (1– x) ln (1– x)+ y ln y + (1– y) ln (1– y)] (1-130)

Eq. (1-130) is identical to Eq. (1-128) apartfrom the factor (1– d ), and gives rise to anequilibrium constant as in Eq. (1-129).Other choices of majority defects result invery similar expressions. The model caneasily be modified to account for otherstoichiometries AmBn , for different num-bers of available interstitial sites, etc., andits extension to multicomponent solutionsis straightforward.

1.10.2 Polymer Solutions

For solutions of polymers in monomericsolvents, very large deviations from simpleRaoultian ideal behaviour (i.e. from Eq. (1-

40)) are observed. This large discrepancycan be attributed to the fact that the indi-vidual segments of the polymer moleculehave considerable freedom of movement.Flory (1941, 1942) and Huggins (1942)proposed a model in which the polymersegments are distributed on the solventsites. A large polymer molecule can thus beoriented (i.e. bent) in many ways, therebygreatly increasing the entropy. To a first ap-proximation the model gives an ideal mix-ing term with mole fractions replaced byvolume fractions in Eq. (1-45):

Lewis and Randall (1961) have com-pared the Flory–Huggins equation with experimental data in several solutions. In general, the measured activities lie be-tween those predicted by Eq. (1-131) andby the Raoultian ideal equation, Eq. (1-45).A recent review of the thermodynamics of polymer solutions is given by Trusler(1999).

1.10.3 Calculation of Limiting Slopesof Phase Boundaries

From the measured limiting slope(dT/dX )XA=1 of the liquidus at the meltingpoint of a pure component A, much infor-mation about the extent of solid solubility,as well as the structure of the liquid, can beinferred. Similar information can be ob-tained from the limiting slopes of phaseboundaries at solid-state transformationpoints of pure components.

Eq. (1-65) relates the activities along theliquidus and the solidus to the Gibbs en-ergy of fusion:

RT ln alA – RT ln as

A = – Dg0f (A) (1-132)

Dgv

v v

vv v

mideal

AA A

0

A A0

B B0

BB B

0

A A0

B A0

=

(1-131)

RT XX

X X

XX

X X

ln

ln

+⎛⎝⎜

++

⎞⎠⎟



In the limit XA Æ 1, the liquidus and solidusconverge at the melting point Tf (A). Let usassume that, in the limit, Raoult’s Law, Eq.(1-40), holds for both phases. That is,al

A = X lA and as

A = X sA. Furthermore, from

Eq. (1-60),

Dg0f (A) Æ Dhf (A) (1 – T /Tf (A))

Finally, we note that

XA Æ1lim (ln XA) =

XA Æ1lim (ln (1– XB)) = – XB

Substituting these various limiting valuesinto Eq. (1-132) yields:

XA Æ1lim (dX l

A/dT – dX sA/dT )

= Dh0f (A) /R (Tf (A))

2 (1-133)

If the limiting slope of the liquidus,

XA Æ1lim (dX l

A/dT ), is known, then the limiting

slope of the solidus can be calculated, orvice versa, as long as the enthalpy of fusionis known.

For the LiF–NaF system in Fig. 1-18,the broken line is the limiting liquidusslope at XLiF =1 calculated from Eq. (1-133) under the assumption that there is nosolid solubility (that is, that dX s

A/dT = 0).

Agreement with the measured limiting liq-uidus slope is very good, thereby showingthat the solid solubility of NaF in LiF is notlarge.

In the general case, the solute B may dis-solve to form more than one “particle”. Forexample, in dilute solutions of Na2SO4 inMgSO4, each mole of Na2SO4 yields twomoles of Na+ ions which mix randomlywith the Mg2+ ions on the cationic sublat-tice. Hence, the real mole fraction of sol-vent, XA, is (1–n XB) where n is the numberof moles of foreign “particles” contributedby one mole of solute. In the present exam-ple, n = 2.

Eq. (1-133) now becomes:

XA Æ1lim (dX l

A/dT – dX sA/dT )

= Dh0f (A) /n R (Tf (A))

2 (1-134)

The broken line in Fig. 1-44 is the limitingliquidus slope calculated from Eq. (1-134)under the assumption of negligible solidsolubility.

It can be shown (Blander, 1964) that Eq.(1-134) applies very generally with the fac-tor n as defined above. For example, add-ing LiF to NaF introduces only one foreign

Figure 1-44. Phase dia-gram of the MgSO4–Na2SO4 system calculatedfor an ideal ionic liquid so-lution. Broken line is thetheoretical limiting liquidusslope calculated for negli-gible solid solubility takinginto account the ionic natureof the liquid. Agreementwith the measured diagram(Ginsberg, 1909) is good.

particle Li+. The F – ion is not “foreign”.Hence, n =1. For additions of Na2SO4 toMgSO4, n = 2 since two moles of Na+ ionsare supplied per mole of Na2SO4. ForCaCl2 dissolved in water, n = 3, and so on.For C dissolving interstitially in solid Fe,n =1. The fact that the solution is intersti-tial has no influence on the validity of Eq.(1-134). Eq. (1-134) is thus very generaland very useful. It is independent of the so-lution model and of the excess properties,which become zero at infinite dilution.

An equation identical to Eq. (1-134) butwith the enthalpy of transition, Dh0

tr , re-placing the enthalpy of fusion, relates thelimiting phase boundary slopes at a trans-formation temperature of a component.

1.10.4 Short-Range Ordering

The basic premise of the regular solutionmodel (Sec. 1.5.7) is that random mixingoccurs even when gE is not zero. To ac-count for non-random mixing, the regularsolution model has been extended thoughthe quasichemical model for short-rangeordering developed by Guggenheim (1935)and Fowler and Guggenheim (1939) andmodified by Pelton and Blander (1984,1986) and Blander and Pelton (1987). Themodel is outlined below. For a more com-plete development, see the last two paperscited above, Degterov and Pelton (1996),Pelton et al. (2000) and Pelton and Char-trand (2000).

For a binary system, consider the for-mation of two nearest-neighbor 1–2 pairsfrom a 1–1 and a 2–2 pair:

(1–1) + (2–2) = 2 (1–2) (1-135)

Let the molar Gibbs energy change forthis reaction be (w –h T ). Let the nearest-neighbor coordination numbers of 1 and 2atoms or molecules be Z1 and Z2. The totalnumber of bonds emanating from an i atom

or molecule is Zi Xi . Hence, mass balanceequations can be written as

Z1 X1 = 2n11 + n12

Z2 X2 = 2n22 + n12 (1-136)

where nij is the number of i– j bonds in onemole of solution. “Coordination equivalentfractions” may be defined as:

Y1 = 1 – Y2 = Z1X1/(Z1X1 + Z2 X2) (1-137)

where the total number of pairs in one moleof solution is (Z1X1 + Z2 X2)/2. Letting Xij

be the fraction of i– j pairs in solution, Eq.(1-136) may be written as:

2Y1 = 2 X11 + X12

2Y2 = 2 X22 + X12 (1-138)

The molar enthalpy and excess entropy ofmixing are assumed to be directly relatedto the number of 1–2 pairs:

Dhm – TsE(non-config)

= (Z1X1 + Z2 X2) X12 (w –h T )/4 (1-139)

An approximate expression for the config-urational entropy of mixing is given by aone-dimensional Ising model:

The equilibrium distribution is calculatedby minimizing Dgm with respect to X12 atconstant composition. This results in a“quasichemical” equilibrium constant forthe reaction, Eq. (1-135):

When (w –h T ) = 0, the solution of Eqs. (1-138) and (1-141) gives a random distribu-

XX X

TRT

122

11 224= (1-141)exp

( )− −⎛⎝⎜

⎞⎠⎟

w h

Ds R X X X XR

Z X Z X

X X Y X X Y

X X Y Y

mconfig =

(1-140)

− +

− +

× ++

( ln ln )

( )

[ ln( / ) ln( / )

ln( / )]

1 1 2 2

1 1 2 2

11 11 12

22 22 22

12 12 1 2

2

2



tion with X11 = Y12, X22 = Y2

2 and 2Y1 Y2, andEq. (1-140) reduces to the ideal Raoultianentropy of mixing. When (w –h T ) be-comes very negative, 1–2 pairs predomi-nate. A plot of Dhm or sE(non-config) versuscomposition then becomes V-shaped and aplot of Dsm

config becomes m-shaped, withminima at Y1 = Y2 =1/2, which is the com-position of maximum ordering, as illus-trated in Fig. 1-45. When (w –h T ) is quitenegative, the plot of gE also has a negativeV-shape.

For Fe–S liquid solutions, the activitycoefficients of sulfur as measured by sev-eral authors are plotted in Fig. 1-46. It isclear in this case that the model should beapplied with ZFe = ZS. The curves shown inFig. 1-46 were calculated from the quasi-chemical model with (w –h T ) expanded as

the following optimized (Kongoli et al.,1998) polynomial:

(1-142)(w – h T ) = – (70 017 + 9T ) – 74 042 YS

– (798 – 15 T ) YS3 + 40 791YS

7 J/mol–1

Far fewer parameters are required than if apolynomial expansion of gE (as in Sec.1.6.2) were used. Furthermore, and moreimportantly, the model permits successfulpredictions of the properties of multicom-ponent systems as illustrated in Fig. 1-47,where measured sulfur activities in quater-nary liquid Fe–Ni–Cu–S solutions arecompared with activities calculated (Kon-goli et al., 1998) solely from the optimizedmodel parameters for the Fe–S, Ni–S and Cu–S binary systems. A pair exchangereaction like Eq. (1-135) was assumed for each M–S pair (M = Fe, Ni, Cu), and an optimized polynomial expansion of(wMS – hMS T ) as a function of YS, similar toEq. (1-142), was obtained for each binary.Three equilibrium constant equations likeEq. (1-141) were written, and it was as-sumed that (wMS – hMS T ) in the quaternarysystem was constant at constant YS. No ad-justable ternary or quaternary parameterswere required to obtain the agreementshown in Fig. (1-47), although the modelpermits the inclusion of such terms if re-quired.

Silicate slags are known to exhibit suchshort-range ordering. In the CaO–SiO2

system, Dhm has a strong negative V-shape,as in Fig. 1-45, but with the minimum atXSiO2

= 1/3 which is the composition corre-sponding to Ca2SiO4. That is, the orderingis associated with the formation of ortho-silicate anions SiO4

4–. In the phase diagram,Fig. 1-14, the CaO-liquidus can be seen todescend sharply near the compositionXSiO2

= 1/3. The quasichemical model hasbeen extended by Pelton and Blander(1984) to treat silicate slags. The diagram

Figure 1-45. Molar enthalpy and entropy of mixingcurves for a system AB calculated at 1000°C withZ1=Z2=2 from the quasichemical model for short-range ordering with (ω – η T ) = 0, – 21, – 42, and– 84 kJ.


Figure 1-46. Activity coefficient of sulfur in liquid Fe – S solutions calculated from optimized quasichemicalmodel parameters and comparison with experimental data (Kongoli et al., 1998).

Figure 1-47. Equilibrium partial pressure of sulfur at 1200 oC over Fe – Ni – Cu – S mattes predicted by thequasichemical model from binary data (Kongoli et al., 1998) and comparison with experimental data (Bale and Toguri, 1996).


shown in Fig. 1-14 is thermodynamicallycalculated (Wu, 1990).

Many liquid alloy solutions exhibitshort-range ordering. The ordering isstrongest when one component is relativelyelectropositive (on the left side of the peri-odic table) and the other is relatively elec-tronegative. Liquid alloys such as Alk–Au(Hensel, 1979), Alk–Pb (Saboungi et al.,1985) and Alk–Bi (Petric et al., 1988a),where Alk = (Na, K, Rb, Cs), exhibit curvesof Dhm and Dsm similar to those in Fig. 1-45 with one composition of maximumordering. For example, in the Au–Cs sys-tem the minima occur near the compositionAuCs; in Mg–Bi alloys the minima occurnear the Mg3Bi2 composition, while inK–Pb alloys the maximum ordering is atK4Pb.

It has also been observed that certain liq-uid alloys exhibit more than one composi-tion of ordering. For example, in K–Te al-loys, the “excess stability function”, whichis the second derivative of Dgm, exhibitspeaks near the compositions KTe8, KTeand K2Te (Petric et al., 1988b) therebyproviding evidence of ordering centred onthese compositions. The liquid might be con-sidered as consisting of a series of mutuallysoluble “liquid intermetallic compounds”.

When (w –h T ) is expanded as a polyno-mial as in Eq. (1-142), the quasichemicalmodel and the polynomial model of Sec.1.6.2 become identical as (w –h T ) ap-proaches zero. That is, the polynomialmodel is a limiting case of the quasichemi-cal model when the assumption of idealconfigurational entropy is made.

When (w –h T ) is positive, (1–1) and(2–2) pairs predominate. The quasichemi-cal model can thus also treat such cluster-ing, which accompanies positive devia-tions from ideality.

Recent work (Pelton et al., 2000; Peltonand Chartrand, 2000) has rendered the

model more flexible by permitting the Zi tovary with composition and by expandingthe (w –h T ) as polynomials in the bondfractions Xij rather than the overall compo-nent fractions. A merger of the quasichem-ical and sublattice models has also beencompleted (Chartrand and Pelton, 2000),permitting nearest-neighbor and second-nearest neighbor short-range-ordering tobe treated simultaneously in molten salt so-lutions.

1.10.5 Long-Range Ordering

In solid solutions, long-range orderingcan occur as well as short-range ordering.In Fig. 1-15 for the Ag–Mg system, atransformation from an a¢ to an a phase isshown occurring at approximately 665 K at the composition Ag3Mg. This is an order–disorder transformation. Below thetransformation temperature, long-range or-dering (superlattice formation) is observed.An order parameter may be defined whichdecreases to zero at the transformationtemperature. This type of phase transfor-mation is not a first-order transformationlike those considered so far in this chapter.Unlike first-order transformations whichinvolve a change of state (solid, liquid, gas)and also involve diffusion over distanceslarge compared with atomic dimensions,order–disorder transformations, at least atthe stoichiometric composition (Ag3Mg inthis example), occur by atomic rearrange-ment over distances of the order of atomicdimensions. The slope of the curve ofGibbs energy versus T is not discontinuousat the transformation temperature. Order-ing and order–disorder transformations arediscussed in Chapter 8 (Inden, 2001).

A type of order–disorder transformationof importance in ferrous metallurgy is themagnetic transformation. Below its Curietemperature of 769 °C, Fe is ferromagnetic.

Above this temperature it is not. The trans-formation involves a change in ordering ofthe atomic spins and is not first order. Ad-ditions of alloying elements will changethe temperature of transformation. Mag-netic transformations are treated in Chapter4 (Binder, 2001). See also Miodownik(1982) and Inden (1982).

1.11 Calculation of Ternary PhaseDiagrams From Binary Data

Among 70 metallic elements 70!/3!67!= 54 740 ternary systems and 916 895 qua-ternary systems are formed. In view of theamount of work involved in measuringeven one isothermal section of a relativelysimple ternary phase diagram, it is very im-portant to have a means of estimating ter-nary and higher-order phase diagrams.

The most fruitful approach to such pre-dictions is via thermodynamic methods. Inrecent years, great advances have beenmade in this area by the international Cal-phad group. Many key papers have beenpublished in the Calphad Journal.

As a first step in the thermodynamic ap-proach, we critically analyze the experi-mental phase diagrams and thermodynamicdata for the three binary subsystems of theternary system in order to obtain a set ofmathematical expressions for the Gibbs energies of the binary phases, as was dis-cussed in Sec. 1.6. Next, interpolation pro-cedures based on solution models are usedto estimate the Gibbs energies of the ter-nary phases from the Gibbs energies of thebinary phases. Finally, the ternary phase di-agram is calculated by computer fromthese estimated ternary Gibbs energies bymeans of common tangent plane or totalGibbs energy minimization algorithms.

As an example of such an estimation of a ternary phase diagram, the experimental

(Ivanov, 1953) and estimated (Lin et al.,1979) liquidus projections of the KCl–MgCl2–CaCl2 system are shown in Fig. 1-48. The estimated phase diagram wascalculated from the thermodynamic prop-erties of the three binary subsystems withthe Gibbs energy of the ternary liquid ap-proximated via the equation suggested by


Figure 1-48. Projection of the liquidus surface ofthe KCl – MgCl2 – CaCl2 system.a) Calculated from optimized binary thermodynamic

parameters (Lin et al., 1979).b) As reported by Ivanov (1953).

1.11 Calculation of Ternary Phase Diagrams From Binary Data 73

Kohler (1960):

gE = (1 – XA)2 gEB/C + (1 – XB)2 gE

C/A

+ (1 – XC)2 gEA/B (1-143)

In this equation, gE is the excess molarGibbs energy at a composition point in theternary liquid phase and gE

B/C, gEC/A and gE

A/B

are the excess Gibbs energies in the threebinary systems at the same ratios XB/XC,XC /XA and XA/XB as at the ternary point. Ifthe ternary liquid phase as well as the threebinary liquid phases are all regular solu-tions, then Eq. (1-143) is exact. In the general case, a physical interpretation ofEq. (1-143) is that the contribution to gE

from, say, pair interactions between A andB particles is constant at a constant ratioXA/XB apart from the dilutive effect of theC particles, which is accounted for by theterm (1– XC)2 taken from regular solutiontheory.

Ternary phase diagrams estimated in thisway are quite acceptable for many pur-poses. The agreement between the experi-mental and calculated diagrams can begreatly improved by the inclusion of one or two “ternary terms” with adjustable coefficients in the interpolation equationsfor gE. For example, the ternary termaXKCl XMgCl2

XCaCl2, which is zero in all

three binaries, could be added to Eq. (1-143) and the value of the parameter awhich gives the “best” fit to the measuredternary liquidus could be determined. This,of course, requires that ternary measure-ments be made, but only a very few (evenone or two in this example) experimentalliquidus points will usually suffice ratherthan the larger number of measurements required for a fully experimental determi-nation. In this way, the coupling of thethermodynamic approach with a few wellchosen experimental measurements holdspromise of greatly reducing the experimen-

tal effort involved in determining multi-component phase diagrams.

Reviews of various interpolation proce-dures and computer techniques for estimat-ing and calculating ternary and higher-or-der phase diagrams are given by Ansara(1979), Spencer and Barin (1979) and Pel-ton (1997).

Other equations, similar to the KohlerEq. (1-143) in that they are based on exten-sion of regular solution theory, are used toestimate the thermodynamic properties ofternary solutions from the properties of thebinary subsystems. For a discussion andreferences, see Hillert (1980). However,for structurally more complex solutions involving more than one sublattice or withsignificant structural ordering, other esti-mation techniques must be used. For a re-view, see Pelton (1997).

An example, the calculation of the phase diagram of the NaCl–KCl–NaF–KFsystem in Fig. 1-31, has already been pre-sented in Sec. 1.10.1.2.

The quasichemical model for systemswith short-range ordering was discussedfor the case of binary systems in Sec.1.10.4. The model has been extended topermit the estimation of ternary and multi-component phase diagrams (Pelton andBlander, 1986; Blander and Pelton, 1987;Pelton and Chartrand, 2000). Very good re-sults have been obtained in the case of sili-cate systems. The liquidus surface of theSiO2–MgO–MnO system, estimated fromoptimized binary data with the quasichem-ical model for the liquid and under the assumption of ideal mixing for the solidMgSiO3–MnSiO3 and Mg2SiO4–Mn2SiO4

solutions, is shown in Fig. 1-49. Agree-ment with the measured phase diagram (Glasser and Osborn, 1960) is within ex-perimental error limits.

1.12 Minimization of GibbsEnergy

Throughout this chapter it has beenshown that phase equilibria are calculatedby Gibbs energy minimization. Computersoftware has been developed in recentyears to perform such calculations in sys-tems of any number of components, phasesand species.

Consider a system in which several stoi-chiometric solid or liquid compounds A, B,C, … could be present at equilibrium alongwith a number of gaseous, liquid or solid

solution phases a, b, g, …. The total Gibbsenergy of the system may be written as:

G = (nA g0A + nB g0

B + …)

+ (na ga + nb gb + …) (1-144)

where nA, nB, etc. are the number of molesof the pure solid or liquidus; g0

A, g0B, etc.

are the molar Gibbs energies of the puresolids or liquids (which are functions of Tand P); na, nb, etc. are the total number ofmoles of the solution phases; ga, gb, etc.are the molar Gibbs energies of the solutionphases (which are function of T, P andcomposition). For a given set of constraints


Figure 1-49. Projection of liquidus surface of the SiO2–MgO–MnO system calculated from optimized binaryparameters with the quasichemical model for the liquid phase.

1.12 Minimization of Gibbs Energy 75

(such as fixed T, P and overall composi-tion), the free energy minimization algo-rithms find the set of mole numbers nA, nB,etc., na, nb, etc. (some may be zero) as wellas the compositions of all solution phaseswhich globally minimize G. This is theequilibrium phase assemblage. Other con-straints such as constant volume or a fixedchemical potential (such as constant pO2

)may be applied.

A discussion of the strategies of such al-gorithms is beyond the scope of the presentchapter. One of the best known generalGibbs energy minimization programs isSolgasmix written by Eriksson (1975) andconstantly updated.

When coupled to a large thermodynamicdatabase, general Gibbs energy minimiza-tion programs provide a powerful tool forthe calculation of phase equilibria. Severalsuch expert database systems have beendeveloped. They have been reviewed byBale and Eriksson (1990).

An example of a calculation performedby the F * A * C * T (Facility for the Analy-sis of Chemical Thermodynamics) expertsystem, which the author has helped to de-velop, is shown in Table 1-2. The program

has been asked to calculate the equilibriumstate when 1 mol of SiI4 is held at 1400 Kin a volume of 104 l. The thermodynamicproperties of the possible product specieshave been automatically retrieved from the database and the total Gibbs energy has been minimized by the Solgasmix algorithm. At equilibrium there will be2.9254 mol of gas of the compositionshown and 0.11182 mol of solid Si willprecipitate. The total pressure will be0.0336 bar.

Although the calculation was performedby minimization of the total Gibbs energy,substitution of the results into the equilib-rium constants of Eqs. (1-10) to (1-12) willshow that these equilibrium constants aresatisfied.

Another example is shown in Table 1-3(Pelton et al., 1990). Here the program hasbeen asked to calculate the equilibrium

Table 1-2. Calculation of equilibrium state when 1 mole SiI4 is held at 1400 K in a volume of 104 l.Calculations performed by minimization of the totalGibbs energy.

SiI4 =2.9254 ( 0.67156 I

+ 0.28415 SiI2

+ 0.24835E – 01 I2

+ 0.19446E – 01 SiI4

+ 0.59083E – 05 SiI+ 0.23039E – 07 Si+ 0.15226E – 10 Si2

+ 0.21462EE – 11 Si3)(1400.0,0.336E – 01,G)

+ 0.11182 Si(1400.0,0.336E – 01,S1, 1.0000)

Table 1-3. Calculation of equilibrium state when re-actants shown (masses in g) are held at 1873 K at apressure of 1 atm. Calculations performed by mini-mization of the total Gibbs energy.

100. Fe + 0.08 O + 0.4 Fe + 0.4 Mn + 0.3 Si + 0.08 Ar =

0.30793 litre ( 99.943 vol% Ar+ 0.24987E – 01 vol% Mn+ 0.24069E – 01 vol% SiO+ 0.82057E – 02 vol% Fe+ 0.79044E – 07 vol% O+ 0.60192E – 08 vol% Si+ 0.11200E – 08 vol% O2

+ 0.35385E – 15 vol% Si2)(1873.0, 1.00 ,G)

+ 0.18501 gram ( 49.948 wt.% SiO2

+ 42.104 wt.% MnO+ 7.9478 wt.% FeO)

(1873.0, 1.00 ,SOLN 2)

+ 100.99 gram ( 99.400 wt.% Fe+ 0.33630 wt.% Mu+ 0.25426 wt.% Si+ 0.98375E – 02 wt.% O2)

(1873.0, 1.00 ,SOLN 3)

state when 100 g Fe, 0.08 g oxygen, 0.4 gFe, 0.4 g Mn, 0.3 g Si and 0.08 g Ar arebrought together at 1873 K at a total pres-sure of 1 bar. The database contains datafor a large number of solution phases aswell as for pure compounds. These datahave been automatically retrieved and thetotal Gibbs energy has been minimized. Atequilibrium there are 0.30793 l of a gasphase, 0.18501 g of a molten slag, and100.99 g of a molten steel of the composi-tions shown.

The Gibbs energies of the solutionphases are represented as functions of com-position by various solution models (Sec.1.10). As discussed in Sec. 1.11, thesemodels can be used to predict the thermo-dynamic properties of N-component solu-tions from evaluated parameters for binary(and possibly ternary) subsystems stored inthe database. For example, in the calcula-tion in Table 1-3, the Gibbs energy of themolten slag phase was estimated by thequasichemical model from optimized pa-rameters for the binary oxide solutions.

1.12.1 Phase Diagram Calculation

Gibbs energy minimization is used tocalculate general phase diagram sectionsthermodynamically using the zero phasefraction line concept (Sec. 1.9.1.1), withdata retrieved from databases of model co-efficients. For example, to calculate the di-agram of Fig. 1-30, the program first scansthe four edges of the diagram to find theends of the ZPF lines. Each line is then fol-lowed from beginning to end, using Gibbsenergy minimization to determine the pointat which a phase is just on the verge of be-ing present. When ZPF lines for all phaseshave been drawn, then the diagram is com-plete. Because, as shown in Sec. 1.9, alltrue phase diagram sections obey the samegeometrical rules, one algorithm suffices to

calculate all types of phase diagrams withany properly chosen variables as axes orconstants.

1.13 Bibliography

1.13.1 Phase Diagram Compilations

The classic compilation in the field of bi-nary alloy phase diagrams is that of Hansen(1958). This work was continued by Elliott(1965) and Shunk (1969). These compila-tions contain critical commentaries. A non-critical compilation of binary alloy phasediagrams is supplied in looseleaf form witha continual up-dating service by W.G. Mof-fatt of the General Electric Co., Schenec-tady, N.Y. An extensive non-critical compi-lation of binary and ternary phase diagramsof metallic systems has been edited byAgeev (1959 – 1978). An index to all com-pilations of binary alloy phase diagrams upto 1979 was prepared by Moffatt (1979). Acritical compilation of binary phase dia-grams involving Fe has been published byKubaschewski (1982). Ternary alloy phasediagrams were compiled by Ageev (1959–1978).

From 1979 to the early 1990s, the Amer-ican Society for Metals undertook a projectto evaluate critically all binary and ternaryalloy phase diagrams. All available litera-ture on phase equilibria, crystal structures,and often thermodynamic properties werecritically evaluated in detail by interna-tional experts. Many evaluations have ap-peared in the Journal of Phase Equilibria(formerly Bulletin of Alloy Phase Dia-grams), (ASM Int’l., Materials Park, OH),which continues to publish phase diagramevaluations. Condensed critical evalua-tions of 4700 binary alloy phase diagramshave been published in three volumes(Massalski et al., 1990). The ternary phase


1.13 Bibliography 77

diagrams of 7380 alloy systems have alsobeen published in a 10-volume compilation(Villars et al., 1995). Both binary and ter-nary compilations are available from ASMon CD-ROM. Many of the evaluationshave also been published by ASM asmonographs on phase diagrams involving aparticular metal as a component.

Each year, MSI Services (http://www.msiwp.com) publishes The Red Book,which contains abstracts on alloy phase diagrams from all sources, notably fromthe extensive Russian literature. MSI alsoprovides a CD-ROM with extensive alloyphase diagram compilations and reports.

Phase diagrams for over 9000 binary, ter-nary and multicomponent ceramic systems(including oxides, halides, carbonates, sul-fates, etc.) have been compiled in the 12-volume series, Phase Diagrams for Ceram-ists (1964–96, Am. Ceramic Soc., Colum-bus, OH). Earlier volumes were non-criti-cal compilations. However, recent volumeshave included critical commentaries.

Phase diagrams of anhydrous salt sys-tems have been compiled by Voskresen-skaya (1970) and Robertson (1966).

An extensive bibliography of binary and multicomponent phase diagrams of alltypes of systems (metallic, ceramic, aque-ous, organic, etc.) has been compiled byWisniak (1981).

1.13.2 Thermodynamic Compilations

Several extensive compilations of ther-modynamic data of pure substances ofinterest in materials science are available.These include the JANAF Tables (Chase et al., 1985) and the compilations of Barinet al. (1977), Barin (1989), Robie et al.(1978) and Mills (1974), as well as the series of compilations of the National Insti-tute of Standards and Technology (Wash-ington, D.C.).

Compilations of properties of solutions(activities, enthalpies of mixing, etc.) aremuch more difficult to find. Hultgren et al.(1973) present the properties of a numberof binary alloy solutions. An extensive bib-liography of solution properties of all typesof solutions was prepared by Wisniak andTamir (1978).

Thermodynamic/phase diagram optimi-zation as discussed in Sec. 1.6.1 has beencarried out for a large number of alloy, ce-ramic and other systems. Many of theseevaluations have been published in theinternational Calphad Journal, publishedsince 1977 by Pergamon Press. Several ofthe evaluations in the Journal of PhaseEquilibria discussed above include ther-modynamic/phase diagram optimizations,as do a number of the evaluations in Vol. 7of Phase Diagrams for Ceramists.

Extensive computer databases of thethermodynamic properties of compoundsand solutions (stored as coefficients ofmodel equations) are available. These in-clude F * A * C * T (http://www.crct.poly-mtl.ca), Thermocalc (http://www.met.kth.se), ChemSage (http://gttserv.lth.rwth-aachen.de), MTS-NPL (http://www.npl.co.uk), Thermodata (http://www.grenet.fr),HSC (http://www.outokumpu.fi), andMALT2 (http://www. kagaku.com). Gibbsenergy minimization software permits thecalculation of complex equilibria from thestored data as discussed in Sec. 1.12 aswell as the thermodynamic calculation ofphase diagram sections. A listing of theseand other available databases is maintainedat http://www.crct. polymtl.ca .

A bibliographic database known asThermdoc, on thermodynamic propertiesand phase diagrams of systems of interestto materials scientists, with updates, isavailable through Thermodata (http://www.grenet.fr).

1.13.3 General Reading

The theory, measurement and applica-tions of phase diagrams are discussed in agreat many texts. Only a few can be listedhere. A recent text by Hillert (1998) pro-vides a complete thermodynamic treatmentof phase equilibria as well as solution mod-eling and thermodynamic/phase diagramoptimization.

A classical discussion of phase diagramsin metallurgy was given by Rhines (1956).Prince (1966) presents a detailed treatmentof the geometry of multicomponent phasediagrams. A series of five volumes editedby Alper (1970–1978) discusses many as-pects of phase diagrams in materials sci-ence. Bergeron and Risbud (1984) give anintroduction of phase diagrams, with par-ticular attention to applications in ceramicsystems, see also Findlay (1951), Ricci(1964) and West (1965).

In the Calphad Journal and in the Jour-nal of Phase Equilibria are to be foundmany articles on the relationships betweenthermodynamics and phase diagrams.

It has been beyond the scope of thepresent chapter to discuss experimentaltechniques of measuring thermodynamicproperties and phase diagrams. For themeasurement of thermodynamic proper-ties, including properties of solutions, thereader is referred to Kubaschewski and Alcock (1979). For techniques of measur-ing phase diagrams, see Pelton (1996),Raynor (1970), MacChesney and Rosen-berg (1970), Buckley (1970) and Hume-Rothery et al. (1952).

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2 Solidification

Heiner Müller-Krumbhaar

Institut für Festkörperforschung, Forschungszentrum Jülich, Germany

Wilfried Kurz

Département des Matériaux, EPFL, Lausanne, Switzerland

Efim Brener

Institut für Festkörperforschung, Forschungszentrum Jülich, Germany

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 822.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.2 Basic Concepts in First-Order Phase Transitions . . . . . . . . . . . . 862.2.1 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.2.2 Interface Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.2.3 Growth of Simple Crystal Forms . . . . . . . . . . . . . . . . . . . . . . . 912.2.4 Mullins–Sekerka Instability . . . . . . . . . . . . . . . . . . . . . . . . . 932.3 Basic Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . 952.3.1 Free Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.3.2 Directional Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.4 Free Dendritic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.4.1 The Needle Crystal Solution . . . . . . . . . . . . . . . . . . . . . . . . . 1012.4.2 Side-Branching Dendrites . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.4.3 Experimental Results on Free Dendritic Growth . . . . . . . . . . . . . . 1152.5 Directional Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.5.1 Thermodynamics of Two-Component Systems . . . . . . . . . . . . . . . 1212.5.2 Scaled Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.5.3 Cellular Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262.5.4 Directional Dendritic Growth . . . . . . . . . . . . . . . . . . . . . . . . 1312.5.5 The Selection Problem of Primary Cell Spacing . . . . . . . . . . . . . . 1352.5.6 Experimental Results on Directional Dendritic Growth . . . . . . . . . . . 1392.5.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1492.6 Eutectic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522.6.1 Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522.6.2 Experimental Results on Eutectic Growth . . . . . . . . . . . . . . . . . . 1582.6.3 Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1632.7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 1642.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165



a distanceA areaA amplitudeA differential operatoraI , aK constantsb arbitrary parameterB self-adjoint differential operatorDC miscibility gapC, C ¢ constantsCE eutectic concentrationCL, CS, Ca, etc. concentrationcp specific heat at constant pressured atomistic length, capillary lengthD diffusion constantd0 capillary lengthDi transport coefficients, diffusion constantsDT thermal diffusion coefficientE1 (P) exponential integralerfc error function complementF Helmholtz energyf scaling functiong Gibbs energy densityDG change in Gibbs energyG Gibbs energy per surface elementG (xi) Gibbs energyGT constant temperature gradientJ probability currentK curvaturek segregation coefficient, wave numberkB Boltzmann constantL natural scalel external length, diffusion lengthl thickness of layer, diffusion lengthLm latent heat of meltingma, mb liquidus or solidus slopeNi particle number of species iP pressure, Péclet number principal valueP (xi) probability of configurationq wave number, inverse lengthQmin minimal workr radius of nucleus, radial distanceR (S) local radius of curvature at point S

82 2 Solidification


R0 radius of curvaturerc critical radiusS position on the interface, entropyDS change in entropyT temperaturet timeDt change in timeT0 reference temperatureTE eutectic temperatureTI interface temperatureTm melting temperatureu dimensionless temperature field or concentration fieldU energyV speedVa maximal speed, absolute stabilityVc critical velocityvR growth ratew probability of fluctuationx space coordinatesXi , Xj extensive variablesYi , Yj intensive variablesZ partition functionz space coordinate

b interface kineticsb4 4-fold anisotropy of the kinetic coefficientg surface tension or surface free energyG ratio of S–L interface energy to specific melting entropyG2 Green’s function integralD supercooling (negative temperature field u at infinity)d local concentration gap, surface tension anisotropye relative strength of anisotropy of capillary lengthz (x, t) deviation of the interfaceh coordinateJ angle of orientation relative to the crystallographic axesQ function of (x, t)k (s) mobilityl interface spacing, wavelengthL (s, e) solvability functionl2 wavelength of side branchlf wavelength of the fastest models stability lengthm chemical potentialn0 effective kinetic prefactorx variable

x i position vectorsDxi small normal displacementr Ivantsov radiuss scaling functiont variableW atomic volume, atomic area

DLA diffusion limited aggregationDS directional solidificationf facetedl.h.s. left-hand sidenf non-facetedPVA pivalic acidr.h.s. right-hand sideSCN succinonitrateWKB Wenzel–Kramers–Brillouin technique for singular pertubations

84 2 Solidification

2.1 Introduction 85

2.1 Introduction

This chapter on solidification discussesthe basic mechanisms of the liquid–solidphase transformation. In particular, we ad-dress the phenomena of cellular and den-dritic patterns formed by the interface be-tween liquid and solid, as the interface, orsolidification front, advances into the liq-uid during the solidification process.

The atomistic processes of the liquid–solid transformation are still not well un-derstood, so we will use a phenomenologi-cal level of description. The processeson very large scales, such as casting orwelding, depend greatly on the experi-mental equipment and are discussed, forexample, by Flemings (1991) and Mordike(1991).

We will therefore restrict our attention tophenomena occurring on some importantintermediate length scales. There is a natu-ral scale L of the order of micrometers (orup to millimeters) that gives a measure forthe microcrystalline structures formed dur-ing the solidification process. In its sim-plest form this natural length is the geomet-ric mean L ~ dl of a microscopic intrinsiclength d defined by typical material proper-ties and an external length l defined by themacroscopic arrangement of the experi-mental equipment. The intrinsic correlationlengths in liquids and solids near the freez-ing point are rather short, of the order ofatomic size, or several Ångströms, becausesolidification is a phase transition of firstorder. By contrast, the experimental equip-ment gives external length scales in therange of centimeters to meters, such thatwe consequently arrive at the mentionedscale of micrometers.

Assuming for the moment that only twolengths are important, why should we ex-pect L to be given by the geometric meanrather than, for example, the arithmetic

mean? An intuitive argument goes as fol-lows: the patterns formed at the solid– liq-uid interface and in both adjacent phasesduring the solidification process resultfrom the competition of two “forces”, onebeing stabilizing for homogeneous struc-tures, the other being destabilizing. Thestabilizing force here clearly must be asso-ciated with the intrinsic atomistic length d,since we have argued that it is related to thelength of correlation or coherence insidethe material or at the interface. In contrast,we must associate the external length l witha destabilizing force. Again this is a quitenatural assumption, as the phase transfor-mation or destabilization of the nutrientphase is induced by the experimental envi-ronment.

It is now obvious that the result of such acompetition of “forces” should be ex-pressed by the product dl of the two repre-sentative quantities rather than by the sum,since the latter would change the relativeimportance of the two lengths when theirvalues become very different.

Admittedly, these arguments look a littleoverstressed considering the many parame-ters controlling the details of a solidifica-tion process. Note, however, that nothinghas been said so far about the precise rela-tion of d, l and L to any specific process,nor have we defined the proportionalityfactor. In principle, d and l could also enterwith different exponents but fortunatelythings are not usually that complicatedright from the start.

If we are still courageous enough tomake one more step on this slightly unsafeground, we may finally assume that the ex-ternal length scale l is related (destabilizingforce!) to the speed V of the solidification,which gives a length when combined witha diffusion constant D for heat or matter asl ~ D/V. From this we immediately obtain arelation between the speed of phase change

V and the length scale L of the resultingpattern:

V L2 ≈ constant (2-1)

Surprisingly enough, we have arrived atabout the most celebrated equation for pro-cesses of dendritic, directional and eutecticsolidification, without even defining any ofthese processes! Two remarks, therefore,may be in order here. First, detailed studiesof the different solidification schemes inrecent years have attempted to extract thecommon and universal aspects of theseprocesses. Such common features indicatea common basis of rather general nature, asoutlined above. Second, we have of courseignored most of the specific aspects of eachindividual process. In directional solidifi-cation, for example, a band of possiblewavelengths for stationary patterns arefound and up to now it is not clear if andhow a specific wavelength from that bandis finally selected. The assumption of justtwo independent length scales in manycases is also a rather gross simplification ofthe actual situation. We will therefore leavethis line of general arguments and look atsome concrete models that are believed tocapture at least some essentials for the fas-cinating patterns produced during solidifi-cation.

Some remarkable progress has beenachieved in the theoretical treatment ofthese phenomena during the recent years.In the list of references, we have concen-trated our attention on recent developmentssince there are some good reviews on olderwork (for example, Langer, 1980a; Kurzand Fisher, 1998).

An experimentalist may finally wonderwhy we have expressed most of the equa-tions in a non-dimensionally scaled formrather than writing all material parametersdown explicitly at each step. One reason isthat the equations then appear much sim-

pler than if we attempted to carry along allprefactors. The second and more importantreason is that the scaled form allows for amuch simpler comparison of experimentsfor different sets of parameters which usu-ally appear only in some combinations inthe equations, thereby leading to cancella-tions and compensations.

Section 2.2 gives a quick summary of theingredients for a theory, starting at nuclea-tion, then deriving boundary conditions fora propagating interface between twophases, and finally discussing some gen-eral aspects of the diffusion equation with apropagating boundary. This is followed byan introduction to basic experimental tech-niques in Sec. 2.3. In Sec. 2.4, the case of asimple solid growing in a supercooled meltis discussed in some detail, leading to thepresent understanding of dendritic growth.In Sec. 2.5 the technically important pro-cess of directional solidification is pre-sented. The evolution of cellular patternsabove a critical growth rate can in principlebe understood without any knowledge ofdendritic growth. Actually, the parameterrange for simple sinusoidal cells is verynarrow so that we usually operate in therange of deep cells or even dendrites,which suggests our sequence of presenta-tion. Finally, these concepts are extendedin Sec. 2.6 to alloys with a eutectic phasediagram and the resulting complex phe-nomena. As this field is currently in rapidtheoretical development, our discussionhere necessarily is somewhat preliminary.The chapter is closed by a summary withcomplementary remarks.

2.2 Basic Conceptsin First-Order Phase Transitions

The different possible phases of a mate-rial existing in thermodynamic equilibrium

86 2 Solidification

2.2 Basic Concepts in First-Order Phase Transitions 87

must be discriminated by some quantity inorder to formulate a theory. Such quantitiesare called “order parameters” and shouldcorrespond to extensive thermodynamicvariables. The difference between a solidand a liquid is defined by the shear mod-ulus, which changes discontinuously at thephase transition. This definition describesthe difference in long-range orientationalcorrelations between two distant pairs ofneighboring atoms. Normally, we use in-stead the more restrictive concept of trans-lational order as expressed through twopoint correlation functions, or Bragg peaks,in scattering experiments.

Although these different order-parame-ter concepts pose a number of subtle ques-tions, particularly in two dimensions wherefluctuations are very important, we willsimply assume in this chapter that thereis some quantity which discriminates be-tween a solid and a liquid in a unique way.Such an order parameter may be the den-sity, for example, which usually changesduring melting, or the composition in atwo-component system. We would like tostress, however, that these are just auxiliaryquantities that change as a consequence ofthe solid– liquid transition but which arenot the fundamental order parameters inthe sense of symmetry arguments. For amore general discussion see the Chapter byBinder (Binder, 2001) and literature on or-der–disorder transitions (Brazovsky, 1975;Nelson, 1983).

First-order transitions are characterizedby a discontinuous change of the order pa-rameter. All intrinsic length scales areshort, typically of the size of a few atomicdiameters. We may therefore assume localthermal equilibria with locally well-de-fined thermodynamic quantities like tem-perature, etc., and smooth variations inthese quantities over large distances. Inter-faces in such systems will represent singu-

larities or discontinuities in some of thequantities, such as the order parameter orthe associated chemical potential, but theywill still leave the temperature as a contin-uous function of the position in space.

2.2.1 Nucleation

A particular consequence of the well-de-fined local equilibrium is the existence ofwell-defined metastable states, correspond-ing to a local, but not global, minimum inthe free energy. But so far we have ne-glected thermal fluctuations. The probabil-ity of a fluctuation (or deviation from theaverage state) of a large closed system is

w ~ exp (DS) (2-2)

where DS is the change in entropy of thesystem due to the fluctuation (see, e.g.,Landau and Lifshitz, 1970). Defining Qmin

as the minimal work necessary to createthis change in thermodynamic quantities,we obtain

DS = Qmin /T0 (2-3)

with T0 being the average temperature ofthe system. Note that this holds even forlarge fluctuations, as long as the change ofextensive quantities in the fluctuation re-gion is small compared with the respectivequantities in the total system.

Considering this system as a metastableliquid within which a fluctuation hasformed a small solid region, and assuming,furthermore, that the liquid is only slightlymetastable, we arrive at the well-known re-sult (Landau and Lifshitz, 1970) in threedimensions for a one-component system:

(2-4)

which together with Eqs. (2-2) and (2-3)gives the probability for the reversible for-

Qr

P P rmin [ ( ) ( )]= L S− − +43

43

2p pW

m m g

mation of a spherical solid nucleus of ra-dius r within a slightly supercooled liquid.Here W is the atomic volume, P the pres-sure, mL > mS are the chemical potentials ofliquid and solid in a homogeneous system,and g is the (here isotropic) surface ten-sion, or surface free energy.

A few remarks should be made here.When deriving Eq. (2-4), we use the con-cept of small deviations from equilibrium,while Eqs. (2-2) and (2-3) are more gener-ally valid (Landau and Lifshitz, 1970). Inthe estimation of the range of validity ofEq. (2-4), it is apparent, however, that itshould be applicable to even very largesupercoolings for most liquids, since thethermal transport is either independent of,or faster than, the kinetics of nucleus for-mation (Ohno et al., 1990). The range ofvalidity of Eq. (2-4) is then typically lim-ited by the approach to the “spinodal” re-gion, where metastable states become un-stable, even when fluctuations are ignored(see the Chapter by Binder and Fratzl, 2001).

Assuming, therefore that we are still inthe range of well-defined metastable states,we may write Eqs. (2-2) and (2-3) as

w = n0 exp (–DG/T0) (2-5)

identifying the change in Gibbs energy DGby Eq. (2-4), with an undetermined prefac-tor n0. Here we do not discriminatebetween surface tension and surface freeenergy, despite the fact that the first is atensorial quantity, and the latter only scalar(although it may be anisotropic, which isignored here). A difference between sur-face tension and surface free energy ariseswhen the system does not equilibratebetween surface and bulk so that, for exam-ple, the number of atoms in the surfacelayer is conserved. Throughout this chap-ter, we will assume perfect local equilibra-tion in this respect, and we may then ignorethe difference.

The extremal value of Eqs. (2-4) and (2-5)with respect to the variation of r gives thecritical radius

(2-6a)

or

(2-6b)

so that for r < rc, the nucleus tends toshrink, while for r > rc it tends to grow andat r = rc it is in an unstable equilibrium.

The same thermal fluctuations causingsuch a nucleus to appear also produce devi-ations from the average spherical shape.This leads to power-law corrections in theprefactor of Eq. (2-5) or logarithmic cor-rections in the exponent (Voronkov, 1983;Langer, 1971).

So far these considerations have dealtwith static aspects only. Since the fluctua-tions vary locally with time, Eq. (2-5) maybe interpreted as the rate at which suchfluctuation occurs, and, consequently, withr = rc we obtain the rate for the formationof a critical nucleus which, after appear-ance, is assumed to grow until the newphase fills the whole system. This is theclassical nucleation theory. A very elegantformulation was given by analytic continu-ation into the complex plane (Langer,1971). Further additions include the defini-tion of the prefactor in Eq. (2-5) and a moredetailed analysis of the kinetics near r = rc

(Zettlemoyer, 1969, 1976).Considering the many uncertainties en-

tering from additional sources such as therange of atomic potentials and the changeof atomic interaction in the surface, we willignore all these effects by absorbing theminto the effective kinetic prefactor n0 in Eq.(2-5), to be determined experimentally.

rc = (2-dim)gmD

W

rc = (3-dim)2g

mDW

88 2 Solidification


2.2.2 Interface Propagation

An interface between two regions inspace of different order parameters (solidvs. liquid) will be treated in this chapter asa jump discontinuity and as an object of in-finitesimal thickness. In this section we de-rive a local equation of motion for the inter-face, which will serve as a boundary condi-tion in the remaining part of the chapter.

For simplicity, consider a one-dimen-sional interface in a two-dimensional sys-tem. Ignoring the atomic structure, assumethat the interface is a smooth continuousline. Marking points at equal distances a onthis line, we may then define velocities ofthe points in the normal direction as

(2-7)

where Dx i is the small normal displace-ment of point i. If we assume these pointsto be kept at fixed positions x i for the mo-ment, a (restricted) Gibbs energy G (x i)may be assigned to this restricted interface.The probability of this configuration is

P (x i) = Z –1 exp (–G/T ) (2-8)

with temperature T in units of kB K and Zthe partition function. Since the total prob-ability is conserved, we obtain the continu-ity equation

(2-9)

with J = Ji as the probability current ini-space

(2-10)

and the divergence taken in the same ab-stract space. The first term in Eq. (2-10) isa drift, the second term the constitutive re-lation with transport coefficients Di , andthe derivative is taken normal to the inter-

J V P DP

i i ii

= − ∂∂x

∂∂

+Pt

idiv =( ) J 0

Vtii=

DDx

face. We now assume local equilibrium toexist on length scales a (i.e. i Æ i ±1) suchthat the probability current is zero

(2-11)

and with Eqs. (2-7) and (2-8), we immedi-ately arrive at

(2-12)

Taking the continuum limit a Æ 0, we ob-tain the final form

(2-13)

This is the time-dependent Ginzburg–Lan-dau equation (Burkhardt et al., 1977). HereS denotes a position on the interface, ∂x (S)is the normal displacement, d/dx (S) thevariational derivative, and k (S) the “mo-bility”, which may depend on position andorientation. G is the Gibbs energy per sur-face element.

We will now make some explicit as-sumptions about G in order to arrive at anexplicit equation of motion. Let nS be thenormal direction on the interface, g (nS) theinterface free energy, J the angle of orien-tation relative to the crystallographic axes,R (S) the local radius of curvature at point Son the interface, and AL,S the areas cov-ered by liquid or solid. The Gibbs free en-ergy of the total solid– liquid system withinterface is

(2-14)

from which the variational derivative inEq. (2-13) is formally obtained by

(2-15)d dd

dG SG

SS

tot = d∫⎡

⎣⎢

⎤

⎦⎥

ˆ( )

xx

G S A

A

Stot

LiquidL L

SolidS S

= d d

d

∫ ∫

∫

+

+

g g

g

( )

( )

x

x

∂∂

−x kx

( )( )

ˆ

( )S

tS

GS

=d

d

∂∂

− ∂∂

xx

i i i

itDT

G=

( )xx

V P DP

i ii

=∂∂x

where dx (S) is a small arbitrary displace-ment of the interface in the normal direc-tion. Explicitly this is written as

(2-16)

with Gibbs energy densities g given hereper atomic “volume” (or area) W. As usual,we have to extract terms dx out of theterms d (dS) and dg. Assuming g to dependonly on local orientation,

(2-17)

and with(2-18)

we obtain

(2-19)

The other term simply gives

(2-20)

Incorporating Eqs. (2-19) and (2-20) intoEq. (2-16) and integrating Eq. (2-19) byparts, we obtain through comparison withEq. (2-15) (2-21)

as an explicit local equation for the ad-vancement of an interface in two dimen-sions with normal velocity V^, anisotropickinetic coefficient k (J ), surface Gibbs en-ergy g (J ), and jump Dg = gL – gS of Gibbsenergy density at position S along the inter-face (Burkhardt et al., 1977). Here, theGibbs energy density g corresponds to an

∂∂

≡ − +⎛⎝⎜

⎞⎠⎟

⎧⎨⎩

⎫⎬⎭

⊥x k g gt

V SR

=dd

( ) ( )JJ

Dg1 2

2

d d( )d = dSR

S1 x

d dg g x( )JJ

=dd

dd

− 1R S

d dJ JJ

J=dd

dd

=dd

dd

dd

=S S S S R

x ; ; − 1

d dg g( )J

JJ=

dd

d d d

d

G S S

S S S S

S

S

tot

L S

= d d

d

∫

∫

+

− −

g g

x

( )

( ) [ ( ) ( )]g g

infinite solid or liquid without influencesfrom curvature terms. For a single-compo-nent system, m = gW, where m is the chemi-cal potential of the respective phase. Thegeneralization to three dimensions adds an-other curvature term into Eq. (2-21), whichthen corresponds to two curvatures and an-gles in the two principal directions (for iso-tropic g , the curvature 1/R is simplychanged to 2/R).

Two useful observations can be made atthis stage. There is a solution with V^ = 0 ifthe term … in Eq. (2-21) is zero. For fi-nite radius of curvature R, this correspondsprecisely to the critical nucleus, Eq. (2-6b),but now with anisotropic g (J ). This equa-tion therefore determines the shape of thecritical nucleus in agreement with the Wulffconstruction (Wulff, 1901). Second, forvery large mobility k (J ) the deviationfrom equilibrium … may be very smallin order to produce a desired normal veloc-ity V^. We will use this simplification ofequilibrium at the interface

(2-22)

wherever possible, but we will comment onthe changes due to finite k (J ). The doubleprime means derivatives as in Eq. (2-21).In many cases this seems to be justifiedby experimental conditions. On the otherhand, very little is known quantitativelyabout k (J ). A last point to mention here isour assumption that the interfaces are notfaceted at equilibrium. If they are faceted,as crystals typically are at low tempera-tures in equilibrium with their vapor, thesituation is considerably more complicatedand not completely understood (Kashubaand Pokrovsky, 1990).

It is now generally believed that facetedsurfaces undergo a kinetic roughening tran-sition even at small driving forces, so that arough surface is present under growth con-

Dg ≅ + ′′1R

( ) (g g 2-dim)

90 2 Solidification


ditions. The concept outlined above thengenerally applies in an effective fashion.Actually, facets may persist over experi-mentally relevant length scales, which arenot covered by this analysis in the kineticregion. A summary of theoretical develop-ments can be found in the article by Krugand Spohn (1992).

To complete this section on the basictheoretical ingredients, we will now dis-cuss the influence of diffusive transport ofheat or matter on the propagation of asolid– liquid interface (see also Langer,1980a).

2.2.3 Growth of Simple Crystal Forms

a) Flat interface

The simplest model for a solidifyingsystem consists of two half-spaces filledwith the liquid and solid of a one-compo-nent material of invariant density and sep-arated by a flat interface. The interface isapproximately in equilibrium (Eq. (2-22))at melting temperature Tm but advancing ata speed V in the positive z-direction into theliquid. During this freezing process, latentheat Lm has to be transported into the su-percooled liquid, the solid remaining at Tm.

The equation of motion is then the ther-mal diffusion equation

(2-23)

with thermal diffusion coefficient DT andappropriate boundary conditions. At infin-ity in the supercooled liquid the tempera-ture T• < Tm is prescribed, and Tm is thetemperature at the interface. It is now veryconvenient to replace the temperature fieldT by a dimensionless u through the trans-formation

(2-24)u tT t T

L cp( , )

( , )( / )

xx

= m

m

−

∂∂

∇t

T t D T tT( , ) ( , )x x= 2

where cp is the specific heat of the liquid atconstant pressure. If, instead of tempera-ture or heat, a second chemical componentis diffusing, a similar transformation tothe same dimensionless equations can bemade. This is described in Section 2.5.2.We would like to stress the importance ofsuch a scaled representation as it allows usto compare at a glance experimental situa-tions with different sets of parameters.

In this dimensionless form, the equationof motion is

(2-25)

and the boundary conditions are

u = u• < 0 for z Æ • (2-26a)

u = 0 at interface (2-26b)

So far we have not specified how the inter-face motion is coupled to the equation ofmotion. Obviously, the latent heat Lm gen-erated during this freezing process at a rateproportional to V has to be carried awaythrough the diffusion field. This requires acontinuity equation at the interface

V = – DT n · —u (2-27)

and since in the model here only the z-axisis important

(2-28)

The growth rate is therefore proportional tothe gradient of the diffusion field at theinterface in the liquid. As the interface ismoving, we conveniently make a coordi-nate transformation from z, t for z¢, t ¢into a frame of reference moving alongwith the interface at z¢ = 0:

(2-29)′ −

′

∫z z V

t t

t

= d

=0

( )t t

V Dz

u z tT= at interface− ∂∂

( , )

∂∂

∇t

u D uT= 2

Performing that transformation, we obtainthe following for the diffusion equation(after having again dropped the primes forconvenience):

(2-30)

which defines a diffusion length

l ∫ 2 DT /V (2-31)

The interface is now always at z = 0, whichmakes the boundary condition Eq. (2-26b)and the continuity Eq. (2-28) definite.

We can obtain a stationary solution bycombining Eq. (2-30) with Eqs. (2-26) and(2-28) when ∂u /∂t = 0:

(2-32)

This equation is consistent with the as-sumption that u = 0 (T = Tm) everywhere inthe solid.

This equation describes the diffusion fieldahead of the interface. It varies exponen-tially from its value at the interface towardthe value far inside the liquid, so that thediffusion field has a typical “thickness” of l.

Note that we have not used Eq. (2-26a)as a boundary condition at infinity, but findfrom Eq. (2-32) that u = –1 implies the so-called “unit supercooling”, which corre-sponds to

T (z = •) = Tm – (Lm/cp) (2-33)

This basically says that the difference inmelting enthalpy Lm between liquid andsolid must be compensated by a tempera-ture difference in order to globally con-serve the energy of the system during thisstationary process. In other words, if Eq.(2-33) is not fulfilled by Eq. (2-26a), theprocess cannot run with a stationary profileof the thermal field, Eq. (2-32). On theother hand, a particular value of the growth

uz

lz= exp ;−⎛⎝

⎞⎠ −

21 0Û

1 22

Dut

ul

uzT

∂∂

∇ + ∂∂

=

rate V (or l) is not specified, but seems tobe arbitrary. This degeneracy is practicallyeliminated by other effects such as inter-face kinetics (Collins and Levine, 1985) orthe density difference between solid andliquid (Caroli et al., 1984), so that in prac-tice a well-defined velocity will be selected.

b) Growth of a sphere

The influence of the surface tension gwhen it is incorporated through Eq. (2-21)is most easily understood by looking at aspherical crystal. This will not lead to sta-tionary growth. In order to make the analy-sis simple, we will invoke the so-called“quasistationary approximation”, by set-ting the left-hand side in Eq. (2-25) equalto zero. The physical meaning is that thediffusion field adjusts itself quickly to achange in the boundary structure which is,however, still evolving in time because thecontinuity equation is velocity-dependent.Of course, this approximation reproducesthe stationary solutions precisely (if theyexist), and in addition, it exactly identifiesthe instability of a stationary solution aslong as it is not of the Hopf type (i.e., thecritical eigenvalue is not complex). It isgenerally assumed that the approximationis good as long as the diffusion length islarge compared with other lengths of theevolving pattern.

In a spherical coordinate system, theequation of motion then becomes

(2-34)

as a simple Laplace equation.The interface is at radius R0(t) and is ad-

vancing with time. The continuity equationrequires

(2-35)vR TR t

Rt

Dur

≡ − ∂∂

dd

==

0

0r ( )

10

22

2Dut r r r

uT

∂∂

≈ ∂∂

+ ∂∂

⎧⎨⎩

⎫⎬⎭

=

92 2 Solidification


Concerning the boundary condition, wenote that for a simple substance the differ-ence in slopes of the chemical potentialsfor solid and liquid at the melting point(Landau and Lifshitz, 1970) is Lm/Tm, giv-ing

Dm = – Lm (T – Tm)/Tm (2-36)

With Eqs. (2-21) and (2-24), the boundarycondition in the case of isotropic g be-comes

u (interface) = – d0 K – b V (2-37)

with curvature

(2-38)

capillary length

d0 = g Tm cp /L2m (2-39)

and interface kinetics

b = k –1 Tm cp /L2m (2-40)

The generalization to non-isotropic g fol-lows from Eq. (2-21) and comments.

Note that in Eq. (2-37) both curvatureand non-equilibrium occur at the boundaryas opposed to the simple case in Eq. (2-26b), shown above. At infinity we finallyimpose

u• = u (•, t) = – D (2-41)

as the now arbitrary dimensionless super-cooling. This equation, together with Eq.(2-24), is the definition of D.

The solution to Eq. (2-34) with Eqs. (2-35), (2-37), and (2-41) in the liquid is thensimply (2-42)

where we have used K = 2/r as the curva-ture of the spherical surface in three dimen-sions.

u rRr

dR

DR

T( ) = − + −⎛⎝⎜

⎞⎠⎟

+⎛⎝⎜

⎞⎠⎟

D D0 0

0 0

21

b

KR

R=

3-dim

2-dim

2

1

/ ,

/ ,⎧⎨⎩

The growth rate therefore is(2-43)

and it is found that for large radii R0(t) thegrowth rate is proportional to the super-cooling D:

(2-44)

It can also be seen that the kinetic coeffi-cient b becomes less important in the limitof large radii and correspondingly smallgrowth rates. This in many cases justifiesour previous equilibrium approximation,Eq. (2-22).

We have not discussed here the solidifi-cation of a binary mixture. As the detailsof the phase diagram will become impor-tant later, we will postpone this topic to thesection on directional solidification. Wewould like to mention, however, that therepresentation of the diffusion field in di-mensionless form, found in Eq. (2-24), hasthe virtue that several results can be carriedover directly from the thermal case to com-positional diffusion, although there aresome subtleties in the boundary conditionswhich make the latter more difficult to an-alyze.

2.2.4 Mullins–Sekerka Instability

We now combine the considerations ofSecs. 2.2.3a) and b) to study the questionof whether an originally flat interface willremain flat during the growth process. Theresults indicate that a flat interface movinginto a supercooled melt will become rip-pled (Mullins and Sekerka, 1963, 1964).

The basic equations are almost exactlythe same as in the previous section a) Eqs.(2-30), (2-27), (2-26a), but instead of theboundary condition Eq. (2-26b), we now

VDR

DRR

T T≈ − +…⎛⎝⎜

⎞⎠⎟0 0

1D b

vRT TR

tDR

dR

DR

≡ −⎛⎝⎜

⎞⎠⎟

+⎛⎝⎜

⎞⎠⎟

dd

=0

0

0

0 0

21D b

have to consider deviations from a flatinterface as we move parallel to the inter-face in the x-direction. This is provided byEq. (2-37) as the boundary condition,where we set b = 0 for simplicity.

Denoting the deviation of the interfacefrom z = 0 as z (x, t), we designate the cur-vature for small amplitudes simply asK = – ∂ 2z /∂x2, where, in agreement withEq. (2-38), the sign of K allows for the cur-vature to be positive for a solid protrusioninto liquid. We follow here the notation ofLanger (1980a).

In a quasistationary approximation, forthe sake of simplicity we set the l.h.s. inEq. (2-30) equal to zero, which of courseallows for the stationary solution, Eq. (2-32). It is not difficult to treat the fully time-dependent problem here, but the modifica-tions do not change the results substan-tially. We now perturb that solution bymaking a small sinusoidal perturbation ofthe flat interface:

z (x, t) = zk exp (i k x + Wk t) (2-45)

Similarly, a perturbation of the diffusionfield is made in the liquid and in the solid

where the unprimed form is for the liquid,and the primed values are for the solid.Inserting Eq. (2-46) into Eq. (2-30) with∂u /∂t = 0, we obtain

– 2q /l + q2 – k2 = 0

2q¢/l ¢ + q¢2 – k2 = 0 (2-47)

Replacing z in Eq. (2-46) by z (x, t) from Eq.(2-45), we can insert this into the boundarycondition Eq. (2-37) for b = 0 to obtain

(2-48)ˆ ˆ / ˆ ˆ′ − + −u l u d kk k k k= =2 02z z

u x z tz

lu i k x q z t

u x z t u i k x q z tk k

k k

( , , ) exp

ˆ exp ( );

( , , ) ˆ exp ( )

= (2-4 )

=

−⎛⎝⎞⎠ −

+ − +′ ′ + ′ +

21 6

WW

where we have linearized the exp (…) withrespect to z. Instead of the velocity in Eq.(2-27), we must now use Vz = 2D/l + ∂z /∂t.With this equation and the same lineariza-tion as before

(2-49)

is obtained for the conservation law, Eq.(2-27) (and for small values of z , practi-cally Eq. (2-28)). Eliminating uk here usingEq. (2-48) and eliminating q using Eq. (2-47)in the limit k l 1 or k≈ q ≈ q¢, we obtain

This formula describes the basic mecha-nism of diffusion-controlled pattern forma-tion in crystal growth. Althoug in general,diffusion tends to favor homogeneousstructures, in the present case it works inthe opposite direction! This is easy tounderstand; the foremost points of a sinu-soidal deformation of the interface can dis-sipate the latent heat of freezing by a largerspace angle into the liquid than the pointsinside the bays. The latter points will there-fore be slowed down. The tip points will beaccelerated in comparison with the averagerate of advancement.

The formula therefore consists of a de-stabilizing part leading to positive Wk anda stabilizing part controlled by the capillar-ity d0. The stabilization is most efficientat large k values or short wavelengths; atlonger wavelengths, the destabilization dueto diffusion into the supercooled liquiddominates.

The dividing line is marked by the com-bination of d0 l, which is most convenientlyexpressed as

(2-51)

assuming D = D¢. This will be called the“stability length”, which gives a measure

lS S= =2 2 0p p/k d l

Wk V q l Dq D q d k

k V D D l d k

=

(2- )

( / ) ( )

[ ( / ) ]

− − + ′ ′≈ − + ′

2

1 1 500

2

12 0

2

Wk k k k kD l Dqu D q uˆ ( / ) ˆ ˆ ˆz z= − + + ′ ′ ′2 2

94 2 Solidification

2.3 Basic Experimental Techniques 95

of the typical lengths for diffusion-gener-ated ripples on the interface, since thefastest-growing mode has a wavelengthlf = ÷-

3lS. This is the first explicit exampleof the fundamental scaling relation Eq.(2-1) mentioned in the introduction to thischapter.

2.3 Basic Experimental Techniques

Before discussing the various growthmodels, a short overview of selected ex-periments that have contributed to ourpresent understanding will be given.

An exhaustive treatment of the subject isnot intended here; only some of the charac-teristic techniques will be described. Nospecific reference will be given to themany solidification processes available.The main emphasis will be on growth ofcrystals, because direct evidence on nucle-ation mechanisms is generally not avail-able. There is an exception; the originaltechnique developed by Schumacher et al.(1998). These authors analysed quenched-in nuclei (inoculants) in glassy Al alloys inthe transmission electron microscope. Inthis state, nucleation kinetics are very slowand in-situ observation of the phases andand their structure, composition and crystalorientation can be measured. The reader

interested in nucleation experiments isreferred to reviews by Perepezko et al.(1987, 1996).

There are two essentially different situa-tions of solidification, or, more generally,of phase transformation (Fig. 2-1):

i) free (undercooled or equiaxed) growth,ii) constrained (columnar on directional)

growth.

In the first case (Fig. 2-1a), the meltundercools, before transforming into acrystal, until nucleation sets in. The cru-cible that contains the melt must be less ef-fectively catalytic to crystallization thanthe heterogeneous particles in the melt.The crystals then grow with an interfacetemperature above the temperature of thesurrounding undercooled liquid. Heat iscarried into the liquid. The temperaturegradient at the solid– liquid interface in theliquid is therefore negative and approxi-mately zero in the solid owing to the in-itially small crystal size relative to the ther-mal boundary layer.

In the second case of constrained growth(Fig. 2-1b), the temperature gradient ispositive in both liquid and solid, and thefirst solid is either formed in contact withthe chill mold or is already present, as insurface treatment by lasers, for example,where growth occurs epitaxially after re-

Figure 2-1. Growth in (a) under-cooled melt (free growth) and (b)superheated melt (directionalgrowth). The arrows on the out-side of the mold represent the heatflux, and the arrows at thesolid/liquid interface, the growthdirection.

melting. In case i), nucleation is essentialin controlling the microstructure (the grainsize), while in case ii), nucleation is of mi-nor importance.

Typical technical processes with respectto these two classes of solidification are asfollows:

i) casting into low conductivity molds(ceramics), producing small temperaturegradients, or adding inoculating agents orstirring the melt, thereby producing manyheterogeneous or homogeneous nuclei;

ii) processes with high heat flux imposedthrough strong cooling of the solid such ascontinuous casting, welding, laser treat-ment, or through heating of the melt andcooling of the solid, such as in single crys-tal growth or Bridgman type directional so-lidification experiments (see also Flem-mings, 1991).

In free growth, the undercooling DT ofthe melt is given and controls the growthrate V and the scale (spacing l or tip radiusR) of the forming microstructure. In direc-tional growth, the rate of advance of theisotherms is imposed by heat flux leadingto an imposed growth rate (constrainedgrowth). This, in turn, controls the inter-face temperature (undercooling) and themicrostructural scales. These three vari-ables (temperature, growth rate, and micro-structural size) have to be measured experi-mentally for a quantitative comparison oftheory and experiment. A number of mate-rial parameters of the alloy system have tobe known, such as the solid– liquid interfa-cial energywith its anisotropy, the diffusioncoefficient, the stable and metastable phasediagram, etc.

The experimental techniques are dividedinto two classes; those which produce rela-tively small growth rates and those whichaim for high solidification rates. The corre-sponding experimental setup is quite dif-

ferent, and some of its important elementswill be described below.

2.3.1 Free Growth

Slow growth rates

In free growth, the undercooling DT orthe temperature of the melt T0 = (Tm – DT )is imposed on the system, and the growthrate V and the microstructural scale are thedependent variables. For low undercool-ings, and therefore slow growth rates, allthree variables can be measured preciselyif the substance is transparent. The crystal-lization of non-transparent metals, how-ever, is the major issue in solidificationstudies relevant to technical applications.

Instead of investigating the crystalliza-tion of metallic systems directly, suitablemodel substances have to be found. Theseare generally organic “plastic crystals”which, like metals, have simple crystalstructures and low melting entropies (Jack-son, 1958). One of the substances whichbehaves very similarly to metals and alsohas well characterized properties is succi-nonitrile (SCN) (Huang and Glicksman,1981). Numerous results have been ob-tained with this material by Glicksman andco-workers. Their careful experimental ap-proach not only produced the most precisemeasurements known at that time but hasalso stimulated new ideas about possiblemechanisms of structure formation throughthe discrepancy found between the obser-vations and the predictions of previous the-ories.

Fig. 2-2 shows the experimental setupdeveloped by Glicksman et al. (1976) tostudy free dendritic solidification. Afterzone melting, the material is introducedinto the growth chamber (C), which is thenslowly undercooled with the heaters (A)and (B) in order to avoid premature crystal-

96 2 Solidification


lization. By careful adjustment of heater(A), growth starts there, and a crystalgrows through the orifice in chamber (C).From then on, the dendrite can grow freelyinto the undercooled liquid and its shape,size and growth rate, depending on super-

cooling, are measured with the aid of pho-tographs. In order to choose a proper pro-jection plane of observation, the entireequipment can be translated, rotated andtilted at (D). A series of similar experi-ments have been performed at low tem-peratures with rare gases (Bilgram et al.,1989).

In this kind of experiment, it is importantto avoid thermal or solutal convection, asthis transport mechanism will change theresults and make them difficult to comparewith diffusional theory. Glicksman et al.(1988) have shown the effect of thermallydriven convection on the growth morphol-ogy of pure SCN dendrites. Fig. 2-3 clearlyshows a strong deviation of the growth rateat undercoolings lower than 1 K, whereconvection is believed to accelerate den-drite growth. Here V0 is the dendrite tiprate of pure diffusion-controlled growth.Recent experiments on free dendriticgrowth of succinonitrile in space under mi-crogravity conditions in the undercoolingrange of 0.064 to 1.844 K confirmed thiseffect clearly (Koss et al., 1999).

Another type of slow dendrite growth inundercooled media has been analyzed byTrivedi and Laorchan (1988). These au-thors made interesting in situ observationsduring the crystallization of glasses. Evenif the driving forces in these systems arevery large, growth is heavily restricted dueto slow diffusion and attachment kinetics.

Figure 2-2. Equipment for free growth of organicdendrites (Glicksman et al., 1976). A and B, controlheaters; C, growth chamber; D, tilting and rotatingdevice; F, tank cover; G and H, zone-refining tubes.

Figure 2-3. Ratio of measuredgrowth rate, V, to predictedrate from diffusion theory, VD,as a function of undercooling(Glicksman et al., 1988).

Fast growth rates

In order to explore solidification behav-ior under very large driving forces, Herlachet al. (1993) and Herlach (1994, 1997)systematically measured dendrite growthrates in a large number of highly under-cooled alloys. Nucleation is avoided bylevitation melting in an ultrahigh vacuumenvironment. Undercooling of more than300 K leading to growth rates of up to70 m/s has been reached. Such large under-coolings have been obtained by several au-thors before, but growth rates have notbeen measured. Interesting results in Ni–Sn alloys have also been obtained by high-speed cinematography of highly under-cooled samples. These results were pro-duced by Wu et al. (1987) by encapsulatingthe melt in glass. They tried to measuretip radii from the photograph. In this case,however, only the thermal images of thedendrites could be seen, their tips beingcontrolled by solute diffusion. The radiifrom the thermal images are thereforebelieved to be much larger than the realradii.

2.3.2 Directional growth

Slow growth rates

Understanding of solidification improvedin the 1950s when the need for better semi-conductor materials stimulated researchusing directional growth techniques suchas zone melting and Bridgman growth.Later, directional casting became an impor-tant topic of research for the productionof single crystal turbine blades. Fig. 2-4shows the essentials of two techniques:Bridgman type and chill plate directionalsolidification. The first of these processes(Fig. 2-4a) has the advantage of being ableto produce a constant growth rate and aconstant temperature gradient over a con-siderable length and to allow for a certainuncoupling of these two most importantvariables. It was this latter advantagewhich, early on in solidification research,provided much insight into the mecha-nisms of growth. With one growth rate, butvarying temperature gradient (or viceversa), plane-front, cellular or dendriticmorphologies could be grown, and theirevolution studied. Important concepts(constitutional undercooling, cell growth,

98 2 Solidification

Figure 2-4. Basic methodsof directional solidification;(a) Bridgman type furnaceand (b) directional casting.(a) (b)


etc.) have been developed with the aid ofobservations made with the Bridgman typeof equipment. The experiments by Hunt etal. (e.g., Burden and Hunt, 1974) deservespecial attention, as the quality of themeasurements was of a very high standard.

Furthermore, most of the work on direc-tional eutectics and their growth mecha-nisms was performed with the aid of thistechnique. In eutectic solidification, themicrostructure after transformation has thesame size (interphase spacing l) as thatfound at the growing interface and thus al-lows for direct conclusions to be drawnconcerning the growth process. This im-portant variable can therefore be easily de-termined in non-transparent metals. This,however, is not possible with the corre-sponding quantity of the dendrite, the tipradius R, as discussed earlier.

One disadvantage of the Bridgman ex-periment is the need for a small diameterdue to heat flux constraints. This is avoidedin the process shown in Fig. 2-4b (direc-tional casting), but in this case a separationof the variables V and G is not possible.Therefore, this method is less interestingfor scientific purposes. However it is usedextensively for directional casting of singlecrystal turbine blades (Versnyder andShank, 1970).

For in-situ observation of microstructureinformation, we can also use plastic crys-tals in an experimental arrangement resem-bling a two-dimensional Bridgman appara-tus, as shown in Fig. 2-5. Two glass slidescontaining the organic analogue are movedover a heating and cooling device produc-ing a constant temperature gradient GT .This can be controlled to certain limits bythe temperature difference and distancebetween heater and cooler (Somboonsukand Trivedi, 1985; Trivedi and Somboon-suk, 1985; Akamatsu et al., 1995; Ginibreet al., 1997; Akamatsu and Faivre, 1998).

A thin thermocouple incorporated intothe alloy allows for measurement of inter-face temperature (Fig. 2-6) and tempera-ture gradient in liquid and solid (Esaka andKurz, 1985; Somboonsuk and Trivedi,1985; Trivedi and Somboonsuk, 1985; Tri-vedi and Kurz, 1986).

Fast growth rates

There have been attempts to increase thegrowth rate in Bridgman type experiments.The first to reach rates of the order of sev-eral mm/s were Livingston et al. (1970).The best way to reach much higher rates isthrough laser resolidification using a stable

Figure 2-5. Schematic diagram of growth cell (a, b)for observation of directional solidification of trans-parent substances under a microscope (Esaka andKurz, 1985). 1, solid; 2, liquid; 3, thermocouple;4, cell; 5, low melting glass; 6, araldite seal; 7 and 8,heaters; 9, cooler; 10, drive mechanism; 11, micro-scope. In (c) the temperature distribution in thegrowth cell is shown.

high-powered laser. In this case, a smallmelt pool with very steep temperature gra-dients is created (for a remelted layer of100 mm thickness, the temperature gradi-ent GT is of the order of 5000 K/mm). Themicrostructure is then constrained to fol-low closely the heat flow, which is perpen-dicular to the isotherms (Fig. 2-7). There-fore, knowing the angle q between the di-rection of growth and the direction and rateof movement of the laser beam alloys forthe growth rate to be obtained locally, evenin the electron microscope (Zimmermannet al., 1989). Note that the growth rate in-creases from zero at the interface with thesubstrate up to a maximum at the surface.The only unknown which would be ex-tremely difficult to measure is the interfacetemperature.

Consequently, in the case of rapid direc-tional growth we can determine the post-

solidification microstructure and its growthrate as well as the bath temperature and thegrowth rate in rapid undercooled growth.In both cases, the measurement of the inter-face temperature is not possible for thetime being and must be evaluated throughtheory alone.

2.4 Free Dendritic Growth

The most popular example of dendriti-cally (= tree-like) growing crystals is givenby snowflakes. The six primary arms oneach snowflake look rather similar but thereis an enormous variety of forms betweendifferent snowflakes (Nakaya, 1954). Fromour present knowledge, the similarity ofarms on the same snowflake is an indica-tion of similar growth conditions over thearms of a snowflake, the variation of struc-ture along each arm being an indication ofa time variation of external growth condi-tions.

In order to formulate a theory, some spe-cific assumptions about these environmen-tal conditions must be made. Unfortu-nately, the growth of a snowflake is anextremely complicated process, involvingstrongly anisotropic surface tension andkinetics and the transport of heat, water va-por, and even impurities. Therefore, wewill initially drastically simplify the modelassumptions. In the same spirit, a numberof precise experiments have been per-formed to identify quantitatively the mostimportant ingredients for this mechanism.

In its simplest form, dendritic growth re-quires only a supercooled one-componentliquid with a solid nucleus inside, so thatthe heat released at the solid– liquid inter-face during growth is transported away intothe liquid. This is precisely the conditiongiven in Sec. 2.2.3 concerning a spheregrowing into the supercooled, and there-

100 2 Solidification

Figure 2-6. Array of dendrites approaching a finethermocouple for measurement of the tip temperatureand temperature gradient. Diameter of the bead ap-proximately 50 µm (Esaka and Kurz, 1985).

2.4 Free Dendritic Growth 101

fore metastable, liquid. A relatively straight-forward stability analysis (Mullins and Se-kerka, 1963, 1964; Langer, 1980a) showsthat the solid tends to deviate from spheri-cal form as soon as its radius R has becomelarger than seven times the critical radiusRc = 2d0/D. A time-dependent analysis hasalso been carried out (Yokoyama and Ku-roda, 1988). During the further evolution,these deformations evolve into essentiallyindependent arms, the primary dendrites.

The growth of such dendrites is a verywidespread phenomenon, as will becomeclearer in Sec. 2.5 on directional solidifica-tion. There are also close relations to pro-cesses called diffusion-limited aggregation(DLA) (for a review see Meakin, 1988; forrelation to crystal growth, see Uwaha and

Saito, 1990; Xiao et al., 1988). For theseprocesses, some specific assumptions aremade about the incorporation of atoms intothe advancing interface which are not eas-ily carried over into the notion of surfacetension, etc. Another line of closely relatedproblems concerns the viscous flow of twoimmiscible liquids (Saffmann and Taylor,1958; Brener et al., 1988; Dombre and Ha-kim, 1987; Kessler and Levine, 1986c). Wewill briefly refer to this in Sec. 2.5.

2.4.1 The Needle Crystal Solution

In this subsection, we will look at an iso-lated, needle-shaped crystal growing understationary conditions into a supercooledmelt. A stationary condition, of course, only

Figure 2-7. Schematic diagrams of a la-ser trace (a, b) and of the local interfacevelocity (c). Vb is the laser scanning ve-locity and Vs is the velocity of the solid/liquid interface which increases from zeroat the bottom of the trace to a maximumat the surface. W and D are the width anddepth of the trace, respectively, and q isthe angle between the growth directionand the scanning direction.

holds in a frame of reference moving at ve-locity V in the positive z-direction. For a one-component crystal, the latent heat of freez-ing must be diffused away into the liquid.

The process is therefore governed by theheat diffusion Eq. (2-25)

(2-52)

where we have used a dimensionless form,Eq. (2-30), for a stationary pattern togetherwith the definition of the diffusion length,Eq. (2-31). If the diffusion constants in thecrystal and the liquid are different, wewould have to use two diffusion lengths.The continuity Eq. (2-27) at the interface isfor normal velocity

V^ = – DT n · —uL + DT n · —uS (2-53)

where the subscripts L and S denote gra-dients taken on the liquid side and on thesolid side of the interface. This is known asthe “two-sided” model (Langer and Turski,1977) and, because of identical diffusionconstants, as the “symmetrical” model. Thegeneralization to different diffusion con-stants is simple.

The boundary condition at the interface is

uI = – d0 K – b V (2-54)

and at infinity

u• = – D (2-55)

as introduced before in Eqs. (2-37) and (2-41), respectively. Note that a constant likeD may be added on the right-hand side ofboth equations without changing the resultsapart from this additive constant in the u-field, as frequently appears in the litera-ture. The definition of D follows easilyfrom Eqs. (2-24) and (2-41).

In general, the capillary length d0 andthe kinetic coefficient b are anisotropic be-cause of the anisotropy of the crystalline

022= ∇ + ∂∂

ul

uz

lattice, but they are not directly related(Burkhardt et al., 1977). For the moment,we will assume b = 0, i.e., the interface ki-netics should be infinitely fast. Even then,the curvature K in Eq. (2-54) is a compli-cated function of the interface profile. Thesimplest approximation for the moment,therefore, is to ignore both terms on ther.h.s. of the boundary condition, which cor-responds to setting the surface tension tozero. The boundary condition is then sim-ply a constant (= 0 in our notation).

A second-order partial differential equa-tion such as the diffusion equation can bedecomposed in the typical orthogonal coor-dinate systems, and we therefore obtain aclosed form solution for the problem with aboundary of parabolic shape: a rotationalparaboloid in three dimensions and asimple parabola in two dimensions. This isthe famous “Ivantsov” solution (Ivantsov,1947).

The straightforward way to look at thisproblem is as a coordinate transformationfrom the cartesian x, z frame, where z isthe growth direction, to the parabolic coor-dinates x, h:

x = (r – z)/r ; h = (r + z)/r

where r is the radial distance ÷---x2 +

---z2 from

the origin, and r is a constant. After trans-forming the differential operators in Eqs.(2-52) and (2-53) to x, h (Langer andMüller-Krumbhaar, 1977, 1978), it can beseen immediately that h (x ) = hS = 1 for theinterface is a solution to the problem con-firming the parabolic shape of the inter-face, with r being the radius of curvature atthe tip.

This Ivantsov radius, r, is now related tothe supercooling D and the diffusion lengthl by the relation

(2-56a)

(2-56b)D =

e 3-dim

e erfc 2-dim

P E P

P P

P

P1 ( )

( )p⎧⎨⎩



which for small D 1 gives

and for D Æ1 asymptotically

where the dimensionless Péclet number Pis defined as the relation

P = r /l = r V /2D (2-57)

between tip radius and growth rate. E1(P)is the exponential integral

(2-58)

and erfc is the complement of the errorfunction.

Eq. (2-56) may be interpreted as an ex-pression of supercooling in terms of thePéclet number. This explanation is impor-tant because the following considerationsof surface tension give only very small cor-rections to the shape of the needle crystal.Therefore, Eq. (2-56) will be also validwith non-zero surface tension at typical ex-perimental undercoolings of D Ó 10–1. Animportant consequence is that the scalingresults derived below then become inde-pendent of the dimensionality (2 or 3), ifthe supercooling is expressed through thePéclet number.

The basic result of Eqs. (2-57) and (2-48) is that the growth rate of this parabolicneedle is inversely proportional to the tipradius, but no specific velocity is selected.For experimental comparison, we make afit of the tip shape to a parabola. The tipradius of that parabola is then comparedwith the Ivantsov radius r. The actual tipradius will be different from the Ivantsovradius of the fit parabola, because of sur-

E Pt

tP

t

1 ( ) =e

d∞ −

∫

D ≈−−

⎧⎨⎩

1 1

1 1 2

/

/

P

P

3-dim

2-dim

D ≈− − …⎧

⎨⎩

P P

P

( ln . )0 5772 3-dim

2-dimp

face tension effects. Before we considersurface tension explicitly we now give anintegral formulation for the problem usingGreen’s functions, which has proved to bemore convenient for analytical and numer-ical calculations (Nash and Glicksman,1974).

The value of a temperature field u (x, t)in space and time is obtained by multiply-ing the Green’s function of the diffusionequation with the distribution for sourcesand sinks for heat and integrating the prod-uct over all space and time. In our case, thisexplicitly gives

(2-59)

(Langer, 1987b), with the Green’s function

(2-60)

for diffusion in an infinite three-dimen-sional medium (symmetrical case), while zis the z coordinate of the interface. Thesource term … in this equation is obvi-ously the interface in the frame of refer-ence moving at velocity V in the z-direc-tion. This equation already contains theconservation law or the continuity equationat the interface. Furthermore, it is valideverywhere in space and, in particular, atthe interface z = z (x, t), where the l.h.s. ofEq. (2-59) is then set equal to Eq. (2-54).Here u is assumed to vanish at infinity, soD must be added on the l.h.s.

In the two-dimensional case and forstationary conditions, this can be rewrittenas

(2-61)D G− dK P P x x0

2rz z , , ( )=

G z tDt

zDt

( , , )( )

exp/xx

=1

4 43 2

2 2

p| |− +⎛

⎝⎜⎞⎠⎟

u z t t G z t

V t t t t Vt

t

( , , ) [ , ( , )

( ), ]

x x x x x= d d−∞∫ ∫′ ′ − ′ − ′ ′

+ − ′ − ′ + ∂∂ ′

⎧⎨⎩

⎫⎬⎭

z

z

(Langer, 1987b), with

In addition, it was shown (Pelce and Po-meau, 1986) that with the parabolic Ivant-sov solution

(2-63)

we can obtain

which is independent of x, with D comingfrom Eq. (2-56b). Note that from Eqs.(2-59) to (2-61) we have replaced x by xretc., which mirrors the scaling form ofEq. (2-57).

We are now ready to consider a non-zerocapillary length d0 Eq. (2-39), which wegeneralize to be anisotropic:

d0 Æ d = d0 (1 – e cos 4J ) (2-64)

where J is the angle between the interface-normal and the z-axis (2-dim, 4-fold an-isotropy), and e > 0 is the relative strengthof that anisotropy. This form arises fromthe assumption

g = g0 (1 + d cos 4J ) (2-65)

for anisotropy surface tension, which givesthrough Eq. (2-22) e = 15d. Note that it isthe stiffness g≤ which dominates the be-havior, not g directly. Along the Ivantsovparabola, the angle J is related to x:

(2-66)

and the deviation from z IV (x) can be ex-pressed as

(2-67)

− −dP

K P x P xr

z z z( ) , , , , = IVG G2 2

d d A x A xx

x= =0

2

2 218

1( ); ( )

( )− +

+e e

P P x xG D2 , , (zIV ) =

z z( )x x→ −IV = 12

2

G20

2 2

12

2

, , ( )

exp ( ) [ ( ) ( ) ]

P x xyy

x

Py

x x x x y

z

z z

=d

d (2-62)p

∞

−∞

+∞

∫ ∫ ′

× − − ′ + − ′ +⎛⎝⎜

⎞⎠⎟

For convenience we combine some param-eters into a dimensionless quantity s :

(2-68)

so that the l.h.s. of Eq. (2-67) becomes

(2-69)

the curvature as usual being

(2-70)

It should be clear at this point that the pa-rameter s in Eq. (2-68) plays an importantrole, because it multiplies the highest de-rivative in Eqs. (2-61) and (2-67). Morespecifically, we can expect that the result-ing structure z (x) of the interface dependson the material properties and experimentalconditions only through this parameter s(within the model assumptions), which be-comes a function of P and e

s = s (P, e) (2-71)

The importance of the paramter s wasrecognized in an earlier stability analysis(Langer and Müller-Krumbhaar, 1978,1980) of the isotropic case. It turned outlater, however, that the anisotropy is essen-tial in determining the precise value of s.This is crucial as e Æ 0 implies s Æ 0, i.e.,no stationary needle solutions exist withoutanisotropy.

We will now briefly describe the analy-sis of Eq. (2-67). The details of this singu-lar perturbation theory are somewhat in-volved, and we therefore have to omit themhere. The basic method was formulated byKruskal and Segur (1985), and the firstscaling relations for dendritic growth wereobtained for the boundary-layer model(Ben Jacob et al., 1983, 1984). A good in-troduction to the mathematical aspects can

Kx

x

/[ ( / ) ] /z z

z= − ∂ ∂

+ ∂ ∂

2 2

2 3 21

− −dP

K A Kr

s=

sr

= =2

20

20

2Dd

V

d

DPV



be found in the lectures by Langer (1987b),on which the following presentation isbased. The most mathematically sound(nonlinear) solution seems to have beengiven by Ben Amar and Pomeau (1986).For convenience, we have sketched aslightly earlier linearized version here,while the nonlinear treatment leads to basi-cally the same result.

Looking for a solution to Eq. (2-67) inlinear approximation, we start by expand-ing to first order in

z1(x) = z (x) – zIV (x) (2-72)

In the limit of the small Péclet number, andwith the substitution

z1(x) = (1 + x2)3/4 Z (x) (2-73)

gives

(B + A) · Z (x) = s /(1 + x2)3/4 (2-74)

where B is a self-adjoint differential opera-tor

(2-75)

and

with denoting the principal value.The integral kernel in Eq. (2-76) is anti-

symmetric apart from a prefactor A (x)–1.An analytic solution to Eq. (2-74) has notyet been found. A necessary condition tobe fulfilled by the present inhomogeneity isthat it should be orthogonal to the null-eigenvectors Z (x) of the adjoint homoge-neous problem:

(B + A+) · Z (x) = 0 (2-77)

ˆ ( )( )

( )

( ) ( )

( ) [ ( ) ]( )

/

/

A ⋅ +

× ′+ ′ + ′

− ′ + + ′′

−∞

+∞

∫

Z xxA x

xx x x

x x x xZ x

= (2-76)

d

12

1

1

2 3 4

2 3 4

14

2

p

ˆ ( )( )

( )/

B =dd

s s2

2

2 1 210

x

xA x

+ + +

such that

(2-78)

In fact, this is already a sufficient conditionfor the solvability of the inhomogeneousequation, but it is not very simple.

A solution for Z (x) can be found by aWKB technique, for which we refer to theliterature (Kessler et al., 1987, 1988; Lan-ger, 1987b; Caroli et al., 1986a, b). Theresult for the solvability condition, Eqs.(2-71) and (2-78), is that the parameter sshould depend on anisotropy e as

s ≈ s0 e7/4 (2-79)

in the limit P Æ 0, e Æ 0, with some con-stant prefactor s0 of order unity.

Eq. (2-79) is the solution for the needle-shaphed crystal with capillary anisotropye > 0, together with Eq. (2-68). Note againthat e is the anisotropy of capillary length,which differs by a factor from surface-ten-sion anisotropy Eq. (2-65). Formally, thereis not just one solution but infinitely many,corresponding to slow, fat needles whichare dynamically unstable. Only the fastestof these needle solutions appears to bestable against tip-splitting fluctuations andmay thus represent an ‘observable’ needlecrystal, as expressed by Eq. (2-79). Forpractical comparison, experimental data isbest compared with numerical solutions ofEq. (2-61), because the applicability of Eq.(2-79) seems to be restricted to very smallvalues of e (Meiron, 1986; Ben Amar andMoussallam, 1987; Misbah, 1987). This willbe discussed further in the next section.

Needle-crystal solutionin three-dimensional dendritic growth

The theory of dendritic growth becomesextremely difficult, however, for three-di-

L( , )˜ ( )

( ) /s e ≡ ′+−∞

+∞

∫ d =xZ x

x102 3 4

mensional (3D) anisotropic crystals. Asimple extrapolation of the 2D case, wherethe surface energy is averaged in the azi-muthal direction (axisymmetric approach,Ben Amar, 1998; Barbieri and Langer,1989), is very important in order to havesome qualitative predictions. But any phys-ical anisotropy will give rise to a non-axisymmetric shape of the crystal. A nu-merical approach to the non-axisymmetricproblem was presented by Kessler and Le-vine (1988) who pointed out the followingaspect of the problem. In either the 2D orthe axisymmetric case, selection of thegrowth velocity follows from the solvabil-ity condition of smoothness of the dendritetip. In the 3D non-axisymmetric case asolvability condition must be satisfied foreach of the azimuthal harmonics. Kesslerand Levine made several approximationsand performed only a two-mode calcula-tion, but the crucial point of their analysisis that they found enough degrees of free-dom to satisfy all solvability conditions.

More recently, an analytic theory ofthree-dimensional dendritic growth hasbeen developed by Ben Amar and Brener(1993). In the framework of asymptoticsbeyond all orders, they derived the innerequation in the complex plane for the non-axisymmetric shape correction to the Ivant-sov paraboloid. The solvability conditionfor this equation provides selection of boththe stability parameter s µe7/4 and theinterface shape. The selected shape can bewritten as

(2-80)

where all lengths are reduced by the tip ra-dius of curvature r. Solvability theory (BenAmar and Brener, 1993) predicts that thenumbers Am are independent of the aniso-tropy strength a, in the limit of small a.For example, the first non-trivial term for

z rr

A r mmm( , ) cosf f= − + ∑

2

2

cubic symmetry corresponds to m = 4 andA4 = 1/96 is only numerically small. There-fore, the shape correction, Eq. (2-80), inunits of the tip radius of curvature, dependsmostly on the crystalline symmetry and isalmost independent of the material andgrowth parameters.

Stability theory for the 3D dendriticgrowth against tip-splitting modes has beendeveloped by Brener and Mel’nikov (1995).

An important aspect of Eq. (2-80) is thatthe shift vector r m cos mf grows at a fasterrate than the underlying Ivantsov solution.This means that only the tip region, wherethe anisotropy correction is still small, canbe described by the usual approximation(Kessler and Levine, 1988; Ben Amar andBrener, 1993), a linearization around theIvantsov paraboloid. This is the crucial dif-ference between the 3D non-axisymmetriccase and the 2D case. In the latter, smallanisotropy implies that the shape of the se-lected needle crystal is close to the Ivant-sov parabola everywhere; in the former,strong deviations from the Ivantsov para-boloid appear for any anisotropy.

Several important questions arise. Howis the tail of the dendrite to be described? Isit possible to match the non-axisymmetricshape (Eq. (2-80)) in the tip region to theasymptotic shape in the tail region? Whatis the final needle-crystal solution? The an-swers to these questions have been givenby Brener (1993).

The basic idea is that the non-axisym-metric shape correction, generated in thetip region, should be used as an ‘initial’condition for a time-dependent two-dimen-sional problem describing the motion ofthe cross-section of the interface in the tailregion. In this reduced description, the roleof time is played by the coordinate z forsteady-state growth in the z direction. Thedeviation from the isotropic Ivantsov solu-tion remains small only during the initial



period of the evolution. This initial devia-tion can be handled by a linear theory start-ing from a mode expansion which takes thesame form as Eq. (2-80) with, in principle,arbitrary coefficients. These amplitudesthen have to be chosen according to thepredictions of selection theory (Ben Amarand Brener, 1993) in order to provide thematching to the tip region. As ‘time’ goeson, the deviation increases owing to theMullins–Sekerka instability and a nonlin-ear theory must take over. We can guesswhat the long-time behavior of the systemwill be. Indeed, this two-dimensional prob-lem is precisely the same as that whichleads to two-dimensional dendritic struc-tures. Four well-developed arms (for cubicsymmetry) grow with a constant speed inthe directions favored by the surface en-ergy anisotropy. Each arm has a parabolicshape, its growth velocity, v2 = 2D P2

2 (D)¥s2*(e)/d0, and the radius of curvature ofits tip, r2 = d0 /(P2s2*), are given by 2D se-lection theory, where the anisotropic sur-face energy again plays a crucial role. ThePéclet number P2 = r2v2 /2D is related tothe undercooling D by the 2D Ivantsov for-mula, which for small D gives

P2 = (D) = D2/p (2-81)

The selected stability parameter s2 =d0 /(P2 r2) depends on the strength of theanisotropy e, and s2*(e) µe7/4 for small eas was explained above. Replacing t by|z |/v and reducing all lengths by r, we canpresent the shape of one-quarter of theinterface (except very close to the dendriticbackbone) in the form

(2-82)

This asymptotic describes the strongly an-isotropy interface shape far behind the tipand does not match, for small D, the shapedescribed by Eq. (2-80). In this case an im-

x y z zy

( , ) = | | vv2

2

22− r

r

portant intermediate asymptotic exists ifthe size of the 2D pattern is still muchsmaller than a diffusion length (Dt)1/2.This 2D Laplacian problem (with a fixedflux from the outside) was solved, both nu-merically and analytically, by Almgren etal. (1993) who were interested in aniso-tropic Hele–Shaw flow. They found thatafter some transition time the systemshows an asymptotic behavior which is in-dependent of the initial conditions and in-volves the formation of four well-devel-oped arms. The length of these arms in-creases in time as t 3/5 and their width in-creases as t 2/5. The basic idea that explainsthese scaling relations is that the stabilityparameter s2 = 2D d0 /(r2

2 v2) is supposed tobe equal to s2*(e) even though both v2 andr2 depend on time. Moreover, the growingself-similar shape of the arms was deter-mined. In terms of the dendritic problem itreads (Brener, 1993)

(2-83a)

where the tip position x tip of the arm isgiven by

x tip(z) = (5 |z |/3)3/5 (s2*/s*)1/5 (2-83b)

The ratio s2*(e)/s*(e) is independent of ein the limit of small e. This means thatthe shape, Eq. (2-83), in the tail region isalmost independent of the material andgrowth parameters, as well as shape, Eq.(2-80), in the tip region (if all lengths arereduced by r).

Recent experiments on 3D dendrites(Bisang and Bilgram, 1995) and numericalsimulations (Karma and Rappel, 1998) arein very good agreement with these theoret-ical predictions.

y x z zx

x

s

s sx x

( , ) ( / )*/

/ /

//

*=

d

tip

tip

5 3

1

2 5

2

1 5 2 3

1

2 3 4

| | ss

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎞

⎠⎟

×−

∫

2.4.2 Side-Branching Dendrites

This section provides a summary of thepresent understanding of dendritic growth.It is centered around numerical simulationsof isolated side-branching dendrites in aone-component system where heat diffu-sion is the relevant dynamical process.Alternatively, it also describes dendriticgrowth from a two-component system atessentially constant temperature. In the lat-ter case, we should also discuss the phasediagram; this will be covered later in thesection on directional solidification. Formany typical cases of growth from a dilutesolution however, the information con-tained in this section should be sufficient.

We start with the definition of the modelresulting from the set of Eqs. (2-52) to (2-55). The dynamics come from the conser-vation law, Eq. (2-53), at the interface. Asin the previous section, we use here theform of dimensionless units introduced inSec. 2.2.3 for the case of heat diffusion.The case of chemical diffusion (tempera-ture then being assumed constant = T0) canbe treated by the same equations. The nor-malization is described in Sec. 2.5.2. Forconvenience, we will simply summarizehere the basic formulas for both cases.

In contrast to Eqs. (2-37) and (2-54), wewill now normalize the following equationto obtain u = 0 at infinity, which resultsfrom adding the constant D to the field-var-iable u in all equations. Then, the diffusionfield becomes

(2-84)

where m is the chemical potential differ-ence between solute and solvent and DC(0 ≤ DC ≤ 1) is the miscibility gap at the op-erating temperature T0.

u

T T L c

C C

p

= thermal diffusion

chemical diffusion

m( ) /( )

( ) /( / )

−

−

⎧

⎨⎪⎪

⎩⎪⎪

∞−

∞

1

m m mD ∂ ∂

The dimensionless supercooling is givenas

(2-85)

The capillary length is then

The quantity ∂m /∂C is not easily measured,but for small DC 1 of a dilute solution, itcan be related to the slope of the liquidusline dT /dC at T0 = Tm by

(2-87)

(Mullins and Sekerka, 1963, 1964; Langer,1980a). Note that the chemical capillarylength can be several orders of magnitudelarger than the thermal length.

The boundary condition, Eq. (2-54), thensimply changes to

uI = D – d K – b V (2-88)

where now the anisotropic capillary lengthd is used. The kinetic coefficient may alsodepend on concentration (Caroli et al.,1988), which we ignore here. Far awayfrom the interface in the liquid, the bound-ary condition becomes

u• = 0 (2-89)

For the chemical case, we may practicallyignore diffusion in the solid.

The diffusion Eq. (2-52) and the conser-vation law, Eq. (2-53), remain unchanged,and the diffusion length is defined as be-fore as l = 2DT /V, with V being the averagevelocity of the growing dendrite.

∂∂

≈mC

LT C

TC

m

m

ddD

d

T c L

C C

p

=thermal (2-86)

chemical

m m

[ ( ) ( )]( / )

g gm

J J+ ′′

⎧

⎨⎪⎪

⎩⎪⎪

−

− −

2

2 1D ∂ ∂

D =thermal

chemical

m m

eq

( ) /( )

( ) /( / )

T T L c

C C

p−

−

⎧

⎨⎪⎪

⎩⎪⎪

∞−

∞

1

m m mD ∂ ∂



There are, of course, important differ-ences between two and three dimensions,as a three-dimensional needle crystal is notnecessarily rotationally symmetric aroundits axis. Snow crystals, for example, showlarge anisotropies in directions orthogonalto the growth direction of the primary den-dritic needle (Yokoyama and Kuroda,1988). In the immediate neighborhood ofthe tip, however, the deviation from this ra-tional symmetry is often small. Therefore,we may work with this two-dimensionalmodel by using an “effective” capillarylength. The scaling relations given beloware expected to be insensitive to this apartfrom a constant prefactor of order unity inthe s (e)-relation (Kessler and Levine,1986b, d; Langer, 1987a).

The numerical simulations were per-formed for a two-dimensional crystal– liq-

uid interface. In Fig. 2-8, we show a typicaldendrite with side branches resulting fromthe time-dependent calculations (Saito etal., 1987, 1988) (compare with the experi-mental result by Glicksman et al. (1976) inFig. 2-9). The profile is symmetric aroundthe axis by definition of the calculation. Anapproximately parabolic tip has beenformed from which side branches begin togrow further down the shaft (only the earlystage of side-branch formation was consid-ered). They have a typical distance which,however, is not strictly regular.

As a first result, the scaling relation, Eq.(2-68), was checked using the Péclet num-ber from Eq. (2-56b). Experimentally, thisrequires the anisotropic capillary lengthand the supercooling to be varied indepen-dently. In Fig. 2-10, the scaled numericalresults are shown as symbols for two dif-

Figure 2-8. Free dendrite in stationary growth com-puted in quasistationary approximation for the two-dimensional case. Capillary anisotropy was e = 0.1(Saito et al., 1988). The parameter-dependence of thegrowth rate, tip radius and sidebranch spacing is con-sistent with analytical scaling results from solvabilitytheory of the needle crystal.

Figure 2-9. Dendrite tip in pure succinonitrile (SCN)at small undercoolings and inscribed parabola formeasuring the tip radius (Huang and Glicksman,1981).

ferent supercoolings and compared withthe results (full lines) for the stationaryneedle crystal. The upper line correspondsto the model with diffusion in the liquidonly (Misbah, 1987), as used in the numer-ical simulation here. The lower line corre-sponds to the symmetrical model withequal diffusion in liquid and solid (BenAmar and Moussallam, 1987). Ap-parently, the two results look the same,apart from a factor of approximately twoin s. Note that in unscaled form (i.e., mul-tiplying by P2) the data for the two super-coolings would differ by about two ordersof magnitude!

From an experimental point of view, it isbetter to use Fig. 2-10 rather than Eq. (2-79) for comparison, as the range of validityof Eq. (2-79) seems to be restricted torather small values of e. For unknown ma-terial parameters such as diffusion con-stant, capillary length and anisotropy, wecan still check the scaling relation of the

growth rate V through the Péclet number,Eq. (2-56), depending on supercooling. Eq.(2-68) should then give a constant, al-though arbitrary, value of s. This scalingresult was confirmed experimentally inthe 1970s, before the full theory existed(Langer and Müller-Krumbhaar, 1978,1980). At that time, it was assumed (“mar-ginal stability” hypothesis) that a universalvalue of s ≈ 0.03 was determined by a dy-namic mechanism independent of aniso-tropy. The results for the needle crystal, to-gether with these numerical simulations,now show that s depends on anisotropy eas shown in Fig. 2-10. Experimental testson the e-dependence (Sec. 2.4.3) are stillrather sparse and do not quite fit that pic-ture, for reasons not well understood.

So far we have only looked at the rela-tion between growth rate, anisotropy andsupercooling. We will now consider thesize of the dendrite, which is approxi-mately parabolic, and which can probablybe characterized by the radius of curvatureat its tip.

This is a subtle point, as the tip radiuscannot easily be measured directly. As analternative, we can try to fit a parabola tothe observed dendrite in the tip region. Thetip radius of this fitted parabola should beinterpreted as the Ivantsov radius r, whichturns out to be slightly larger than the truetip radius R of the dendrite. The deviationof R for r does not depend on supercoolingD but on anisotropy e. This is shown in Fig.2-11, where a comparison is made betweenthe dynamic numerical simulations (Saitoet al., 1988) and the needle crystal solutionin the limit of small Péclet number (BenAmar and Moussallam, 1987). It can beseen that there is excellent agreement andthat the actual tip radius R becomes smallerthan the Ivantsov radius r at increasing e.

We now can relate the growth rate V andthe tip radius R or the Ivantsov radius r in


Figure 2-10. Scaling parameter s (e) for free den-dritic growth depending on capillary anisotropy eand for two-dimensional supercoolings D. Averagecapillary length is d0, diffusion constant D, andPéclet number P. Comparison of numerical results(circles and squares, Fig. 2-8, one-sided model) withsolvability results: (a) one-sided model (Misbah,1987), (b) two-sided model (Ben Amar and Moussal-lam, 1987). The agreement is excellent, the indepen-dence upon supercooling is seen to work at least upto D = 0.5.


order to check the scaling form Eq. (2-68)involving the radius rather than the Pécletnumber. The Ivantsov parabola and its ra-dius r basically originate from a globalconservation law for the quantity (heat) re-leased at the interface, while the tip radiusR is a local geometric quantity. In practicalexperiments, by fitting a parabola to thetip, we can interpolate between these twonumbers, the result depending on how fardown the shaft the fitting parabola is used.Using the actual radius R rather than theIvantsov radius r, perfect scaling can beseen in Fig. 2-12 with respect to supercool-ing D, even up to the very large value ofD = 0.5. Since for smaller supercoolings,D Ù 0.1, the difference between r and R be-comes negligible, as shown in Fig. 2-11,and we may safely use Eq. (2-68) as

(2-90)

independent of supercooling D, to interpretexperiments and to make predictions. Theterm “constant” here means that the prod-uct V R2 depends on material parameters

V R VDd2 2 02≈ r

s e= = constant

( )

only. This is precisely the relation, Eq. (2-1), derived from qualitative considerationsin the introduction to this chapter. This re-lation has been confirmed by the analysisof many experiments (Huang and Glicks-man, 1981).

The final point to be discussed here con-cerns the side branches and their origin,spacing and amplitudes. This issue is theo-retically not completely resolved, becausenone of the available analytical approxima-tions can correctly handle the long-wave-length limit of side-branch perturbations.Moreover, the subject is somewhat tech-nically involved. Therefore, we will onlysummarize the main arguments below andrefer to the above-mentioned numericalsimulations (Saito et al., 1988) for compar-ison with experiments.

An important quantity which charac-terizes the stability of flat moving interfaceripples is the so-called stability length

(2-91)

where d0 is the capillary length, and l thediffusion length. Perturbations of wave-

lS = 2 0p l d

Figure 2-11. Tip radius of free dendrite over Ivant-sov radius plotted versus anisotropy as a function ofdimensionless supercooling. The numerical results(see also Fig. 2-8, 2-10) are consistent with the pre-dictions from the needle solution (Ben Amar andMoussallam, 1987).

Figure 2-12. Numerical scaling result confirmingV R2 = const. for free dendritic growth independentof supercooling D, depending on anisotropy e only(Saito et al., 1988).

lengths l longer than lS will grow, whileshorter wavelengths will decay with time.This quantity characterizes the competitionbetween the destabilizing diffusion fieldthrough l against the stabilizing surfacetension through d0. A derivation of thisMullins–Sekerka instability was given inSec. 2.2.4.

It is natural to assume that this lengthscale is related to the formation of sidebranches. A direct estimate of the typicalwavelength l2 of the side branches is

(2-92)

The remarkable result of the numericalsimulation is shown in Fig. 2-13. Appar-ently, the ratio l2 /lS is a constant of ap-proximately 2.5, which is independent ofsupersaturation and anisotropy. This seemsto be in quite good agreement with experi-ments (Glicksman et al., 1976; Doughertyet al., 1987; Honjo et al., 1985; Huang andGlicksman, 1981).

The experimental comparison was made,in fact, with an older theoretical concept(Langer and Müller-Krumbhaar, 1978),which did not correctly consider aniso-tropy. By accident, however, the experi-

l l r s2 02 2 S = =p pl d

mental anisotropy of the material succino-nitrile (Huang and Glicksman, 1981),e ≈ 0.1, gave the same s-value as the theo-retical concept, and since e cannot be var-ied easily, there was no discrepancy.

To summarize these results, it appearsthat the scaling relation, Eq. (2-92), shownin Fig. 2-13 from the numerical solution ofthe model in two dimensions, is in agree-ment with the experimental results.

We will now give a somewhat qualitativeexplanation of the mechanism of side-branch formation as far as this can be de-duced from the theoretical approaches. Alinear stability analysis (Langer and Mül-ler-Krumbhaar, 1978, 1980; Kessler andLevine, 1986a; Barber et al., 1987; Bar-bieri et al., 1987; Bensimon et al., 1987;Caroli et al., 1987; Kessler et al., 1987;Pelce and Calvin, 1987) indicates that therelevant modes for side-branch formationin the frame of reference moving with thetip consist of an almost periodic sinusoidalwave travelling from the tip down the shaftsuch that they are essentially stationary inthe laboratory frame of reference (Langerand Müller-Krumbhaar, 1982; Deissler,1987). The amplitude of these waves is notconstant in space, but first grows exponen-tially in the tip region (Barbieri et al.,1987; Caroli et al., 1987). The exponentialincrease of that envelope in the tip regiondepends on the “wavelength” of the oscil-latory part (Bouissou et al., 1990).

In contrast to the earlier analysis byLanger and Müller-Krumbhaar, all thesemodes are probably stable, so that withouta triggering source of noise, they woulddecay, and a smooth needle crystal wouldresult. Some driving force in the formof noise due to thermal or hydrodynamicfluctuations is needed to generate sidebranches, but apparently this is usuallypresent. Estimates of the strength of thesefluctuations (Langer and Müller-Krumb-


Figure 2-13. Scaled sidebranch spacing l = l2, nor-malized with Ivantsov radius R0 and s (e)1/2, plottedversus capillary anisotropy for two supercoolings.No dependence on e or D is found, as expected (Saitoet al., 1988).


haar, 1982; Barbieri et al., 1987; Langer,1987a) are still somewhat speculative.

Given such a small noise at the tip, theexponentially increasing envelope over theside branches into the direction of the tailthen amplifies that noise so that the sidebranches become visible. This happensover a range of about two to ten side-branch spacings. The actual selected wave-length of the side branches in that tip re-gion (assuming a white noise, triggering allmodes equivalently), according to theseconsiderations, is defined by the mode withthe largest amplitude at a distance of aboutone “wavelength” away from the tip. Thisis the product of the average amplitude dueto noise at the tip and the amplification fac-tor from the envelope.

Langer (1987a) described the side-branching deformation as a small (linear)perturbation moving on a cylindricallysymmetric needle crystal (Ivantsov parabo-loid). The noise-induced wave packetsgenerated in the tip region grow in ampli-tude, spread and stretch as they move downthe sides of the dendrite producing a trainof side branches. In the linear approxima-tion, the amplitude grows exponentiallyand the exponent is proportional to |z |1/4.These results are in approximate, qualita-tive agreement with available experimentalobservations (Huang and Glicksman, 1981;Dougherty et al., 1987; Bisang and Bil-gram, 1995), but experimentally observedside branches have much larger amplitudesthan explicable by thermal noise in theframework of the axisymmetric approach.This means that either the thermal fluctua-tion strength is not quite adequate to pro-duce visible side-branching deformations,or agreement with experiment would re-quire at least one more order of magnitudein the exponential amplification factor.

The description of the side-branchingproblem, which takes into account the ac-

tual non-axisymmetric shape of the needlecrystal, defined by Eqs. (2-83a, b), wasgiven by Brener and Temkin (1995). Theyfound that the root-mean-squared ampli-tude for the side branches generated bythermal fluctuations is

(2-93)

where the function x (z) is given by theunderlying “needle” solution, Eq. (2-83b),and the fluctuation strength Q

–is given by

Langer (1987a), Q–2 = 2kB T 2cp D/(L2

m v r4).The root-mean-squared amplitude for

the side branches increases with the dis-tance from the tip, |z |. This amplitudegrows exponentially as a function of(|z |2/5/s1/2). The important result is thatthe amplitude of the side branches for theanisotropic needle grows faster than forthe axisymmetric paraboloid shape. Inthe latter case x (z) = 2 |z |1/2 and the ampli-tude grows exponentially as a functionof (|z |1/4/s1/2). This effect resolves thepuzzle that experimentally observed sidebranches have much larger amplitudes thancan be explained by thermal noise in theframework of the axisymmetric approach.Agreement with experiment now is indeedvery good (Bisang and Bilgram, 1995).

Far down from the tip the side-branchingdeformations grow out of the linear regimeand eventually start to behave like den-drites themselves. It is clear that thebranches start to grow as free steady-statedendrites only at distances from the tipwhich are of the order of the diffusionlength which, in turn, is much larger thanthe tip radius r in the limit of small P. Thismeans that there is a large range of z,1 |z | 1/P, where the side branches al-ready grow in the strongly nonlinear re-gime, but they do not yet behave as freedendrites. We can think of some fractal ob-

⟨ ⟩⎧⎨⎩

⎫⎬⎭

xs1

2 1 23 2

1 22

3 3( , ) ~ exp/

/

/Z Y Qx

z| |

ject where the length and thickness of thedendrites and the distance between themincrease according to some power lawswith the distance |z | from the tip. The den-drites in this object interact, owing to thecompetition in the common diffusion field.Some of them die and some continue togrow in the direction prescribed by the an-isotropy. This competition leads to thecoarsening of the structure in such a waythat the distance between the survivingdendrites l (z) is adjusted to be of the sameorder of magnitude as the length of the den-drites, l (z). The scaling arguments givel (z) ~ l (z) ~ |z | (Brener and Temkin, 1995).The morphology measurements on SCNcrystals yield a good quantitative agree-ment with this linear law (Li and Becker-mann, 1998).

We now summarize the presently estab-lished findings for free dendritic growthwith respect to their experimental signifi-cance. A discussion of additional effectssuch as faceting will be given in Sec. 2.5.7in the context of directional solidification.

For a given material with fixed D, d0 ande, the growth rate V depends upon super-cooling D through Eq. (2-68), and withPéclet number P taken from Eq. (2-56).The dimensionless parameter s is given inFig. 2-10. This is demonstrated for variousmaterials in Fig. 2-14. The size or tip ra-dius of the dendrite is related to its velocityby Eq. (2-90) and can be taken from Fig.2-12. The typical wavelength of the sidebranches is then given by Eq. (2-92) andcan be taken from Fig. 2-13. This providesall the basic information that should bevalid in the tip region.

Beyond the understanding of steady-state growth of the tip, the major new con-cept that has emerged over the last fewyears is that complex pattern formationprocesses occurring on the much largerscale of an entire dendrite grain structure

can be described by remarkably simple“scaling laws”. The whole dendritic struc-ture with side-branches looks like a fractalobject on a scale smaller than the diffusionlength and as a compact object on a scalelarger than the diffusion length (Brener etal., 1996).

The new steady-state growth structuresthat have been identified are the so-called“doublons” in two dimensions (Ihle andMüller-Krumbhaar, 1994; Ben Amar andBrener, 1995), first observed in the form ofa doublet cellular structure in directionalsolidification (Jamgotchian et al., 1993),and the “triplon” in three dimensions (Abelet al., 1997). Both structures have beenshown to exist without crystalline aniso-tropy, unlike conventional dendrites. Thedoublon has the form of a dendrite split


Figure 2-14. Dimensionless growth rate V =Vd0/2Dversus dimensionless undercooling D. The scalingquantity for the full curve (Langer and Müller-Krumbhaar, 1977, 1978) was taken as s = 0.025 (co-incidentally in agreement with the anisotropy of suc-cinonitrile). For references to the experimental pointssee Langer (1980a). Excellent agreement betweentheory (solid line) and experiment is found.


into two parts about its central axis with anarrow liquid groove between the twoparts, and triplons in three dimensions aresplit into three parts. For a finite aniso-tropy, however, these structures only existabove a critical undercooling (or supersat-uration for the isothermal solidification ofan alloy), such that standard dendritesgrowing along ·100Ò directions are indeedthe selected structures in weakly aniso-tropy materials at low undercoolings, inagreement with most experimental obser-vations in organic and metallic systems.From a broad perspective, the existence ofdoublons and triplons is of fundamentalimportance because it has provided a basison which to classify the wide range of pos-sible growth morphologies that can form asa function of undercooling and anisotropy(Brener et al., 1996).

We have so far ignored the influence ofthe kinetic coefficient b in Eq. (2-88). Thisomission is not likely to be very importantfor low growth rates, but for fast growthrates, as in directional solidification, bshould be taken into account. We will re-turn to this point in Sec. 2.5.

2.4.3 Experimental Resultson Free Dendritic Growth

The answer to the question of whetherdendritic growth is diffusion-controlledor controlled by anisotropic attachment ki-netics, was sought by Papapetrou (1935),who was probably the first to make system-atic in situ experiments on free dendriticgrowth. He examined dendritic crystalsof transparent salts (KCl, NaCl, NH4Br,Pb(NO3)2, and others) under a microscopein aqueous solutions and proposed that thetip region should be close to a paraboloidof rotational symmetry.

Many years later, the extensive andsystematic experiments by Glicksman and

his co-workers made an essential contribu-tion to our understanding of dendriticgrowth in pure undercooled melts (Glicks-man et al., 1976; Huang and Glicksman,1981). This research was initially con-cerned mainly with highly purified succi-nonitrile (SCN). It was extended to cyclo-hexanol (Singh and Glicksman, 1989), wa-ter (Fujioka, 1978; Tirmizi and Gill, 1989),rare gases (Bilgram et al., 1989), and toother pure substances with a crystal aniso-tropy different from SCN such as pivalicacid (PVA) (Glicksman and Singh, 1989).Work on free growth of alloys includesNH4Cl–H2O (Kahlweit, 1970; Chan et al.,1978), NH4Br–H2O (Dougherty and Gol-lub, 1988), SCN with acetone and argon(Glicksman et al., 1988; Chopra et al.,1988), PVA–ethanol (Dougherty, 1990),and others.

The specific merit of the work of Glicks-man et al. was that the systems for whichthey characterized all the properties, in-cluding surface energy, diffusion coeffi-cient, phase diagram etc., have been exam-ined. This led to clear evidence in the mid-1970s that the theory of that time (usingextremum arguments for the operatingpoint of the tip) was not able to describethe results quantitatively.

At the same time, Müller-Krumbhaarand Langer worked on precisely the sameproblem and proposed a theory based onthe stability of the growing dendrite tip,called the marginal stability criterion (Lan-ger and Müller-Krumbhaar, 1977, 1978).Most of the existing experimental datacould be fitted using this criterion. Despitethe fact that this theory incorrectly ignoredthe important role of anisotropy (as weknow now), it inspired a number of newexperiments and also attracted the interestof other physicists.

As has been said before, today’s theoryis consistent with the older approximate

theory if we allow for a s (e) value that var-ies with the anisotropy of the capillarylength. The corresponding central equationfor dendritic growth (Eq. (2) in Kurz andTrivedi, 1990) should therefore still apply.

Pure substances (thermal dendrites)

Fig. 2-15 shows dendrites of two differ-ent transparent materials with cubic crystalstructure: face-centered cubic PVA andbody-centered cubic SCN (Glicksman andSingh, 1989). Qualitatively, both dendriteslook similar, but their branching behaviorshows some important differences. The un-

perturbed tip of PVA is longer, with asharper delineation of the crystallographicorientations. Glicksman and Singh (1989)found that PVA has a ten-fold larger sur-face energy anisotropy than SCN (see Ta-ble 2.1). The tip radii and growth rates as afunction of undercooling for both sub-stances scale well when using the values0.22 and 0.195 for s , respectively (Fig. 2-16). According to solvability theory, thegreat difference in the anisotropy constante should make a larger difference in s (e)(compare with Fig. 2-10). The reason forthis discrepancy is not known, and we haveto leave this point to future research.


Figure 2-15. Dendrite morphologies of two transparent materials with small melting entropies and cubic crys-tal structures (plastic crystals); (a) pivalic acid (PVA) and (b) succinonitrile (SCN) (Glicksman and Singh,1989).

(a) (b)


behind the tip is delayed up to about sevenradii in PVA. This is quite consistent withthe recent calculations discussed in Sec.2.5.4. The ratio of initial secondary armspacing l2 over tip radius R is also indi-cated in Table 2-1.

Koss et al. (1999) have shown that evenin microgravity environment there is asmall but significant difference betweentransport theory and experiment. Againthis has to be left to the future.

The secondary branch formation whichstarts in SCN at a distance of three tip radii

Table 2-1. Experimentally determined dendrite tip quantities.

System Growth s* R2 V l2/R d Referencetype [µm3/s]

Thermal dendrites Pure

Succinonitrile (SCN) Free 0.0195 3 0.005 Huang andGlicksman (1981)

Pivalic acid (PVA) Free 0.022 7 0.05 Glicksman andSingh (1986, 1989)

Cyclohexanol Free 0.027 Singh andGlicksman (1989)

Solutal dendrites Alloy

NH4Br–49 wt.% H2O Free 0.081 ± 0.02 18 ± 3 5.2 0.016 ± 0.004 Dougherty andGollub (1988)

SCN–1.3 wt.% ACE Directional 0.032* 1300 2.1 ± 0.2 Esaka andKurz (1985)

SCN–4 wt.% ACE Directional 0.037* 441 ± 30 2.2 ± 0.3 Somboonsuket al. (1984)

CBr4–7.9 wt.% C2Cl6 Directional 0.044* 978 ± 8 3.18 Seetharamanet al. (1988)

C2Cl6–89.5 wt.% CBr4 Directional 0.038* 124 ± 13 3.47 Seetharamanet al. (1988)

Thermal and solutal Alloydendrites

SCN–ACE Free See reference Chopra et al.(1988)

SCN–argon Free See reference Chopra et al.(1988)

PVA–1 wt.% ethanol Free 0.05 ± 0.02 6 ± 1 0.006 ± 0.002 Dougherty(1991)

PVA–2/4 vol.% ethanol Free 0.032 ± 0.006 35 6.8 Bouissou et al.(1989)

* Due to differences in the definitions of s* these values, as given in the corresponding literature, are smallerby a factor of 2 with respect to the values used by Dougherty and Gollub (1988) and defined in this paper. Thevalues given here have been obtained by multiplying the original data by a factor of 2 in order to compare withthe same (one-sided) model. (See also Fig. 2-10.)

Free alloy growth(thermal and solutal dendrites)

In the free dendritic growth of alloys, aninteresting observation has been made byvarious authors. For constant undercool-ing, the growth rate first increases whensmall amounts of a second substance areadded to a pure material, then reaches amaximum, and finally drops and convergeswith the pure solutal case. Early experi-ments in this area by Fujioka and Linde-meyer were first successfully analyzed byLanger (1980c). Fig. 2-17 shows some re-sults on SCN–ACE alloys from Chopra etal. (1988). The increase in V is accompa-nied by a decrease in the tip radius, whichsharpens due to the effect of solute. Theexperimental findings can be comparedto two models: Karma and Langer (1984)(broken line) and Lipton et al. (1987) (fullline). Both models provide at least qualita-tively good predictions of the observed be-havior. In their more recent calculations,Ben Amar and Pelce (1989) concluded thatthe simple model by Lipton et al. (1987)is consistent with their more rigorous ap-proach.

Table 2-1 gives a summary of represen-tative results of in situ experiments con-cerning the dendrite tip.

Large undercoolings

Interesting experiments have also beenperformed with pure and alloyed systemsunder large driving forces, which reachvalues beyond unit undercooling (for ex-ample by Wu et al. (1987) and especially byHerlach et al. (1990–1999), see Table 2-2).

Some of these are reproduced in Fig.2-18 together with predictions from IMS(Ivantsov–marginal stability) theory (Lip-ton et al., 1987; Trivedi et al., 1987; Boet-tinger et al., 1988) (with s (e) = 0.025) andincluding interface attachment kinetics.


Figure 2-16. Effect of undercooling on (a) tip radiusand (b) growth rate of the two organic materialsshown in Fig. 2-15 (a = D). The experimental resultssuperimpose as they are plotted with respect to dimen-sionless parameters (Glicksman and Singh, 1989).

(a)

(b)


These results are specifically of interest be-cause two different techniques, one for thedetermination of the diffusive speed and theother for the undercooling–growth rate re-lationship of the dendrites, have been cou-pled in order to make the analysis as freefrom adjustable parameters as possible. Upto undercoolings of 200 K there is reason-able agreement between experiment andtheory using the best fit for the measureddiffusive speed and the liquid diffusivity,VD = 26 m/s and DL = 2.7 ¥ 10–9 m2/s (Ar-nold et al., 1999). The lower V (∆T )-curveis for a constant (equilibrium) value of thedistribution coefficient k (V) = ke showingthe importance of the appropriate velocitydependence of the distribution coefficient.

At higher undercoolings other phenom-ena take over.

One example is the grain refinement be-yond a certain undercooling. This structurehas been explained by dendrite fragmenta-tion due to morphological instability of thefine dendrite trunks (Schwarz et al., 1990;Karma, 1998).

Table 2-2. Dendritic growth velocity measurements inhighly undercooled melts; comparison between experi-ment and theory (Herlach and coworkers, 1999–2000).

Metals and alloys

Co–Pd Volkmann et al. (1998)Co–V Tournier et al. (1997)Cu Li et al. (1996)Ni Eckler and Herlach (1994)Ni–Al Barth et al. (1994),

Assadi et al. (1998)Ni–B Eckler et al. (1991a, 1992, 1994)Ni–C Eckler et al. (1991b)Ni–Si Cochrane et al. (1991)Ni–Zr Schwarz et al. (1997),

Arnold et al. (1999)

Intermetallics

CoSi Barth et al. (1995)FeAl Barth et al. (1997)FeSi Barth et al. (1995)Ni3Al Assadi et al. (1996)NixSny Barth et al. (1997)NiTi Barth et al. (1997)NixTiyAlz Barth et al. (1997)

SemiconductorsGe Li et al. (1995a, 1996)Ge–Cu Li and Herlach (1996)Ge–Si Li et al. (1995b)Ge–Sn Li and Herlach (1996)

Figure 2-17. Effect of dimensionless composition at constant undercooling of 0.5 K (2.1% of unit undercool-ing) on (a) dimensionless growth rate and (b) on dimensionless tip radius for free dendritic growth in SCN–ace-tone alloys (Lipton et al., 1987). Points: experiments (Chopra et al., 1988); solid line: LGK model (Lipton et al.,1987); interrupted line: Karma and Langer (1984) model.

(a) (b)

2.5 Directional Solidification

Directional solidification is the most fre-quent way in which a material changes itsstate from solid to liquid. The necessary re-moval of the latent heat of freezing usuallyoccurs in a direction prescribed by the lo-cation of heat sinks: for a freezing lake, it

is the cold atmosphere above it, in castingiron in a foundry, it is the cold sand mold,into which the heat flow is directed.

At first, it may seem surprising to thinkthat anything interesting should happen atthe solid– liquid interface during this pro-cess. In contrast to the situation describedearlier, in Secs. 2.4 and 2.2.4, the solid in acasting process is cold and the liquid is hot,so that we would expect the interface to bestable against perturbations.

However, so far we have just consideredthe solidification of a one-component ma-terial, while in reality a mixture of materi-als is almost always present, even if one ofthe components is rather dilute. If, there-fore, we assume that material diffusion isthe rate-determining (slow) mechanism,while heat diffusion is much faster, the ori-gin of a destabilization of the flat interfacecan be easily understood on a qualitativebasis. We may consider one of the twocomponents of the liquid as an “impurity”,which, instead of being fully incorporatedinto the solid, is rejected at the interface.Such excess impurities have to be diffusedaway into the liquid in much the same wayas latent heat has to be carried away in thecase of a pure material as a rate-determin-ing mechanism. Accordingly, precisely thesame destabilization and subsequent for-mation of ripples and dendrites should oc-cur.

Based on these qualitative argumentswe can expect the following modificationof the Mullins–Sekerka instability (Sec.2.2.4) to occur in the present situation ofdirectional solidification. The diffusion ofmaterial together with capillary effects pro-duces a spectrum for the growth rates or thedecay rates similar to Eq. (2-50), while thetemperature field acts as a stabilizer, inde-pendent of the curvature of the interface,when a constant term (independent of V )inside the brackets of Eq. (2-50) is sub-


Figure 2-18. Dependence on total bath undercoolingof Ni–1 wt.% Zr alloy. (a) Dendrite growth velocity,as measured (dots), and results from Ivantsov–mar-ginal stability dendrite growth theory using the val-ues VD = 26 m/s and Dl = 2.7 ¥ 10–9 m2/s. The lowercurve in (a) is for local equilibrium partition. (b) Thecalculated dendrite tip radius, (c) the computed inter-face compositions and (d) the individual undercool-ing contributions: thermal undercooling DTt , consti-tutional undercooling DTc, curvature undercoolingDTr , and attachment kinetic undercooling DTk (Ar-nold et al., 1999).

2.5 Directional Solidification 121

tracted. At low solidification rates, the flatinterface is stable; above a critical speed, itbecomes unstable against the formation ofripples, cells and dendrites.

In the next section, a few thermody-namic questions related to interface prop-erties in two-component systems are con-sidered, before describing patterns in direc-tional solidification.

2.5.1 Thermodynamicsof Two-Component Systems

There is a vast amount of literature avail-able on the thermodynamics of solidifica-tion and on multi-component systems (forexample Callen, 1960; Baker and Cahn,1971). Despite this fact, to further the clar-ity of presentation, we would like to at leastsketch the tools that may be used to gener-alize some approximations which will bemade in the next sections.

The fundamental law of thermodynam-ics defines entropy as a total differential inrelation to energy and work:

dU = T dS – P dV + Si

mi dNi (2-94)

with energy U, entropy S, volume V, pres-sure P, particle numbers Ni for each speciesand chemical potential mi . The energy is ahomogeneous function of the extensivevariables

(2-95)U (bS, bV, bNi , …) = bU (S, V, Ni , …)

with an arbitrary scale parameter b > 0.With the differentiation rule d(XY ) =

X dY + Y dX, other thermodynamic poten-tials U are obtained from U by Legendretransformations

U = U – Sj

Xj Yj (2-96)

where Xj are some extensive variables, andYj the corresponding intensive variables.

The Helmholtz energy F is then

F (T, V, Ni , …) = U – T S ;

dF = – S dT – P dV + Si

mi dNi (2-97)

and the often-used Gibbs energy G is

G (T, P, Ni , …) = U – T S + PV = Si

mi dNi ;

dG = – S dT + V dP + Si

mi dNi (2-98)

At atmospheric pressure in metallurgicalapplications, the differences between F andG often can even be neglected, but gener-ally the Gibbs energy, Eq. (2-98), is mostfrequently used. It follows immediately thatthe chemical potentials mi are defined as

(2-99)

The thermodynamic equilibrium for a sys-tem is defined by the minimum of the re-spective thermodynamic potential with re-spect to all unconstrained internal parame-ters of the system. If the system consistsof two subsystems a and b in contact witheach other, then in thermal equilibrium thetemperatures, pressures, and the chemicalpotentials for each particle type i must beequal:

Ta = Tb , Pa = Pb , mi,a = mi,b (2-100)

For the case under consideration we have asolid phase a (with assumed low concen-tration of B atoms) and a liquid phase b(with higher concentration of B atoms).For simplicity, we further assume that theatomic volumes of both species are thesame and unchanged under the solid– liq-uid transformation.

For a system with a curved interfacebetween a solid and a liquid of differentcompositions, the chemical potentials canbe calculated as follows. Assuming that NA

particles of solvent and NB particles ofsolute are given, there will be an unknown

∂∂

⎛⎝⎜

⎞⎠⎟

GNi T P

i,

= m

number of NS particles in the solid and NL

particles in the liquid, whose compositionis still undetermined. Define the number nas the number of B particles in the solid.Keeping NS and n initially fixed, the Gibbspotential is obtained as

G (NA, NB, NS, n) = NS gS(CS) + NL gL(CL)

+ 4p R2 g (2-101)

where g is the surface free energy density,R is the radius of the solid sphereNS = –43 p R3 (unit atomic volume), and gS

and gL are free energy densities for homo-geneous solids or liquids at concentrationsCS and CL. Removing the constraints on NS

and n, we obtain thermodynamic equilib-rium by minimizing G with respect to NS

and n, so that G = G (NA, NB). Togetherwith the chemical potentials from Eq. (2-99), this gives

(2-102)

with curvature K = 2/R and m = mB – mA be-ing the difference in chemical potentialsbetween solute and solvent. The values m0,C 0

L, C 0S correspond to equilibrium at g = 0

as a reference, around which g (C ) was lin-earized. Here g was assumed to be inde-pendent of curvature and concentrations,but this can easily be incorporated (for ef-fects of surface segregation, for example).

This equation is the boundary conditionfor the chemical potential on the surface ofa solid sphere of surface tension g coexist-ing with a surrounding liquid of higherconcentration CL of B atoms. From a prac-tical point of view, the formation in termsof chemical potentials does not look veryconvenient, as they are not directly mea-surable. From a theoretical point of view,this has advantages, as the chemical poten-tial is the generalized force controllingmatter flow and phase changes. In particu-

m m g− −−0 0 0=

L S( )C CK

lar, the spatial continuity of chemical po-tentials together with continuous tempera-ture and pressure guarantees local thermalequilibrium, which we will assume to holdin most of the following discussions.

We now come to the discussion of a typ-ical phase diagram for a two-componentsystem (Fig. 2-19). The vertical axis de-notes the temperature, the horizontal axisthe relative concentration of “solute” in a“solvent”, or, more generally, B atoms rel-ative to A atoms.

At high temperatures, T > T0, the systemis liquid, regardless of concentration. As-suming a concentration C• to be given in-itially, we lower the temperature to T1. At


Figure 2-19. Typical solid– liquid phase diagram fora two-component system with the possibility of eu-tectic growth. CL, CS are the liquidus and soliduslines with a solid– liquid two-phase region inbetween. For a relatively low concentration C• of so-lute in the liquid, a single solid phase at the samecomposition CS

0 = C• can show stationary growth. Indirectional solidification, a positive temperature gra-dient ∂T /∂z is assumed to be given perpendicular tothe solid– liquid interface and to advance in + z-di-rection. The advancing interface chooses its positionto be at temperature T0, the concentration in the liq-uid at the interface is then at CL

0. Ahead of the inter-face (z = 0), the concentration profile C (z) decaystowards C• at z = • (dashed curve). As long asC (z) > C L(T (z)) with T (z) = T0 + z ∂T /∂z, a flat inter-face remains stable.


this temperature, we first hit the liquidusline CL(T ), and the system begins to solid-ify, producing a solid of very low concen-tration marked by the solidus line CS(T ).When the temperature is slowly lowered,solidification becomes complete at T0. Atlower temperatures, the whole system issolid.

The region between CS(T ) and CL(T ) isthe two-phase region: if we prepare asystem at a concentration between C 0

S andC 0

L at high temperatures and then quicklyquench it to T0, the system starts to sepa-rate into a solid phase of concentration C 0

S

and a liquid phase of C 0L. In practice, this is

a very slow process, with lengths varyingwith time t approximately as t1/3 (Lifshitzand Slyozov, 1961; Wagner, 1961).

In the case of directional solidification, athermal gradient in the system defines a di-rection such that the liquid is hot and thesolid is cold. A flat interface may then bepresent at a position in space at tempera-ture T0. For equilibrium between solid andliquid at that temperature, the concentra-tion in the solid must be at C 0

S = C•, and inthe liquid at C 0

L = CL(T0). We now assumethat the liquid at infinity has concentrationC•. Clearly there must be a decrease inconcentration as we proceed from the inter-face into the liquid. In order to maintainsuch an inhomogeneous concentration, theinterface must move toward the liquid.

In other words, when the liquid of com-position C 0

L freezes, the solid will onlyhave a concentration C 0

S. The difference inconcentrations

DC = C 0L – C 0

S (2-103)

is not incorporated but is pushed forwardby the advancing interface and must be car-ried away through diffusion into the liquid.This is equivalent to the latent heat gener-ated by a pure freezing solid, discussed inSec. 2.2.3. Therefore, we expect a spatial

concentration profile ahead of the interfacewhich decays exponentially from C 0

L at theinterface to C• far away from the interface.But why should it decay to C• (or whyshould the interface choose a temperatureposition such that C 0

S = C•)?The answer is quite simple, and again

was given in similar form in Sec. 2.2.3, Eq.(2-33) for the pure thermal case: if C 0

S werenot identical to C•, then during the solid-ification process there would be either a to-tal increase (or decrease) of concentration– which clearly is impossible – or at leastthe concentration profile could not be sta-tionary.

This is a rather strict condition, whichwe can reformulate as follows: if we im-pose a fixed temperature gradient ∂T /∂zand move this at fixed speed V0 over asystem of concentration C• at infinity inthe liquid (toward the liquid in the positivez-direction), then the interface will choosea position such that its temperature is at T0,the concentration in the solid will be C 0

S

(averaged parallel to the interface), and theliquid concentration at a flat interface willbe at C 0

L. This follows from global conser-vation of matter together with the imposedstationary solidification rate.

As a final point, we can even derive acondition for the stability of the interface.The concentration profile in the liquid willdecay exponentially with distance z awayfrom the interface as

CL(z) = (C 0L – C•) e–2z /l + C• (2-104)

by analogy to Eq. (2-32). Since we assumethe temperature gradient

GT = ∂T /∂z > 0 (2-105)

to be fixed, the temperature varies linearlywith distance z from the interface. This maybe written (by GT = (T – T0)/z) as

(2-106)CL(T ) = (C 0

L – C•) e–2(T –T0) / (l GT) + C•

and incorporated into Fig. 2-19 as adashed-dotted line. Note that the diffusionlength l is again defined as l = 2D/V0, withD being the diffusion constant of soluteatoms in the solvent, and V0 the interfacevelocity imposed by the advancement rateof the temperature gradient.

From Eq. (2-106), it is obvious that thedashed-dotted concentration line in Fig. 2-19 converges very quickly to C• for highsolidification speeds V0. As long as thatconcentration line is fully in the liquid re-gion of the phase diagram, nothing specifichappens. But if the dashed-dotted linepartly goes through the two-phase regionbetween CS(T ) and CL(T ), the liquid infront of the interface is supercooled! Thisimplies the possibility of an instability ofthe solid– liquid interface, which is com-pletely analogous to the discussion in Sec.2.2.4.

A sufficient condition for stability of theinterface in directional solidification istherefore

(2-107)

so that the dashed-dotted curve remainsoutside the two-phase region (Mullins andSekerka, 1963; Langer, 1980a). Here wehave assumed that material diffusion in thesolid can be ignored. In fact, in practicalsituations, violation of this condition typi-cally means “instability” of the interface,so that cellular or dendritic patterns areformed. The reason for this latter conclu-sion is that the effect of stabilization due tocapillarity (or surface tension) is ratherweak for typical experiments at threshold.

In summary, in this section we havederived both the boundary condition – interms of chemical potential – for a curvedinterface and a basic criterion for interfacestability during directional solidification.

DCV

DGCTT

< dd

L˜

2.5.2 Scaled Model Equations

A theoretical analysis of practical situa-tions of directional solidification suffers –among other problems – from the many rel-evant parameters entering the description.The usual way to proceed in such cases isto scale out as many parameters as pos-sible, writing the problem in dimensionlessvariables. We have done this already in thediscussion of free dendritic growth by in-troducing the dimensionless diffusion fieldu; in hydrodynamic applications, it is com-mon practice to use Reynolds and Rayleighnumbers (Chandrasekhar, 1961).

For our present problem, we will pro-ceed in an analogous way. The first stepis to express all experimental parameters(wherever possible) in length units (i.e.,diffusion length, capillary length, etc.). Forpresenting results, we divide these lengthsby the thermal length introduced below, asthis is a macroscopic length which will ap-proximately set the scale at the onset of theinstability.

Directional solidification involves chem-ical diffusion of material together with heatdiffusion. As heat diffusion is usuallyfaster by several orders of magnitude, wemay often assume constant temperaturegradients to exist in the liquid and in thesolid. Furthermore, we will also assumethat there are equal thermal diffusivities inliquid and solid, which is often the casewithin a few percent, but this has to bechecked for concrete applications. The dif-fusion field to be treated dynamically thencorresponds to the chemical concentration.

It is clear from the discussion in the pre-vious section that for a flat interface mov-ing at constant speed there is a concentra-tion jump DC = C 0

L – C 0S across the inter-

face, while in the liquid and in the solid,the term C• is approached asymptoticallybecause of the condition of stationary



movement, together with the global con-servation of matter. We therefore normalizethe diffusion field in the liquid to

(2-108)

so that it varies from one to zero in the pos-itive z-direction from the interface at z = 0to z = •. If the interface is not at positionz = 0 but at z, we must require uL = 1 – z /lT ,because at a distance

(2-109)

the liquidus concentration has reached theasymptotic value. This is the thermallength which we assume to be fixed by thethermal gradient GT , the concentrationjump DC, and the liquidus line C (T ),which is here assumed to be a straight linein the T vs. C diagram.

The equation of motion in quasi-station-ary approximation then becomes, in anal-ogy to Eq. (2-52) with l = 2DL/V

(2-110)

This equation applies equivalently to thesolid but with a different chemical diffu-sion length l ¢ due to different chemicaldiffusion constants. The boundary condi-tion in analogy to Eq. (2-88) obviously be-comes

uL (z ) = 1 – dK – z (x, t)/lT – b V^ (2-111)

where we now have D =1 as the first termon the r.h.s., with curvature K being posi-tive for the tip of a solid nose pointing intothe liquid. The capillary length d is dis-cussed below and interface kinetics withb ≠ 0 will be discussed in Sec. 2.5.4. Thesolid boundary condition is simply

uS = k (uL – 1) (2-112)

10

22

Du u

luzt

LL L

L=∂ ≈ ∇ + ∂∂

lC

GT

CT

T=

d

d L

D˜

uC C

CL =(Liquid) − ∞

D

with segregation coefficient k (equilibriumvalues assumed) defined as

(2-113)

through the slopes of the liquidus and sol-idus lines. When they intersect at Tm,C = 0 this is equivalent to the conventionaldefinition k = CS/CL, but in the above for-mulation, k =1 may also be true for a con-stant jump in concentrations, independentof temperature.

The conservation law at the interfacez = z finally becomes

V^ 1 + (1 – k) (uL – 1)

= – DL n · —uL + DS n · —uS (2-114)

where V^ is the interface velocity in direc-tion n normal to the interface. For k =1, thebrackets … give 1, corresponding to aconstant concentration jump, while fork = 0, they give uL, since for a solid in Eq.(2-112), uS = 0.

This standard model for directional so-lidification (Saito et al., 1989) thereforeconsists of Eqs. (2-108) to (2-114), to-gether with an additional diffusion equa-tion as Eq. (2-110) inside the solid phase.

The open point is finally the relation ofthe capillary length d (Eq. (2-111)) with ex-perimentally measurable material parame-ters. As a first step, we interpret the u-field,Eq. (2-108), as a scaled form of the chemi-cal potential m (see Sec. 2.5.1) near T0

(2-115)

assuming that linearizing m around its equi-librium value at the liquidus line CL(T0) issufficient to describe its dependence uponC. By the definitions in Eq. (2-102) to-gether with Eq. (2-111), we now obtain thecapillary length in the form given in Eq.(2-86) for the chemical case. Here we have

uC C

LL

=m m

m−∂ ∂

∞

D ( / ˜ )

kT

C

T

C=

d

d

d

dL S˜ ˜

generalized to anisotropic g as derivedin Eq. (2-21). It can finally be related tomeasurable quantities using the Clausius–Clapeyron equation for the latent heat Lm

of freezing of a solution at Tm

Lm = – Tm DC (dm /dT )coex (2-116)

where (dm /dT )coex is the slope of the coex-istence line when m is plotted against T.Together with

(2-117)

this gives for the chemical capillary length,d, in the limit of small DC

(2-118)

Certain approximations used here, such asthe linearization involved in Eq. (2-115) orneglecting of ∂m /∂T in Eqs. (2-118), maynot be safe for the case of a large segrega-tion coefficient k ≈1 or when DC is large(Langer, 1980a). In most practical applica-tions, however, this is a minor source of er-ror in comparison with other experimentaluncertainties. Furthermore, the concentra-tion jump DC in Eq. (2-118) is kept fixed,while in reality it should correspond to thetemperature-dependent difference in con-centration between the liquidus and soliduslines. Both for slow and fast growth rates,however, this only gives a minor correc-tion, and we will ignore its effect in orderto facilitate comparison with free dendriticgrowth.

In summary, with this model we nowhave all the ingredients to discuss some ba-sic features of directional solidification byanalytical and numerical tools. The presen-tation in the scaled form may not seem atfirst to be the most convenient means of di-rect comparison with experiments. Its greatadvantage over an explicit incorporation of

dT

C L T C= m

m L

[ ( )]/ ˜g g+ ′′

∂ ∂J

D | |

dd

=ddcoex

Lm m mT C

CT T

⎛⎝

⎞⎠

∂∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂

˜

all parameters is that qualitatively differentbehavior always corresponds to differentratios of length scales or time scales ratherthan some differences in absolute mea-sures, and, consequently, this presentationallows for a more intuitive formulation ofresults.

2.5.3 Cellular Growth

A plane interface between the solidphase and the liquid phase of a two-compo-nent system tries to locate its position in athermal gradient so that the chemical po-tentials of both components are continuousacross the interface. Under stationarygrowth conditions, that position corre-sponds to a temperature, such that the con-centration in the solid (solidus line of thephase diagram) is equal to the concentra-tion in the distant liquid. This growth modepersists for velocities up to a critical veloc-ity, above which the interface undergoes aMullins–Sekerka instability toward cel-lular structures. A necessary condition forthis instability to occur follows from Eq.(2-107), which can be written in terms ofchemical diffusion length l = 2D/V andthermal diffusion length lT , Eq. (2-109), as

l /lT 2 (2-119)

The 2 comes from the specific definition ofl, and the inequality for instability is onlyapproximate because minor surface tensioneffects have not yet been considered here.Incorporation of surface tension revealsthat the instability first occurs for a criticalwavelength lc larger than the stabilitylength ls= 2p ÷--

dl. Slightly above the criti-cal pulling speed Vc, the interface makesperiodic structures of finite amplitude. Thiswas analyzed by Wollkind and Segel(1970), and for other specific cases, byLanger and Turski (1977). A more generaltreatment was given by Caroli et al. (1982).



The result of these investigations is thatin a diagram of pulling speed V versuswavelength l there exists a closed curve ofneutral stability, Fig. 2-20. At fixed V, asmall amplitude perturbation of the inter-face at a wavelength on that curve neithergrows nor decays. Perturbations at wave-lengths outside that curve decay, inside thecurve they grow to some finite amplitude.This is similar to periodic roll patterns inthe Rayleigh–Benard system of a fluidheated from below (Chandrasekhar, 1961),but here a maximal speed, Va, is present,above which a flat interface is absolutelystable. For normal alloys, this speed is veryhigh, while for liquid crystals, it is moreeasily accessible in controlled experiments(Bechhoefer et al., 1989). The diffusionlength at Va is of the order of the capillarylength.

At low speeds, but slightly above thecritical velocity, Vc sinusoidal “cells” willbe formed for systems with segregation co-efficients k near unity. For small segrega-tion coefficients, however, the neutral

curve does not define such a normal bifur-cation but rather an inverse bifurcation.This means that immediately above Vc

large amplitude cells with deep grooves areformed. A time sequence of the evolutionof a sinusoidal perturbation into elongatedcells at 1% above Vc due to inverse bifurca-tion is shown in Fig. 2-21. This can beunderstood theoretically (Wollkind andSegel, 1970; Langer and Turski, 1977; Ca-roli et al., 1982) by means of an amplitudeequation valid near Vc:

(2-120)

where A is the (possibly complex) ampli-tude of a periodic structure exp (ikx) withk = 2p /lc. The coefficient a1 is called theLandau coefficient. If it is positive, we have

∂∂

−⎛⎝⎜

⎞⎠⎟

− +…At

V VV

A a A A= c

c1

2| |

Figure 2-20. Neutral stability curve for a flat solid–liquid interface in directional solidification (sche-matic). The dependence of the growth rate (pullingspeed) V on the wavelength l of the interface pertur-bation is approximately V ~ l–2, both for the shortand long wavelength part of the neutral stabilitycurve, Vc and Va are the lower critical velocity andupper limit of absolute stability, respectively.

Figure 2-21. Time evolution of an interface from si-nusoidal to cellular structure slightly above the criti-cal threshold Vc for the case of inverse bifurcation. Asecondary instability quickly leads to a halving of thewavelength.

a normal bifurcation with | A | ~ ÷-----V – Vc,

while for a1 < 0, the third order term doesnot stabilize the pattern but allows verylarge amplitudes leading to elongated cells(Fig. 2-21), which will be stabilized bysome higher-order effects.

A second phenomenon is usually asso-ciated with this inverse bifurcation, namely,the splitting of the wavelength lc Æ lc /2.Qualitatively, this is understandable fromnonlinear corrections since the squaring ofthe original pattern ~ exp (ik x) producesterms ~ exp (i 2k x). This effect has clearlybeen observed in experiments (de Che-veigne et al., 1986).

We will discuss some aspects of the veryhigh speed region in Sec. 2.5.7 but devotethe main part of the following discussionsto the most interesting region for practicalpurposes, which is not too close to theupper and lower bounds of the growth rateVa and Vc.

Approximating by straight lines the neu-tral curve of the logarithmic plot Fig. 2-20in the intermediate velocity region, we findfor both the small and the large l limits therelation

V l2 ≈ constant (2-121)

Again we have recovered the form of Eq.(2-1) mentioned in the introduction as ascaling law where l here is the cell spac-ing. This suggests that the cellular patternformed in actual experiments would alsofollow this behavior. Unfortunately, thisproblem has not yet been settled to a satis-factory degree from a theoretical point ofview. This is partly due to the difficulty offinding good analytical approximations tothe cellular structures, which makes nu-merical calculations necessary to a largedegree. We will return to this point in Sec.2.5.5.

For small amplitude cells obtainableunder normal bifurcation, some progress

has been made (Brattkus and Misbah,1990). A phase-diffusion equation has beenderived describing the temporal evolutionof a pattern without complete periodic vari-ation of the interface. The basic idea is toreplace the periodic trial form exp (ik x)by a form exp (iQ (x, t)) so that q (x, t) =∂Q /∂x is now no longer a constant but isslowly varying in space along the interfaceand evolving with time. We can derive anonlinear phase diffusion equation

∂t q = ∂x D (q) ∂ x q (2-122)

with a diffusion coefficient D (q) depend-ing in a complicated way on q. The proce-dure is well known in hydrodynamics andit is associated there with the so-calledEckhaus instabilility (Eckhaus, 1965). Thisinstability eventually causes an (almost)periodic spatial structure to lose or gainone “period”, thereby slightly changing theaverage wavelength. In directional solidifi-cation, the result (Brattkus and Misbah,1990) is shown in Fig. 2-22, where velocityis plotted against wavenumber in a smallinterval above the critical velocity. The fullline is the neutral curve, the full trianglesmark the Eckhaus boundary of phasestability. A periodic (sinusoidal) pattern isstable against phase slips only inside theregion surrounded by triangles, thereby al-lowing for an Eckhaus band of stationaryperiodic solutions with a substantially re-duced spread in wavenumbers as comparedto the linear stability results. Note also thatthe results for phase stability (dashed line)from the amplitude equation only hold inan extremely small region above Vc, whileonly 20% above Vc, it shows no overlapwith the result from the present analysis(triangles). The short-wavelength branchhas a very complicated structure, while thelong-wavelength branch far from thethreshold scales as lEck ~ V –1/2, again likethe neutral curve. This also seems to be in



agreement with experimentally observedresults, as discussed later (Billia et al.,1987, 1989; Somboonsuk et al., 1984;Esaka and Kurz, 1985; Eshelman and Tri-vedi, 1987; Faivre et al., 1989; Kurowsky,1990).

At higher velocities, the cells quickly be-come elongated (Fig. 2-23) with deepgrooves forming bubbles. This was firstobtained through numerical calculations byUngar and Brown (1984a, b, 1985a, b).Calculations with a dynamical code inquasi-stationary approximation (Saito etal., 1989) confirmed the stability of thesestructures with respect to local deforma-tions and short-wavelength perturbations.The long-wavelength Eckhaus stability hasnot been investigated yet for these cells.All calculations were made in two dimen-sions, which are believed to be appropriatefor experiments of directional solidifica-tion in a narrow gap between glass plates.

At higher velocities and wavelengths (orcell sizes) not much smaller than the dif-fusion length, the grooves become very

narrow, similar to Fig. 2-21 (Ungar andBrown, 1984a, b, 1985a, b, Karma, 1986,Kessler and Levine, 1989, McFadden andCoriell, 1984, Pelce and Pumir, 1985).

Figure 2-22. Stability diagram V vs. 2p /l near the lower critical threshold for a flat moving interface in direc-tional solidification. The solid line is the neutral stability curve Fig. 2-20, the dotted curve is the most danger-ous mode, the dashed curve is the limit of the Eckhaus stability from the amplitude equation. The triangles markthe Eckhaus limit as obtained from the full nonlinear analysis (Brattkus and Misbah, 1990), with stable cellularinterfaces possible only inside that region. The band width of possible wavelengths for cellular interfaces ac-cordingly is smaller by a factor of ≈0.4, as compared to the band width given by the neutral (linear) stabilitycurve.

Figure 2-23. Computed example for a deep cellularinterface at V ≈ 5 Vc with bubble formation at the bot-tom of the groove (Saito et al., 1990).

If the velocity is fixed and the wavelengthl is reduced significantly below the diffu-sion length l, the Saffmann–Taylor limit isreached (Brener et al., 1988; Dombre andHakim, 1987; Kessler and Levine, 1986c),which is equivalent to a low-viscosity fluidbeing pushed into a channel of width lfilled with a high-viscosity fluid. The low-viscosity fluid forms a finger just like thesolid in directional solidification. Near thetip, the width of the finger lf corresponds to

lf = D l

with cell spacing l, where D <1 is the ac-tual supercooling at the tip (remember thatD =1 for a flat interface at z = 0, and D = 0for a flat interface at z = lT). This servesto verify the consistency of numericalcalculations (Saito et al., 1989). An evenmore detailed analysis was carried out byMashaal et al. (1990).

For comparison with experiments, it isuseful to draw a V vs. l diagram (Fig. 2-24). Here the full line is again the neutralcurve, the broken line is the most danger-ous (or most unstable) mode, and the dot-ted line denotes the relation l = l, where thediffusion length is equal to the imposedwavelength. The asterisks mark some de-tailed numerical investigations (Saito et al.,1989). The asterisk furthest to the left isclose to the above-mentioned Saffmann–Taylor limit. At slightly higher wave-lengths, where l < l still holds, we are in ascaling region, where the radius of curva-ture at the tips of the cells is about 1/5 ofthe cell spacing, as also found experimen-tally (Kurowsky, 1990). All these consider-ations give sufficient confidence that thenumerical calculations may also provideinsight into the mechanism of directionalsolidification for the most interesting case


Figure 2-24. “Phase”-diagram log (V ) vs. log (l) for interface patterns in directional solidification. The solidand dashed curves denote the neutral stability curve, and the dashed-dotted curve the most dangerous mode.Asterisks mark fixed parameter values discussed hereafter. The lower critical threshold here is Vc≈1, lc≈ 0.5 forvelocity and cell spacing. For other parameters see text. At low pulling speeds and high wavelengths cellularpatterns with narrow grooves are found (a). At very short wavelengths and moderate speeds cellular patternswith wide grooves are found, consistent with theories for viscous fingering. At high pulling speeds, such thatthe cell spacing l is significantly wider than the diffusion length l, side-branching dendritic arrays are formed(c) (Saito et al., 1990). The speeds are still much smaller than the absolute stability limit Va.


of dendritic arrays formed at higher growthrates.

A few words on numerical methods andsystem parameters may be in order beforewe discuss the dendritic region. The nu-merical code is equivalent to the one usedfor the free dendritic case with the modifi-cation that it is necessary to integrate overseveral cells to arrive safely outside thediffusion length. Furthermore, in principle,diffusion has to be considered in both theliquid and the solid. Since the diffusion co-efficient for material in the solid is usuallymuch lower than it is in the liquid, it isfound that diffusion in the solid alloy canusually be neglected on time scales for theformation of cells. For long durations, ofcourse, microsegregation takes place, andsolid diffusion then becomes important(Kurz and Fisher, 1998).

A more serious difficulty in directionalsolidification is the large number of param-eters defining the system. We will concen-trate here on typical parameter values usedin experiments performed for some trans-parent materials between glass plates. Sev-eral tests and specific calculations alsodone for alloys, however, indicate that alarge part of the results can be carried overto these more relevant situations from ametallurgical point of view without qual-itative changes.

2.5.4 Directional Dendritic Growth

The diagram in Fig. 2-24 showing veloc-ity versus l in logarithmic form indicatesthat qualitatively different behavior may beexpected depending on whether the diffu-sion length is larger or smaller than the cellspacing. In the previous section, we dis-cussed the first case. When the diffusionlength becomes smaller than the cell spac-ing we expect that the individual cells be-come more and more independent of each

other, until finally they may behave like in-dividual isolated dendrites.

In order to test this hypothesis, a seriesof numerical experiments were performedat a fixed cell spacing and increasing pull-ing velocity (Saito et al., 1989). The nu-merical parameters of the model were rep-resentatively taken to correspond to steelwith Cr–Ni ingredients (Lesoult, 1980). Indimensionless units, the critical velocityand wavelength for the plane-front in-stability were Vc = 1.136, lc = 0.514. Theanisotropy of the capillarity length was notknown and was taken as e = 0.1 to allow forcomparison with the previously mentionedcalculations on the free dendritic case. Thecellular wavelength was fixed to l = 0.36,corresponding to the asterisks at increasingvelocity and constant l in Fig. 2-24.

At the lowest velocity still below thel = l dividing line, rounded cells were ob-served; the tip was not well approximatedby a parabola. At higher speeds (V =12) theparabolic structure of the tip became vis-ible, Fig. 2-25, and at even higher speeds(V = 20) the dendritic structure with sidebranches was fully developed, Fig. 2-26.

We can now compare the resulting tip ra-dius with the predictions from free den-dritic growth. Note that in the present casethe velocity is fixed rather than the super-cooling, so that the dendrite now uses asupercooling corresponding to the givenvelocity. This means that the tip of the den-drite is no longer at a position in the tem-perature-gradient field like a flat interface,but has advanced toward the warmer liq-uid.

Fig. 2-27 contains the ratio of the tip ra-dius divided by the radius from scaling, Eq.(2-68) (where the Péclet number P wasused in the original form as the ratio of tipradius to diffusion length). Furthermore,this figure gives the ratio of the tip radiusto the Ivantsov radius, which comes from

the Péclet number through the relation forthe supercooling at the tip, Eq. (2-56).The data are instantaneous measurementsrather than time-averaged measurements.It is obvious from Fig. 2-27 that the scalingrelation, Eq. (2-68), holds very well atrather low speeds, where neighboring cellsstill interact substantially through the dif-fusion field, while the relation from theIvantsov formula for the Péclet numberonly holds at higher velocities. The obviousreason for the latter deviation at low veloc-ities is that the Ivantsov relation representsa global conservation law for an isolatedparabolic structure, which is clearly notvalid when several cells are within a diffu-sion length.

The observation that the scaling relation,Eq. (2-68), is very robust obviously has todo with the fact that it results from a solv-ability condition at the tip of the dendrite,which is only very weakly influenced bydeformations further down the shaft.

In the same study, it was also confirmedthat the side branches fulfilled preciselythe same scaling relation (Fig. 2-13) as the


Figure 2-25. Transition from needle-shaped to dendritic cells at increasing pulling speeds. V = 4 is below thedotted line in Fig. 2-24, and V = 12 above it. Parabolas adjusted to the tip radius are not a good fit to the profiles.

Figure 2-26. Time sequence of a dendritic array atV = 40, e = 0.1, corresponding to point (c) in Fig. 2-24. The starting structure corresponds to V = 12, sim-ilar to Fig. 2-25. The cellular array quickly convergesto a stationary side-branching mode of operation.


free dendrites at relatively low speeds ofV =12 shown in Fig. 2-25. In this case, theside branches are just beginning to showup, while the tip is not very noticeably par-abolic.

Up to this point these investigationswere done at a constant anisotropy ofe = 0.1 of the capillarity length. In Fig.2-28 the normalized supercooling D hasbeen plotted against velocity V, where

Figure 2-27. Ratios Rtip /Rx of tip radius computed numerically (Fig. 2-24) over two theoretical predictions,where Rx is either the Ivantsov radius (circles) or the radius from solvability theory (asterisks). See also Fig.2-12. The result is in nearly perfect agreement with the solvability theory down to very low speeds in the cellularregion. The Ivantsov radius (for free growth) is not a good approximation there, as the diffusion fields of neigh-boring cells strongly overlap. At high speeds, essentially free dendritic growth is confirmed (Saito et al., 1990).

Figure 2-28. Supercooling at the tip of a cell or dendrite vs. pulling speed as obtained from numerical simula-tion. For a flat interface at low speeds, the global conservation law forces D = 1, then it decreases until l ≈ l, andfinally approaches the slowly increasing relation D (V ) obtained for the free dendritic case (see also Fig. 2-40).The expected dependence on capillary anisotropy e is also recovered.

D = 1 for a flat interface at stationarygrowth. Two sets of data for e = 0.1 ande = 0.2 are shown. If we increase the pull-ing speed above the critical value V ≈1, thesupercooling at the tip of the cellular pat-terns first decreases, because the forwardbulges come into a range of higher tem-perature. At intermediate velocity, D goesthrough a minimum and finally approachesthe broken lines corresponding to the scal-ing relation, Eq. (2-68), together with theIvantsov relation, Eq. (2-56). At intermedi-ate velocities, the supercooling D is abovethe corresponding curve meaning that thePéclet number, and therefore, the Ivantsovradius is larger than expected from the freedendritic scaling.

This is in agreement with Fig. 2-27shown above. The minimum of the D ver-sus V relation is in the range where the dif-fusion length is comparable to the cellspacing, as expected from Fig. 2-24.

As a final example, Fig. 2-29 shows adendritic array at the relatively high veloc-ity V = 40 at anisotropy e = 0.2. As in freedendritic growth, the structure appearssharper than the structure in Fig. 2-26 atsmaller anisotropy.

The opposite case of extremely small an-isotropies has not yet been analyzed ingreat detail, and it is rather unclear whathappens both from a theoretical and an ex-perimental point of view. It is likely that atzero anisotropy e = 0, the cells will tend tosplit if the cell spacing becomes muchlarger than the diffusion length, which af-fords the possibility for chaotic dynamicsat high speeds. However, this is still specu-lative.

Let us take a quick look at the kinetic co-efficient b in Eq. (2-111). As can be con-cluded from its multiplication by V^, b be-comes more and more important at highgrowth rates. For the free dendritic casewith kinetic coefficient b and 4-fold an-

isotropy b4 of the kinetic coefficient, ascaling relation similar to Eq. (2-68) wasderived by Brener and Melnikov (1991):

(2-123)

with a constant prefactor sb ≈ 5 and withPéclet number P as used before in Eq. (2-68). The scaling relation, Eq. (2-123), con-sists of several non-trivial power laws;only the one with P = R/l relating tip radiusto velocity has been confirmed (Classenet al., 1991). With regard to the generalagreement between analytical and numeri-cal results obtained so far, however, thereis little doubt that these scaling results (andothers given by Brener and Melnikov,1991) will also hold for the dendritic re-gion l l in directional solidification.

b sb

bbVDd

P= L2

0

9 211 2

47 2⎛

⎝⎜⎞⎠⎟

// /


Figure 2-29. Pronounced parabolic directional den-drites at speed V = 40, e = 0.2, as used in Fig. 2-28.Note that the tip-radius here is about 0.03 and theshort-wavelength limit of neutral stability 0.05 inunits of the cellular spacing. This is qualitativelyconsistent with experimental observations of largeinterdendritic spacings in units of tip radii. Tip split-ting was only observed at much lower values of cap-illary anisotropy.


A final point to be kept in mind is thatthe tip supercooling D in directional solid-ification is not small, as required by the ap-proximations used for the derivation of thescaling relation. On the other hand, P =1corresponds to a supercooling as large asD≈ 0.75; 0.6 for 2-dim and 3-dim, respec-tively, and the scaling relations can be ex-pected to hold over a large range of veloc-ities, as already indicated from the otherfree dendritic case, Fig. 2-12.

In summary, these investigations haveshown that there appears to be a smoothtransition from cellular to dendritic struc-tures. The dendritic growth laws are verywell represented by the scaling relationsfor the free dendritic case. This scalingshould hold in the region

d0 < R ≈ l < l < lT (2-124)

where l is the primary cell spacing. It wasproposed by Karma and Pelce (1989a, b)that the transition from cells to dendritescould occur via an oscillatory instability,for which the present investigations underquasi-stationary approximation have shownno evidence. A fully time-dependent calcu-lation is possible in principle with Green’sfunction methods (Strain, 1989).

2.5.5 The Selection Problemof Primary Cell Spacing

An important question from an engineer-ing point of view appears to be the follow-ing: suppose we know all the material pa-rameters and the experimentally controlla-ble parameters like thermal gradient andpulling speed for a directional solidifica-tion process – can we then predict the dis-tance between the cells and dendrites?

A positive answer to this question is de-sirable because the mechanical propertiesof the resulting alloy are improved with adecrease in the primary cell spacing (see

Kurz and Fisher (1998) and referencestherein).

In a rigorous sense the answer is stillnegative, but at least arguments can begiven for the existence of some boundarieson the wavelengths (or cell spacings)which can be estimated with the use of sim-plifications.

The situation here shows some similarityto the formation of hydrodynamic periodicroll patterns (Newell and Whitehead, 1969;Kramer et al., 1982; Riecke and Paap,1986). In a laterally infinite system, awhole band of parallel rolls is presentabove the threshold for roll formation, theso-called Eckhaus band. This was men-tioned in Sec. 2.5.3 for directional solidifi-cation.

The reason for the stability of these rollsis that an infinitesimal perturbation is notsufficient to create or annihilate a roll, buta perturbation must exceed a thresholdvalue before such an adjustment can occur.

In directional solidification, the situationis different insofar as the envelope over thetips of the cells could make a smooth defor-mation of very long wavelength, therebybuilding up enough deformation energy sothat a cell could be created or annihilated atisolated points. One indication for such aprocess is the oscillatory instability of cellspostulated by Karma and Pelce (1989) andRappel and Brener (1992).

The only hard argument for the selectionof a unique wavelength comes from ananalysis of a spatially modulated thermalgradient acting on a cellular pattern ofsmall amplitude (normal bifurcation)which imposes a ramp on the pattern(Misbah, 1989; Misbah et al., 1990). Theidea originally proposed for the hydrody-namic case (Kramer et al., 1982) is to havea periodically varying thermal gradientparallel to the interface, which keeps theinterface flat in some regions and allows

for the formation of cells in between(Fig. 2-30). For such a specific setup it wasshown (Misbah et al., 1990) that a uniquewavelength must be selected in the centerof the small-gradient area. The reason forthis special construction is that it allows forthe formation of cells at arbitrarily smallamplitudes (and therefore small pinningforces) in the region of strong thermal gra-dient.

In general, however, the boundary con-ditions on the other sides of the cells, dueto the container walls, are not well speci-fied and typically will not provide such aramp structure (see Misbah (1989), how-ever, for growth in a rotating vessel). Forthe time being, we can therefore try to atleast find some boundary similar to theEckhaus band for the limits of large andsmall wavelengths in the cell spacing.

It is not easy to extend the correspond-ing analysis of small-amplitude cells(Brattkus and Misbah, 1990) to cells withdeep grooves, as these essentially infinitegrooves present a kind of topological con-straint on the number of longitudinal cellsin a given lateral interval. The creation or

annihilation of cells is therefore likely tobe a discrete process.

A natural mechanism for the local reduc-tion of cell spacings (or creation of a newcell) is either a nucleation in one of thegrooves (the liquid is supercooled), or evenmore likely, the formation of a new cell outof a side branch in such a groove. Alterna-tively, tip splitting of a cell may give thesame result (Fisher and Kurz, 1978, 1980).

The opposite mechanism for the increaseof cell spacing (or annihilation of an exist-ing cell) could occur through the competi-tion of neighboring cells for the diffusionfield, such that one cell finally moves at aslightly slower speed than the neighboringcells and, consequently, will be supressedrelative to the position of the moving front.

These two mechanisms have been con-jectured by many authors in the past. Someprogress has been made recently by theconfirmation of the scaling relations inthe dendritic region. It seems, therefore,worthwhile to reformulate those conjec-tures with the help of these scaling rela-tions. Let us first consider the short-wave-length l (cell spacing) argument. Assumethat we are at dendritic growth speeds, Eq.(2-124), ignoring here kinetic coefficients.The solidification front then looks like anarray of individual dendrites which onlyweakly interact with each other through thediffusion field l < l.

The solidification front z = 0 will beunderstood here as a smooth envelopetouching all the dendrite tips, so that defor-mations of the front have a smallest wave-length l equal to the cell spacing. There arenow basically two “forces” acting on de-formations ∂z (x, t)/∂t of that front. If someof the tips are trailing a little behind theothers, they will be screened through thediffusion field of the neighboring tips,as in the conventional Mullins–Sekerkainstability, but now without a stabilizing


Figure 2-30. Numerical study of cellular wave-length selection at the interface by introducing aramp in the thermal gradient field. A high thermalgradient on the side approximately normal to theinterface keeps the interface flat, the smaller gradientin the center allows for cells to develop. At fixedramp profile a unique cell spacing is selected in thecenter, starting from different initial conditions (Mis-bah, 1990, unpublished).


surface tension interacting between neigh-boring tips. Taking this into account, thedestabilizing force is Fd≈ Wk

(d) zk withWk

(d) =V |k | for a sinusoidal perturbation ofamplitude zk of a plane interface withoutsurface tension moving at velocity V andwavenumber k. The maximum lies atk = 2p/l. The actual area under this pertur-bation zk contained in the solid cells issmaller by a factor ≈ 2R /l. We thus arriveat a maximal destabilizing force of

Fd ≈ W (d) zl ; W (d) ≈ 4 p V R /l2 (2-125)

corresponding to a depression or enhance-ment of every second dendrite.

On the other hand, each of these individ-ual dendrites knows its operating point,and through the given velocity its super-cooling at the tip. We approximate thisby the asymptotic form of Eq. (2-56)D ≈1–1/P since basically the variationof D with P enters below, even through Pmay not be very large compared to unity.Capillary effects do not seem to be veryimportant in this region and are thus ne-glected here for simplificity. By definition,D =1– z tip /lT , and the two expressions forD can be evaluated: z tip≈ lT l /R. The stabi-lizing force Fs≈W (s)zl follows from theobvious relation W (s) = dVtip /dz tip as

(2-126)

As expected, the sign of this “force” is op-posite to the destabilizing force, Eq. (2-125). Setting the sum of the two equal tozero, we expect an instability to occur firstat cell spacings

(2-127)

for segregation coefficients around one.We cannot say much about small segrega-tion coefficients because the nonlinearityin Eq. (2-114) replacing Eq. (2-56) be-

l l lT

FV Rl lT

ss s≈ ≈ −W W( ) ( );zl 2

comes important there. Of course, a num-ber of rough approximations were usedspecifically in the treatment of the destabi-lizing force, but this argument should atleast qualitatively capture the competitionmechanism between neighboring den-drites. A more detailed analysis (Warrenand Langer, 1990) is quite promising atlarge velocities in comparison with experi-ments (Somboonsuk et al., 1984) (see alsoKessler and Levine, 1986c; Bechhoeferand Libchaber, 1987).

Let us now look at the large-wavelengthlimit l. The initial growth conditions areassumed to be just as before, Eq. (2-124),but now at possibly large cell spacings l. Inthe numerical calculations, it was found(Saito et al., 1989) that for fixed cell spac-ing l at increasing velocity, a tail instabil-ity occurs (Fig. 2-31). A side branch in the

Figure 2-31. After a sudden increase in the growthrate in the dendritic region at fixed cell spacing, tailinstability occurs. One of the side branches near thetip produces a proturberance in the forward direction,which then becomes a new primary cell (the imposedmirror symmetry is not present in reality of course).In accordance with the stability of dendritic cellsagainst tip splitting (compare Fig. 2-29), this tail in-stability appears to be an important selection mecha-nism for primary cell spacing. See also Fig. 2-43.

groove between two dendrites splits off aternary branch, which then moves so fast inthe strong supersaturation in the groovethat it finally becomes a new primarybranch. In the plot, the imposed mirrorsymmetry is, of course, artificial, but iteven acts opposite to this effect, makingthe process more plausible in reality. Infact, this is also observed experimentally(Esaka and Kurz, 1985), in particular whenthe solid consists of slightly misaligned re-gions separated by grain boundaries, sothat the growth direction of two neighbor-ing dendrites is slightly divergent. Our ba-sic assumption now is that this tail instabil-ity occurs when the intersection of para-bolic envelopes over neighboring dendritesoccurs at a point z 0, where, theoretically,D >1. In this case, there is no need forlong-range diffusion around a side branch,for its dynamics become local. Of course,this assumption ignores geometrical com-petition between neighboring side branchesto a certain extent, but for the momentthere seems to be no better argument tohand.

Taking into account the point that neigh-boring parabolas with tip position at z tip > 0cannot intersect further down to the coldside than at z = 0, we obtain z tip = lT l /Rfrom the two relations for D, just as in theprevious case of small wavelengths. Butnow we must use the parabolic relationz tip = l2/8R for the intersection of two pa-rabolas of radius R at z = 0, which are a dis-tance l apart. The tail instability is then ex-pected to occur for

(2-128)

again with a prefactor roughly of the orderof unity. In comparison with Eq. (2-127), itcan be seen that in both cases the samescaling relation results. The scaling withthe inverse growth rate l follows the neu-tral stability curve at velocities safely in

l l lT

between the two critical values and againrecovers Eq. (2-1) by noting that l ~ V –1

and therefore V l2≈ constant. The results,Eqs. (2-127) and (2-128), seem to be inagreement with experiments (Somboonsuket al., 1984; Kurowsky, 1990; Kurz andFisher, 1981, 1998) concerning the scalingwith respect to diffusion length l and ther-mal length lT . The limit k Æ 0 for the seg-regation coefficient as a singular point isnot reliably tractable here.

The previously given relation l ~ l1/4

(Hunt, 1979; Trivedi, 1980; Kurz andFisher, 1981) seems to be valid in an inter-mediate velocity region (Fig. 2-28) whereD does not vary significantly, so thatz tip /lT ≈1/2 (see also Sec. 2.6.1).

A serious point is the neglect of surfacetension and anisotropy in these derivations.In the experiments analyzed so far the rela-tion V l2≈ const. seems to hold approxi-mately, but what happens for the capillaryanisotropy e going to zero? Numerically,tip splitting occurs at lower velocities forsmaller e. In a system with anistropy e = 0(and zero kinetic coefficient) the structuresprobably show chaotic dynamics at veloc-ities where the diffusion length l is smallerthan the short wavelength limit of the neu-tral stability curve (Fig. 2-24), but this israther speculative (Kessler and Levine,1986c).

In considering whether the tail instabil-ity (large l) or the competition mechanism(small l) will dominate in casting pro-cesses, we tend to favor the former. If thesolidification front consists of groups ofdendrites slightly misoriented against eachother due to small-angle grain boundaries,cells will disappear at points where the lo-cal growth directions are converging andnew cells will appear through the tail in-stability at diverging points at the front.

To summarize, the most likely scalingbehavior of the primary cell spacing l, de-



pending on pulling velocities, follows Eqs.(2-1) and (2-128) as a consequence of thearguments presented in this section. Thisconclusion is supported by a number of ex-periments (Billia et al., 1987; Somboonsuket al., 1984; Kurowsky, 1990; Esaka andKurz, 1985), but more work remains to bedone.

2.5.6 Experimental Resultson Directional Dendritic Growth

Since 1950, in situ experiments on direc-tional solidification (DS) of transparentmodel systems have been performed (Kof-ler, 1950). However, it was some time be-fore such experiments were specificallyconceived to support microstructural mod-els developed in the 1950s and early 1960s.The work of Jackson and Hunt (1966) is amilestone in this respect (see also Hunt etal., 1966). Their experimental approach todendritic growth has been developed fur-ther by several groups: Esaka and Kurz(1985), Trivedi (1984), Somboonsuk et al.(1984), Somboonsuk and Trivedi (1985),Eshelman et al. (1988), Seetharaman andTrivedi (1988), Seetharaman et al. (1988),de Cheveigne et al. (1986), Akamatsu et al.(1995), Akamatsu and Faivre (1998), andothers. Substantial progress has been madeduring these years, especially due to resultsobtained by Faivre and coworkers in verythin (~ 15 µm) cells that constrain the pat-terns to two dimensions. This research isstill producing interesting new insights intothe dynamics of interface propagating dur-ing crystallization.

The specific interest of DS is that growthmorphologies can be studied not only fordendritic growth but also for cellular andplane front growth. We will discuss thesephenomena in the sequence of their appear-ance when the growth rate is increasedfrom Vc, the limit of first formation of

Mullins–Sekerka (MS) instabilities (alsocalled limit of constitutional undercool-ing), to rates where plane front growthagain appears at velocities above Va, theabsolute stability limit.

Morphological instabilities. The onsetof plane front interface instability is ob-served to start at defects such as grainboundaries, subgrain boundaries and dislo-cations, forming a more or less pronounceddepression at the intersection with thesolid– liquid boundary, as shown in Fig.2-32 (Fisher and Kurz, 1978). It is inher-ently difficult to quantitatively observe thebreak-down because the growth rates aresmall, and a long period of time is requiredto reach steady state.

The amplitude and wavelength of theperturbations as a function of V developingin systems with small distribution coeffi-cients k (of the order of 0.1) are shown inFig. 2-33. In CBr4–Br2 (de Cheveigne etal., 1986) and SCN–ACE (Eshelman andTrivedi, 1987), the bifurcation is of a sub-critical type, i.e., there are two criticalgrowth rates, one for increasing growthrate, Vc

+, and another lower value for de-creasing growth rate, Vc

–. Therefore, a peri-odic interface shape with infinitesimallysmall amplitudes cannot form in thesesystems. As has been discussed in Sec.2.5.3, only systems with k near unity willgive supercritical (normal) bifurcationwith a single, well-defined critical growthrate Vc. The evolution of the wavelengthfor two temperature gradients (70 and120 K/cm) is shown in Fig. 2-33b. At theonset of instability the experimentally de-termined wavelength is smaller than thecritical wavelength by a factor of 2–3 com-pared with linear stability analysis. In-creasing the rate above the threshold leadsto a decrease in the wavelength proportionalto V –0.4 (de Cheveigne et al., 1986; Kurow-sky, 1990). Once stability has started, the

structure evolves to a steady-state cellular ordendritic growth mode. Which one of thesestructures will finally prevail is a question ofthe growth conditions.

Cells and dendrites. Losert et al. (1998a)developed an interesting technique inwhich a spatially periodic UV laser pulse isdirected onto the solidification front of the


Figure 2-32. Morphological instabilities of a planar solid/liquid interface as seen on an inclined solidificationfront between two glass plates (arrows). Photograph (a) was taken at an earlier stage than (b). The beginning ofthe breakdown at defects such as dislocations, subgrain boundaries, or grain boundaries intersecting with thesolid/liquid interface is evident. The widths of the photographs correspond to 100 µm.

(a)

Figure 2-33. Amplitude, A, and wavelength, l, of a periodic deformation of the solid/liquid interface ofCBr4–Br2 solution versus pulling speed (de Cheveigne et al., 1986). Diagram (a) shows the hysteresis betweenappearance and disappearance of perturbations (typical for an inverse bifurcation) for a temperature gradient of12 K/mm. The lines in diagram (b) represent the calculated neutral stability curves for the two gradients indi-cated. Open circles are experimental results for 7 K/mm, and crosses, for 12 K/mm. See also Fig. 2-21.

(b)

(a) (b)

J

J

j

j


transparent succinonitrile–coumarin sys-tem. These experiments allowed for asystematic investigation of the dynamic se-lection and stability of cellular structures.

Through an increase in V (or C• , or a de-crease in GT), a columnar dendritic struc-ture can be formed out of a cellular array(Fig. 2-34). All three morphologies (in-

stabilities, cells, dendrites) appear to havetheir own wavelength or array spacing.Owing to competition between neighboringcrystals, the mean spacing of large ampli-tude cells seems to be always larger thanthat of the initial perturbations of the planefront, and the mean trunk distance (primaryspacing l) of the dendrites, larger than the

Figure 2-34. Time evolution of the solid/liquid interface morphology when accelerating the growth rate from0 to 3.4 µm/s at a temperature gradient of 6.7 K/mm. Magnification 41¥. (a) 50 s, (b) 55 s, (c) 65 s, (d) 80 s,(e) 135 s, (f) 740 s (Trivedi and Somboonsuk, 1984).

spacing between smooth cells. The reasonfor this change in typical spacing is not yetclear. We will come back to this subjectlater. Before we do so, some relevant obser-vations on tip growth need to be discussed.

The tip is the “head” of the dendritewhere most of the structural features are in-itiated. Fig. 2-35 shows a dendrite tip ofsuccinonitrile (SCN) with 1.3 wt.% ace-tone (ACE) in an imposed temperature gra-dient, GT = 16 K/cm and a growth rate,V = 8.3 µm/s. The smooth tip of initiallyparabolic shape (Fig. 2-35b) is clearly vis-ible. In contrast of free thermal dendrites(Fig. 2-9), in DS of alloys the secondaryinstabilities start forming much closer tothe tip. The imposed temperature gradientalso widens the dendrite along the shaftrelative to a parabola fitted to the tip. Thiseffect increases with an increasing temper-ature gradient (Esaka, 1986).


Figure 2-35. Dendrite tip ofSCN–1.3 wt.% acetone so-lution in directional growth.(a) For V = 8.3 µm/s andG = 1.6 K/mm; (b) parabolafitting the tip growingat V = 33 µm/s and G =4.4 K/mm (Esaka and Kurz,1985; Esaka, 1986).

(b)

(a)


The sequence of steady-state growthmorphologies, from well-developed largeamplitude cells to well-developed den-drites, is shown in Fig. 2-36. Besides theinformation on the form and size of the cor-responding growth morphologies, this fig-ure also contains indications specifying thediffusion length l = 2D/V and the ratio ofthe half spacing over tip radius. The char-acteristic diffusion distance decreases morerapidly than the primary spacing of the den-drites (Fig. 2-36). When l Û l the ratio l /2Rof directionally solidified SCN–1 wt.% ACEalloys is between 5.5 and 6, in agreementwith numerical calculations (Sec. 2.5.3).

These observations are summarized inFig. 2-37. Three areas of growth can be dif-ferentiated for this alloy:

i) at low speeds, cells are found showingno side branches and a non-parabolic tip;

ii) at intermediate rates (over a factor 5 inV ), cellular dendrites are formed with weaklydeveloped secondary arms, and they showan increasingly sharpened parabolic tip;

iii) at large rates dendritic arrays growwith well-developed side branches and atip size much smaller than the spacing.

It is difficult at present to judge the influ-ence of the width of the gap of the experi-

Figure 2-36. Cellular and den-dritic growth morphologies inSCN–1.3 wt.% acetone; thermalgradient 8 K/mm < G < 10.5 K/mm(Esaka, 1986). The growth rate Vin µm/s, the diffusion length(2D/V ) in mm, and the ratio of theprimary trunk spacing to the tipdiameter are as follows: A = 1.6,1.6, 2.0; B = 2.5, 1.0, 2.5; C = 8.3,0.3, 5.5; D = 16, 0.16, 6.0; E = 33,0.08, 7.5; F = 83, 0.03, 9.0.

mental cell, which had approximately thesame size as the diffusion distance, whenl = l. The corresponding growth rate alsomarks the transition from two- to three-di-mensional growth of dendrites, as can beseen in Fig. 2-36. Under conditions A to C,no secondary arms are observed perpendic-ular to the plane of observation, while con-ditions D to F show well-developed 3Ddendrites even far behind the tip. There-fore, the gap might somewhat influence thevalues of the transition rates but not thequalitative behavior of the transition. Asmentioned above, the theory (in quasi-sta-tionary approximation) only indicates avery gradual change in morphology. It isinteresting to compare Fig. 2-36 with Figs.2-25 and 2-26. It can be seen that both thetheoretical and the experimental ap-proaches show qualitatively the same be-havior, even if the material constants usedare not the same.

The critical role of crystalline anisotropyin interface dynamics has been demon-strated experimentally in directional growthof transparent CBr4–C2Cl6 alloys grown assingle crystals in thin samples (Akamatsu

et al., 1995). With the ·100Ò directionoriented along the heat flow direction,characteristic cellular/dendritic arrays suchas shown in Fig. 2-36 are obtained. Withother orientations, where growth is ren-dered “effectively isotropic” a “seaweed”structure is observed, the tips of whichform what has been called on the basis oftheoretical predictions a doublon (Ihle andMüller-Krumbhaar, 1994).

The formation of doublons has also beensuggested to play an important role in theformation of “feathery” grains in technicalaluminum alloys. Recent detailed observa-tions with electron back-scattered diffrac-tion (EBSD) combined with optical andscanning electron microscopy (Henry etal., 1998) have clearly shown that featherygrains are made of ·110Ò columnar den-drites, whose primary trunks are alignedalong and split in their center by a (111)coherent twin plane. The impingement ofsecondary ·110Ò side arms gives rise to in-coherent wavy twin boundaries. The switchfrom ·100Ò to ·110Ò growth direction wasattributed to the small anisotropy of thesolid– liquid interfacial energy of alumi-


Figure 2-37. Primary trunk spacing, diffusionlength and tip radius of SCN–1.3 wt.% ace-tone dendrites as a function of the growth ratefor a temperature gradient of 9.7 K/mm. Thevarious points correspond to the conditionsand microstructures given in Fig. 2-36 (Esaka,1986). In region A no side arms are observedand the tips are of non-parabolic shape; in Bthe tip becomes parabolic and some side armsappear; in C well-developed side arms and aparabolic tip are the sign of isolated tips.


num, which can be changed by the additionof solute (Henry et al., 1998).

The scaling of the initial side-branchspacing l2 with respect to the tip radius isshown in Fig. 2-38. Both quantities scaleclosely with L2 V = const. or L2 C• = const.(where L ~ R or l2). Fig. 2-38b also showsthe quantity lp, the distance from the tipdown the shaft, where the first signs of tipperturbations in the SCN–ACE system canbe observed. Here lp is of the same orderas l2.

The ratio l2/R obtained by Esaka andKurz (1985) for SCN–1.3 wt.% ACE is in-dependent of growth rate according to theprecision of the measurements and takes avalue of 2.1 ± 0.2 (see Fig. 2-39 and Table2-1). This is in good agreement with themeasurement of Somboonsuk and Trivedi(1985) who found a value of 2.0 on thesame system over a wide range of growthrates, compositions, and temperature gra-dients (Trivedi and Somboonsuk, 1984).This ratio increases with crystal anisotropy

(a)

(b)

Figure 2-38. Characteristic dimensions(tip radius, R, initial secondary arm spac-ing, l2, and distance of appearance ofsecondary instabilities, lp) as a functionof growth rate (a), and as a function ofthe alloy concentration (b) (Esaka, 1986).

in agreement with numerical calculations(Figs. 2-26, 2-29, 2-13) and decreases withincreasing temperature gradient. On theother hand, its value is larger in the case offree dendritic growth (Table 2-1).

The tip undercooling of the dendrite is ameasure of the driving force necessary forits growth at the imposed rate V. Fig. 2-40shows the variation of the tip temperaturewith V, the undercooling being defined bythe difference between Tm and the tip tem-perature. During an increase in growth rate,

the cellular growth region is characterizedby a decrease in undercooling, while to-ward the dendritic region, the undercoolingincreases again (Fig. 2-28).

One of the characteristics of directionalsolidification of cells/dendrites is the for-mation of array structures with a primarytrunk spacing l. For cellular and cellu-lar–dendritic structures the growth direc-tion/crystal orientation relationship playsan important role in the establishment ofstable or unstable array patterns with a


Figure 2-39. Ratio of initial arm spac-ing to tip radius as a function of growthrate (Esaka and Kurz, 1985).

Figure 2-40. Tip temperatures ofthe dendrites of Fig. 2-36 as afunction of growth rate (Esaka andKurz, 1985). See also Fig. 2-28.


characteristic spacing, as has been shownby Akamatsu and Faivre (1998). Thegrowth-rate dependence of dendritic pat-terns has been shown in the log– log plot ofFig. 2-37. We can see that the slope of l vs.V is initially lower than the slope of R vs.V. If we take the three points aroundV (l = l), where the tip temperature is ap-proximately constant, the slope is about0.25, while a mean slope through all mea-surements shown gives 0.4. This is fullyconsistent with data by Kurowsky (1990).Furthermore, it is in agreement with our ar-guments in Sec. 2.5.5.

Most of the measurements give a rate ex-ponent somewhere between the two limit-ing values, 0.25–0.5. Taking the lowervalue expressed by temperature gradientand concentration gives a relationshipl ~ GT

–0.5 C•0.25 (Hunt, 1979; Kurz and

Fisher, 1981; Trivedi, 1984). Therefore, anormalized spacing l4 G 2

T V /k DT0 is plot-ted vs. V in Fig. 2-41, showing that differ-ent materials behave similarly except for aconstant factor (note that the solidus– liq-uidus interval of an alloy, DT0, is propor-tional to C•).

Primary spacings, however, are notuniquely defined but form a rather widedistribution. This is shown in Fig. 2-42 forone superalloy which was directionally so-lidified under different conditions. This be-havior can be understood by examining themechanism of wavelength (spacing) reduc-tion through tail instability, which is acomplicated process. A series of competi-tive processes between secondary andtertiary arms in a region behind the tip (ofthe order of one primary spacing) finallycauses one tertiary branch to grow throughand to become a new primary trunk (Esakaet al., 1988), as shown in Figs. 2-43 and2-31. The range of stability of dendriticarrays has been analysed by Losert et al.(1998b). Hunt and Lu (1996) approached

Figure 2-41. Normalized primary trunk spacing, l,as a function of growth rate for various alloy systems(Somboonsuk et al., 1984). Note that the variationwith growth rate at these high velocities is in agree-ment with the discussion in Sec. 2.5.5.

Figure 2-42. Distribution of nearest dendrite–den-drite separations (primary trunk spacings) measuredon a transverse section of a directionally solidifiedNi-base superalloy (Quested and McLean, 1984).Solid lines for 16.7 µm/s and broken lines for83.3 µm/s.

this problem by numerical modeling usinga minimum undercooling criterion forthe selection of the lower limit of arraygrowth.

Effects at high growth rates

As the final topic in this section, we dis-cuss some interesting effects which havebeen observed at very high growth rates.In laser experiments of the type shown in

Fig. 2-7, the interface may be driven to ve-locities of several m/s. Under such highrates, the structures become extremelyfine. Primary spacings as small as 20 nmhave been measured in Al–Fe alloys (Gre-maud et al., 1990). Fig. 2-44 representsmeasured primary spacings (black squares),measured secondary spacings (opensquares), and the calculated tip radius ofthe dendrites, using Ivantsov’s solutionand the solvability-scaling criterion (Kurzet al., 1986, 1988). l and l2 vary as V –0.5,as does the tip radius when the Péclet num-ber of the tip is not too large. At largePéclet numbers (or small diffusion dis-tances), capillary forces become dominant,which is the reason for the limit of absolutestability, Va. Ludwig and Kurz (1996a) de-termined cell spacing and amplitude closeto absolute stability in succinonitrile–argon alloys and found for both l3/2 V =const.

Since the classical paper by Mullins andSekerka (1964), it has been known thatplane front growth should also be observedat interface rates where the diffusion lengthreaches the same order as the capillarylength. This critical rate, called the limit ofabsolute stability, can be calculated fromlinear stability analysis (for temperature


Figure 2-43. Mechanism for the formation of a newprimary trunk by repeated branching of side arms at agrain boundary (gb). The increasing spacing at the gballows a ternary branch to develop and to compete inits growth with other secondary branches (hatchedarms) (Esaka et al., 1988).

Figure 2-44. Experimentally determinedprimary spacings (black squares), secon-dary arm spacings (open squares) andcalculated tip radius (line) for Al–4 wt.%Fe alloy rapidly solidified by laser treat-ment. The minimum in tip temperatureor the maximum in undercooling is dueto the decreasing curvature undercoolingwhen the dendrite approaches the limitof absolute stability, Va (Gremaud et al.,1990).


gradients which are not too high) approxi-mately as

Va = DT0 D /kG

Here G is the ratio of solid– liquid interfaceenergy to specific melting entropy. Typicallimits are of the order of m/s. The precisevalue depends also on the effect of a vary-ing solute diffusion coefficient (due to thelarge undercoolings), on the variations of kand DT0 with the growth rate associatedwith the loss of local equilibrium, and, fi-nally, on interface kinetics, which cannotbe ignored. According to an extensivestudy of several alloy systems, it may bestated that at Va we generally do not ob-serve a simple plane front growth butrather an oscillating interface, which pro-duces bands of plane front and cellulardendritic morphology. Fig. 2-45a showssuch a transition from columnar dendriticgrains to bands (Carrard et al., 1992).Fig. 2-45b shows more details of thebanded structure. It is visible that the darkband is cellular/dendritic and the clearband structure-less (supersaturated planefront growth). The possibility of chaoticinterface motion was shown by Misbah etal. (1991), and oscillatory motion of aplane interface with time was suggested byCorriell and Sekerka (1983) and Temkin(1990). The full theoretical analysis of thebanding problem has been given by Karmaand Sarkissian (1993), see also Kurz andTrivedi (1996). Growth rates much higherthan Va are needed in order to definitelyproduce an absolutely stable plane solid–liquid interface (Fig. 2-46).

Under steady-state growth conditionsusing a transparent organic system, Ludwigand Kurz (1996b) observed the onset of ab-solute stability but no sign of banding. Thelatter was due to the fact that the experi-ments had to be undertaken with highly di-luted alloys. This was necessary as one had

to work under heat flow limited conditionsallowing a growth rate of not more thansome mm/s. In order to reach absolutestability at these low rates the composition(and DT0) had to be kept very small (typi-cally below 100 mK).

2.5.7 Extensions

In this section, we summarize a few im-provements on the theory of solidificationregarding various effects that are importantin the practical experimental situation.They are observed when experimental pa-rameters are outside the range of the simplemodels considered in this chapter, or when

Figure 2-45. Transmission electron micrographs ofthe banded structure in laser-remelted alloys. (a)General view of the abrupt transition from columnareutectic grains (lower part) to a banded structure(upper part) in a eutectic Al–33 wt.% Cu alloy witha solidification rate, V = 0.5 m/s. (b) Enlarged viewof the dark and light bands in a cellular dendriticAl–4 wt.% Fe alloy with V = 0.7 m/s (Carrard et al.,1992).

additional factors influence the growth ofthe solid.

The models in this chapter are minimalin the sense that they were intended to cap-ture the essence of a phenomenon with the

smallest possible number of experimen-tal parameters. Despite this simplification,however, the results appear to be of generalimportance.

The discussion of free dendritic growthhas been restricted so far to small Pécletnumbers or low supercoolings. In direc-tional solidification, however, we are usu-ally at moderate or even high Péclet num-bers (Ben Amar, 1990; Brener and Melni-kov, 1990). It was shown by Brener andMelnikov (1990) that in this case a devia-tion from dendritic scaling occurs if Pe1/2

increases approximately beyond unity.At high growth rates, furthermore, kineticcoefficients can no longer be ignored atthe interface. A proposed scaling relation(Brener and Melnikov, 1991) was con-firmed numerically (Classen et al., 1991),and combined anisotropy of surface ten-sion and kinetic coefficient was treated an-alytically (Brener and Levine, 1991; Ben-Jacob and Garik, 1990). A general treat-ment of kinetic coefficients on interfacestability was given by Caroli et al. (1988).For eutectic growth, this was formulated byGeilikman and Temkin (1984). For higheranisotropies than considered so far, we en-counter facets on the growing crystals.This was analyzed for single dendrites(Adda Bedia and Ben Amar, 1991; Maureret al., 1988; Raz et al., 1989; Yokoyamaand Kuroda, 1988) and for directional so-lidification (Bowley et al., 1989).

In our treatment of directional solidifica-tion, only one diffusion field was treated ex-plicitly, namely the compositional diffusion.If a simple material grows dendritically(thermal diffusion), small amounts of impur-ities may become a matter of concern. Thiswas reconsidered by Ben Amar and Pelce(1989), confirming the previous conclusion(Karma and Langer, 1984; Karma and Kot-liar, 1985; Lipton et al., 1987) that impur-ities may increase the dendritic growth rate.


Figure 2-46. Experimental velocities at the transi-tions from dendrites to bands (squares) and frombands to microsegregation-free structures (circles)for the (a) Al–Fe, (b) Al–Cu and (c) Ag–Cu sys-tems. The triangles indicate experimentally deter-mined transitions from dendrites to microsegregation-free structure, and the crosses the values of the abso-lute stability limit. The full and dashed lines are thetheoretical predictions of the model for dilute and con-centrated solutions, respectively (Carrard et al., 1992).


A subject of appreciable practical impor-tance concerns the late stages of growth,where coarsening of the side branch struc-tures occurs together with segregation(Kurz and Fisher, 1998). If elastic forcesare not of primary importance, it is nowgenerally accepted (see Chapter 6, Sec.6.4.1) that a relation L* = A + B t1/3 (Huse,1986) is a good representation of typicallength scales L* varying with time t duringdiffusional coarsening processes. This con-firms the classical Lifshitz–Slyozov–Wagner theory (Lifshitz and Slyozov,1961; Wagner, 1961). The result, however,is not specific about the geometric details,for example, in directional solidification,and it also assumes constant temperature.

We have only briefly mentioned the tran-sition from dendritic arrays back to a planefront during directional solidification atvery high speeds. The importance of ki-netic coefficients was demonstrated byBrener and Temkin (1989), and recently,the possibility of chaotic dynamics in thisregion similar to those described in the Ku-ramoto–Sivashinsky equation was sug-gested by Misbah et al. (1991).

A richness of dynamic phenomena wasobtained in a stability analysis of eutectics(Datye et al., 1981). The possibility oftilted lamellar arrays in eutectics was dem-onstrated by Caroli et al. (1990) and, simi-larly, for directional solidification at highspeeds by Levine and Rappel (1990). Thiswas observed in experiments on nematicliquid crystals (Bechhoefer et al., 1989)and in eutectics (Faivre et al., 1989).

Finally, the density difference betweenliquid and solid should have some markedinfluence on the growth mode (Caroliet al., 1984, 1989). For dendritic growth,forced flow was treated in some detail byBen Amar et al. (1988), Ben Amar and Po-meau (1992), Bouissou and Pelce (1989),Bouissou et al. (1989, 1990) and Rabaud et

al. (1988). Other sources of convective in-stability (Corriell et al., 1980; Sahm andKeller, 1991) cannot be discussed herein any detail, as the literature is too exten-sive in order to be covered within thischapter.

Phase-field models

In recent years, the phase-field metho-dology has achieved considerable impor-tance in modeling and numerically simulat-ing a range of phase transitions and com-plex growth structures that occur duringsolidification. Phase-field models havebeen applied to a wide range of materialssuch as pure melts, simple binary alloys,eutectic and peretectic phase transitionsand grain growth structures in situations asdiverse as dendritic growth and rapid solid-ification.

In the phase-field formulation a mathe-matically sharp solid– liquid interface issmeared out or regularized and treated asa boundary layer, with its own equationof motion. The resulting formulation nolonger requires front tracking and the im-position of boundary conditions, but mustbe related to the sharp interface model byan asymptotic analysis. In fact, there aremany ways to prescribe a smoothing anddynamics of the sharp interface consistentwith the original sharp-interface model. Sothere is no unique phase-field model, butrather a family of related models. The firstphase-field model was developed by Lan-ger (1986) in ad hoc manner. Subsequently,models have been placed on a more securebasis by deriving them within the frame-work of irreversible thermodynamics (Pen-rose and Fife, 1990; Wang et al., 1993).Caginalp (1989) presented for the first timethe relation between phase-field modelsand sharp interface models or free boun-dary problems by considering the sharp

interface limit in which the interface thick-ness is allowed to tend to zero.

However, the technique also has somedisadvantages. The first is related to the ef-fective thickness of the diffuse interface ofalloys (1–5 nm) which is three to four or-ders of magnitude smaller than the typicallength scale of the microstructure. Sincethe interface must spread over severalpoints of the mesh, this limits considerablythe size of the simulation domain. The sec-ond problem arises from the attachment ki-netics term, which plays a significant rolein the phase-field equation, unlike micro-structure formation of metallic alloys atlow undercooling. These two factors haveso far limited phase-field simulations ofalloy solidification with realistic solid-state diffusivities to relatively large super-saturations. Recent mathematical and com-putational advances, however, are rapidlychanging this picture. Some of the recentadvances include: 1) a reformulated asymp-totic analysis of the phase-field modelfor pure melts (Karma and Rappel, 1996,1998) that has (i) lowered the range of ac-cessible undercooling by permitting moreefficient computations with a larger widthof the diffuse interface region (as com-pared to the capillary length), and (ii) madeit possible to choose the model parametersso as to make the interface kinetics vanish;2) a method to compensate for the grid an-isotropy (Bösch et al., 1995); 3) an adaptivefinite element method formulation that re-fines the zone near the diffuse interface andthat has been used in conjunction with thereformulated asymptotics to simulate 2Ddendritic growth at low undercooling (Pro-vatas et al., 1998); 4) the implementationof the method for fluid flow effects (Diep-ers et al., 1997; Tonhardt and Amberg,1998; Tong et al., 1998), and 5) the extensionof the technique to other solidification phe-nomena including eutectic (Karma, 1994;

Elder et al., 1994; Wheeler et al., 1996) andperitectic reactions (Lo et al., 2000).

2.6 Eutectic Growth

2.6.1 Basic Concepts

Eutectic growth is a mode of solidifica-tion for a two-component system. Operat-ing near a specific point in the phase dia-gram, it shows some unique features (Kurzand Fisher, 1998; Lesoult, 1980; Hunt andJackson, 1966; Elliot, 1983).

The crucial point in eutectic growth isthat the solidifying two-component liquidat a concentration near CE (Fig. 2-19) cansplit into two different solid phases. Thefirst phase consists of a high concentrationof A atoms and a low concentration of B at-oms; the second solid phase has the oppo-site concentrations. These two phases ap-pear alternatively as lamellae or as fibers ofone phase in a matrix of the other phase.

One condition for the appearance of aeutectic alloy is apparently a phase dia-gram (as sketched in Fig. 2-19) with a tem-perature TE. The two-phase regions meet at(TE, CE), and the two liquidus lines inter-sect before they continue to exist as meta-stable liquidus lines (dash-dotted line) attemperatures somewhat below TE. This is amaterial property of the alloy (Pb–Sn forexample). The other condition is the ex-perimental starting condition for the con-centration in the high-temperature liquid.Assume that we are moving a containerfilled with liquid at concentration C• in athermal gradient field where the solid iscold, the liquid is hot, and the solidificationfront is proceeding towards the liquid inthe positive z-direction. As a stationary so-lution, we find a similar condition to thesimple directional solidification case; thatis the concentration in the solid C

–S aver-


2.6 Eutectic Growth 153

aged across the front must be equal to C• tomaintain global mass conservation togetherwith a stationary concentration profile nearthe front.

Assume now that the liquid concentra-tion at infinity, C•, is close to CE and thetemperature at the interface is at TE. Thesolid may split into two spatially alternat-ing phases now on the equilibrium (bino-dal) solidus lines CS(TI), one located nearC Æ 0, and one located near C Æ1 atTI < TE. To see what happens, let us lookat the situation with the concentration inthe liquid C• being precisely at eutecticcomposition C• = CE. At TI < TE, there maynow be alternating lamellae formed ofsolid concentration CS

a and CSb (Fig. 2-47),

the corresponding metastable liquidus con-centrations being at CL

a and CLb. For both

solid phases now the liquid concentrationC• = CE is in the metastable two-phaseregions at TI : CS

a < C• < CLa, CL

b < C• < CSb.

In principle, we have to consider the pos-sibility of metastable solid phases (Tem-kin, 1985), which we ignore here for sim-plicity.

Diffusion of excess material not incorpo-rated into one of the lamellae does not haveto continue up to infinity but only to theneighboring lamella, which has the oppo-site composition relative to C•. Assumingagain for simplicity a symmetric phase dia-gram, we may write this flux balance as

(2-129)

where the left-hand side describes the fluxJ of material to be carried away from eachlamellar interface during growth at veloc-ity V, which then must be equal to thediffusion current (concentration gradient)over a distance l/2 to the neighboring la-mella (across the liquid, as we ignore diffu-sion in the solid). DC = CL

(a)– CL(b) is the

concentration difference in the liquid at theinterfaces of the lamellae, and k <1 is thesegregation coefficient.

With the help of the liquidus slopesdT/dCL as material parameters, we can ex-press the undercooling DTI = TE – TI as

(2-130)

(symmetrical phase diagram assumedabout CE) and thus

(2-131)

Again, l = 2 D/V is the diffusion length.So far we have ignored the singular

points on the interface where the liquid andtwo lamellae a and b meet (Fig. 2-47). Atthis triple point in real space (or three-phase junction), the condition of mechani-cal equilibrium requires that the surfacetension forces exerted from the three inter-faces separating a, b and the liquid cancelto zero. In the simplest version, this condi-tion defines some angles Ja, Jb of the two

DTT

CC k

lIL

E=d

d2 1˜ ( )− l

D DT CT

CI

L

=d

d˜

˜2

V C k DC

E =( )˜

( / )1

2− D

l

Figure 2-47. Sketch of symmetrical eutectic phasediagram (a) and eutectically growing lamellar cells(b), where a is the solid phase with concentrationCS

a, and b the solid phase with concentration CSb. The

average temperature at the interface is around TI be-low the eutectic temperature TE. The liquid at theinterface is at metastable concentrations (dashedlines in (a)). The solid– liquid interfaces in (b) arecurved and they meet with a – b interfaces at three-phase coexistence points.

(a) (b)

solid– liquid interfaces relative to a flatinterface. Accordingly, the growth frontwill have local curvature. What is worse,there is no guarantee that the a–b solid–solid interfaces are really parallel to thegrowth direction. Making a symmetry as-sumption for simplicity again, we canignore this problem for the moment. Thecurvature K of the growth front at each la-mella is then proportional to l–1. This cur-vature requires, through the Gibbs–Thom-son relation, Eq. (2-102), another reduc-tion, DTK, of the interface temperature be-low TI :

DTK = TE d K (2-132)

where d is an effective capillary length, de-pending on surface tensions, with K nowtaken as K ~l–1.

The total reduction of temperature belowTE during eutectic growth can thus be writ-ten as (2-133)

with dimensionless constants aI and aK.Plotting this supercooling as a function

of lamellar spacing, we find a minimalsupercooling ∂DT/∂l = 0 at

(2-134)

Again, this is the fundamental scaling rela-tion conjectured in Eq. (2-1) and encoun-tered in various places in this chapter,where the origin is a competition betweendriving force and stabilizing forces, mostintuitively expressed in Eq. (2-133).

A more thorough theoretical analysis ofeutectic growth was given in the seminalpaper by Jackson and Hunt (1966), whichis still a standard reference today. One ba-sic approximation in this paper was to av-erage the boundary conditions on flux andtemperature over the interface. This led toEq. (2-133), and it was argued that the min-

l = K Ia a l d/

D D DT T T T al

ad

= + =I K E I Kl

l+⎧

⎨⎩

⎫⎬⎭

imum undercooling would serve as the op-erating point of the system with spacinggiven by Eq. (2-134).

This stationary calculation was extendedby Datye and Langer (1981) to a dynamicstability analysis, where the solid–solid–liquid triple points could move parallel andperpendicular to the local direction ofgrowth, coupled however to the normal di-rection of the local orientation of the front.It was found that the marginal stability co-incided exactly with the point of minimumundercooling.

The basic model equations for eutecticgrowth in a thermal gradient field can bewritten in scaled form as follows (Brattkuset al., 1990; Caroli et al., 1990). Neglectingdiffusion in the solid (one-sided model)and assuming a single diffusion coefficientD for solute diffusion near eutectic concen-tration CE (Fig. 2-47), we assume a sym-metric phase diagram about CE for simplic-ity.

We define a relative (local) concentra-tion gap as

(2-135)

with DCa = CE – CSa(T ), DCb = CS

b(T ) – CE

and DC= DCa + DCb .The dimensionsless composition is then

defined as

(2-136)

and (in contrast to the previous definitions)we express lengths and times in units of adiffusion length l and time t

l = D/V , t = D/V 2 (2-137)

(Note that l here differs by a factor of 2from previous definitions.) Restricting ourattention to two dimensions correspondingto lamellar structures, we now define the

uC C

C T= E

E

−D ( )

d =DD

aCC



diffusion equation as

(2-138)

The conservation law at the interface z =z (x, t) is

– n · u = D (x, t) (1 + z·) nz (2-139)

with the unit vector n normal to the inter-face and (2-140)

The Gibbs–Thomson relation for theboundary becomes

(2-141)

(kinetic coefficients were considered byGeilikman and Temkin (1984). Finally, andthis is a new condition in comparison withthe simple directional solidification, thetriple point where three interfaces meetshould be in mechanical equilibrium

gaL + gbL + gab = 0 (2-142)

with the surface tension vectors g orientedso that each vector points out of the triplepoint and is tangent to the correspondinginterface. As before, K is the local curva-ture of the interface, being positive for asolid bulging into the liquid. The dimen-sionless capillary and thermal lengths asso-ciated with the a-phase are

(2-143)

(2-144)ll

m CGT

Ta

a D=

1˜

( )

dl

Tm C La

a

a aD= L E1

˜ ( )gg

u t

d Kl

d Kl

T

T

( , )

/

/z

z a l

z b l=

in -regions

in -regions

− −

− −

⎧

⎨⎪⎪

⎩⎪⎪

aa

bb

D( , )/

/x t

x

x=

for in -front regions

for in -front regions

d a ld b l−

⎧⎨⎩ 1

∂∂

∇ + ∂∂

ut

uz

u= 2

and are equivalent for the b phase, withma = |dT /dCL

a| as the absolute liquidusslope, GT as the fixed thermal gradient andLa as the specific latent heat. At infinity,z Æ •, the boundary condition to Eq. (2-138) is u•, depending on the initial concen-tration in the liquid. This then forms aclosed set of equations.

Ahead of the eutectic front, the diffusionfield can be thought of as containing twoingredients: a diffusion layer of thickness lassociated with global solute rejection andmodulations due to the periodic structureof the solid of the extent l (l l ). Whenthe amplitude of the front deformations issmall compared to these lengths, the aver-aging approximation by Jackson and Hunt(1966) (and also by Datye and Langer,1981) seems justified.

This point was taken up by Caroli et al.(1990), who found that the approximationis safe only in the limit of large thermalgradients

GT V m (DC)/D. (2-145)

This approximation appears difficult toreach experimentally, though.

In an attempt to shed some light onwavelength selection, Datye et al. (1981) andLanger (1980b) considered finite ampli-tude perturbations of the local wavelength(Fig. 2-48). This type of approach was usedin a somewhat refined version by Brener etal. (1987). They derived an approximatepotential function for wavelengths l andargued that under finite amplitude of noise,the wavelength selected on average is de-fined by a balance in the creation rates andthe annihilation rates of lamellae. In otherwords, if lamellae disappear through sup-pression by neighboring lamellae and ap-pear through nucleation, then an equal rateof these processes leads to a selection ofan average spacing l

–, because both depend

on l. The operating point was found in a

limited interval near the wavelength cor-responding to minimal supercooling (ormaximal velocity in an isothermal process)and, accordingly, is described by Eq. (2-134).

More recent extensions of the theory(Coullet et al., 1989) gave indications thatthe orientation of the lamellae (under iso-tropic material parameters) are necessarilyparallel to the growth direction of the frontbut may be tilted and travel sideways atsome specific angles (Caroli et al., 1990;Kassner and Misbah, 1991). Finally, it wasfound (Kassner and Misbah, 1991b) thatthe standard model of eutectic solidifica-tion has an intrinsic scaling structure

(2-146)

with a scaling function f depending only onl /lT so that l ~V –1/2 for 2 l lT or for highenough velocities, while at lower velocitiesthe exponent should be smaller: l ~V –0.3.This is in good agreement with the argu-ments given for ordinary directional solid-ification and also explains a lot of experi-mental data (Lesoult, 1980).

Numerical simulations of the full contin-uum model of eutectic growth were subse-quently carried out by Kassner and Misbah(1990, 1991a, b, c). They gave a fairlycomplete description of the steady-statestructures to be expected in the eutectic

l ~ ( / )l d f l lT

system, revealing the similarities in thestructure of solution space between eutec-tic and dendritic growth by showing theexistence of a discrete set of solutionsfor each parameter set (Kassner and Mis-bah, 1991b). Axisymmetric solutions wereshown to cease to exist beyond a certainwavelength by merging in a fold singular-ity. For larger wavelengths, extended tiltedstates appear. They were characterized indetail, both numerically (Kassner and Mis-bah, 1991c) and analytically (Misbah andTemkin, 1992; Valance et al., 1993). Thesupercritical nature of the tilt bifurcationwas demonstrated and a prediction wasmade as to how extended tilted domainscould be produced rather than the localizedstructures found so far. This prediction wasborne out by subsequent experiments bythe Faivre group (Faivre and Mergy,1992a). Amplitude equations were derivedthat explained how a supercritical bifurca-tion could give rise to localized tilted do-mains and predicted a dynamical wave-length selection mechanism (Caroli et al.,1992; Kassner and Misbah, 1992a) as wellas describing the influence of crystallineanisotropy (Kassner and Misbah, 1992a).Again, this was verified in detail experi-mentally (Faivre and Mergy, 1992b). Asimilarity equation was derived bothfor axisymmetric (Kassner and Misbah,


Figure 2-48. Turbulent behav-ior of lamellar spacings withtime in a simplified model ofeutectic growth (Datye et al.,1981).


1991b) and tilted states (Kassner and Mis-bah, 1992b), giving the corrections to theaforementioned scaling law. Anomalousasymmetric cells bearing resemblance toasymmetric cells in experiments on dilutealloys (Jamgotchian et al., 1993) werefound in the numerical solution of the eu-tectic growth problem and described ana-lytically (Kassner et al., 1993).

To conclude the discussion of steadystates, it was shown numerically that anumber of periodicity-increasing bifurca-tion exist, greatly enlarging the space ofpossible stationary solutions (Baumann etal., 1995a). These results strongly suggestthat the question of wavelength selectionshould be considered from a different pointof view. It is known that a band of steady-state solutions exists in directional solidifi-cation beyond the instability threshold anda similar statement holds for eutectic (eventhough there is no threshold in this case).Moreover, careful experiments (Faivre,1996) have clarified that the observedwavelength depends on the history of thesample. Therefore, there is no wavelengthselection in the strict mathematical sense.This is at least true in the absence of noise,i.e. for the deterministic equations of mo-tion. However, the investigations of Bau-mann et al. (1995a) render it plausible thata whole attractor of steady-state solutionswith accumulation points exists. One ofthese accumulation points is the point ofmarginal stability. Since the temporal evo-lution of any structure in the vicinity of thisattractor will become slow, it seems likelythat observed wavelengths will correspondto points of increased density of this attrac-tor. There have been objections (Karmaand Rappel, 1994) that many of the newlyfound steady-state solutions are on the low-wavelength side of the point of marginalstability, hence they are unstable and there-fore irrelevant for pattern formation. This

objection was countered (Baumann et al.,1995b) by pointing out that there is quite anumber of steady states on the high-wave-length side of this point and that the ques-tion of irrelevance for pattern formation isa question of time-scales. Since these be-come slow near the accumulation point, therelevance of a density of additional statescannot be easily discarded. It should bekept in mind that tilted lamellar states arealso unstable with respect to phase diffu-sion near the tilt bifurcation (Fauve et al.,1991). Nevertheless, they survive experi-mental time-scales, because phase diffu-sion is very slow.

The first dynamic simulation of the con-tinuum model of eutectic growth (in thequasi-stationary approximation) appears tohave been done by Kassner et al. (1995),but the only extensive study so far is due toKarma and Sarkissian (1996). This beauti-ful work was done in close collaborationwith experimentalists and thus validatedquantitively. It pinpointed a number of newshort-wavelength oscillatory instabilitiesand made quantitative predictions for theCBr4–C2Cl6 system allowing these in-stabilities to be reproduced in experiments(Faivre, 1996; Ginibre et al., 1997). There-fore, a pretty complete picture exists todayof the possible stationary and oscillatorypatterns in lamellar eutectic growth andtheir stability range.

A recent development is the stabilityanalysis of ternary eutectics by Plapp andKarma (1999), based on an extension of theDatye–Langer theory. It shows how thepresence of a third component in the sys-tem leads to the formation of eutectic colo-nies; an effective description of the eutecticfront on length scales much larger than thelamellar spacing is provided, which yieldsa simple means of calculating an approxi-mate stability spectrum.

2.6.2 Experimental Resultson Eutectic Growth

Eutectic growth was a subject of muchinterest to experimentalists in the late1960s and early 1970s. Substantial re-search has been motivated by the possibil-ity of developing new high-temperaturematerials. The in situ directional solidifica-tion of two phases of very different proper-ties is an interesting method of producingcomposite materials with exceptional prop-erties. However, since these materials couldnot outperform the more conventional, di-rectionally solidified dendritic superalloysin the harsh environment of a gas turbine,the interest dropped. Therefore, most of theresearch on eutectics was performed before1980 (for a review, see Kurz and Sahm,1975; Elliott, 1983). One exception is theongoing research concerning casting of eu-tectic alloys such as cast iron (Fe–C orFe–Fe3C eutectic) and Al–Si.

Casting alloys are generally inoculatedand solidify in equiaxed form (free eutecticgrowth) (see Flemings, 1991). This fact,however, does not make any substantialdifference to their growth behavior becausegrowth is solute diffusion controlled innearly all cases owing to the high concen-tration of the second element. The modelsdescribed above therefore apply to both di-rectional and free solidification.

The different alloys can be classifiedinto four groups of materials (Kurz andFisher, 1998): lamellar or fibrous systems,and non-faceted/non-faceted (nf/nf) ornon-faceted/faceted (nf/f) systems.

The distinction between f and nf growthbehavior can be made with the aid of themelting entropy. Small entropy differencesDSf between liquid and solid (typical formetals and plastic crystals such as SCN,PVA, etc.) lead to nf growth with atomi-cally rough interfaces. Materials with large

DSf values are prone to form atomicallysmooth interfaces, which lead to the forma-tion of macroscopically faceted appear-ance.

In the case of nf/nf eutectics, volumefractions (of one eutectic phase) of lessthan 0.3 lead generally to fibers, whileat volume fractions between 0.3 and 0.5,lamellar structures prevail. The micro-structures of nf/nf eutectics (often simplemetal/metal systems) are considered regu-lar and those of nf/f eutectics (mostly theabove-mentioned casting alloys) are con-sidered irregular. Fig. 2-49 shows schemat-ically the morphology of the growth frontin both cases. It can be easily understoodthat growth in nf/nf eutectics is much moreof a steady-state type than it is in f/nf eu-tectics.

Regular structures

Applying a criterion such as growth atthe extremum to the solution of the capil-


Figure 2-49. Solid/liquid interface morphologies of(a) regular (nf/nf) eutectics and of (b) irregular (nf/f)eutectics during growth.


lary-corrected diffusion equations (Jacksonand Hunt, 1966), Eq. (2-133), we obtain fornf/nf eutectics the well-known relation-ships (Eq. (2-134))

l2 V = C

DT 2/V = 4C ¢

where C and C ¢ are constants. Fig. 2-50shows that this behavior has been observedglobally in many eutectic systems, some ofthem having been studied over five ordersof magnitude in velocity. The situation ismuch less clear when it comes to analyzing

eutectoid systems. (Eutectoids are “eutec-tics” with the liquid parent phase replacedby a solid.) Often a l3V relationship isfound in these systems (Fig. 2-50b) oversome range of the variables (see Eq. (2-146)).

In general, it may be said that the field ofeutectic growth is under-represented in ma-terials research, and many more carefulstudies are needed before a clearer insightinto their growth can be gained. Trivediand coworkers began such research in theearly 1990s, and some of their results are

Figure 2-50. Lamellar spacings of (a) directionally solidified eutectics and (b) directionally transformed eutec-toids as a function of growth rate (Kurz and Sahm, 1975).

(a)

(b)

presented here. Fig. 2-51 indicates that ineutectics the spacings are also not at alluniquely defined. There is a rather widedistribution around a mean value for eachrate (Trivedi et al., 1991). The operatingrange of eutectics is determined by the per-

manent creation and movement of faults(see below). This process is three-dimen-sional and cannot be realistically simulatedin two-dimensional calculations.

In Fig. 2-52, the mean spacings are plot-ted as points, and the limits of the distribu-


Figure 2-51. Eutectic spacing distribution curves as a function of velocity for Pb–Pd eutectics (Trivedi et al.,1991).

Figure 2-52. A comparison of the ex-perimental results on the interlamellarspacing variation with velocity forCBr4–C2Cl6 with the theoretical values(solid lines) for two marginally stablespacings (Seetharaman and Trivedi,1988).


tion are given by the extension of thebars. From the calculated range of stability,which was discussed by Jackson and Hunt(1966), it can be seen that the minimumof the experimental values coincides withthe theoretical prediction (see also Sec.2.6.1). This, however, does not providedefinitive proof of this prediction, due tothe fact that several physical parametersof the system are not precisely known. Onthe other hand, it is clear that eutectic spac-ings do not explore the upper range ofstability (catastrophic breakdown), at leastnot in nf/nf systems. Some other mecha-nism limits the spacing at its upper bound.The adjustment of spacings is a rapid pro-cess, and its rate increases when the spac-ing increases (Seetharaman and Trivedi,1988).

Irregular structures

The above relationship for eutectic spac-ing and undercooling as a function ofgrowth rate are also useful in the case of ir-regular systems such as the nf/f casting al-loys Fe–C or Al–Si (Fig. 5-49b). Jonesand Kurz (1981) introduced a factor, j,which is equal to the ratio of the meanspacing, ·lÒ, of the irregular structure tothe spacing at the extremum. This leads tothe following relationships:

·lÒ2V = j2C

·DT Ò2/V = [j + (1/j)]2C ¢

Faults

Defects in the ideal lamellar or fibrousstructure are an essential ingredient of eu-

Figure 2-53. Eutectic fault struc-tures in directionally solidifiedAl–CuAl2 alloy (Double, 1973).

tectic growth. They allow the two-phasecrystal to rotate into crystallographically(energetically) favorable orientations (Ho-gan et al., 1971) and to adapt its spacing tothe local growth conditions. In lamellarsystems, there are different types of faults(Double, 1973): single or extended faults,with or without net mismatch (Fig. 2-53).They mostly represent sub-boundaries of aeutectic grain and could develop throughpolygonization of dislocations which formbecause of the stresses created at the inter-phase boundaries of the composite. Fig. 2-54 shows distributions of fault spacings, L,for different growth rates indicated by dif-ferent lamellar spacings, l (Riquet and Du-rand, 1975). In the case of non-faceted fi-brous structures, the faults are formed bysimple fiber branches or terminations.

Oscillations

Periodic oscillations have been observedas a morphological instability in severalsystems grown under various conditions(Hunt, 1987; Carpay, 1972; Zimmermannet al., 1990; Gill and Kurz, 1993, 1995).These morphological instabilities form inoff-eutectic alloys even at growth ratesof several cm/s, as is shown in Fig. 2-55.


Figure 2-54. Eutectic fault spacing distributioncurves for Al–CuAl2 directionally solidified withdifferent growth rates and spacing values as indi-cated (Riquet and Durand, 1975).

Figure 2-55. Periodic oscillations in hypoeutecticAl–CuAl2 eutectic under rapid laser resolidificationconditions: (a) experimental observation and (b) sim-ulation (Zimmermann et al., 1990; Karma, 1987).

(a)

(b)


There is good correspondence of the ob-served structures with the results of theo-retical modeling by Datye and Langer(1981) and by Karma (1987). In recent ex-periments a complete stability diagram forthe lamellar CBr4–C2Cl6 eutectic has beendetermined by Ginibre et al. (1997) (Fig.2-56a). It shows the observed structuresfor the variables dimensionless spacing(L) and concentration (h). In the limits1< L < 2 the basic state is stable, but overa decreasing composition range whenL = l /lmin is increased. Outside this re-gime, three different states and combina-tions of these have been found; tilt mode(T ), 1l oscillations (period-preserving os-cillatory or optical mode) and 2l oscilla-tions (period-doubling oscillatory mode,Fig. 2-56). These results are in good quan-titative agreement with numerical resultsby Karma and Sarkissian (1996).

As was the case with dendrites, eutecticsalso form extremely fine microstructureswhen they are subject to rapid solidifica-tion. A value of l = 15 nm (each phase issome 20 atoms wide) seems to be the mini-mum spacing that can be achieved withgrowth rates of the order of 0.2 m/s (Zim-mermann et al., 1989). In highly under-cooled alloys anomalous eutectic struc-tures have been found which, as with den-drites, seem to be the result of the largecapillary forces involved with such finescales (Goetzinger et al., 1998).

2.6.3 Other Topics

Peritectics

A wide spectrum of microstructures canbe found in peritectic alloys. Recently thishas produced a great deal of interest (Kerrand Kurz, 1996). In directional solidifica-tion, when the G/V ratio is high enough,i.e., in the range of the limit of constitu-tional undercooling of the phase with the

smaller distribution coefficient, more com-plex microstructures can form in the two-phase region. New phases may appear dur-ing transient or steady-state growth via nu-cleation at the solidification front. The ac-

Figure 2-56. Stability diagram of directionally so-lidified CBr4–C2Cl6 alloys. (a) Symbols: experimen-tally observed periodic patterns (Ginibre et al.,1997). Lam. Term.: lamellar termination. (b) Curves:numerically calculated diagram (Karma and Sarkis-sian, 1996).

(a)

(b)

tual microstructure selection process isthus controlled by nucleation of the phases,and by growth competition between the nu-cleated grains and the pre-existing phaseunder non-steady-state conditions. In thiscase nucleation in the constitutionallyundercooled zone ahead of the growthfront has to be taken into account in orderto determine the microstructure selection(Ha and Hunt, 1997; Hunziker et al., 1998).Further, new structures have been observedwhich are controlled by convection. Moredetailed discussions of this subject can befound in Karma et al. (1998), Park and Tri-vedi (1998), Trivedi et al. (1998), Van-dyoussefi et al. (2000), Mazumder et al.(2000).

Phase/microstructure selection

Phase and microstructure selection is ofutmost importance to applications of solid-ification theory as it controls the propertiesof materials. Nucleation and growth haveto be modeled in order to make predictionsof structures as a function of compositionand the solidification variables; growthrate and temperature gradient for direc-tional solidification, undercooling for freegrowth. The interested reader is referred to arecent overview by Boettinger et al. (2000)where these phenomena are discussed.

Micro/macro modeling

The last decade has experienced signifi-cant progress in numerical modeling ofcombined macroscopic and microscopicsolidification phenomena. Nucleation andgrowth models have been implemented in2D and 3D heat flow and fluid flow pro-grams providing a substantial improvementof our analytical tools for optimized andnew materials processes such as singlecrystal casting of superalloys for gas tur-

bines. The interested reader is referred toRappaz (1989), Rappaz et al. (2000), Wanget al. (1995), Wang and Beckermann (1996),Beckermann and Wang (1996), Boettingeret al. (2000).

2.7 Summary and Outlook

During the recent years, very substantialimprovements in our understanding of thepattern-forming processes in solidificationhave been achieved. Although the basicmodel equations have been known for sev-eral decades, it is only during recent yearsthat the mathematical and numerical toolswere extended to allow for a reliable analy-sis of the complicated expressions. In addi-tion, careful experiments have been per-formed, mostly on model substances, whichhave provided an impressive amount ofprecise, quantitative data. This combinedeffort has basically solved the problem offree dendritic growth with respect to veloc-ity selection and side branch formation.

In the process of directional solidifica-tion, a consistent picture is now emerging,relating the growth mode to free dendriticgrowth and, at the same time, to viscousfingering and growth in a channel. At veryhigh growth rates that approach the limitof absolute stability, the situation is stillsomewhat unclear, for non-equilibriumeffects like kinetic coefficients then be-come of central importance. These quan-tities are difficult to determine experimen-tally. Furthermore, the selection of the pri-mary spacings of the growing array of cellsand dendrites is still subject to discussion.One such point of contention, for example,is the typically observed increase in spac-ing when moving from cellular to dendriticstructures in model substances.

In eutectic growth, the situation is evenless understood, for good reason. The


2.8 References 165

three-phase junctions at the solid– liquidinterface enter as additional conditions andfurther details of the phase diagram be-come important. The dynamics of the sys-tem seem to show a richer structure thanordinary directional solidification. The se-lection of spacings between the differentsolid phases in materials of practical im-portance occurs through three-dimensionaldefect formation. In addition, nucleationand faceting of the interfaces should beconsidered.

A number of problems common to allof these growth modes have only beentouched upon so far. These problems in-clude, for example, the redistribution ofmaterial far behind the tip regions, thetreatment of elastic effects, and the interac-tion with hydrodynamic instabilities due tothermal and compositional gradients.

In summary, we expect the field to re-main very active in the future, as it is at-tractive from a technological point of view.It will certainly provide some surprises andnew insights into the general concepts ofpattern formation in dissipative systems.

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3 Diffusion Kinetics in Solids

Graeme E. Murch

Department of Mechanical Engineering, University of Newcastle, N.S.W. 2308, Australia

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 1733.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1753.2 Macroscopic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1753.2.1 Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1753.2.2 Types of Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . 1783.2.2.1 Tracer or Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1783.2.2.2 Impurity and Solute Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 1783.2.2.3 Chemical or Interdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1783.2.2.4 Intrinsic or Partial Diffusion Coefficients . . . . . . . . . . . . . . . . . . 1793.2.2.5 Surface Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 1803.2.3 Phenomenological Equations of Irreversible Thermodynamics . . . . . . . 1813.2.3.1 Tracer Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1823.2.3.2 Chemical Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1843.2.3.3 Einsteinian Expressions for the Phenomenological Coefficients . . . . . . 1853.2.3.4 Relating Phenomenological Coefficients to Tracer Diffusion Coefficients . 1853.2.4 Short-Circuit Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1863.3 Microscopic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1893.3.1 Random Walk Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1893.3.1.1 Mechanisms of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1893.3.1.2 The Einstein Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923.3.1.3 Tracer Correlation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 1933.3.1.4 Impurity Correlation Factor . . . . . . . . . . . . . . . . . . . . . . . . . 1963.3.1.5 Correlation Factors for Concentrated Alloy Systems . . . . . . . . . . . . 1993.3.1.6 Correlation Factors for Highly Defective Systems . . . . . . . . . . . . . 2013.3.1.7 The Physical or Conductivity Correlation Factor . . . . . . . . . . . . . . 2043.3.1.8 Correlation Functions (Collective Correlation Factors) . . . . . . . . . . . 2063.3.2 The Nernst–Einstein Equation and the Haven Ratio . . . . . . . . . . . . 2113.3.3 The Isotope Effect in Diffusion . . . . . . . . . . . . . . . . . . . . . . . 2133.3.4 The Jump Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2163.3.4.1 The Exchange Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 2173.3.4.2 Vacancy Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2183.4 Diffusion in Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2193.4.1 Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2203.4.1.1 Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2203.4.1.2 Impurity Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2203.4.2 Diffusion in Dilute Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . 221


3.4.2.1 Substitutional Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.4.2.2 Interstitial Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2243.4.3 Diffusion in Concentrated Binary Substitional Alloys . . . . . . . . . . . 2263.4.4 Diffusion in Ionic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 2273.4.4.1 Defects in Ionic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2273.4.4.2 Diffusion Theory in Ionic Crystals . . . . . . . . . . . . . . . . . . . . . 2283.5 Experimental Methods for Measuring Diffusion Coefficients . . . . . . 2313.5.1 Tracer Diffusion Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2313.5.2 Chemical Diffusion Methods . . . . . . . . . . . . . . . . . . . . . . . . 2323.5.3 Diffusion Coefficients by Indirect Methods . . . . . . . . . . . . . . . . . 2333.5.3.1 Relaxation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2333.5.3.2 Nuclear Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2343.5.4 Surface Diffusion Methods . . . . . . . . . . . . . . . . . . . . . . . . . 2343.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2353.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

172 3 Diffusion Kinetics in Solids



a lattice parameterb1, b2 solvent enhancement factorsB1, B2 solute enhancement factorsC, Ci concentration: particles per unit volume of species icA, cB, ci , cv, cp mole fractions of metal A, B, species i, vacancies, paired interstitialsD diffusion coefficient (in m2 s–1)D¢ short-circuit diffusion coefficientD collective diffusion coefficient or interdiffusion coefficientD* tracer diffusion coefficientDI

A, DIB, etc. intrinsic or partial diffusion coefficient of metal A, B, etc.

Dl lattice diffusion coefficientDs diffusion coefficient derived from the ionic conductivityD0 pre-exponential factore electronic chargeEv

f energy of vacancy formationEb, EB binding energy of impurity to dislocation (impurity-vacancy binding energy)f, fI( fc) tracer correlation factor, physical correlation factorFv

f Helmholtz free energy for vacancy formationg coordinationGm Gibbs free energy of migrationHv

f enthalpy of vacancy formationH m enthalpy of migrationHR Haven ratioJ flux of atomsk Boltzmann constantK equilibrium constantL phenomenological coefficientsl spacing between grain boundaries (average grain diameter)l distance between dislocation pinning pointsm massP–AV vacancy availability factor

Q activation energy for diffusionR vector displacementDRi total displacement of species ir, rq jump distance of atom, of charge qSv

f entropy of vacancy formationS m entropy of migrationt timeT temperatureTM.Pt. temperature of melting pointu mobility·vÒ average drift velocityV volume

x distance, coordinateXi , Xv driving force on species i, vacanciesZi number of charges on i

g activity coefficientG, Gq , Gi jump frequency (of charge q; of species i)m chemical potentialn vibration frequencys ionic conductivityw exchange frequency

CASCADE computer codeDEVIL computer codeerf Gaussian error functionPPM path probability methodSIMS secondary ion mass spectrometry


3.2 Macroscopic Diffusion 175

“The elementary diffusion process isso very fundamental and ubiquitousin the art and science of dealing withmatter in its condensed phase that itnever ceases to be useful but, at thesame time, is a problem which isnever really solved. It remains impor-tant by any measure.”

D. Lazarus, 1984

3.1 Introduction

Many phenomena in materials sciencedepend in some way on diffusion. Commonexamples are sintering, oxidation, creep,precipitation, solid-state chemical reac-tions, phase transformations, and crystalgrowth. Even thermodynamic propertiesand structure are sometimes dependent ondiffusion, or rather the lack of it. Many ofthese phenomena are the subjects of othercontributions in this series. This contribu-tion is concerned with the fundamentals ofthe diffusion process itself.

The depth of subject matter is generallyintroductory, and no prior knowledge ofsolid-state diffusion is assumed. Wherepossible, the reader is directed to more de-tailed texts, reviews, and data compila-tions.

3.2 Macroscopic Diffusion

3.2.1 Fick’s Laws

Although diffusion of atoms, or atomicmigration, is always occurring in solids attemperatures above absolute zero, for mac-roscopically measurable diffusion a gradi-ent of concentration is required. In thepresence of such a concentration gradient∂C/∂x (where C is the concentration in, say,particles per unit volume and x is the dis-

tance) in one direction of a certain speciesof atom, a flux J of atoms of the same spe-cies is established down the concentrationgradient. The law relating flux and concen-tration gradient is Fick’s First Law, which,for an isotropic medium or cubic crystalcan be expressed as

(3-1)

The proportionality factor or “coefficient”of ∂C/∂x is termed the “diffusion coeffi-cient” or less commonly the “diffusivity”.The recommended SI units for D are m2 s–1

but much of the literature is still in theolder c.g.s. units cm2 s–1. The negative signin Eq. (3-1) arises because the flux is in theopposite direction to the concentration gra-dient. This negative sign could, of course,have been absorbed into D, but it is moreconvenient for D to be a positive quantity.

When an external force such as anelectric field also acts on the system a moregeneral expression can be given:

(3-2)

where ·vÒ is the average velocity of thecenter of mass arising from the externalforce on the particles. The first term in Eq.(3-2) is thus the diffusive term, and the sec-ond term is the drift term. Note the inde-pendence of these terms. The external forcehere is assumed to be applied gradually sothat the system moves through a series ofequilibrium states. When the force is sud-denly applied the system can be thrown outof equilibrium. These matters and the typesof external force are considered by Flynn(1972) in a general consideration of diffu-sion under stress.

By itself, Fick’s First Law (Eq. (3-1)) isnot particularly useful for diffusion mea-surements in the solid state since it is virtu-

J DCx

C= − ∂∂

+ ⟨ ⟩v

J DCx

= − ∂∂

ally impossible to measure an atomic fluxunless steady state is reached. Further, be-cause solid-state diffusivities are generallysmall, the attainment of steady state in a macroscopic specimen can take a verylong time. In a few cases, where the solid-state diffusivity is high, for example car-bon diffusion in austenite (Smith, 1953),the steady-state flux and the concentrationgradient can be measured and the diffusiv-ity obtained directly from Eq. (3-1).

In order to produce a basis for measuringthe diffusion coefficient, Eq. (3-1) is usu-ally combined with the equation of con-tinuity:

(3-3)

to give Fick’s Second Law:

(3-4)

If the diffusion coefficient is independentof concentration and therefore position,then Eq. (3-4) reduces to

(3-5)

The second-order partial differential equa-tion Eq. (3-5) (or (3-4)) is sometiems calledthe “diffusion equation”. Eq. (3-2) can alsobe developed in the same way to give

(3-6)

and, if ·vÒ and D are independent of C, then

(3-7)

In order to obtain a solution to the diffu-sion equation, it is necessary to establishthe initial and boundary conditions. Once a

∂∂

∂∂

− ⟨ ⟩ ∂∂

Ct

DC

x

Cx

=2

2 v

∂∂

∂∂

∂∂

⎛⎝⎜

⎞⎠⎟

− ∂∂

⟨ ⟩Ct x

DCx x

C= ( )v

∂∂

∂∂

Ct

DC

x=

2

2

∂∂

∂∂

∂∂

⎛⎝⎜

⎞⎠⎟

Ct x

DCx

=

∂∂

− ∂∂

Jx

Ct

=

solution, C (x, t), of the diffusion equationhas been established, the diffusion coeffi-cient itself is obtained as a parameter byfitting the experimental C (x, t) to the ana-lytical C (x, t). In the following we shall fo-cus on some analytical solutions for somewell-known initial and boundary condi-tions used in experimental diffusion stud-ies.

Let us first examine some solutions forEq. (3-5). In a very common experimentalarrangement for “self” and impurity diffu-sion a very thin deposit of amount M of ra-dioactive isotope is deposited as a sand-wich layer between two identical samplesof “infinite” thickness. After diffusion for atime t the concentration is described by

(3-8)

which is illustrated in Fig. 3-1. If the de-posit is left as a surface layer rather than asandwich, C (x, t) is doubled.

In another experimental arrangement,the surface concentration Cs of the diffus-ing species is maintained constant for timet, perhaps by being exposed to an atmo-sphere of it. Again, the substrate is thick ormathematically infinite. The solution of thediffusion equation is

(3-9)

where C0 is the initial or background con-centration of the diffusing species in thesubstrate and erf is the Gaussian error func-tion defined by

(3-10)

This function is now frequently availableas a scientific library function on mostmodern computers. Accurate series expan-

erf = dz u uz2

0

2

p ∫ −exp ( )

C x t CC C

x Dt( , )

( / )−

−s

s= erf

02

C x tM

Dtx Dt( , ) exp ( / )=

242

p−



sions can be found in mathematical func-tion handbooks.

The total amount of diffusant S taken up(or, in fact, lost, depending on the relativevalues of Cs and C (x, t)) from the substrateis given by

S (t) = [2 /C0 – Cs)] /A (D t /p)1/2 (3-11)

where A is the surface area of the sample.A further common experimental ar-

rangement is the juxtaposition of two “infi-nite” samples, one of which has a uniformconcentration C0 and the other a concentra-tion C1 of the diffusing species. After timet the solution is

(3-12)

This gives a time evolution of the concen-tration profile as shown in Fig. 3-2.

On many occasions where this experi-mental arrangement is used, we are con-cerned with diffusion in a chemical compo-sition gradient. The relevant diffusion co-efficient (see Sec. 3.2.2.3) is frequently de-pendent on concentration, so Eq. (3-12) isthen inappropriate, and a solution to Eq.(3-4) where D = D (C ) must be sought. Awellknown technique for this is the graphi-cal integration method, usually called theBoltzmann–Matano analysis. The generalsolution for the concentration-dependent

C x t CC C

x Dt( , )

( / )]−

−−0

1 0

12

1 2= [ erf

diffusion coefficient can be expressed as(Boltzmann, 1894; Matano, 1933)

(3-13)

where

(3-14)

A very detailed description of the use ofthis analysis is given by Borg and Dienes(1988).

In many practical situations of interdif-fusion the relevant phase diagram traversedwill ensure that diphasic regions or newphases will appear. This does not imply in-cidentally that all equilibrium phases willthus appear. The Boltzmann–Matano anal-ysis can still be applied within single phaseregions of the concentration profile. Thegrowth of a new phase, provided it is dif-fusion controlled, can usually be describedby a parabolic time law. These matters aredealt with in detail by Philibert (1991).

The solutions of the diffusion equationgiven here are among the more commonlyencountered ones in solid-state diffusionstudies. Numerous others have been givenby Carslaw and Jaeger (1959) and Crank(1975).

C

C

x C1

0

0∫ d =

D Ct

xC

x CC C

C

( )′ − ∂∂

⎛⎝⎜

⎞⎠⎟ ′

′

∫= d12

1

Figure 3-1. Time evolution of Eq. (3-8). Figure 3-2. Time evolution of Eq. (3-12).

3.2.2 Types of Diffusion Coefficients

For the non-specialist, the meaning andsignificance of the various diffusion coeffi-cients used can be confusing because of in-consistent terminology in the literature. Anattempt will be made here to clarify the sit-uation as much as possible.

3.2.2.1 Tracer or Self-Diffusion

Consider a well-annealed sample of apure metal. Although atoms are diffusingabout in the sample at a rate depending ontemperature, from a macroscopic point ofview nothing appears to be happening. Inorder to observe diffusion macroscopicallywe must impose a concentration gradient.For the case of a pure metal, a radioactivetracer of the same metal is used. The result-ing diffusion coefficient is termed thetracer diffusion coefficient with the symbolD*. Because the tracer is chemically thesame as the host, this diffusion coefficientis also termed the self-diffusion coefficient,although sometimes this terminology is re-served for the tracer diffusion coefficientdivided by the tracer correlation factor, f(see Sec. 3.3.1.3). The same idea is readilyextended to alloys and compounds. How-ever, care must always be taken that achemical composition gradient is not un-wittingly imposed. For example, in order tomeasure the oxygen self-diffusion coeffi-cient in a nonstoichiometric compoundsuch as UO2+x , we can permit 18O (a stableisotope that can be probed later by nuclearanalysis or Secondary Ion Mass Spectrom-etry) to diffuse in from the gas phase, pro-vided that the partial pressure of oxygen isalready in chemical equilibrium with thecomposition of the sample. Alternatively, alayer of U18O2+x can be deposited on thesurface of a sample of UO2+x provided thatit has exactly the same chemical composi-tion as the substrate.

3.2.2.2 Impurity and Solute Diffusion

In order to measure the impurity diffu-sion coefficient, the tracer is now the im-purity and is different chemically from thehost. However, the concentration of impur-ity must be sufficiently low that there is nota chemical composition gradient. Strictly,of course, the tracer impurity should bepermitted to diffuse into the sample alreadycontaining the same concentration of im-purity. In practice, the concentration oftracer impurity is normally kept extremelylow, thereby making this step unnecessary.Because the impurity is always in stablesolid solution (unless implanted), is oftentermed the solute and the impurity diffu-sion coefficient is sometimes also termedthe solute diffusion coefficient at infinitedilution. However, the terminology solutediffusion coefficient is often “reserved” fordilute alloys, where, in addition, we oftenmeasure the solvent diffusion coefficient.In the context of those experiments bothsolute and solvent diffusion coefficientsfrequently depend on solute content (seeSec. 3.4.2). In all of these experiments, asin self-diffusion, the chemical compositionof the sample must remain essentially un-changed by the diffusion process, other-wise it is a chemical diffusion experiment.

3.2.2.3 Chemical or Interdiffusion

So far, we have discussed diffusion coef-ficients which are measured in the absenceof chemical composition gradients. Chemi-cal diffusion is the process where diffusiontakes place in the presence of a chemicalcomposition gradient. It is the diffusion co-efficients describing this process whichgenerate the greatest amount of confusion.It is helpful to look at several examples.

Consider first diffusion in a pseudo-one-component system. One example is the dif-fusion of an adsorbed monolayer onto a



clean section of surface. Another is the dif-fusion between two metal samples differ-ing only in their relative concentrations ofa highly mobile interstitial species such asH. A further example is diffusion betweentwo nonstoichiometric compounds, e.g.,Fe1–dO and Fe1–d¢O. In all of these casesonly one species of atom is involved in thediffusion process which brings the systemto a common composition (which is why itcan be considered a pseudo-one-compo-nent system). Often the process can be pic-tured as the interdiffusion of vacant sitesand atoms. The diffusion coefficient de-scribing this process is usually called thechemical diffusion coefficient, sometimesthe interdiffusion coefficient, and occasion-ally, the collective diffusion coefficient.Generally the symbol used is D. For thesepseudo-one-component systems, the pre-ferred name is chemical diffusion coeffi-cient. In general, the chemical diffusion co-efficient does not equal the self-diffusioncoefficient because of effects arising fromthe gradient of chemical composition, seeEq. (3-99).

Chemical diffusion in binary substitu-tional solid solutions is frequently calledinterdiffusion. In a typical case pure metalA is bonded to pure metal B and diffusionis permitted at high temperature. Althoughboth A and B atoms move, only one con-centration profile, say of A, is established(the profile from B contains no new infor-mation). The resulting diffusion coefficientwhich is extracted from the profile, by theBoltzmann–Matano analysis (e.g., see Sec.3.2.1) is termed the interdiffusion coeffi-cient and is given the symbol D. Not infre-quently, the diffusion coefficient for thissituation is loosely called the chemical dif-fusion coefficient or the mutual diffusioncoefficient. This single diffusion coeffi-cient is sufficient to describe the concentra-tion profile changes of the couple. Because

of its practical significance the interdiffu-sion coefficient is the one often quoted inmetal property data books (Brandes, 1983;Mehrer, 1990; Beke 1998, 1999). See alsothe following section for further discussionof D.

3.2.2.4 Intrinsic or Partial Diffusion Coefficients

In contrast to the pseudo-one-componentsystems described above where the dif-fusion rates of the atoms and vacant sitesare necessarily equal, in the substitutionalbinary alloy the individual diffusion ratesof A and B are not generally equal since thecorresponding self-diffusion coefficientsare not. In the interdiffusion experimentthis implies that there is a net flux of atomsacross any lattice plane normal to the diffu-sion direction. If the number of lattice sitesis conserved, each plane in the diffusion re-gion must then shift to compensate. Thisshift with respect to parts of the sampleoutside the diffusion zone, say the ends ofthe sample, is called the Kirkendall effect.This effect can be measured by observingthe migration of inert markers, usually fineinsoluble wires which have been incorporat-ed into the sample before the experiment.The assumption is that the wires follow themotion of the lattice in their vicinity.

The intrinsic diffusion coefficients of Aand B, DI

A and DIB, are defined with refer-

ence to the fluxes of A and B relative to thelocal lattice planes:

(3-15)

and

(3-16)

The diffusion coefficients are sometimesalso termed partial diffusion coefficients.

′ − ∂∂

J DCxB B

I B=

′ − ∂∂

J DCxA A

I A=

If v is the velocity (Kirkendall velocity)of the lattice plane measured with respectto parts of the sample outside the diffusionzone then the fluxes with respect to it are

JA = JA¢ + v CA (3-17)

and

JB = JB¢ + v CB (3-18)

where CA and CB are the concentrations atthe lattice planes. It should be noted thatthe interdiffusion coefficient is measuredin this frame of reference. Since ∂CA/∂x= –∂CB/∂x, we easily deduce that

(3-19)

and

(3-20)

where cA and cB are the mole fractions of A and B. The term in parentheses in Eq.(3-20) is the interdiffusion coefficient D :

D = cA DIB + cB DI

A (3-21)

The interdiffusion coefficient is seen to bea weighted average of the individual (in-trinsic) diffusion coefficients of A and B.With the aid of Eqs. (3-19) and (3-21) theintrinsic diffusion coefficients can be de-termined if D and v are known. The compo-sition they refer to is the composition at theinert marker. Note that if there is no markershift, Eq. (3-19) implies that the intrinsiccoefficents are equal. The relationship ofthe intrinsic diffusion coefficient to thetracer (self) diffusion coefficient is ex-plored in Sec. 3.2.3.2.

From the form of Eq. (3-21) whencAÆ 0, it can be seen that D Æ DI

A. Itshould also be noted that in this limit DI

A

reduces to the impurity diffusion coeffi-cient of A in B.

J c D c DCxA A B

IB A

I A= − + ∂∂

( )

v = AI

BI A( )D D

Cx

− ∂∂

3.2.2.5 Surface Diffusion Coefficients

Surface diffusion refers to the motion ofatoms, sometimes molecules, over the sur-face of some substrate. The diffusing spe-cies can be adsorbed atoms, e.g. impuritymetal atoms on a metal substrate (this isnormally referred to as hetero-diffusion) orof the same species as the substrate (this isusually referred to as self-diffusion).

As a field, surface diffusion has evolvedsomewhat separately from solid-state dif-fusion, perhaps because of the very differ-ent techniques employed. This separate-ness has resulted in some inconsistencies innomenclature between the two fields. It isappropriate here to discuss these briefly.For both self- and hetero-diffusion it isusual to refer to the motion of atoms forshort distances, where there is one type ofsite, as “intrinsic diffusion”. When the con-centration of diffusion species is very lowthe relevant diffusion coefficient is called a “tracer” diffusion coefficient. This is notin fact the diffusion coefficient obtainedwith radioactive tracers but a single parti-cle diffusion coefficient. (A single particlecan be followed or traced.) At higher con-centrations, the relevant diffusion coeffi-cient is called a chemical diffusion coeffi-cient. Thus “intrinsic” diffusion is formallythe same as diffusion in the pseudo-one-component system, as described in Sec.3.2.2.3.

When the surface motion of atoms ex-tends over long distances, and many typesof site are encountered (this is macroscopicdiffusion in contrast with the microscopicor “intrinsic” diffusion discussed above)the relevant diffusion coefficient is called a mass transfer diffusion coefficient. Fur-ther discussion of these and relationshipsamong these diffusion coefficients are re-viewed by Bonzel (1990).



3.2.3 Phenomenological Equationsof Irreversible Thermodynamics

An implication of Fick’s First Law (Eq.(3-1)) is that once the concentration gradi-ent for species i reaches zero, all net flowfor species i stops. Although frequentlycorrect, this is rather too restrictive as acondition for equilibrium. In general, netflow for species i can cease only when alldirect or indirect forces on species i arezero. This is conveniently handled by pos-tulating linear relations between each fluxand all the driving forces. We have for fluxi in a system with k components

(3-22)

where the Ls are called phenomenologicalcoefficients and Xi is the driving force oncomponent i and is written as – grad mi ,where mi is the chemical potential of spe-cies i. Xi can also result from an externaldriving force such as an electric field. Inthis case Xi = Zi e E, where Zi is the numberof changes on i, e is the electronic charge,and E is the electric field. For ionic con-ductors Zi is the actual ionic valence. Foralloys Zi is an “effective” valence which isusually designated by the symbol Zi*. Zi*consists of two parts, the first, Zi

el, repre-sents the direct electrostatic force on themoving ion (Zi

el is expected to be the ion’snominal valence), and the second, Zi

wd, accounts for the momentum transfer be-tween the electron current and the diffusingatom. A comprehensive review of all as-pects of Zi* and the area of electromigra-tion has been given by Huntington (1975).Xq is a driving force resulting from a tem-perature gradient (if present). Xq is givenby – T –1 grad T. When referring to diffu-sion in a temperature gradient it is usual tolet Liq be expressed as

L Q Liqk

n

k ik==1∑ *

J L X L Xik

ik k iq q= ∑ +

where Qk* is called the “heat of transport”for species k (Manning, 1968). This can be further related to the heat of transportderived from the actual heat flow. Thissubject is dealt with further by Manning(1968) and Philibert (1991). Sometimes itis convenient to discuss the flux in anelectric field occurring simultaneously withdiffusion in a chemical potential gradient.A simple example would be diffusion froma tracer source in an electric field – the so-called Chemla experiment (Chemla, 1956).In such a case, we write Xi as

Xi = – grad mi + Zi e E (3-23)

Now the phenomenological coefficientLiq (Eq. (3-22)) describes the phenomenonof thermal diffusion, i.e., the atom flow re-sulting from the action of the Xq , i.e., thetemperature gradient (Soret effect). Thereis also an equation analogous to Eq. (3-22)for the heat flow itself:

(3-24)

Lqk is a heat flow phenomenological coef-ficient, i.e., the heat flow which accompa-nies an atom flow (the Dufour effect). Lqq

refers directly to the thermal conductivity.The other phenomenological coeffi-

cients, Lik in Eq. (3-22), are concernedwith the atomic transport process itself.The off-diagonal coefficients are con-cerned with “interference” between the at-oms of different types. This may arise frominteractions between A and B atoms and/orthe competition of A and B atoms for thedefect responsible for diffusion, see Sec.3.3.1.8. A most important property of thephenomenological coefficients is that theyare independent of driving force. Further,the matrix of L coefficients is symmetric:this is sometimes called Onsager’s theoremor the reciprocity condition. That is,

Lij = Lji (3-25)

J L X L Xqk

qk k q q q= ∑ +

The set of equations represented by Eq.(3-22) are normally called the “phenomen-ological equations” or Onsager equationsand were postulated as a central part of thetheory of irreversible processes. Details ofthe theory as it applies to diffusion can befound in many places but see especially thereviews by Howard and Lidiard (1964),Adda and Philibert (1966), and Allnatt andLidiard (1993).

The phenomenological coefficients aresometimes said to have a “wider meaning”than quantities such as the diffusion coeffi-cients or the ionic conductivity. The widermeaning comes about because the pheno-menological coefficients do not depend ondriving force but only on temperature andcomposition. In principle, armed with a fullknowledge of the Ls, the technologistwould have the power, if not to control thediffusive behavior of the material, at leastto predict the diffusive behavior no matterwhat thermodynamic force or forces orcombination thereof were acting.

Unfortunately, the experimental determi-nation of the Ls is most difficult in the solidstate. This is in contrast, incidentally, toliquids, where, by use of selectively perme-able membranes, measurement of the Ls ispossible. The reader might well ask thenwhy the Ls were introduced in the firstplace when they are essentially not amen-able to measurement! There are severalreasons for this. First, relations can be de-rived between the Ls and the (measurable)tracer diffusion coefficients – we will dis-cuss this further in Sec. 3.2.3.4. Further,the Onsager equations provide a kind of ac-counting formalism wherein an analysisusing this formalism ensures that once thecomponents and driving forces have allbeen identified nothing is overlooked, andthat the whole is consistent; this will bediscussed in Secs. 3.2.3.1 and 3.2.3.2. Finally, many “correlations” in diffusion,

e.g., the tracer correlation factor and thevacancy-wind effect, can be convenientlyexpressed in terms of the Ls. This cansometimes aid in understanding the natureof these complex correlations. A most im-portant contribution to this was made byAllnatt (1982), who related the Ls directlyto the microscopic behavior – see Sec.3.3.1.8.

In the next sections we shall restrict our-selves to isothermal diffusion and focus onthe relations which can be derived amongthe Ls and various experimentally acces-sible transport quantities.

3.2.3.1 Tracer Diffusion

Let us first consider tracer diffusion in apure crystal. The general strategy is (1) todescribe the flux of the tracer with the On-sager equations (Eq. (3-22)) and (2) to de-scribe the flux with Fick’s First Law (Eq.(3-1)) and then equate the two fluxes tofind expressions between the diffusion co-efficient and the phenomenological coeffi-cients. We shall deal with this in detail toshow the typical procedure (see, for exam-ple, Le Claire (1975)). We can identify twocomponents, A and its tracer A*. We writefor the fluxes of the host and tracer atoms,respectively,

JA = LAA XA + LAA* XA* (3-26)

and

JA* = LA*A* XA* + LA*A XA (3-27)

Strictly, the vacancies should enter as acomponent but it is unnecessary here be-cause of the lack of a vacancy gradient.

Since the tracer atoms and host atomsmix ideally (they are chemically identical),it can easily be shown that the drivingforces are

(3-28)Xk Tc

cxA

A

A= − ∂∂



and

(3-29)

where ci is the mole fraction. This leadsimmediately to

(3-30)

It is convenient to introduce the quantity Ci (= N ci /V ) which is the number of atomsof type i per unit volume and N is the num-ber of entities (A + A*), and V is the vol-ume. Since there must be a symmetry con-dition ∂cA/∂x = – ∂ cA*/∂x, we now find that

(3-31)

However, Fick’s First Law, Eq. (3-1), statesthat

(3-32)

and so the tracer diffusion coefficient isgiven by

(3-33)

When cA* Æ 0, which is the usual situationwhen tracers are used experimentally, wefind that

DA* = kT V LA*A* /cA* N cA* Æ 0 (3-34)

Since the fluxes JA and JA* are alwaysequal but opposite in sign, then goingthrough the above procedure, but now forJA, leads to the relation

(3-35)

Now let us relate the diffusion coefficientto the ionic conductivity in order to obtain

L Lc

L Lc

AA A*A

A

AA* A*A*

A*=

+ +

Dk TV

NLc

LcA*

A*A*

A*

A*A

A= −⎛

⎝⎜⎞⎠⎟

J DC

xA* A*A*= − ∂

∂

JC

xk TV

NLc

LcA*

A* A*A

A

A*A*

A*=

∂∂

−⎛⎝⎜

⎞⎠⎟

J Lk Tc

cx

Lk Tc

cxA* A*A*

A*

A*A*A

A

A= − ∂∂

− ∂∂

Xk Tc

cxA*

A*

A*= − ∂∂

an expression for the Haven ratio HR (seealso Sec. 3.3.2).

We consider the system of A and A* inan electric field. We will assume that the A and A* are already mixed. We will nowlet XA = XA* = Z e E, where Z is the numberof charges carried by each atom. From Eq.(3-27) the flux of A* is

JA* = ZA* e E (LA*A* + LA*A) (3-36)

The drift mobility uA* is related to the fluxby

JA* = CA* uA* E (3-37)

and so

(3-38)

The mobility uA is equal to uA* since A andA* are chemically identical.

By means of the so-called Nernst–Ein-stein equation (Eq. (3-130)), the mobilitycan be converted to a dimensionally correctdiffusivity, Ds , i.e.,

(3-39)

As is discussed in detail in Sec. 3.3.2, Dsdoes not have a meaning in the sense ofFick’s First Law (Eq. (3-1)). Its meaning isthe diffusion coefficient of the assembly ofions as if the assembly itself acts like a sin-gle (hypothetical) particle. It is conven-tionally related to the tracer diffusion coef-ficient by the Haven ratio HR which is de-fined as:

(3-40)

Using the equation for DA* we find, afterletting cA* Æ 0, that

(3-41)HL

L LRA*A*

A*A* A*A=

+

HDDR

A*≡s

Dk T VN c

L Ls =A*

A*A* A*A( )+

uZ eV

NL L

cA*A* A*A* A*A

A*=

+⎛⎝⎜

⎞⎠⎟

For a pure crystal and where the vacancyconcentration is very low, HR can be iden-tified directly with the tracer correlationfactor f (see Sec. 3.3.2) and

(3-42)

Note that if LA*A= 0, i.e., there is no inter-ference between tracer and host, thenHR =1. More generally, when the defectconcentration is high and there are corre-lations in the ionic conductivity (see Sec.3.3.1.7), we find that

(3-43)

and fI is the physical correlation factor.

3.2.3.2 Chemical Diffusion

For chemical diffusion between A and Bwhen, say, the vacancy mechanism is oper-ating (see Sec. 3.3.1.1), the Onsager equa-tions are written as (Howard and Lidiard,1964)

JA = LAA XA + LAB XB (3-44)

JB = LBB XB + LBA XA (3-45)

Strictly, the vacancies should enter here asanother species, so that we would write

JA = LAA XA + LAB XB + LAV XV (3-46)

JB = LBB XB + LBA XA + LBV XV (3-47)

and, because of conservation of latticesites,

JV = – (JA + JB) (3-48)

The usual assumption is that the vacanciesare always at equilibrium and XV = 0. Forthis to happen the sources and sinks for va-cancies, i.e., the free surface, dislocations,or grain boundaries, must be effective andsufficiently numerous.

H f fL

L LR IA*A*

A*A* A*A= =/

+

H fL

L LRA*A*

A*A* A*A= =

+

Following the same kind of procedure asbefore (see Sec. 3.2.3.1), but recognizingthat generally A and B do not ideally mix,we find that rather than by Eq. (3-28), Xi isnow given by

(3-49)

where g is the activity coefficient of either Aor B. We find that the diffusion coefficientof, say A, actually an intrinsic diffusion co-efficient (see Sec. 3.2.2.4), is given by

(3-50)

If we want to find a relation between DIA

and the tracer diffusion coefficient, sayDA*, we would need to introduce A* for-mally into the phenomenological equationswhich would now become (Stark, 1976;Howard and Lidiard, 1964)

JA = LAA XA + LAA* XA* + LAB XB (3-51)

JA* = LA*A* XA* + LA*A XA + LA*B XB (3-52)

JB = LBB XB + LBA XA + LBA* XA* (3-53)

These equations can be developed to give

(3-54)

and

(3-55)

When these two equations are combined we find that

(3-56)D Dc

cL c

Lc

Lc

AI

A*

A*

A*A* A

A*A

A*

AB

B

= 1

1

+ ∂∂

⎛⎝⎜

⎞⎠⎟

× + −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

lnln

g

Dk TV

NL L

cL L

c

c

AI AA A*A

A

AB A*B

B=

+ − +⎛⎝⎜

⎞⎠⎟

× + ∂∂

⎛⎝⎜

⎞⎠⎟

lnln

1g

Dk TV

NLc

LcA*

A*A*

A*

A*A

A= −⎛

⎝⎜⎞⎠⎟

Dk T V

NLc

Lc cA

I AA

A

AB

B= −⎛

⎝⎜⎞⎠⎟

+ ∂∂

⎛⎝⎜

⎞⎠⎟

1lnln

g

Xk Tc

cx ci

i

i= − ∂∂

+ ∂∂

⎛⎝⎜

⎞⎠⎟

1lnln

g



This is in fact the so-called Darken equa-tion (Darken, 1948), relating the intrinsicdiffusion coefficient to the tracer diffusioncoefficient but with the addition of the termin brackets. This term is sometimes calledthe “vacancy-wind-term” and does notvary much from unity. Note that it resultsfrom non-zero cross terms LA*A and LAB.The approach from the phenomenologicalequations has told us only of the formal ex-istence of this term, but an evaluation of itrequires consideration of the detailed mi-croscopic processes which generate thecorrelations contained in the cross terms.This is considered in Sec. 3.3.1.8.

These examples suffice to show how thephenomenological equations are useful inpresenting a consistent and unified pictureof the diffusion process, no matter how com-plex. For further information on the subject,the reader is directed to the classic review byHoward and Lidiard (1964) and also manyother treatises, such as those by Adda andPhilibert (1966), Le Claire (1975), Stark(1976), Kirkaldy and Young (1987), Phili-bert (1991), and Allnatt and Lidiard (1993).

3.2.3.3 Einsteinian Expressions for the Phenomenological Coefficients

An important development in the area ofsolid-state diffusion was the fact that thephenomenological coefficients can be ex-pressed directly in terms of atomistic Ein-steinian formulae (Allnatt, 1982).

(3-57)

where DR i (t) is the total displacement ofspecies i in time t, V is the volume, and theDirac brackets denote a thermal average A.It is important to note that DR(i) is the sumof the displacements of the individual par-ticles of type i. In effect, DR(i) is the dis-

L V k T t

t t

ijV t

i j

= lim lim ( )

( ) ( )( ) ( )→ ∞ → ∞

−

× ⟨ ⋅ ⟩

6 1

D DR R

placement of the system of species i as ifthat system were a particle itself. The diag-onal phenomenological coefficients derivefrom correlations of the system of species iwith itself, i.e., self-correlations. This doesnot mean tracer correlations here: they arecorrelations in the random walk of an atom.The correlations here are in the randomwalk of the system “particle”. The off-diag-onal terms derive from interference of thesystem of species i with the system of spe-cies j. In effect it is mathematically equiva-lent to two systems i and j being treatedlike two interfering “particles”.

Eq. (3-57) has been very useful for cal-culating the Lij by means of computer sim-ulation – see the pioneering calculationsby Allnatt and Allnatt (1984). Much of thatmaterial has been reviewed by Murch andDyre (1989). A brief discussion is given inSec. 3.3.1.8.

3.2.3.4 Relating PhenomenologicalCoefficients to Tracer DiffusionCoefficients

For the case of multi-component alloys,Manning (1968, 1970, 1971) derived expres-sions relating the phenomenological coeffi-cients to tracer diffusion coefficients. The ex-pressions were developed for a particularmodel for concentrated alloys, the so-calledrandom alloy model. In this model the atomiccomponents are randomly distributed overthe available sites, the vacancy mechanism isassumed and the exchange frequencies of theatomic components with the vacancies de-pend only on the nature of the atomic compo-nents and not on their environment.

For a binary system the relations are

LN

k Tc D

ff

c D

c D c D

AA A A

A A

A A B B

= (3-58)*

*

* *

( )

( )× + −

+

⎡

⎣⎢⎢

⎤

⎦⎥⎥

11 0

0

(3-59)

where f0 is the tracer correlation factor inthe lattice of either pure component, seeSec. 3.3.1.3. The relation for LBB is ob-tained from LAA by interchanging B with A.

Lidiard (1986) showed that Eqs. (3-58)and (3-59) can in fact be obtained withoutrecourse to the random alloy model, on thebasis of two microscopic assumptionswhich, although intuitive in nature, are in-dependent of any microscopic model ex-cept for the inclusion of the tracer correla-tion factor in the pure lattice f0, which isdependent on mechanism and structure.Lidiard’s findings immediately suggestedthat Eqs. (3-58) and (3-59) are probably ap-propriate for a much wider range of alloysthan are reasonably represented by the ran-dom alloy model itself. Bocquet (1987)also found relations of the same form forthe random alloy when interstitial mecha-nisms are operating. With Monte Carlocomputer simulation, Zhang et al. (1989a)and Allnatt and Allnatt (1991) explored thevalidity of Eqs. (3-58) and (3-59) in thecontext of the interacting binary alloymodel described in Sec. 3.3.1.5. It wasfound that Eqs. (3-58) and (3-59) applyvery well except, perhaps surprisingly, atcompositions approaching impurity levels.The breakdown there is readily traced tothe fact that these equations have built intothem the requirement that w4 = w0 (see Sec.3.3.1.4 for the impurity frequency nota-tion). For many cases w3 also equals w0

and so this is equivalent to a condition ofno vacancy-impurity binding. This is notsurprising given that the equations origi-nated with the random alloy model wherethis requirement is automatically fulfilled.

Eqs. (3-58) and (3-59) have also beenderived for B1 and B2 ordered alloys (Be-lova and Murch, 1997).

LN

k Tf

fc D c D

c D c DAB

A A B B

A A B B

=* *

* *

( )

( )

1 0

0

−+

A most important consequence of Eqs.(3-58) and (3-59) is the Darken–Manningexpression which relates the interdiffusioncoefficient D, the tracer diffusion coeffi-cients DA* and DB* and the thermodynamicfactor (1 + ∂ lng /∂c):

(3-60)

where the vacancy wind factor S is given by

(3-60a)

Eq. (3-60) is not only appropriate forrandom alloys, but also for ordered alloys/intermetallic compounds with B1 and B2structures. Recent computer simulationshave also shown that Eq. (3-60) is a goodapproximation for ordered alloys/intermet-allic compounds with L12, D03, and A15structures (Murch and Belova, 1998).

Further discussion on relations betweenphenomenological coefficients and tracerdiffusion coefficients, and impurity diffu-sion coefficients, can be found in the re-views by Howard and Lidiard (1964), LeClaire (1975), and Allnatt and Lidiard(1987, 1993).

3.2.4 Short-Circuit Diffusion

It is generally recognized that the rate atwhich atoms migrate along grain boundar-ies and dislocations is higher than thatthrough the lattice. From an atomistic pointof view it is very difficult to discuss, in aprecise way, the diffusion events in suchcomplex and variable situations. Much ofthe understanding has been gained fromcomputer simulations using the MolecularDynamics and Lattice Statics methods; see, for example, Kwok et al. (1984) andMishin (1997). In this section we shall dis-

Sf c c D D

f c D c D c D c D= A B A* B*

A A* B B* B A* A B*1

1 02

0+ − −

+ +( ) ( )

( ) ( )

˜ ( )ln

D c D c Dc

S= B A* A B*+ + ∂∂

⎛⎝⎜

⎞⎠⎟

1g



cuss short-circuit diffusion from a pheno-menological point of view.

Fisher (1951) was probably the first tointroduce a distinct diffusion coefficientfor material migrating along the short-cir-cuit path. For grain boundaries, each one isconceived as a slab of width d for whichthe average diffusion coefficient is D¢. Fordislocations, each one is conceived as apipe of diameter 2a for which the averagediffusion coefficient is D¢.

For conventional diffusion experimentsit has been usual to distinguish three dis-tinct regimes, depending on the magnitudeof the lattice diffusion length ÷---

Dl t, whereDl is the lattice diffusion coefficient and t istime (Harrison, 1961).

Regime A: ÷---Dl t is much larger than the

spacing l between the short-circuit paths.For grain boundaries l is the average diam-eter of the grains and for dislocations l isthe distance between dislocation pinningpoints. Diffusion from adjacent short cir-cuits overlap extensively. This condition is met for small-grained materials or verylong diffusion times. Hart (1975) proposedthat an effective diffusion coefficient Deff

can be introduced which still satisfies solu-tions to Fick’s Second Law and can bewritten as

Deff = D¢ + (1 – ) Dl (3-61)

where is the fraction of all sites whichbelong to the short-circuit paths. Eq. (3-61)is usually called the Hart Equation. LeClaire (1975) has given a rough calculationusing Eq. (3-61). For typical dislocationdensities in metals, = 10–8. The disloca-tion contribution to a measured or effectiveD will then exceed 1% when D¢/D > 106.Because the activation energy for latticediffusion in metals is about 34 TM.Pt. ,where TM.Pt. is the melting temperature and the activation energy for short-circuitdiffusion is generally about half that of

volume diffusion, then this conditionD¢/D > 106 is usual for temperatures belowabout half the melting temperature. This iswhy experiments where lattice diffusiononly is of interest tend to be made at tem-peratures above about half the melting tem-perature.

Impurities are often bound to short-cir-cuit paths, in which case the Hart Equation(Eq. (3-60)) is written as

D¢ = D¢ exp (Eb /kT ) + (1 – ) Dl (3-62)

where Eb is the binding energy of the im-purity and the diffusion coefficients referto impurities. The Hart Equation can beshown to be fairly accurately followedwhen l /÷---

Dl t ≤ 0.3 and also for l /÷---Dl t ≥100

(this corresponds to Regime C – see thenext paragraph) (Murch and Rothman,1985; Gupta et al., 1978).

Regime C: It is convenient to discuss theother limit out of sequence. When ÷---

Dl t ismuch smaller than the distances l betweenthe short-circuit paths, we have Regime Ckinetics. In this instance, all materialcomes down the short-circuit paths and themeasured diffusion coefficient is given en-tirely by D¢.

Regime B: In this intermediate case, it isassumed that ÷---

Dl t is comparable to l sothat material which is transported down ashort-circuit path and which diffuses outinto the lattice is unlikely to reach anothershort-circuit path.

There are various solutions of Fick’sSecond Law available to cope with tracerdiffusion in the presence of short-circuitpaths, but space prevents us from dealingwith these in any detail: they are discussedat length by Kaur, Mishin and Gust (1995).Suzuoka (1961) and Le Claire (1963) havegiven a solution for the grain boundaryproblem for the usual case when there is afinite amount of tracer originally at the sur-face, see Eq. (3-8). It is found, among other

things, that

(3-63)

Hence the product D¢d can be found by de-termining the slope of the linear region(penetrations reached by grain boundarydiffusion) in a plot of lnC versus x 6/5 (notthe usual x 2) and also with a knowledge ofDl itself. It can be difficult, however, to ob-tain the two diffusion coefficients in oneexperiment; the practicalities of this and alternatives are discussed by Rothman(1984). An example of a tracer penetrationplot with a clear contribution from grainboundary diffusion (x 6/5 dependence) isshown in Fig. 3-3.

Le Claire and Rabinovitch (1984) haveaddressed the dislocation pipe problem

dd

=ln

( ) ./

/ /C

x

Dt

Dl6 5

5 3 1 214

0 66⎛⎝

⎞⎠

⎛⎝

⎞⎠ ′ −d

and provided near-exact solutions to Fick’sSecond Law for both an isolated disloca-tion pipe and arrays of dislocation pipes.The profiles generally are not unlike thegrain boundary ones except that a linear re-gion in a plot of lnC versus x is now found,i.e.,

(3-64)

where A is a slowly varying function ofa/(D t)1/2. An example of such a tracer pen-etration plot with a clear contribution fromdislocation pipe diffusion (x dependence)is shown in Fig. 3-4.

For further details on grain boundary diffusion we refer to reviews by Peterson(1983), Mohan Rao and Ranganathan(1984), Balluffi (1984), Philibert (1991),

dd

=ln ( )

[( / ) ] /C

xA a

D D al−

′ − 1 2 1 2


Figure 3-3. Tracer concentrationprofiles of 203Pb into polycrystal-line Pb showing a contributionfrom grain boundaries (Gupta andOberschmidt, 1984).

3.3 Microscopic Diffusion 189

Kaur et al. (1995), Mishin et al. (1997). Fordislocation pipe diffusion we refer to LeClaire and Rabinovitch (1984) and Phili-bert (1991).

3.3 Microscopic Diffusion

3.3.1 Random Walk Theory

We have already seen in Sec. 3.2 that thephenomenological or continuum theory ofdiffusion is set up in terms of quantitiessuch as the diffusion coefficients, the ionicconductivities, or more generally, the phen-omenological coefficients, Lij . For the ma-terials technologist this theory provides aperfectly suitable formalism to describe themacroscopic features arising from the dif-fusion of atoms in solids. By its very na-ture, this macroscopic theory makes no ref-

erence to the discrete atomic events whichgive rise to macroscopically observablediffusion. To describe the atomic events, awholly separate theory, termed “RandomWalk Theory” has been formulated. Thistheory is based on the premise that macro-scopic diffusion is the net result of manyindividual atomic jumps. The theory at-tempts to relate the quantities such as thediffusion coefficients, the ionic conductiv-ities, or, more generally, the Lij , in terms oflattice and atomic characteristics, notablyjump frequencies. Although originallyquite precise in use (Howard, 1966), theterm “Random Walk Theory” is now usedrather loosely to describe any mathematicalapproach that focuses on the sequence ofjumps of atoms in the solid state. Inextri-cably linked with this theory is the exis-tence of correlated random walks of atoms,in other words, walks where there is an ap-parent memory between jump directions.Much of random walk theory is concernedwith describing such correlations.

3.3.1.1 Mechanisms of Diffusion

As a prelude to a discussion of randomwalk theory, in this section we will brieflydiscuss the common mechanisms of solid-state diffusion.

Interstitial Mechanism

In the interstitial mechanism, see Fig. 3-5, sometimes called the direct intersti-tial mechanism, the atoms jump from oneinterstice to another without directly in-volving the remainder of the lattice. Sincethe interstitial atom does not need to “wait”to be neighboring to a defect in order tojump (in a sense it is always next to a va-cancy), diffusion coefficients for atoms mi-grating with this mechanism tend to befairly high. As would be expected, atoms

Figure 3-4. Tracer concentration profiles of 22Nainto single crystal NaCl showing a contribution fromdislocations (Ho, 1982).

such as H, N, O, and C diffuse in this wayin metals. It should also be noted that with-out the defect being required to affect thejump, no defect concentration term andtherefore defect formation energy entersthe activation energy for diffusion; see Sec.3.3.4.

Interstitialcy Mechanism

In the interstitialcy mechanism, see Fig.3-6, sometimes called the indirect intersti-tial mechanism, two atoms, one an intersti-tial and the other an atom on a regular lat-tice site, move in unison. The interstitialatom moves to a regular site, whereas theregular site atom moves to an interstice.Collinear and non-collinear versions arepossible depending on constraints imposedby the lattice.

Because the interstitial formation energyis generally very high in metals, the equi-librium concentration of interstitials is very

small and their contribution to diffusion isunimportant in most cases. However, forplastically deformed or irradiated metalsthe concentration of interstitials (besidesvacancies) can be appreciable. It should benoted that the interstitial thus formed is notlocated on an interstice, but in a dumbbellsplit configuration. The migration possibil-ities of dumbbells have been discussed bySchilling (1978).

As a result of measurements of theHaven ratio (see Sec. 3.3.2), the interstis-tialcy mechanism appears to be highlylikely for silver diffusion in AgBr (Friauf,1957). Note that with the interstitialcymechanism, say the collinear version, atracer atom moves a distance r whereas thecharge apparently moves a distance 2r.This needs to be taken into account in theinterpretation of the Haven ratio. The inter-stitialcy mechanism probably occurs rela-tively frequently in ionic materials, espe-cially those with open lattices such as theanion lattice in the fluorite structure orthose which are highly defective. Nonethe-less, there is often so much attendant localrelaxation around the interstitial that farmore complex quasi-interstitialcy mecha-nisms could well operate. An example isoxygen diffusion in UO2+x , where the ex-cess oxygen is located as di-interstitialswith relaxation of two oxygen atoms fromregular sites to form two new interstitialsand two new vacancies. The actual mecha-nism of oxygen transport is not known but a quasi-interstitialcy mechanism is un-doubtedly responsible (Murch and Catlow,1987).

We should also mention that the interstit-ialcy mechanism appears to be very impor-tant in self-diffusion in silicon and possiblycertain substitutional solutes in silicon arealso transported via this mechanism (Franket al., 1984).


Figure 3-5. Interstitial diffusion mechanism.

Figure 3-6. Interstitialcy diffusion mechanism.


Vacancy Mechanism

The most important diffusion mecha-nism of all is undoubtedly the vacancymechanism, shown in Fig. 3-7. A substitu-tional solute atom or an atom of the solventdiffuses simply by exchanging places withthe vacancy. There can be attractive or re-pulsive interactions of the solute with thevacancy, which can have a profound in-fluence on the diffusion coefficient of thesolute and to a lesser extent the solvent by way of correlation effects. This is dis-cussed in Secs. 3.3.1.4 and 3.4.2. The va-cancy mechanism is definitely the pre-ferred mechanism for metals and alloys forboth host and substitutional solutes. Inmost other materials the vacancy appearsto play the major diffusion role exceptwhen the concentration of interstitials pro-duced by nonstoichiometry or doping (inoxides) or irradiation (in metals) is so highthat contributions from interstitialcy orsimilar mechanisms become dominant.

Aggregates of vacancies such as thedivacancy or trivacancy can also contributeto diffusion. These appear to be importantat high temperatures in metals where theircontribution has been largely inferred fromcurvatures of the Arrhenius plot (logD vs.1/T ); see, for example, the f.c.c. metals(Peterson, 1978) (see Sec. 3.4.1.1). Boundvacancy pairs, i.e., a cation vacancy boundto an anion vacancy, are important contrib-

utors to tracer diffusion in alkali halides (Bénière et al., 1976).

Interstitial-Substitutional Mechanism

On occasion, solute atoms may dissolveinterstitially and substitutionally. These solute atoms may diffuse by way of the dissociative mechanism (Frank and Turn-bull, 1956) and/or the kickout mechanism(Gösele et al., 1980). In both mechanismsthe interstitial solute diffuses rapidly by theinterstitial mechanism. In the dissociativemechanism the interstitial combines with avacancy to form a substitutional solute. Ata later time this substitutional can dissoci-ate to form a vacancy and an interstitial so-lute (really a Frenkel defect). The anoma-lously fast diffusion of certain solutes, e.g.,Cu, Ag, Au, Ni, Zr, and Pd in Pb, appearsto have a contribution from the dissocia-tive mechanism (Warburton and Turnbull,1975; Bocquet et al., 1996). In the kickoutmechanism, on the other hand, the soluteinterstitial uses the interstitialcy mecha-nism to involve the regular site lattice. Inthe process a host interstitial is formed andthe interstitial solute then occupies a sub-stitutional site. This process can be re-versed at a later stage. The kickout mecha-nism appears to operate for rapid diffusionof certain foreign atoms, such as Au in Si (Frank et al., 1984).

Exchange Mechanism

The exchange mechanism, in which twoneighboring atoms exchange places, has inthe past been postulated as a possible diffu-sion mechanism. The existence of the Kir-kendall effect in many alloy interdiffusionexperiments (this implies that the respec-tive intrinsic diffusion coefficients are un-equal, which is not possible with the ex-change mechanism, see Sec. 3.2.2.4) andthe very high theoretical activation ener-Figure 3-7. Vacancy mechanism.

gies in close-packed solids suggest that thismechanism is unlikely. It may, however,occur in liquids and in quenched-liquidssuch as amorphous alloys (Jain and Gupta,1994). Ring versions of the exchangemechanism certainly have lower theoreti-cal activation energies but require substan-tial cooperation among the atoms, whichseems unlikely.

Surface Diffusion Mechanisms

A number of mechanisms for surfacediffusion on metals (by far the most studiedclass of material) have been postulated.They include activated hopping of ad-sorbed atoms from one surface site to an-other, where the jump distance is simplythe distance between sites. Similarly, a va-cancy in a terrace, i.e., surface vacancy,can also move in the same sort of waywithin the terrace. At low temperatures andrough surfaces, exchange between an ad-sorbed atom and an atom in the substrate ispredominant. At high temperatures non-lo-calized diffusion and surface melting arealso possible. The mechanisms for surfacediffusion are discussed further in the bookedited by Vu Thien Binh (1983) and in thereviews by Rhead (1989) and Bonzel(1990).

3.3.1.2 The Einstein Equation

Let us consider atoms (of one chemicaltype) diffusing in their concentration gradi-ent in the x direction. It can readily beshown (see, for example, Adda and Phili-bert, 1966; Manning, 1968; Le Claire,1975) that the net flux of atoms across agiven plane x0 is given by

(3-65)

J C xt

Cx t

C xCx C t

x = ( )

( )

0

2

0

2

2

2

⟨ ⟩ − ∂∂

⟨ ⟩

− ∂∂

∂∂

⟨ ⟩

X X

X

plus higher order terms. In Eq. (3-65), C (x0)is the concentration of the diffusing atomsat x0, ·XÒ is the mean displacement or drift,and ·X2Ò is the mean squared displacement.The Dirac brackets · Ò denote an averageover all possible paths taken in time t.

In a situation where diffusion propertiesdo not depend on position, e.g., diffusionof tracer atoms in a chemically homogene-ous system, the third term in Eq. (3-65) iszero. The term containing the drift ·XÒ isalso zero because in a chemically homoge-neous system the probability that, say, atracer atom migrates some distance +X intime t (starting from x0) equals the prob-ability that an atom migrates a distance –X.Similarly, other odd moments are zero.Provided that other higher-order terms canbe neglected, Eq. (3-65) reduces to

(3-66)

By comparison with Fick’s First Law, Eq.(3-1), Eq. (3-66) immediately gives

(3-67)

Eq. (3-67) is called the Einstein Equation,probably the single most important equa-tion in the theory of diffusion kinetics. Thesuperscript * indicates that the diffusioncoefficient refers to tracer atom diffusionin a chemically homogeneous system.Equations for Dy* and Dz* have the sameform as Eq. (3-67). For three-dimensionalisotropic crystals the tracer diffusion coef-ficient is the same in every direction and

(3-68)

where R is the vector displacement of anatom in time t. For two-dimensional situa-tions the factor 6 in Eq. (3-68) is replacedby 4.

Dt

* =⟨ ⟩R2

6

Dtx* =

⟨ ⟩X2

2

JCx tx =

∂∂

⟨ ⟩X2

2



3.3.1.3 Tracer Correlation Factor

Either Eq. (3-67) or (3-68) can providethe starting point for a discussion of corre-lation effects in the diffusion walk. Forconvenience we shall use Eq. (3-68). Let usconsider the atomistic meaning of the dis-placement R. It is simply the sum of n suc-cessive jump vectors, r1, r2, r3, …, rn

(3-69)

The squared displacement R2 is then simply

(3-70)

The average of the squared displacement·R2Ò equals of course the sum of the aver-ages and we have

(3-71)

For a complete random walk, i.e., whereeach direction is unrelated to the previousone, the second term in Eq. (3-71) is zerobecause for any product ri · ri+ j there willalways be another of opposite sign. Wenow have

(3-72)

Because in most cases atoms require theassistance of point defects in order to moveabout (see Sec. 3.3.1.1), there is generallyan unavoidable memory or correlation ef-fect between jump directions. In order toappreciate this, let us focus on Fig. 3-8awhere the vacancy mechanism is shown. Inthis figure one atom symbol is shown filledto indicate that it is a tracer and can be fol-lowed. Let us assume that the tracer and thevacancy have just exchanged places. Be-cause the vacancy is still neighboring to thetracer the next jump of the tracer is quitelikely to cancel out the previous jump. The

⟨ ⟩ ⟨ ⟩∑R r2

1

2==i

n

i

⟨ ⟩ ⟨ ⟩ + ⟨ ⋅ ⟩∑ ∑ ∑− −

+R r r r2

1

2

1

1

12=

= = =i

n

ii

n

j

n i

i i j

R r r r2

1

2

1

1

12=

= = =i

n

ii

n

j

n i

i i j∑ ∑ ∑+ ⋅− −

+

R r==i

n

i1

∑

probability of doing this is in fact exactly2/g where g is the lattice coordination (Kelly and Sholl, 1987). On the other hand,the tracer has a reduced probability of con-tinuing in the direction of the first jump,since this requires the vacancy to migrateto point A. In other words, there is a mem-ory or correlation between directions oftracer jumps. In no way does this implythat the vacancy somehow favors the traceratom. In fact, in this example atoms sur-rounding the vacancy jump randomly withthe vacancy so that the vacancy itselfmoves on an uncorrelated random walkwith no memory or correlation whatsoeverbetween its jump directions. It should benoted that weaker correlations in the tracerjumps also come about as long as the va-

Figure 3-8. (a) Correlation effects arising from thevacancy mechanism – see text. (b) Correlation ef-fects arising from the interstitialcy mechanism (seetext).

cancy remains in the vicinity of the tracerand can approach the tracer from a non-random direction.

Let us consider Fig. 3-8b, where the col-linear interstitialcy mechanism is assumed.Let us assume that the tracer is in the inter-stitial position. The first jump (a pair of atoms move) will take the tracer immedi-ately to a regular lattice site and a newatom, formerly at a regular lattice site,takes up a position as the interstitial. Thepair of atoms which next moves possiblyagain involves the tracer atom and in doingso will cancel out the previous tracer jump.Like the vacancy mechanism, the memoryor correlation between successive traceratom jump directions comes about purelybecause of the proximity of the defect. Un-like the vacancy diffusion mechanism,however, in this case the correlation comesabout only for consecutive pairs of tracerjumps of the type interstitial site Æ regularlattice site Æ interstitial site.

In the presence of correlation betweenjump directions, the sum of the dot prod-ucts ·ri · ri+ jÒ no longer averages out tozero. A convenient way of expressing thesecorrelations quantitatively is to form theratio of the actual ·R2Ò to the ·R2Ò result-ing from a complete random walk, i.e.,·R2Òrandom:

(3-73)

where the quantity f is called the tracer cor-relation factor or simply the correlationfactor. Sometimes the terminology “corre-lation coefficient” is used, but this is to bediscouraged. With Eqs. (3-71) and (3-72)we have

(3-74)fn

i

n

j

n i

i i j

i

n

i

= = =

=

lim→∞

− −

+

+⟨ ⋅ ⟩

⟨ ⟩

∑ ∑

∑1

21

1

1

1

2

r r

r

f =random

⟨ ⟩⟨ ⟩

RR

2

2

It should be noted that the limit n Æ • isapplied to Eq. (3-74) to ensure that all pos-sible correlations are included.

There have been numerous publicationsconcerned with the calculation of the tracercorrelation factor. The earlier work has beenreviewed in detail by Le Claire (1970),more recent work has been covered by All-natt and Lidiard (1993). Table 3-1 givessome values of f for various mechanismsand lattices. It should be noted from Table3-1 that the correlation factor for the inter-stitial mechanism is unity. For this mecha-nism the interstitials, which are consideredto be present at a vanishingly small concen-tration, move on an uncorrelated randomwalk, very much like the vacancy in the va-cancy-assisted diffusion mechanism. Accord-ingly, the second term in Eq. (3-71) drops outand f = 1. If the interstitial concentration isincreased, the interstitials impede one an-other and correlations are introduced; as aresult, the tracer correlation factor de-creases from unity. In fact, it continues todecrease until only one interstice is left va-cant. The situation now is identical to thatfor vacancy-assisted diffusion. The varia-tion of f with vacancy (or vacant interstice)concentration in the f.c.c. lattice is shownin Fig. 3-9. Of course, in the unphysical sit-uation where the interstitials do not “see”one another, i.e., multiple occupancy of asite is permitted, the interstitials continueto move on a complete random walk at allconcentrations, and f always equals unity.

For cubic lattices all the jumps are of thesame length. Then we have that | ri | = r :

(3-75)

Then we have, with Eqs. (3-68), (3-71),and (3-75)

(3-76)Dn r f

tr f

* = =2 2

6 6G

i

n

i nr=

=1

2 2∑ ⟨ ⟩r



where G is the jump frequency. The jumpfrequency is further discussed in Sec. 3.3.4.This partitioning of the diffusion coeffi-cient into its uncorrelated (G r2/6) and cor-related ( f ) parts is very basic to randomwalk theory. Other partitionings are cer-tainly possible, e.g., only jumps which arenot immediately cancelled contribute to thejump frequency, i.e., an “effective jump”frequency, but the partitioning here seemsto be the most natural.

The tracer correlation factor itself can beexpressed as

(3-77)

This equation has been very useful for di-rect computer simulation calculations of f;see the review by Murch (1984a).

We see that f normally acts to decreasethe tracer diffusion coefficient from its ran-dom walk ( f =1) value. The inclusion of fin the expression for D* is necessary for acomplete description of the atomic diffu-sion process. From Table 3-1, however, itcan be seen that for many 3D lattices freally only decreases D* by some 20–30%. This is not much more than the preci-sion routinely obtainable in measurementsof the tracer diffusion coefficient (Roth-man, 1984) and, given the difficulty in cal-culating the jump frequency G (see Sec.3.3.4), may not appear to be particularlysignificant. There are, however, many rea-sons why a study of correlation effects indiffusion is sufficiently important that ithas consumed the energies of many re-searchers over almost a 40-year period.

fR

nrn= lim

→∞

⟨ ⟩2

2

Table 3-1. Some correlation factors (at infinitely low defect concentrations) from Le Claire (1970), Manning(1968), and Murch (1982d).

Lattice Mechanism f

Honeycomb Vacancy 1/3Square planar Vacancy 1/(p – 1)

Triangular Vacancy 0.56006Diamond Vacancy 1/2

B.c.c. Vacancy 0.72714Simple cubic Vacancy 0.65311

F.c.c. Vacancy 0.78146F.c.c. Divacancy 0.4579 ± 0.0005

All lattices Interstitial 1NaCl structure Collinear interstitialcy 2/3

CaF2 structure (F) Non-collinear interstitialcy 0.9855CaF2 structure (Ca) Collinear interstitialcy 4/5CaF2 structure (Ca) Non-collinear interstitialcy 1

Figure 3-9. Tracer correlation factor vs. vacancyconcentration for non-interacting vacancies in thef.c.c. lattice; after Murch (1975).

First, as can be seen from Table 3-1, thetracer correlation factor is quite sensitive tothe mechanism of diffusion operating. Al-though f, by itself, is not measurable, mea-surements of the Haven ratio (see Sec.3.3.2) and the isotope effect (see Sec. 3.3.3)(which are closely related to f ) in favorablecases can throw considerable light on themechanism(s) of diffusion that are operat-ing. Identification of the diffusion mecha-nism is surely the most important ingredi-ent in understanding the way atoms mi-grate in solids and how it can be controlled.

Much of our discussion so far has beenconcerned with pure solids with few de-fects. There are, however, a large numberof solids where the apparent defect concen-tration can be fairly high, e.g., highly non-stoichiometric compounds, fast ion con-ductors, and certain concentrated intersti-tial solid solutions. In such cases the de-fects interact and correlation effects tend tobe magnified and become highly tempera-ture dependent. Then the apparent activa-tion energy for tracer diffusion includes animportant contribution directly from thecorrelation factor. Clearly, an understand-ing of f is a very important part of theunderstanding of the diffusion process insuch materials. We shall deal further withthis subject in Sec. 3.3.1.6. In ordered binary alloys, the correlation factor can become very small since atoms whichmake a jump from the “right” lattice to the“wrong” lattice tend to reverse, i.e., cancelthat jump. As a result, the tracer diffusioncoefficient can become abnormally small,largely because of this strong memory ef-fect. We shall deal further with this subjectin Sec. 3.3.1.5.

Another important case of correlation ef-fects is impurity diffusion. The correlationfactor for the impurity is highly dependenton the relative jump frequencies of the im-purity and host atoms in the vicinity of the

impurity. We shall deal with this subject inthe following section.

3.3.1.4 Impurity Correlation Factor

We first consider that the impurity con-centration is sufficiently dilute that there isnot a composition-dependent impurity dif-fusion coefficient. Accordingly, each im-purity atom is considered to diffuse in purehost. The jump frequency of the impurity isgiven by w2 whereas the jump frequency ofthe host atoms is given by w0. The signifi-cance of the subscripts will be apparentlater. The vacancy mechanism is assumed.

The “tracer” now is the impurity. The result for the impurity correlation factor f2

in the f.c.c. lattice as a function of w2 /w0

is shown in Fig. 3-10. When w2 < w0 (uppercurve), the impurity motion becomes considerably decorrelated since, after animpurity/vacancy exchange, the vacancy does not trend to remain in the vicinity ofthe impurity. When the impurity is next


Figure 3-10. Impurity correlation factor in the f.c.c.lattice as a function of w2/w0: points by computersimulation (Murch and Thorn, 1978), solid linesfrom the formalism of Manning (1964); after Murchand Thorn (1978).


approached by the vacancy, it will tend to be from a random direction, with the result that the impurity correlation factorf2 Æ1.0, thereby signifying a less corre-lated random walk.

Conversely, when w2 > w0 (lower curve),the impurity and vacancy tend to continueexchanging places in a particular configu-ration. Many impurity jumps are therebyeffectively cancelled and f2 Æ 0.0, i.e., the jumps are more correlated. Of course,when w2 = w0 the “impurity” is now atracer in the host, and in effect it can beconsidered to have a different “color” fromthe host. The impurity correlation factornow is the tracer correlation factor, whichin this example is 0.78146 (see Table 3-1).

The diffusion coefficient can be consid-ered to be the product of an uncorrelatedand a correlated part (Eq. (3-76)). In thisexample, when w2 > w0, the reduced im-purity correlation factor acts to reduce theimpurity diffusion coefficient from that ex-pected on the basis of w2 alone. Similarly,when w2 < w0 the raised impurity correla-tion factor acts to increase the impurity dif-fusion coefficient from that expected onthe basis of w2 alone.

In general, the presence of the impurityatom in the host also influences the hostjump frequencies in the vicinity of the im-purity so that they differ from w0. For thef.c.c. lattice the usual model adopted is theso-called five-frequency model and is de-picted in Fig. 3-11. The frequency w1 is the host frequency for host atom/vacancyjumps that are both nearest neighbors to theimpurity. This jump is often called the “ro-tational jump” because in effect the host canrotate around the impurity. The frequencyw3 refers to a “dissociative” jump, i.e., ahost atom jump which takes the vacancyaway from the impurity. Finally w4 (notshown) is the “associative” jump, which isa host jump that brings the vacancy to the

nearest neighbor position of the impurity,i.e., the reverse of w3. All other host jumpsoccur with the frequency w0.

We have mentioned “associative” and“dissociative” jumps of the vacancy. Theseare directly related to the impurity–va-cancy binding energy EB at the first nearestneighbor separation by

w4 /w3 = exp (– EB/k T ) (3-78)

Note that EB is negative for attractionbetween the impurity and the vacancy.

Manning (1964) has shown that the im-purity correlation factor f2 is given rigor-ously for the five-frequency model by

f2 = (2w1 + 7 F w3)/(2w2 + 2w1 + 7 F w3)(3-79)

where F is the fraction of dissociating va-cancies that are permanently lost from asite and are uncompensated for by return-ing vacancies. F is given to a very good ap-proximation by

and a = w4/w0.

7 7

10 180 5 927 13412 40 2 254 597 436

4 3 2

4 3 2

F = (3-80)

..

− + + ++ + + +

⎡

⎣⎢

⎤

⎦⎥

a a a aa a a a

Figure 3-11. Five-frequency model for impurity dif-fusion in the f.c.c. lattice (see text), figure taken fromMurch and Thorn (1978).

Other more accurate expressions for Fhave been reviewed by Allnatt and Lidiard(1987, 1993). For many practical purposesthe extra rigor is probably unwarranted.Extension to the case where there are va-cancy–impurity interactions at secondnearest neighbors has been considered byManning (1964).

In the b.c.c. lattice the second nearestneighbor is close to the nearest neighbor (within 15% in fact). In a formal sense weneed to consider dissociative jumps for avacancy, jumping from the first to the sec-ond, third (not the fourth), and fifth nearestneighbors: these are denoted by w3, w¢3,and w≤3.

The corresponding reverse associativejumps are w4, w¢4, and w≤4. We also need toconsider dissociative jumps from the sec-ond nearest neighbor to the fourth (w5) andthe reverse of this (w6). There is no rota-tional, i.e., w1, jump.

The binding energy EB1 between the im-purity and the vacancy at first nearestneighbor is expressible as (see, for exam-ple, Bocquet et al., 1996)

w¢4 /w¢3 = w≤4 /w≤3 = exp (– EB1/k T ) (3-81)

The binding energy EB2 between the im-purity and the vacancy at second nearestneighbors can be expressed as

w6 /w5 = exp (– EB2 /k T ) (3-82)

We also have that

w4 w4 /w5 w3 = w¢4 /w¢3 (3-83)

In order to proceed it has been usual to re-duce the large number of jump frequenciesby making certain assumptions. There aretwo basic models. In the first, usuallycalled Model I, it is assumed that w6 = w0 =w¢4 = w≤4. This implies that w¢3 = w≤3 andw3w5 = w¢3w4. In the second, Model II, on the other hand, the impurity–vacancyinteraction is in effect restricted to the first

nearest neighbor by means of the assump-tions w3 = w¢3 = w≤3 and w5 = w6 = w0. Thisalso implies that w4 = w¢4 = w≤4.

Model I leads to the expression (Man-ning, 1964; Le Claire, 1970)

(3-84)

with F given by

(3-85)

and a = w3/w¢4.Model II leads to the expression

(3-86)

where F is given by(3-87)

and a = w4/w0. Correlation factors for impurity diffu-

sion via vacancies in the simple cubic lat-tice (Manning, personal communication,cited by Murch, 1982a) and the diamondstructure have also been calculated (seeManning, 1964), as well as the h.c.p. struc-ture (see Huntington and Ghate (1962) andGhate (1964)). Correlation factors for im-purity diffusion by interstitialcy jumpshave also been reported for the f.c.c. latticeand the AgCl structure (Manning, 1959).

In many cases of impurity diffusion theexpression for the impurity correlation fac-tor is of the form

(3-88)

where u contains only the host frequencies;e.g., for vacancy diffusion in the f.c.c. lat-tice u is given by

(3-89)u F= w w w w13

4 02+ ( / )

fu

u22

=w +

73 33 43 97 38 66 06

8 68 18 35 9 433

3 2

3 2F =a a aa a a

+ + ++ + +

. . .. . .

fF

F23

2 3

72 7

=w

w w+

72 5 1817 2 476

0 8106

2

F =a a

a+ +

+. .

.

fF

F23

2 3

72 7

=′

+ ′w

w w



This form for the impurity correlation fac-tor is especially relevant to the isotope ef-fect in diffusion (see Sec. 3.3.3) where forself-diffusion the isotopes are tracers of thehost but, being isotopes, actually haveslightly different jump frequencies fromthe host and can be classified as “impur-ities”. We might also mention that for im-purity diffusion experiments the isotopescan additionally be of the impurity.

As a closing remark for this section, itshould be remembered that because of itsmathematical form, the impurity correla-tion factor will be temperature dependent.Over a fairly small temperature range thetemperature dependence can frequently beapproximated by an Arrhenius-like expres-sion, i.e.,

f2 ≈ f20 exp (– Q¢/kT ) (3-90)

where Q¢ is some activation energy for thecorrelation process (this has no particularphysical meaning), k is the Boltzmann con-stant, and f2

0 is a “constant”. It should benoted that Q¢ will be unavoidably includedin the activation energy for the entire im-purity diffusion process and is not neces-sarily unimportant.

3.3.1.5 Correlation Factors forConcentrated Alloy Systems

The impurity correlation factors dis-cussed in the previous section are probablyappropriate for impurity concentrations upto about 1 at.%. At higher concentrationsthe impurities can no longer be consideredindependent in the sense that the correla-tion events themselves are independent.This presents a special difficulty becausethere is no easy way of extending, say, thefive-frequency model into the concentratedregime without rapidly increasing the num-ber of jump frequencies to an unworkablenumber. As a result, models have been in-

troduced which limit the number of jumpfrequencies but only as a result of someloss of realism.

The first of these is the “random alloy”model introduced by Manning (1968,1970, 1971). The random alloy is of con-siderable interest because, despite its sim-plicity, it seems to describe fairly well thediffusion behavior of a large number of al-loys. The atomic components (two ormore) are assumed to be ideally mixed andthe vacancy concentration is assumed to bevery low. The atomic jump frequencies,e.g., wA and wB, in a binary alloy are expli-citly speficied and neither changes withcomposition or environment. In the verydilute limit this model formally corre-sponds, of course, to specifying a hostjump frequency w0 and an impurity jumpfrequency w2 with all other host jump fre-quencies equal to w0. However, it shouldbe appreciated for this model that physi-cally, although not mathematically, thejump frequencies wA and wB are conceivedto be “average” jump frequencies. The average jump frequency of the vacancy, wv,was postulated to be given by

wv = cA wA + cB wB (3-91)

where cA and cB are the atomic fractions of A and B. Manning (1968, 1970, 1971)finds that

(3-92)

where

S = [(cB – x0) wB/wA + (cA – x0)]2

+ 4 x0 (1 – x0) wB/wA1/2 (3-92a)

and

x0 = 1 – f (3-92b)

and f is the correlation factor for vacancydiffusion in the pure metal A or B.

fS x c c x

S x xAB B A A

B A= + − + + −

+ − +( ) / ( )

( ) / 2 32 1

0 0

0 0

w ww w

Early Monte Carlo calculations indi-cated that Eq. (3-92) was a very good ap-proximation over a wide range of wA/wB

but more recently it has been found thatthis was largely illusory (Belova andMurch, 2000). Eq. (3-92) is actually only areasonable approximation when the ex-change frequency ratio is within about anorder of magnitude of unity. A much betterapproximation to the problem is that givenby Moleko et al. (1989), but unfortunatelythe equations are much more cumbersometo use.

A different approach to diffusion in con-centrated systems has been taken by Kiku-chi and Sato (1969, 1970, 1972). They de-veloped their Path Probability Method(PPM) to cope specifically with the prob-lems of diffusion in concentrated systems.The method can be considered to be a time-dependent statistical mechanical approach.They started with the so-called binary alloyanalogue of the Ising antiferromagnetmodel. In this model, sometimes called the“bond” model, interactions EAA, EBB, andEAB are introduced between nearest neigh-bor components of the type A-A, B-B, andA-B, respectively. Equilibrium propertiesof this model are well known; see for ex-ample, Sato (1970). It is convenient to fo-cus on the ordering energy E defined by

E = EAA + EBB – 2 EAB (3-93)

(note that in the literature there are otherdefinitions of the ordering energy whichdiffer from this one by either a negativesign or a factor of 2). When E > 0, there isan ordered region in the b.c.c. and s.c. lat-tices (the f.c.c. lattice is more complicatedbut has not been investigated by the PPM).The ordered region is symmetrically placedabout cA= 0.5. When E < 0, there is a two-phase region at lower temperatures (thisside has not been investigated by the PPM).The exchange frequencies wA and wB are

not given explicitly as in the random alloymodel but are expressed in the followingway in terms of the interaction energies (bonds) for a given atom in a given config-uration:

wi = ni exp (– Ui /k T )

¥ exp [(gi Eii + gj Ejj)/k T ], (3-94)

i = A, B i( j

where Eij < 0, ni is the lattice vibration fre-quency, Ui is the reference saddle point en-ergy in the absence of interactions, gi is thenumber of atoms of the same type whichare nearest neighbors of the given atom andgj is the number of atoms of the other typewhich are nearest neighbors of the givenatom. Other forms of wi are also possiblebut have not been developed in this con-text.

In the PPM a path probability function is formulated and maximized, a processwhich is said to be analogous to mini-mizing the free energy in equilibrium sta-tistical mechanics. The first calculationswere made using what was called a time-averaged conversion. The results were inpoor agreement with later Monte Carlocomputer simulation results. Subsequently,substantial improvements were made (Sato, 1984) and the PPM results are nowin fairly good agreement with computersimulation results. A typical comparison ofthe earlier results with Monte Carlo resultsfrom Bakker et al. (1976) is shown in Fig.3-12. Note the bend at the order/disordertransition. The lower values of the correla-tion factors on the ordered side can be ascribed generally to the higher probabilityof jump reversals as an atom which jumpsfrom the “right” sublattice to the “wrong”sublattice tends to reverse the jump whilethe vacancy is still present. This cancella-tion of jumps obviously leads to small val-ues of f.



Although there is no particular reasonfor assuming that the correlation factorshould follow an Arrhenius behavior, e.g.,Eq. (3-90), these results nonetheless showit quite well. The contribution to the totalactivation energy is difficult to determinedirectly because Ui is a “spectator quan-tity” in these calculations and the vacancyformation energy has not been calculatedin the same work, although the latter can bedetermined separately in this model (Limet al., 1990). In a recent application of thismodel to self-diffusion in b-CuZn (Belovaand Murch, 1998), the contribution to thetotal activation energy for, say, Cu tracerdiffusion was variously estimated at 22–40%. It is clear that for a complete analysisof diffusion in such materials the assess-ment of the contribution of the tracer corre-lation factor to the activation energy is es-sential.

We have mentioned that cancellation ofjumps leads to small values of f. There are,however, special sequences of jumps in theordered structure which lead to effectivediffusion. The most important of these se-quences is the so-called six-jump cycle

mechanism, which we shall now brieflydiscuss. This sequence is interspersed, ofcourse, with jump reversals.

It became clear at an early stage that dif-fusion in fully ordered structures poses cer-tain difficulties. A purely random walk ofthe vacancy will lead to large amounts ofdisorder and inevitably to a great increasein the lattice energy of the solid. Hunting-ton (see Elcock and McCombie, 1958)seems to have been the first to suggest that netmigration of atoms in, say, the B2 structurecould occur by way of a specific sequenceof six vacancy jumps with only a relativelyslight increase in energy. This sequence isillustrated in Fig. 3-13a for the orderedsquare lattice (Bakker, 1984), and sche-matic energy changes for the sequence areshown in Fig. 3-13b. The starting and fin-ishing configurations have the same energybut there has been a net migration of atoms.

The correlation factor for the six-jumpcycle sequence alone has been calculatedby Arita et al. (1989). The six-jump cyclesequence is contained in the earlier pathprobability method calculations and theMonte Carlo computer simulations (Bak-ker, 1984). The correlation factors calcu-lated in those calculations are statistical av-erages over all possible sequences in addi-tion to the jump reversals which tend topredominate.

Further detailed discussion of the corre-lation factors in concentrated alloy systemscan be found in the detailed reviews byBakker (1984), Mehrer (1998), Murch andBelova (1998).

3.3.1.6 Correlation Factors forHighly Defective Systems

Many solids, such as nonstoichiometriccompounds, intercalation compounds, andfast ion conductors, appear to have a veryhigh concentration of defects. Manning

Figure 3-12. Arrhenius plot of the tracer correlationfactor for B in the alloy A0.6B0.4 for various values ofU = (EAA– EBB)/E. Points from Bakker et al. (1976),lines from Kikuchi and Sato (1969, 1970, 1972); af-ter Murch (1984a). The abscissa is in units of E/k.

(1968) was probably the first to note that thecorrelation factor could increase if morevacancies are present. In effect, the extravacancies decorrelate the reverse jump of atracer atom; see the discussion in Sec.3.3.1.3. As more vacancies are added wereach the limit of a single atom remaining(for convenience, the tracer) and the corre-lation factor is unity as befitting a completerandom walk of an interstitial. Between the

extremes, provided the vacancies are ran-domly distributed, roughly linear behaviorof the correlation factor is found (Fig. 3-9).

When the vacancies are not randomlydistributed, which is usually the case be-cause of atom–atom repulsion, the behav-ior of the correlation factor becomes morecomplicated. The calculations that havebeen performed are for the so-called latticegas model. In this, the atoms occupy dis-crete sites of a rigid lattice, typically anearest neighbor interaction being speci-fied between the atoms. For nearest neigh-bor attractive interactions, at low tempera-tures a two-phase region develops symmet-rically about the concentration 0.5. Con-versely, with nearest neighbor repulsion anordered region centered about a concentra-tion 0.5 develops for the simple latticessuch as honeycomb, square planar, or sim-ple cubic. The face centered cubic and tri-angular lattices are rather more compli-cated, but have not been investigated. Weshall focus on repulsion here because it isthe most likely. In a manner similar to thatfor the concentrated alloy, the exchangefrequency of a given atom with a vacancyis written for a given configuration as

w = n exp (– U/kT ) exp (gnn fnn /kT ) (3-95)

where gnn is the number of atoms which arenearest neighbors to the given atom, U isthe activation energy for diffusion of anisolated atom, n is the vibration frequency,and fnn is the atom–atom interaction en-ergy. It can be seen that nearest-neighborrepulsion between atoms works to diminishthe activation energy. Other forms of theexchange frequency are also possible, butmost calculations have used this one.

Obviously lattice gas models cannotgenerally be very realistic in their diffusionbehavior. The interest in them comes aboutprimarily because they exhibit behaviorwhich is rich in physics and likely to occur


Figure 3-13. (a) Six-jump cycle in the ordered squarelattice: the upper figure shows the path of the va-cancy, the lower figure shows the displacements re-sulting from the complete cycle, after Bakker (1984).(b) Schematic representation of energy changes dur-ing the six-jump vacancy cycle, after Bakker (1984).


to a greater or lesser extent in real materi-als. Surprisingly, some materials such asinterstitial solid solutions are describedfairly well by the lattice gas model evenwith only nearest neighbor interactions.This must be a result of short-range inter-actions being by far the most important.

Sato and Kikuchi (1971) have employedthe PPM to good effect in the lattice gasmodel. Extensive calculations have beenmade, especially for the honeycomb lattice.Equally extensive Monte Carlo calcula-tions have also been made both for this lat-tice and many others; see the review byMurch (1984a). A typical result is shownin Fig. 3-14. The pronounced minimum inthe correlation factor is due to the effects ofthe ordered structure. This ordered struc-ture consists of atoms and vacant sites ar-ranged alternately. The wide compositionvariability of the ordered structure at thistemperature (indicated by the arrows) is ac-commodated by vacant sites or “intersti-tials” as appropriate. When an atom jumps

from the right lattice to the wrong lattice ittends to reverse that jump, thereby givinglow values of the correlation factor. As inthe ordered alloy, there are sequences ofjumps which lead to long-range diffusion.By and large these sequences are basedaround interstitialcy progressions, butrather than two atoms moving in unison asin the actual interstitialcy mechanism, herethey are separated in time.

The temperature dependence of f in thelattice gas, like the alloy, is fairly strong.Again, it is usual to write f in an Arrheniusfashion, e.g., Eq. (3-90), but there is noknown physical reason for assuming that fmust really take this form. Fig. 3-15 showsthe behavior in the square planar latticegas. The Arrhenius plot is in fact curvedabove and below the order/disorder tem-perature, though in an experimental studythis would be overlooked because of amuch smaller temperature range. This acti-vation energy will be included in the ex-perimental tracer diffusion activation en-ergy. This is likely to be a significant con-tribution which (like in the ordered alloy)cannot be ignored.

The subject of tracer correlation effectsin defective materials with high defect con-

Figure 3-14. PPM results for the dependence of thetracer correlation factor on ion site fraction ci in thehoneycomb lattice with nearest neighbor repulsion.T* = k T /fnn ; after Murch (1982d).

Figure 3-15. Arrhenius plot of the tracer correlationfactor gained from Monte Carlo simulation of asquare planar lattice gas with 50% particles and 50%vacant sites (Zhang and Murch, 1990).

tent is dealt with at length in a number ofplaces; see, for example, Murch (1984a)and Sato (1989).

3.3.1.7 The Physical or ConductivityCorrelation Factor

In Sec. 3.3.1.3 we have seen that tracercorrelation effects are conventionally em-bodied in the so-called tracer correlationfactor. In this way the correlation factor ap-pears as a correction factor in the randomwalk expression for the tracer diffusion co-efficient Eq. (3-76).

In 1971 Sato and Kikuchi showed thatthe ionic conductivity should also include acorrelation factor, now called the physicalor conductivity correlation factor andgiven the symbol fI , sometimes fc. In theusual hopping model expression for thed.c. conductivity s (0), we have

s (0) = C (Z e)2 Gq rq2 f I /6 k T (3-96)

where C is the concentration of charge car-riers per unit volume, Z is the number ofcharges, e is the electronic charge, Gq is thejump frequency, rq is the jump distance ofthe charge, k is the Boltzmann constant,and T is the absolute temperature.

It should be noted that fI does not be-come nontrivial, i.e., 71, until a relativelyhigh defect concentration is present andion– ion interactions and/or trapping sitesare present. A model containing these wasexplored by Sato and Kikuchi (1971) intheir pioneering work. Their results for fI inthe lattice gas model of nearest neighborinteracting atoms diffusing on a honey-comb lattice with inequivalent sites ar-ranged alternately are shown in Fig. 3-16.

This work was the starting point for the development of the new area of correla-tion effects in ionic conductivity. This areahas been reviewed by Murch and Dyre(1989).

Historically, the atomistic meaning of fI

has taken quite some time to determine,principally because the calculations (pathprobability and Monte Carlo methods)were based on the calculation of a flow ofcharge.

It is now known that the physical corre-lation factor can be expressed as

(3-97)

where DR is the total displacement of thesystem after a total of n jumps in time t.This differs from Eq. (3-77) only in the im-portant points that DR refers to the dis-placement of the entire system and not asingle particle. DR is simply the sum of theindividual particle displacements R occur-ring in time t:

(3-98)

We can now see that whereas f encom-passes correlation effects of a single (tracer)

DR R= all

particle

∑ i

f nrn

I = lim /→∞

⟨ ⟩DR2 2


Figure 3-16. PPM results for the dependence of thephysical correlation factor on ion site fraction ci inthe honeycomb lattice with nearest neighbor repul-sion and site inequivalence. T* = k T /fnn and the sitedifference a priori is 5fnn ; after Murch (1982d).


particle, fI encompasses correlation effectsof the entire system in a collective sense.The correlation factor fI of the system ismanifested in the d.c. ionic conductivity,Eq. (3-96). It is also manifested in chemi-cal diffusion (see below), but not in tracerdiffusion.

What is sufficient to make fI71? First, a relatively high concentration of defects is required (see Fig. 3-16). Next, inequalityof lattice sites (and therefore a variablejump frequency), or mutual interactionsamong the ions, or differences in acces-sibilities of ions: all are sufficient, with ahigh concentration of defects, to give anontrivial value for fI .

The importance of fI in materials withhigh defect concentrations cannot be over-stated. For example, in a calculation of fI

in a model for CeO2 doped with Y2O3

(to obtain a high concentration of anion va-cancies), it was found that at 455 K fI

changed from unity at low Y3+ content toonly 0.05 at high Y3+ content, correspond-ing to 14% of the anion sites being vacant(Murray et al., 1986). This very low valueof fI came about primarily because of thetrapping effects of the Y3+ ions. The phys-ical correlation factor is also temperaturedependent, but there is no known physicalreason why fI should take an Arrheniusform. Fig. 3-17 shows a typical result for alattice gas above and below the order/dis-order transition. Both regions are slightlycurved in fact, although this could be over-looked in an experiment over a small tem-perature range. The activation energy asso-ciated with fI will be included in the overallactivation energy for d.c. ionic conduction.This contribution cannot be ignored. In themodel for CeO2 doped with Y2O3, the con-tribution is of the order of 30% very highY3+ contents.

Most of the contributions to the applica-tion of fI have centered around ionic con-

duction. However, fI also occurs in the ex-pression for the chemical diffusion coeffi-cient in the one-component system, i.e.,

D = G r2 fI (1 + ∂ lng /∂ lnc)/6 (3-99)

or, with Eq. (3-76)

where g is the activity coefficient of theparticles with the term in parentheses be-coming the “thermodynamic factor”. Thisequation applies to interstitial solid solu-tions and intercalation compounds such asTiS2 intercalated with Li where the atomicmobility on the defective lattice is rate determining. It does not apply to ionic conductors, however, since – for changebalance reasons – compositional changesare controlled by atomic mobility or hole-electron hopping on another lattice. For example, in the oxygen ion conductor calcia stabilized zirconia, compositionalchanges of the oxygen ion vacancy concen-tration are probably controlled by the dopant calcium ion mobility on the cationlattice.

= IDf

fc

* lnln

1 + ∂∂

⎛⎝⎜

⎞⎠⎟

g

Figure 3-17. Arrhenius plot of the physical correla-tion factor gained from Monte Carlo simulation of asquare planar lattice gas with 50% particles and 50%vacant sites (Zhang and Murch, 1990).

3.3.1.8 Correlation Functions(Collective Correlation Factors)

There are a number of other correlationphenomena in diffusion. In order to discussthese in a unified fashion, say for a binarysystem, it is convenient first to introducecorrelation functions or collective correla-tion factors. They represent the correlatedparts of the phenomenological coefficients(see Sec. 3.4). For the diagonal coefficientswe write (Allnatt and Allnatt, 1984)

Lii = L0ii fii i = A, B (3-100)

where L0ii is the uncorrelated part of Lii ,

i.e.,

Lii(0) = Ci Gi r2/6 k T i = A, B (3-101)

where Ci is the number of species i per unitvolume, Gi is the jump frequency of speciesi, and r is the jump distance, k is the Boltz-mann constant, and T is the absolute tem-perature. For the off-diagonal terms (wenote the Onsager condition LAB = LBA), wewrite

LAB = L(0)AA fAB

(A) = LBB(0) fAB

(B) (3-102)

This partitioning of the Lij into these partsmight appear arbitrary. The choice comesabout largely by analogy with the tracerdiffusion coefficient which is partitioned asa product of an uncorrelated part, contain-ing the jump frequency etc., and a corre-lated part, containing the tracer correlationfactor (see Eq. (3-76) and associated dis-cussion).

Some physical understanding of the fij isappropriate here. There are two equivalentexpressions for the fij . First, Allnatt (1982)and Allnatt and Allnatt (1984) showed thatthe fij have Einsteinian forms reminiscentof the expression for the tracer correlationfactor (see Eq. (3-76)). Thus they write fora binary system A, B

fAA = ·DR2AÒ/nAr2 (3-103)

where DRA is the displacement of thesystem of particles of type A, i.e., DRA isthe vector sum of the displacements of allthe atoms of type A and nA is the totalnumber of jumps of the species A. Thus thegroup of particles of type A in the system istreated itself like a (hypothetical) particle,and fAA is its correlation factor.

Similarly,

fAB(i) = ·DRA · DRBÒ/nAr2 i = A, B (3-104)

Thus fAB(i) expresses correlations between

the vector sum of the displacements of allthe atoms of type A with the correspondingquantity for atoms of type B.

An alternative way of looking at the fijis in terms of the drifts in a driving force.Let us consider a “thought experiment” for a binary system AB where an externaldriving force Fd is directly felt only by the A atoms. It is straightforward to show(Murch, 1982a) that fAA is given by

(3-105)

where ·XAÒ is the average drift distance ofthe A atoms in the driving force and nA isthe number of jumps of a given A atom in time t. However, although only the A atoms feel the force directly, the B atomsfeel it indirectly because they are compet-ing for the same vacancies as the A atoms. Accordingly, there is also a drift of the B atoms, smaller than the drift of the A atoms. The correlation function fAB

(i) is re-lated to this drift by

(3-106)

where ·XBÒ is the average drift distance ofthe B atoms.

fk T X

F t ri

iAB

B

d=( ) 2

2⟨ ⟩

G

fk T X

F n rAA

A

d A=

22

⟨ ⟩



Calculation of the Correlation Functions

Calculations have been made for variousimpurity models, the random alloy modeland interacting or (bond) models for alloys.For impurity systems, the area has been re-viewed exhaustively by Allnatt and Lidiard(1987, 1993). Here we shall restrict our-selves to a discussion of the results for thefive-frequency impurity model in the f.c.c.lattice; see Fig. 3-11 and associated text fordetails of the model.

Inspection of Eq. (3-103) indicates thatwhen the A atoms (say) are the infinitelydilute impurities in B, i.e., cAÆ 0, thesystem of atoms of type A reduces to a single A atom. Accordingly the diagonalcorrelation function fAA reduces to the im-purity correlation factor, fA. We have al-ready given Manning’s (1964) expressionfor fA for the five-frequency model in thef.c.c. lattice (see Eq. (3-78)). Manning(1968) showed that the cross correlationfunction fAB

(A) can be calculated by a carefulanalysis of the various impurity jump tra-jectories in a field. He found that fAB

(A) isgiven by

where F is the fraction of dissociating va-cancies that are permanently lost from asite and are uncompensated for by return-ing vacancies and is given by Eq. (3-80).

Let us move on to the binary random alloy model described in detail in Sec.3.3.1.5; explicit expressions have been de-rived by Manning (1968). He found thatfAA is given by

(3-108)

f f c f M c fk

k k kAA A A A A=A,B

= 1 2 0+⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

∑w w

f f

FF

ABA

A= (3-107)( )

[ ( ) ( ) ]

2

3 7 1 22 7

3 4 0 41

1

1 3× − − − −

+

−w w w w ww w

and

(3-109)

where M0 = 2 f (1 – f ) and f is the tracer cor-relation factor in the pure lattice of eithercomponent.

The correlation functions have also beencalculated in interacting bond models (seeSec. 3.3.1.5), especially with the PPM andby computer simulation. Some results ofthe latter are shown in Figs. 3-18 and 3-19for the simple cubic lattice with orderingbetween A and B.

A brief discussion of the behavior ofthese quantities is appropriate here. First,with respect to fAA, as cAÆ 0, the A atomsbehave like impurities in the B matrix, andfAA does in fact reduce to the impurity cor-relation factor fA. At cA≈ 0.5, the minimaare further manifestations of the prepon-derance of jump reversals in the diffusionprocess in the ordered alloy. As cAÆ 1, allcurves converge on unity and the correla-

f c c f f

c M c fk

k k k

ABA

A B A B A B

A A=A,B

=( ) 2

0

1w w

w w×⎛

⎝⎜⎞

⎠⎟∑

−

Figure 3-18. Monte Carlo results for the diagonalcorrelation factor fAA in the simple cubic alloy as afunction of cA at various values of the ordering en-ergy; after Zhang et al. (1989a).

tion effects contained in fAA disappear en-tirely. Next, let us discuss fAB

(A). As cAÆ 0,again A behaves like an impurity in B. Thephysical meaning of fAB

(A) has been dis-cussed earlier, see Eq. (3-106). At this limitthe B atoms have maximum interference on A. Similar to fAA, minima develop atcA≈ 0.5 as a result of jump cancellations.As cAÆ 1, all curves converge on zero asthe B atoms, now in the minority, no longerhave much effect on the diffusion of A.

With the correlation functions in hand,we can form the various correlation termsthat commonly occur in ionic conductivityand chemical diffusion. We shall restrictourselves to binary systems. Unary sys-tems are dealt with in Sec. 3.3.1.7, where itis seen that the physical correlation factoris really a correlation function for a one-component system.

Correlations in Ionic Conductivity(Binary Systems)

When ionic conductivity occurs in a ma-terial where two or more ionic speciessharing the same sublattice (and thereforecompeting for the same defects) carry the

current there are correlations or interfer-ence between the two ionic conductivities.When there are two ionic species and thevacancy mechanism is operating, the ex-pressions for the d.c. ionic conductivitiesare (Murch and Dyre, 1989)

(3-110)

and(3-111)

where e is the electronic charge, ZA(B) is thenumber of charges on A(B), CA(B) is theconcentration of A(B), GA(B) is the jumpfrequency of A(B) and r is the jump dis-tance.

Sometimes the bracketed terms are calledthe binary conductivity correlation factors;they are, in fact, formally binary analoguesof fI in Eq. (3-96). Eqs. (3-110) and (3-111)become

sA = e2 Z2A CA Gi r2 fIA/6 k T (3-112)

sB = e2 Z2B CB Gi r2 fIB /6 k T (3-113)

For those readers familiar with Man-ning’s (1968) treatment of impurity ionicconductivity, fIA (where A is the impurityin B) is expressed as

(3-114)

where fA is the impurity correlation factorand ·npÒ is a complex kinetic parameter introduced by Manning. Eq. (3-114) as en-visaged by Manning (1968) applies only tosituations where the vacancy concentrationis very low. Often the term in parenthesesin Eq. (3-114) is loosely called a vacancy-wind factor, although this terminology isfrequently applied to the whole of fIA. Inthe special case where the impurity is a

f fZZ

npIA AB

A= 1 + ⟨ ⟩⎛

⎝⎜⎞⎠⎟

s BB B B

BBA

BAB

B=e Z C r

k Tf

ZZ

f2 2 2

6G +⎛

⎝⎜⎞⎠⎟

( )

sAA A A

AAB

AAB

A=e Z C r

k Tf

ZZ

f2 2 2

6G +⎛

⎝⎜⎞⎠⎟

( )


Figure 3-19. Monte Carlo results for the off-diago-nal correlation factor f A

AB in the simple cubic alloy asa function of cA at various values of the ordering en-ergy; after Zhang et al. (1989a).


tracer of the host and where, of course,ZA = ZB, the term in parentheses reduces tofA

–1 with the result that fIA = 1 and there areno correlations in the ionic conductivity.However, it is emphasized that this resultapplies only where the defect concentrationis very low. For high concentrations of de-fects which in general cannot be randomlydistributed there are still residual correla-tions arising from non-ideal effects. Theseare encompassed in fI (see Eq. (3-96)).

For the five-frequency model, see Sec.3.3.1.4, for certain combinations of thejump frequencies it is possible to make fIA

(Eq. (3-114)) negative. Specifically thiscan occur when the vacancy and impurityare tightly bound together, i.e., whenw4 > w3. Vacancies that are bound to im-purities can be transported around the im-purity by the host atom flux. The probabil-ity of the impurity moving upfield can thenbe larger than the probability of movingdownfield. The impurity may then actuallymove upfield, opposite to its “expected”direction. For further details, see Manning(1975). It is also possible in the five-fre-quency model to make fIA for the impurityexceed unity, even when ZB = ZA.

For concentrated interacting systems, thecalculation of the fIA (and fIB) is of someinterest, especially in the case of mixed fastion conductors such as Na,K b-alumina.An example of the results of calculation offIA and fIB, by way of the PPM, is shown inFig. 3-20. This lattice gas model containstwo species A and B (ZA= ZB) with 20%vacancies on a honeycomb lattice. Themodel approximates the fast ion conductorNa,K b-alumina, although it is probablyalso a reasonable description of some sili-cate glasses. Note the minima in fIA andfIB. These are caused by what is called a“percolation difficulty” in the flow createdby ordered arrangements. Physically, manyjumps are reversed. This effect seems

stronger as the dimensionality is lowered(in fact all correlation effects do), and maybe a major contributor to the so-calledmixed-alkali effect. This effect is charac-terized by a substantial decrease in the d.c.ionic conductivity at intermediate mixedcompositions without any obvious physicalcause. This subject is dealt with further byMurch and Dyre (1989) and Sato (1989).

Correlations in Chemical Diffusion(Binary Systems)

The usual equation of practical interestfor chemical diffusion in a binary system isthe Darken equation. This equation relatesthe intrinsic diffusion coefficient of a par-ticular component to its tracer diffusion coefficient. The original Darken equation(which neglects correlations) is written as

(3-115)D DcA

IA*

A

A= 1 + ∂

∂⎛⎝⎜

⎞⎠⎟

lnln

g

Figure 3-20. PPM results for the quantities fIA andfIB as a function of concentration rA in the two-com-ponent conductor with 20% vacancies at half the or-der–disorder temperature. The dashed line shows thecase where the development of long-range order isartificially suppressed; after Murch and Dyre (1989).

where gA is the activity coefficient of A.However, it is straightforward to show thatthe rigorous equation is Eq. (3-56) whichcan be rewritten in terms of the correlationfunctions (Le Claire, 1975; Murch, 1982a;Stark, 1976):

(3-116)

The fA in the denominator is the correlationfactor for the actual composition of the al-loy and not some impurity limit. In the lit-erature the term derived from correlations [ ] is sometimes called a vacancy-wind fac-tor and is given the symbol rA. As it turnsout, however, this term cannot vary muchfrom unity since the original Darken equa-tion is reasonably well obeyed. The behav-ior of rA is not especially transparent. Thevacancy flux in chemical diffusion is al-ways in the same direction as that of theslower moving species (and opposite tothat of the faster moving species). The va-cancy-wind effect always tends to providea given atom with an enhanced probabilityof flowing in a direction opposite to that ofthe vacancy flow. When A is less mobilethan B, the vacancy-wind factor rA effec-tively makes DI

A even smaller. Conversely,when A is more mobile than B, the va-cancy-wind factor rA effectively makes DI

A

even larger. But how much? This can bepartially answered in the following.

Although we have given Manning’s(1968) expressions for the individual fijand the correlation factor fA for the randomalloy (Eqs. (3-108) and (3-109)), it may notbe immediately obvious that rA is in factgiven in this model by (see also Sec.3.2.3.4)

(3-117)rf D c D D

f c D c DAB* A A* B*

A A* B B*=

+ −+

( )( )

D Dc

f fcc

f

AI

A*A

A

AA AB(A) A

BA

= 1 + ∂∂

⎛⎝⎜

⎞⎠⎟

× −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

lnln

g

where f is the tracer correlation factor fordiffusion in the pure crystal, i.e., pure A orpure B. It turns out that the maximum in rA

in this model occurs when cAÆ 1. In theextreme case when the jump frequency ofA is much larger than that of B, thenrAÆ f0

–1. Therefore, when f0 = 0.78146 (va-cancy diffusion on the f.c.c. lattice, see Ta-ble 3-1), rA can only enhance chemical dif-fusion by a factor of 1.28.

Manning’s equation (Eq. (3-117)) seemsto have more general significance than application to the random alloy. Murch(1982a) showed that this equation performsvery well indeed for an interacting alloymodel that can exhibit short and long-rangeorder. This suggests that this equation canbe used almost with impunity. This subjectarises again in the “Manning relations”which relate the phenomenological coeffi-cients to the tracer diffusion coefficients(see Sec. 3.2.3.4).

The subject of the experimental check ofrA has been discussed by Bocquet et al.(1996). They pointed out that in most casesthe Manning formulation for rA and rB

(Eq. (3-117)) improves the agreement be-tween experimental and calculated valuesof the Kirkendall shift and the ratio of the intrinsic diffusion coefficients DI

A/DIB.

However, the individual experimental val-ues of DI

A and DIB often tend to be quite a

bit higher than the calculated values. Theexperimental Kirkendall shift also has atendency to be higher than the calculatedvalue. Carlson (1978) points out that theproblem could be due to the random alloyassumption in the Manning formulation.However, the success of the Manning for-mulation for the interacting alloy modelmentioned above seems to vindicate therandom alloy assumption and so the reasonfor the discrepancy probably should besought elsewhere.



3.3.2 The Nernst–Einstein Equationand the Haven Ratio

The Nernst–Einstein equation relatesthe d.c. ionic conductivity to a diffusioncoefficient. Probably no other equation indiffusion has generated more misunder-standing than this one. Let us consider thestandard derivation (see, for example,Batchelor (1976)), and discuss its implica-tions in detail.

We consider a pseudo-one-componentsystem in a situation where the flux result-ing from an applied force on the particles (which are completely noninteracting) ex-actly counterbalances the flux due to diffu-sion. That is, from Eq. (3-2)

(3-118)

It is important to note that the diffusion co-efficient here refers to a chemical composi-tion gradient and is most definitely con-ceived as a chemical diffusion coefficient,not a tracer or self-diffusion coefficient.Lack of appreciation of this fact leads tomisunderstandings and inconsistencies.

The external force Fd is a result of a po-tential so that

(3-119)

At equilibrium the distribution of com-pletely noninteracting particles follows aBoltzmann distribution such that

C (x) = C0 exp [– f (x)/k T ] (3-120)

Eq. (3-120) must be the solution of Eq. (3-120) at steady state. We then have that

(3-121)

With Eq. (3-118) we find that

(3-122)⟨ ⟩vD

Fk T

= d

dd

=dd

= dCx

Ck T C

C Fk T

− f

Fxd =

dd

− f

⟨ ⟩v C DCx

=dd

˜

When the external force is the result of anelectric field E we have

Fd = Z e E (3-123)

where Z is the number of charges (ionic valence) and e is the electronic charge.

The mobility u is defined as the velocityper unit field and so we have

(3-124)

In the solid-state diffusion literature Eq.(3-124) is generally called the Nernst–Ein-stein relation. Because the d.c. ionic con-ductivity is related to the mobility by s =C Z e u, Eq. (3-124) can be rewritten as

(3-125)

More generally, interactions are presentbetween the particles and it can be shownthat the general form of the Nernst–Ein-stein equation is in fact (Murch, 1982b)

(3-126)

where m is the chemical potential of theparticles and c is the site fraction.

Now let us discuss this equation in detailby exploring some particular cases. Whenthe distribution of particles is completelyideal, meaning that the particles do not feelone another, not even site blocking, thethermodynamic factor drops out of Eq. (3-126) and s /D reduces to Eq. (3-125). In thisvery special case, and only in this case, thetracer diffusion coefficient D* equals D, sothat

(3-127)

When the particles are ideally distributedbut subject to the condition that no morethan one particle can occupy one site, then

sD

C Z ek T*

=2 2

sm˜

ln

D

C Z ek T

c=

2 2 ∂∂

⎛⎝⎜

⎞⎠⎟

sD

C Z ek T

=2 2

u

D

Z ek T˜ =

the thermodynamic factor in Eq. (3-126)equals (1 – c)–1. However, for this situationthe tracer diffusion coefficient is related toD by (see, for example, Murch (1982c))

D* = D (1 – c) f (3-128)

where f is the tracer correlation factor, sothat

(3-129)

Ionic solids having virtually a perfect lat-tice of particles (charge carriers) fall intothis category and Eq. (3-129) is appropriateto such solids. Similarly, when the particlesare extremely dilute, Eq. (3-129) is againappropriate.

In the solid-state diffusion literature wevery often see Eq. (3-125) used directly to calculate another diffusion coefficient,sometimes called the “charge” diffusion co-efficient and given the symbol Ds. Thus weencounter the following equation, which isalso called the Nernst–Einstein equation

(3-130)

Ds is dimensionally correct but it does notcorrespond to any diffusion coefficient thatcan actually be measured by way of Fick’slaws for a solid system. Recall that theidentical Eq. (3-125) requires that the parti-cles are completely noninteracting for it tobe meaningful. We frequently then see thefollowing equation relating D* and Ds

D* = f Ds (3-131)

with Ds often being called a self-diffusioncoefficient or the diffusion coefficient ofthe (random walking) charge carriers. Eq.(3-131) is generally given as if it is self-ev-ident or similar to Eq. (3-73). Eqs. (3-130)and (3-131) are then combined to give

(3-132)sD

C Z ek T f*

=2 2

ssD

C Z ek T

=2 2

sD

C Z ek T f*

=2 2

i.e., formally the same as Eq. (3-129). Thisroute to Eq. (3-132) is clearly a case of twowrongs ending up making a right. How-ever, the real danger lies in the fact that Eq.(3-132) obtained in this way blinds us to itslimitations, limitations which are clearlystated in the derivation of Eq. (3-129).

In practice, Eq. (3-132) has often beenused to describe situations where the par-ticles are interacting and many sites are vacant, such as in fast ion conductors. Insuch cases, we cannot necessarily expect ameaningful interpretation. What is the cor-rect way to proceed? We already have theNernst–Einstein equation to cover the situ-ation of interacting particles (Eq. (3-126)).However, this normally cannot be appliedto real materials because local charge neu-trality prevents composition gradients be-ing set up in ionic conductors by the con-ducting ions themselves. What can be doneis to return once again to Eq. (3-130) and useit purely as a definition of Ds , recognizingat the same time that Ds has no Fickianmeaning. Eq. (3-129) is then being usedpurely as a means of changing s to a quan-tity which has the dimensions of a diffusioncoefficient. It is clear that it would be inap-propriate in these circumstances to call thisequation the Nernst–Einstein equation.

We can now define the Haven ratio,which is simply the ratio of D* to Ds :

(3-133)

In view of what has been said above aboutDs , it is appropriate to ask whether HR hasany physical meaning. This can be partiallyanswered by examining hopping modelsfor conduction. It can be shown quite gen-erally for hopping models using the va-cancy mechanism that

(3-134)Hff

ff cR

I= =

+ g

HDDR ≡ *

s



where fI is the physical or conductivity cor-relation factor and g is a two-particle corre-lation factor (see Sec. 3.3.1.7). For specificmodels of interacting particles fI ≤ 1 (seeMurch and Dyre (1989) for a detailed re-view). At the limits of an almost full orempty lattice of charge carriers, fI Æ1 andHR = f. This is compatible, of course, withEq. (3-131) for these conditions.

Accordingly, a measurement of the Haven ratio (obtained by measuring s andD*, preferably in the same sample) can insome cases give f alone and therefore themechanism of diffusion can be exposed.However, the interstitialcy mechanism addsa minor complication because the “charge”moves two jump distances whereas thetracer moves only one (see Fig. 3-6). Thisis easily taken care of in the analysis so thatHR is now written generally as (see for ex-ample, Murch (1982d))

(3-135)

where r and rq are the distances moved bythe tracer and charge respectively. Pro-vided we focus on cases where fI =1 (eitheralmost empty or full lattice of charge car-riers), then HR still has a unique value foreach mechanism. The classic example ofthe value of HR in identifying the mecha-nism is Ag motion in AgBr (Friauf, 1957).It was found that HR varied from 0.5 at lowtemperature to 0.65 at high temperature.Frenkel defects (see Sec. 3.4.4) are ex-pected and therefore a temperature depen-dence of HR. But HR is not consistent witha combination of vacancy and direct inter-stitial jumps. This would require that HR

varies from 0.78 to unity. The lower valuesof HR obtained require a contribution frominterstitialcy jumps. It was found that bothcollinear and noncollinear interstitialcyjumps were required to fit the behavior ofHR.

Hff

rrq

RI

=⎛

⎝⎜⎞

⎠⎟

2

Much less satisfactorily interpreted arefast ion conductors where the defect con-centration is high and fI must be includedin the analysis of HR. A review of HR forsuch materials has been provided by Murch(1982d). Further comments on the subjectcan be found in the review on correlationeffects in ionic conductivity by Murch andDyre (1989). An introduction to the subjectmay be found in the book by Philibert(1991).

3.3.3 The Isotope Effect in Diffusion

The isotope effect, sometimes called themass effect, is of considerable importancein diffusion because it provides one of thetwo experimental means of gaining accessto the tracer correlation factor, f (the otheris the Haven ratio; see Sec. 3.3.2). Since fdepends on mechanism (see Sec. 3.3.1.3), ameasurement of the isotope effect can, inprinciple, throw light on the diffusionmechanism operating.

The measurement of the isotope effectdepends on accurately measuring a smalldifference between the diffusion coeffi-cients of two tracers a and b (Peterson,1975). This small difference can be relatedto f in the following way. For isotropictracer and impurity diffusion, the diffusioncoefficients of a and b can be written as

Da = Awa fa (3-136)

Db = Awb fb (3-137)

where A contains geometrical terms anddefect concentrations which do not dependon the atom/defect exchange frequency w.

The correlation factors fa and fb have themathematical form of impurity correlationfactors because the tracers (i.e. isotopes) inreal experiments are formally impurities inthe sense that their jump frequencies differfrom the host atoms. Most impurity corre-lation factors for impurity diffusion in oth-

erwise pure and nearly perfect crystalshave the mathematical form

fa = u /(wa + u) (3-138)

fb = u /(wb + u) (3-139)

where u contains exchange frequenciesother than the tracer. After taking the ratioof Eqs. (3-136) and (3-137) and eliminat-ing u and fb (say) from Eqs. (3-136), (3-138), and (3-139), we find that

(Da – Db) /Db = f [(wa – wb) /wb ] (3-140)

At this point, the formal distinction be-tween fa , fb and the tracer correlation fac-tor f, the latter referring to a hypotheticaltracer with the same jump frequency as thehost, can be dropped since they are verysimilar in numerical value. Knowledge ofthe experimental jump frequencies, wa andwb , is usually not available, of course. Ac-cordingly, they are expressed in terms ofquantities which are known, such as themasses of the tracers.

Application of classical statistical me-chanics to describe the dynamics of thejumping process leads to the result

(3-141)

where ma and mb are the masses of thetracers a and b, DK is the fraction of thetotal kinetic energy, associated with thewhole motion in the jump direction at thesaddle point of a jump, that actually be-longs to the diffusing atom. The conversionof jump frequencies to masses has unfortu-nately led to the introduction of this newquantity DK, about which detailed informa-tion is difficult to obtain. If the remainderof the lattice is not involved in the jump,DK = 1 (this is the upper limit for DK).More usually, there is a certain amount ofcoupling between the diffusing atom andthe remainder of the lattice and DK <1.

( )//

w w wa b bb

a− ⎛

⎝⎜⎞⎠⎟

−⎡

⎣⎢⎢

⎤

⎦⎥⎥

= DKm

m

1 2

1

From Eqs. (3-140) and (3-141), we findthat

(3-142)

Eq. (3-142) refers to a process where onlyone atom jumps. The more general case ofn atoms jumping simultaneously, e.g., theinterstitialcy mechanism (where n = 2) (seeFig. 3-6), is described by

(3-143)

where m0 is the average mass of the non-tracers. It is common to refer to f DK inEqs. (3-142), (3-143) as the “isotope ef-fect” E (n). Because f and DK are ≤ 1 thenthis also applies to E (n). Accordingly, al-though a measurement of the isotope effectmay not uniquely determine the mecha-nism, it can be invaluable in showingwhich mechanisms (through their values off and n) are not consistent.

There are some mechanisms for whichthe appropriate correlation factor does nothave the simple form of Eq. (3-136), forexample a mechanism which has severaljump frequencies. Nonetheless, it is alwayspossible to derive equations equivalent to Eqs. (3-142), (3-143) for such cases (seethe review by Le Claire (1970) for details).

In the discussion above, there has per-haps been the implication that the theorydescribed here applies only to tracer diffu-sion in “simple” materials, e.g., pure met-als, and alkali metal halides. Certainly thetheory was originally developed with thesecases in mind, but it can be used, with cau-tion, in other more complex situations. Oneof the earliest of these was application toimpurity diffusion. In this case the tracersa and b now refer to isotopes of an impur-ity which is chemically different from thehost. When the mechanism is not in any

f KD D D

m n m m n m

D

=[( )/ ]

[( ( – ) ) /( ( – ) )] –/a b b

b a

−+ +1 1 10 0

1 2

f KD D D

m mD =

( )/

( / ) /a b b

b a

−−1 2 1



doubt, and frequently it is not to the experi-enced researcher in the field, a measure-ment of the isotope effect for self-diffusionyields DK. If the impurity is substitutional,and provided that DK is assumed to be thesame as for self-diffusion, then an isotopemeasurement for the impurity can give thecorrelation factor for impurity diffusion.Since the impurity correlation factor is afunction of local jump frequencies, e.g., thefive-frequency model in the f.c.c. lattice(see Fig. 3-11), then a knowledge of f com-bined with other knowledge of the samejump frequencies can lead to a knowledgeof some of their ratios (see, for example,Rothman and Peterson (1967), Bocquet(1972), and Chen and Peterson (1972,1973)).

The “impurity form” of the tracer corre-lation factor is also implied for Manning’stheory for vacancy diffusion in the randomalloy model (see Sec. 3.3.1.5). Therefore,isotope effect measurements in concen-trated alloys which are reasonably well described thermodynamically by the ran-dom alloy model can be interpreted alongtraditional lines. Similarly, it has beenshown by Monte Carlo computer simula-tion that the impurity form of f is also validfor vacancy diffusion in a lattice containinga high concentration of randomly distrib-uted vacancies, up to 50% in fact (Murch,1984b).

When order is increased among the com-ponents, the impurity form for f even forvacancy diffusion is increasingly not fol-lowed. For example, in the stoichiometricbinary alloy AB with order, the correlationfactor, forced to follow the impurity form,diverges from the actual f below the or-der/disorder temperature (Zhang et al.,1989b) (see Fig. 3-21). A very similar situ-ation arises for the case of ordered atomsdiffusing by the vacancy mechanism on ahighly defective lattice such as is often en-

countered at low temperature in fast ionconductors (Zhang and Murch, 1997). Theimplication of these and other findings isthat the isotope effect measured in orderedmaterials does not contain the usual tracercorrelation factor but some other correla-tion factor defined only by way of the im-purity form, Eq. (3-136). This does notmean that isotope effect measurements inordered materials are intrinsically withoutmeaning, but simply that they cannot beeasily interpreted.

The actual measurement of the isotopeeffect in diffusion requires the very accu-rate measurement of (Da – Db)/Db . Thisquantity typically lies between 0.0 and0.05. Because of the inaccuracies in mea-surements of D it is not feasible to measureDa and Db in separate experiments. Nor-mally the isotopes a and b are co-de-posited in a very thin layer and permitted to diffuse simultaneously into a thick sam-ple. The geometry of the experiment per-mits Eq. (3-8) to be used for each isotope.

Figure 3-21. Arrhenius plot of Monte Carlo resultsfor the tracer correlation factor in the simple cubiclattice. represents the actual tracer correlation fac-tor; represents the tracer correlation factor forcedto follow the impurity form; after Zhang et al.(1989b).

With a little manipulation it is soon foundthat

(3-144)

This means that by plotting ln (Ca /Cb) vs.lnCa , (Da – Db)/Db can in fact be obtainedvery accurately, avoiding errors due to timeand temperature of the diffusion anneal anderrors in sectioning. The separation of theisotopes is normally achieved by half-life,energy spectroscopy, or the use of differentkinds of radiation. Details of how to carryout careful experiments in this area can befound in the very comprehensive review ofexperimental techniques in diffusion byRothman (1984). A more detailed reviewof the isotope effect in diffusion has beenprovided by Peterson (1975); see also LeClaire (1970).

3.3.4 The Jump Frequency

In Sec. 3.3.1.2 we showed that a tracerdiffusion coefficient is, in essence, a prod-uct of an uncorrelated part containing the jump frequency and a correlated partcontaining the correlation factor, see Eq.(3-76). The Arrhenius temperature depen-dence of the diffusion coefficient ariseslargely from the jump frequency, althoughsome temperature dependence of the corre-lation factor in some situations, e.g., al-loys, cannot be ignored (see Secs. 3.3.1.4to 3.3.1.6). In this section we will study themake-up of the atomic jump frequency.

First, for an atom to jump from one siteto another, the defect necessary to providethe means for the jump must be available.For the interstitial diffusion mechanismthere is essentially always a vacancy (really a vacant interstice) available exceptat high interstitial concentrations. For othermechanisms, however, the atom has to

ln ln [( )/ ]CC

C D D Da

ba a b b

⎛

⎝⎜⎞

⎠⎟− −= const.

“wait” until a defect arrives. The probabil-ity of a defect, say a vacancy, arriving at aparticular neighboring site to a given atomis simply the fractional vacancy concentra-tion cv. However, in certain situations suchas alloys showing order, or fast ion conduc-tors showing relatively high defect order-ing, the availability of vacancies to the atoms can be enhanced or depressed fromthe random mixing value which cv signi-fies. This topic is discussed further in Secs.3.4.2 and 3.4.3.

The probability of a given atom beingnext to a vacancy is thus gcv where g is thecoordination number. The jump frequencyG can we decomposed to

G = g cv w (3-145)

where w is usually called the “exchange”frequency to signify the exchange betweenan atom and a neighboring vacant site andto distinguish it from the actual jump fre-quency G. It is permissible at low defectconcentrations to call w the defect jumpfrequency, since the defect does not have to“wait” for an atom.

In, say, f.c.c. crystals where the jumpdistance r is given by a /÷–

2, where a is thelattice parameter and g =12, the tracer dif-fusion coefficient from Eq. (3-76) can nowbe written as

D* = 12cvw (a2/2) f /6 = cvw a2 f (3-146)

The same equation is also valid for b.c.c.crystals. Other examples have been givenby Le Claire (1975).

In many solids, such as metals, alloysand ionic crystals, cv is said to be “intrin-sic” with an Arrhenius temperature depen-dence. The vacancy formation enthalpy iscontained in the measured activation en-thalpy for the tracer diffusion process (seeEq. (3-149)). In some solids, notably ioniccrystals and certain intermetallic com-pounds, apart from the inevitable tempera-



ture-dependent intrinsic concentration, thevacancy concentration can also be manipu-lated by doping or by changing the degreeof nonstoichiometry. In such cases this va-cancy concentration is sometimes said tobe “extrinsic”, and this contribution to thetotal vacancy concentration can easilyswamp the intrinsic vacancy contribution.The extrinsic vacancy concentration is in-dependent of temperature. However, in thecase of a change in stoichiometry, tempera-ture independence of cv requires adjust-ment of the external partial pressure (seeSec. 3.4.4).

3.3.4.1 The Exchange Frequency

With the defect immediately available,the atom can jump when it acquires suffi-cient thermal energy from the lattice for itto pass over the energy barrier between itspresent site and the neighboring site. Theprobability of the atom having this thermalenergy is given by the Boltzmann probabil-ity exp (– Gm/kT ), where Gm (a Gibbs en-ergy) is the barrier height, k is the Boltz-mann constant, and T is the absolute tem-perature. The attempt frequency, i.e., thenumber of times per second the atom on itssite is moving in the direction of the neigh-boring site, is the mean vibrational fre-quency n–. Accordingly, the “jump rate”,i.e., number of jumps per second, w, isgiven by

w = n– exp (– Gm/kT ) (3-147)

Because Gm is a Gibbs energy, the ex-change frequency can be partitioned as

w = n– exp (S m/k) exp (– H m/kT ) (3-148)

where S m is the entropy of migration andH m is the enthalpy of migration. H m issometimes loosely called the “activation”enthalpy, but this can lead to misunder-standings, as we shall now see.

In Sec. 3.3.4.2 we show that the defectconcentration in many cases depends ontemperature in an Arrhenius fashion (Eq.(3-159)). In such cases, the atomic jumpfrequency G, being the product of w, thedefect concentration, and the coordinationnumber, is written as from Eq. (3-145)

G = g n– exp [(S m + Svf)/k]

¥ exp [– (H m + Hvf)/kT ] (3-149)

The sum of H m and H f is more accuratelycalled the activation enthalpy. We empha-size again that the activation enthalpymeasured in a diffusion experiment can, forsome solids such as ordered alloys, containother contributions such as from correla-tion effects or defect availability terms.

The enthalpy of migration and the en-thalpy of defect formation are now rou-tinely calculated by Lattice or MolecularStatics methods; see Mishin (1997) andreferences therein for examples of suchcalculations in grain boundaries.

Although it is straightforward to couchthe argument above concerning w in statis-tical mechanical terms, this would still beinadequate because we have neglected theparticipation of the remainder of the lattice,especially the neighbors of the jumpingatom, in the jump process. A more satisfac-tory treatment has been provided by Vine-yard (1957). This puts n– on a sounder basisand gives physical significance to the en-tropy of migration S m.

Vineyard (1957) took a classical statisti-cal mechanical approach by consideringthe phase space of the system comprisingall of its N atoms and one vacancy. Eachpoint in this phase space represents anatomic configuration of the system. Allpossible configurations are represented.We are concerned with two neighboringenergy minima in this phase space. Theseminima are centered on two points, P andQ. Point P is the phase space point repre-

senting an atom adjacent to a vacancy; allatoms are in their equilibrium positions.Point Q is the corresponding point after theatom–vacancy exchange. Between thepoints P and Q there is a potential energybarrier. In order to jump, the atom mustpass over this barrier at its lowest point, asaddle point.

Vineyard (1957) was able to calculatethe jump rate, i.e., the number of crossingsper unit time at the “harmonic approxima-tion”. This entailed expanding the potentialenergy at point P and also at the saddlepoint in a Taylor series to second order togive for w

(3-150)

where the ni are normal frequencies for vi-bration of the system at point P, ni¢ are thenormal frequencies at the saddle point andU is the difference in potential energybetween the saddle point and the equilib-rium configuration point, P.

It is possible to transform Eq. (3-150)into the form of Eq. (3-148) by writing

(3-151)

where the entropy of migration can beidentified with

(3-152)

and

(3-153)nn

n= =

=

i

N

i

i

N

i

1

3

2

30

∏

∏

S ki

N

i im

==

2

30∑ ′ln ( / )n n

i

N

i

i

N

i

S k T=

=

m=1

1

1

∏

∏−

′

n

nn exp ( / )

wn

n= =

=

i

N

i

i

N

i

U k T1

1

∏

∏ ′

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

−exp ( / )

where ni0 is the frequency of the ith normal

mode with the system constrained to movenormal to the direction joining point P tothe saddle point.

There have been many further considera-tions of the detailed dynamics of the jumpprocess, the most notable of these being thework carried out by Jacucci, Flynn and co-workers (see reviews by Jacucci (1984)and Pontikis (1990)).

For general purposes it is often sufficientto focus on Eq. (3-148) and to note that theDebye frequency is an adequate represen-tation of the mean vibration frequency n–.

3.3.4.2 Vacancy Concentration

The change in Helmholtz energy F fV as-

sociated with the formation of one vacancyis given by

F fV = E f

V – T S fV (3-154)

where E fV and S f

V are the energy and en-tropy of formation, respectively. The for-mation process itself is conceived to be theremoval of an atom from the interior of thecrystal to the surface. The entropy part hastwo contributions: a vibrational or thermalpart S f

vib arising from the fact that atomsclose to the vacancy have a different vibra-tional frequency from those far from thevacancy, and a configurational part S f

config

which is usually thought to be an ideal mix-ing entropy, at least for pure metals – but itis rather more complicated for alloys. Theconfigurational part for pure metals is eas-ily found from elementary classical statisti-cal mechanics (see, for example, Peterson(1978)). The number of different ways Wof putting n vacancies and N atoms on N + nsites assuming indistinguishability withineach group is

(3-155)W =( )!

! !N nN n

+


3.4 Diffusion in Materials 219

The configurational entropy is given by

(3-156)

which, with Stirling’s approximation lnN != N lnN – N for large N, results in

(3-157)The total change in free energy when n va-cancies are produced is

n F fV = n E f

V – T (n S fconfig) (3-158)

After substitution of Eq. (3-157) and put-ting the derivative ∂F/∂n = 0 to obtain theequilibrium number of vacancies, we soonfind that

(3-159)

where cv is the site fraction of vacancies. Note that the configurational term has dis-appeared. The vibrational contribution S f

vib

is considered to be small and the leadingterm in Eq. (3-159) is usually ignored, and

cv = exp (– E fV /kT ) (3-160)

For low pressures we can also assume (notalways safely) that the energy of formationE f

V can be approximated by the enthalpy offormation H f

V.Divacancies are handled in a similar way

(see, for example, Peterson (1978)). Fren-kel and Schottky defects in ionic crystals(see Sec. 3.4.4) can also be analysed alongsimilar lines (Kofstad, 1972). Alloys, how-ever, present a special problem because ofuncertainties about reference states includ-ing the typical surface site where an atomis to be symbolically placed (see, for exam-ple, Lim et al. (1990) and Foiles and Daw(1987)).

Calculations of defect formation ener-gies are frequently handled by lattice relax-

nN n

c S k E k T+

−= =v vibf

Vfexp( / ) exp( / )

S k NN

N nn

nN nconfig

f = −+

⎡⎣⎢

⎤⎦⎥

++

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

ln ln

S k kN nN nconfig

f = =ln ln( )!

! !W +⎡

⎣⎢

⎤⎦⎥

ation techniques with computer codes suchas DEVIL and CASCADE (these tech-niques are described in the book edited byCatlow and Mackrodt, 1982) and latticedynamics and Monte Carlo methods (see,for example, Jacucci, 1984).

Vacancy concentrations in metals are con-ventionally measured by quenching, dila-tometry, and positron annihilation. An intro-duction to these techniques has been pro-vided by Borg and Dienes (1988), and a de-tailed review is also available (Siegel, 1978).

3.4 Diffusion in Materials

Probably every type of solid has, at onetime, been investigated for its diffusion be-havior. It is impossible in the space here tocover even the major findings. In this sec-tion we propose to discuss, at an introduc-tory level, diffusion in metals and alloys,and, as an example of ionic crystals, oxides. Our emphasis here is on the usualtheoretical framework for a description ofdiffusion in these materials rather than ondata compilations or reviews. Diffusiondata compilations for metals and alloys canbe found in the Smithells Metals ReferenceBook (Brandes, 1983) and in the extensivecompilations in Landolt-Börnstein (Meh-rer, 1990). The older tracer diffusion dataup to 1970 on metals and oxides has beencollected by Askill (1970) and data on oxides up to 1970 on metals and oxides has been collected by Askill (1970) anddata on oxides up to 1980 by Freer (1980).Extensive compilations of diffusion data in nonmetallic solids have been publishedin Landolt–Börnstein (Beke, 1998 1999).The journal Defect and Diffusion Form,formerly Diffusion and Defect Data, regu-larly publishes abstracts of all diffusion-re-lated papers and extensive indices are reg-ularly provided.

3.4.1 Diffusion in Metals

3.4.1.1 Self-Diffusion

We have seen (Sec. 3.3.4) that the tracerdiffusion coefficient in solids normally fol-lows an Arrhenius form. This is usuallywritten empirically as

D = D0 exp (– Q /RT ) (3-161)

where D0 is called the pre-exponential fac-tor, or sometimes the “frequency factor”, Qis the activation energy for diffusion, R isthe ideal gas constant, and T is the absolutetemperature. In most pure metals self-dif-fusion is characterized by a pre-exponen-tial D0 which falls in the range 10–3 to5 ¥10–6 m2 s–1. This corresponds to activa-tion entropies which are positive and of theorder of k, the Boltzmann constant. The ac-tivation energy Q is given fairly closely(± ≈10%) by Q = 34 TM.Pt. , where TM.Pt. isthe melting point. These values lead to avalue of the self-diffusion coefficient at themelting point of about 10–12 m2 s–1.

Careful dilatometric and quenching ex-periments on a large number of f.c.c. met-als indicate the vacancy as being the defectresponsible for “normal” diffusion in f.c.c.metals. For vacancy diffusion the tracerdiffusion coefficient is written for cubiclattices as

D* = g cvw r2 f /6 (3-162)

where g is the coordination, cv is the va-cancy concentration, w is the vacancy–atom exchange frequency, r is the jumpdistance, and f is the tracer correlation factor. The decomposition of cv into its enthalpy/entropy parts is given by Eq. (3-159). The decomposition of w is givenin Eq. (3-148). The tracer correlation factoris discussed in Sec. 3.3.1.3.

For some f.c.c. metals, e.g., silver (Roth-man et al., 1970; Lam et al., 1973), where

diffusion measurements have been madeover a very wide temperature range, thereis a slight curvature in the Arrhenius plots.This has been variously attributed to a con-tribution to diffusion from divacancies (see,for example, Peterson (1978)), or from va-cancy double jumps (Jacucci, 1984). Be-cause of the high activation energy, how-ever, the latter is really only a candidate forexplaining curvature very close to meltingtemperature and not over a wide tempera-ture range.

There is still controversy over the causeof the curvature in Ag and other f.c.c. met-als, with one view favoring a temperature-dependent activation energy and diffusionby single vacancies (see the review byMundy (1992)), and the other tending to fa-vor, in part, a contribution from divacan-cies at high temperatures (Seeger, 1997).The behavior of self-diffusion in b.c.c.metals is quite varied. Some b.c.c. metalssuch as Cr show linear Arrhenius plots.The alkali metals show slightly curvedplots. Finally, there is a large group of“anomalous” metals including b-Zr, b-Hf,b-Ti, g-U, and e-Pu, which show a strongcurvature and show values of D0 and Qboth of which are anomalously low. Therehas been considerable controversy herealso. Again, the monovacancy mechanismand a temperature-dependent activation en-ergy have been strongly supported (Mundy,1992) but this view alone does not seem tobe entirely consistent with the nature of thecurvature. Other processes such as ringmechanisms and Frenkel pair formation/re-combination may be operative (Seeger,1997).

3.4.1.2 Impurity Diffusion

The appropriate diffusion coefficient foran impurity A (at infinite dilution) in a hostof B when the vacancy mechanism is oper-



ating is (for cubic lattices)

DA = g pAV wA r2 fA/6

= D0A exp (– QA/RT ) (3-163)

where g is the coordination, pAV is the va-cancy availability factor to the impurity,wA is the impurity–vacancy exchange fre-quency, r is the jump distance, and fA is theimpurity correlation factor (for details seeSec. 3.3.1.4). Again, as for self-diffusion inmetals, in many cases we can discern “nor-mal” behavior wherein the Arrhenius plotsare linear and D0

A and QA do not differgreatly from the self-diffusion values. Aspointed out by Le Claire (1975, 1978), therelative values are determined principallyby DQ = QA – QB, where QB is the self-dif-fusion activation energy for the host. WhenDA> DB, the impurity is a fast diffuser andDQ is negative. It is often found that this iscorrelated to a situation where the valenceof the impurity is greater than that of thehost. The converse is true when DA< DB.

The similarity between impurity and hostdiffusivities suggest the vacancy mecha-nism for both. For this mechanism it caneasily be shown that (see, for example, LeClaire, 1978)

DQ = (HAm – HB

m) + EB – Q ¢ (3-164)

where HAm, HB

m are the activation enthal-pies of migration for the impurity and host,EB is the vacancy–impurity binding en-thalpy, and Q ¢ is the activation enthalpyarising from the temperature dependenceof fA (see Eq. (3-90)). This is normally afairly small contribution.

There are “anomalous” impurity diffus-ers which have diffusion coefficients muchhigher than the host. Well-known examplesare the diffusion of noble metals in Pb. Ashas been discussed in conjunction with thesubstitional-vacancy diffusion mechanism,the impurities probably exist partly as

interstitials and partly as substitutionals(see Sec. 3.3.1.1). For a full discussion ofall aspects of impurity diffusion we refer toLe Claire (1978).

3.4.2 Diffusion in Dilute Alloys

3.4.2.1 Substitutional Alloys

Alloys containing something less thanabout 1–2% solute concentrations are con-sidered to be sufficiently dilute that the dif-fusion of solute atoms can be considered interms of isolated atoms or isolated group-ings of atoms such as pairs. For many bi-nary dilute alloys, measurements of the solute diffusion coefficient D2 (measuredas a tracer diffusion coefficient) have beenmade as a function of solute concentration(we use the subscript 2 to denote the soluteand 0 to denote the solvent). It is usual torepresent the solute diffusion coefficientempirically as

D2(c2) = D2(0) (1+ B1c2 + B2 c22 …) (3-165)

where D2(0) is the solute diffusion coeffi-cient for c2 Æ 0, c2 is the atomic fraction ofsolute and B1, B2, … are termed solute en-hancement factors. D2(0) is also called theimpurity diffusion coefficient (at infinitedilution) depending on the context of theexperiment.

Similarly, solvent diffusion coefficientsD0, also measured as tracer diffusion coef-ficients, are represented as

D0(c2) = D0(0) (1+ b1c2 + b2 c22 …) (3-166)

and b1, b2, … are termed solvent enhance-ment factors.

Of interest are the solute and solvent en-hancement factors. Naturally, also of inter-est is the composition (a) of values of thesolute and solvent diffusion coefficientsand (b) of values of their respective activa-tion energies.

In Eq. (3-166) D2(0) arises from isolatedsolute atoms, while the term containing B1

arises from pairs of solute atoms which aresufficiently close that the solute jump fre-quency differs from the isolated solute–va-cancy exchange frequency w2. B2 arisesfrom triplets of solute atoms.

The solute enhancement factor B1 hasbeen calculated for the f.c.c. lattice by ex-tending the five-frequency model (see Sec.3.3.1.4) to include three new frequencies.These frequencies describe solute jumpswhich create, i.e., associate, a new solutepair (w23), dissociate a solute pair (w24),and reorient an existing pair (w21). Pairs ofsolute atoms do not occur randomly whensolute atoms interact. When an interactionenergy E22 between solute atoms is definedit is straightforward to show that

w24 exp (– E22 /kT ) = w23 exp (– E2B /kT )

where E2B is the impurity–vacancy bind-ing energy. If it is assumed that the impur-ity correlation factor does not depend onsolute concentration (Stark, 1972, 1974),then B1 is given by (Bocquet, 1972; LeClaire, 1978)

If it is assumed that E22 = 0 and thatw21=w23 =w2 then B1 is reduced to

B1 = 18 [exp (– E2B /kT ) – 1] (3-167a)

There have been no calculations along sim-ilar lines of B2.

The solvent enhancement factor b1 canbe calculated by noting the number of jumpfrequencies close to the solute. For in-stance, in the five-frequency model for thef.c.c. lattice we need to count the numberof solvent frequencies which differ from

B E k TE k T

E k T

1 22

21

222

23

2

6 12

4 14

= (3-167)

2B

− + −− −

× − −⎡⎣⎢

⎤⎦⎥

[ exp( / )]exp( / )

exp( / )ww

ww

w0, the solvent exchange frequency farfrom the solute. For the f.c.c. lattice the re-sult for b1 is (Howard and Manning, 1967)

(3-168)

where c1 and c2 are termed partial correla-tion factors and are known functions of theratios w2/w1, w1/w3 and w4/w0, and f0 isthe correlation factor in the pure lattice.

Eq. (3-168) can be re-expressed as

(3-169)

thereby showing the effect of an altered va-cancy concentration near a solute by virtueof the solute–vacancy binding energy E2B.

There are similar relations for the b.c.c.lattice. For the impurity model described in Sec. 3.3.1.4 (Model I), b1 is given by (Jones and Le Claire, 1972)

(3-170)

and for Model II (see Sec. 3.3.1.4)

(3-171)

where u1, n1 (w2/w¢3, w3/w¢3) and u2, n2

(w2/w3, w4/w0) are called mean partialcorrelation factors and are known functionsof the frequencies given.

It is possible to express b1 in terms of theratio of the solute and solvent diffusion co-efficients. This aids in the discussion of thenumerical values taken by the enhance-ment factors. For the f.c.c. lattice, b1 isgiven by Le Claire (1978).

buf

fE k T

12

3

0

2

20 6

14

=

2B

− +

+ −exp( / )ww

n

buf

fE k T

11

3

0

1

20 14

6

=

2B

− +

+ −exp( / )ww

n

bf f

E k T

11 1

0

2

0

3

018 4 14=

2B

− + +⎡⎣⎢

⎤⎦⎥

× −

c ww

c ww

exp( / )

bf f1

4

0

1

0

1

3

2

018 4 14= − + +⎡

⎣⎢⎤⎦⎥

ww

c ww

c



(3-172)

where F has been given in Sec. 3.3.1.4 (seeEq. (3-80)) and f0 is the tracer correlationfactor in the pure lattice. The term in brack-ets is roughly unity and the impurity corre-lation factor f2 is about 0.5. The greater thesolute diffusion coefficient is comparedwith the solvent diffusion coefficient, thenthe more likely it is that b1 is greater thanzero. On the other hand, for slow solutediffusers b1 will probably be negative butnot less then –18. A few examples takenfrom Le Claire (1978) are given in Table 3-2.

The solvent enhancement factor b2 re-sults from the change in the solvent jumpfrequencies in the vicinity of pairs of solute atoms. Expressions for b2 for thef.c.c. lattice have been given by Bocquet(1972).

As noted by Le Claire (1978), the ex-pressions for b1 and B1 are similar in form.Provided that the frequency ratios in oneequation do not greatly differ from theother, as might be expected from relativelyweak perturbations caused by the solute,then b1 and B1 tend to have the same signand roughly comparable magnitudes, apartfrom the exp (– E22/kT ) term in the expres-

bff

DD f F1

0

2

2

0

1 1 3 2

0 1 318

41

4 144 14

= − +−

++

⎡⎣⎢

⎤⎦⎥

c w w cw w

( / )( / )

sion for B1. These comments are borne outby the data in Table 3-2.

The relative values of the diffusion coef-ficients of solute and solvent are dictatedlargely by the difference in activation ener-gies for solute and solvent diffusion ratherthan by differences in the pre-exponen-tial factors. The difference DQ = Q2 – Q0

(where Q2 and Q0 are the activation ener-gies for solute and solvent respectively) of-ten seems to be closely related to the differ-ence in valencies of solute and solvent, Z2

and Z0. When DQ is negative (fast solutediffusion), Z2 > Z0. On the other hand,when DQ is positive Z2 < Z0. These mattersare discussed further in a detailed reviewby Le Claire (1978). The reader is also di-rected to a recent commentary (Le Claire,1992).

An approximate approach which acts asa transition between the models describedabove and the concentrated alloy models ofSec. 3.4.3 is the complex model initiatedfirst by Dorn and Mitchell (1966) and de-veloped by Faupel and Hehenkamp (1986,1987). When impurities (B) have a positiveexcess charge, vacancy–impurity com-plexes of one vacancy and i impurity atomsare likely to form. Assuming that the bind-ing free energy (– GBi) is independent ofthe configuration of the impurity atoms,Dorn and Mitchell (1966) wrote for the

Table 3-2. Some values of solute and solvent enhancement factors, from Le Claire (1978).

Alloy system Lattice D2/D0 b1 B1 T (°C)(Solvent–solute)

Ag–Pd f.c.c. 0.04 – 8.2 – 7.5 730Ag–Au f.c.c. 3.7 7.0 5.5 730Ag–Cd f.c.c. 5.3 13.0 8.5 730Ag–In f.c.c. 8.4 37 30 730Ag–Tl f.c.c. 0.26 – 1.2 – 0.56 730b-Ti–V b.c.c. 1.26 – 4.3 – 2.5 1400V–Ti b.c.c. 1.66 16 27 1400

a-Fe–Si b.c.c. 1.64 20.4 12.4 1427

mole fraction of vacancies in the ith com-plex cvi

(3-173)

where cv(0) is the mole fraction of vacan-cies in the pure metal (A) and c0 and c2 arethe mole fractions of solvent (A) and solute(B) respectively, and g is the coordination.Bérces and Kovács (1983) modified Eq. (3-173) to take into account possible configu-rations of the complexes.

Within this complex model frameworkFaupel and Hehenkamp (1986, 1987) haveintroduced average effective jump fre-quencies per solvent (A) atom ·wA

effÒi andaverage effective jump frequencies per solute (B) atom ·wB

effÒi in the ith complex(the latter specifically includes correlationeffects – which are always important forthe solute). The normalized solvent diffu-sion coefficient can now be written as

(3-174)

Similarly, the normalized solute diffusioncoefficient can be written as

These equations for the normalized soluteand solvent diffusion coefficients can beexpanded in terms of c2 to give explicit ex-pressions for the various solute and solventenhancement factors by way of comparisonwith Eqs. (3-165) and (3-166); see LeClaire (1992).

It has been found, in a number of carefulstudies in f.c.c. systems (Ag–Sb, Ag–Sn)where the impurities have excess positivecharge, that solvent diffusion on alloying is

D cD

ci

f i

c c G G k T

i

i

ii

0 2

00

1

2 2 2

01

21

1

0( )( )

exp[( )/ ]

= (3-175)=

effB

B B

gg

g

gg−

− −

− ⟨ ⟩ ⎛⎝⎜

⎞⎠⎟

× −

∑ ww

D cD

cf i

c c G k T

i

i

i ii

0 2

00

1

1

1

0 0

01

2

0

1( )( )

exp( / )

==

effA

B

gg

g

g−−

− −

− ⟨ ⟩ −⎛⎝⎜

⎞⎠⎟

×

∑ ww

c c c c G k Tii i

iv v B= ( ) exp( / )00 0 2g g⎛

⎝⎜⎞⎠⎟

−

enhanced in a nonlinear way (Hehenkampet al., 1980; Hehenkamp and Faupel, 1983).Moreover, independent measurements ofthe vacancy concentration on alloying haveshown a linear dependence between nor-malized vacancy concentration and thenormalized solvent diffusion coefficient(Hehenkamp et al., 1980; Hehenkamp andFaupel, 1983). The nonlinear enhancementof solvent diffusion upon alloying is thusapparently due to a corresponding nonlin-ear increase in the vacancy concentrationon alloying. These findings are very nicelydescribed by the complex model.

Despite its obvious limitations, the com-plex model can be useful quantitatively upto 5 at.% of solute and qualitatively evenhigher; see Le Claire (1992).

3.4.2.2 Interstitial Alloys

Elements such as H, N, O, and C dis-solve interstitially in metals and diffuse bythe interstitial mechanism. The diffusioncoefficients here are often measured by re-laxation techniques or outgassing. Tracerdiffusion is difficult to measure in the caseof N and O because of the lack of suitableradioisotope; however, other techniquesare available (see Sec. 3.5.1).

For interstitials at infinite dilution, theinterstitials move independently (by theinterstitial mechanism, see Fig. 3-5). Thevacancy concentration (vacant interstice) isunity, the correlation factor is unity (a com-plete random walk), and the tracer diffu-sion coefficient is given simply by (fromEqs. (3-76) and (3-145))

D* = g w0 r2/6 (3-176)

where w0 is the interstitial/vacant interstice“exchange” frequency.

In a number of cases the interstitial con-centration can become sufficiently high forthe interstitials to interfere with one an-



other and this interference may have an im-portant effect on diffusion. An example isC in austenite where the carbon interstitialscan fill roughly 8% of the octahedral voids.The two theoretical approaches taken tocope with the problem are reminiscent ofthe situation found in the modeling of sub-stitutional alloys (see Sec. 3.4.2.1). In thefirst, the exchange frequencies for the so-lute are specified. In the second, the ex-change frequencies are specified indirectlyby way of interaction energies.

For the first approach, McKee (1980a,b) and Le Claire (1981) conceived a four-frequency model for interstitial solute dif-fusion in the f.c.c. lattice (the octahedralvoids of the f.c.c. lattice). Two distinct spe-cies are considered: the isolated interstitialwith a jump frequency of k0, and pairedinterstitials, which rotate with frequencyu1, dissociate with frequency u3, and asso-ciate with frequency u4. The following ex-pression for the tracer diffusion coefficientof the interstitial solute can then be derived(Le Claire (1981) using Howard’s (1966)random walk method)

where cp is the site fraction concentrationof paired interstitials and c is the site frac-tion of interstitials. Using the pair associa-tion method, McKee (1980a, b) derived asimilar result valid for weak binding of thepaired interstitials. Explicit expressions forthe tracer correlation factor can be found inthe original papers.

It was also possible to determine an ex-pression for the chemical diffusion coeffi-cient, D, relating to diffusion of the solutein its own concentration gradient. In theweak binding limit the expression for D is

D r kc

cu u

u uu u

k kuu

*

( )( )

= (3-177)p46

4 14

2 32 7

12 7

20 1 3

1 32

1 30 0

3

4

+⎡⎣⎢ +⎛

⎝⎜

− −+

− −⎞⎠⎟

⎤

⎦⎥⎥

(McKee, 1981)

D = –13

r2 [12 k0 + 12 u4/u3]

¥ [4 u1 + 7 u3 – 12 k0 + 7 (u4 – k0) u3/u4]

¥ c 1 + c [1 + 12 (1 – u4/u3)] (3-178)

where r is the jump distance. Eq. (3-178)contains a thermodynamic factor.

The above model, where the frequenciesare specified explicitly, is probably onlyvalid for 1–2% of interstices occupied. Athigher concentrations many more frequen-cies are required with the result that thisapproach becomes unwieldy. For the sec-ond approach a lattice gas model is used torepresent the solute interstitials and theirlattice of interstices. High concentrationsof interstitials can in principle be handledwith this model. In this rigid model the at-oms are localized at sites and pairwiseinteractions among atoms are specified.The exchange frequency of a solute atom toa neighboring site can be written as (Satoand Kikuchi, 1971)

w = n exp(gnn fnn /kT ) exp(–U/kT ) (3-179)

although many other choices are possible.In Eq. (3-179) n is the vibration frequency,gnn is the number of occupied intersticesthat are nearest neighbors to a solute, fnn isthe solute–solute “binding” energy (posi-tive or negative), and U is the migration en-ergy for an isolated solute atom. In effectthe solute neighbors can assist or impedethe diffusion process depending on the signof fnn .

By its nature this model requires a statis-tical mechanical approach for its analysis.One method used is the PPM (Sato andKikuchi, 1971; Sato, 1989, see also Sec.3.3.1.5); another is the Monte Carlo com-puter simulation method (Murch, 1984a).It is convenient to decompose the tracerdiffusion coefficient into

D* = g r2 V W f exp (– U/kT )/6 (3-180)

where V is the vacancy availability factor(the average availability of neighboring va-cant interstices to an interstitial soluteatom, and is a more general quantity thancv), and W represents the effect of the envi-ronment on the jump frequency. It is in ef-fect the statistical average of the first termin Eq. (3-179). The same lattice gas modelhas also been applied extensively to fastion conductors (see, for example, Murch(1984a)).

Chemical diffusion can readily be ex-pressed along similar lines (Murch, 1982c)as

D = g r2 V Wn fI (3-181)

¥ exp (– U/kT ) (∂m /kT/∂ lnc)/6

where m is the chemical potential of the solute and fI is the physical correlation fac-tor (see Sec. 3.3.1.7). The physics for fI

are not contained in the first approach (Eq.(3-178)).

These approaches seem to describe Ctracer and chemical diffusion in g-Fe fairlywell (McKee, 1980a, b, 1981; Murch andThorn, 1979) but other applications havenot been made because of the lack of suit-able data, especially for D*.

Quantum effects are an important ingre-dient in the description of H diffusion inmetals; see the reviews by Völkl and Ale-feld (1975), Fukai and Sugimoto (1985),and Hempelmann (1984).

3.4.3 Diffusion in Concentrated BinarySubstitional Alloys

Dilute alloy models, see Sec. 3.4.2.1,cannot be extended very far into the con-centrated regime without rapidly increas-ing the number of jump frequencies to anunworkable level. As a result, models havebeen introduced which limit the number ofjump frequencies but only as a result of

some loss of realism. The first of these isthe random alloy model introduced byManning (1968, 1970, 1971). The secondis the interacting bond model which hasbeen extensively developed by Kikuchiand Sato (1969, 1970, 1972).

In the random alloy model, the atomiccomponents are assumed to be mixed ran-domly and the vacancy mechanism is as-sumed. The atomic jump frequencies wA

and wB for the two components A and Bare specified and these do not change withcomposition or environment. The tracerdiffusion coefficient of, say, A is given by

D*A = g wAr2cv fA/6 (3-182)

where g is the coordination, r is the jumpdistance, and cv is the vacancy site fraction(see Sec. 3.3.4.2). The only theoretical re-quirement here is to calculate the tracercorrelation factor fA; this has been de-scribed in Sec. 3.3.1.5 on correlation ef-fects in diffusion. The random alloy modelperforms very well in describing the diffu-sion behavior of many alloy systems whichare fairly well disordered. This subject isdiscussed extensively in the review byBakker (1984) to which we refer. A muchmore accurate method for dealing with fAhas been given by Moleko et al. (1989).

For alloys which exhibit order, Man-ning’s theory, based on the random alloymodel, still performs reasonably well indescribing certain aspects of diffusion be-havior, notably correlation effects whichcan be expressed in terms of the tracer dif-fusion coefficients, see Sec. 3.3.1.8 and thereview by Murch and Belova (1998). Awholly different approach pioneered byKikuchi and Sato (1969) is the PPM (seeSec. 3.3.1.5) in which the tracer diffusioncoefficient is expressed in terms of quan-tities which are statistically averaged overthe atomic configurations encountered inthe alloy. The tracer diffusion coefficient



of, say, A is now written as

D*A = g w–A r2 p–Av f–A/6 (3-183)

where w–A is an averaged exchange fre-quency which includes the effect of the en-vironment on the jump frequency, p–Av is thevacancy availability factor, i.e., the prob-ability of finding a vacancy next to an Aatom (see below), and f

–A is the tracer corre-

lation factor. We have discussed the corre-lation factor f

–A in Sec. 3.3.1.5, to which we

refer. The quantity w–A is the average of theexchange frequency for an A atom with avacancy in particular configuration:

wA = nA exp (– UA/kT ) (3-184)¥ exp [(gA EAA + gB EAB)/kT]

where nA is the vibration frequency, UA isthe migration energy (referred to some ref-erence) for a jump, gA is the number of Aatoms that are nearest neighbors to a givenA atom (itself next to a vacancy), gB is thenumber of B atoms which are likewisenearest neighbors to the A atom, and EAA

and EAB are nearest neighbor interactions(assumed negative here). This equation isessentially the binary analogue of Eq. (3-179). We see that the neighbors of the atomA can increase or decrease the apparent mi-gration energy.

The use of cv alone in the expression forD*A in Eq. (3-182) implies that either com-ponent “sees” the vacancy equally. Thisobviously cannot generally be true. The in-troduction of the quantity p–Av in Eq. (3-183) (and p–Bv) is a recognition that vacan-cies are somewhat apportioned between thetwo atomic components. This is not a smalleffect, particularly in alloys which showseparate sublattices.

The overall activation energy for diffu-sion is now rather complicated. Apart fromthe usual contribution from the vacancyformation energy and reference migrationenergy, we have a complicated contribution

from w–A because the configurations changewith temperature. We also have contribu-tions from p–Av and the tracer correlationfactor. In order to make sense of the experi-mental activation energies, we need tomodel the system in question; there seemsto be no alternative. Few detailed applica-tions of this model have yet been made.One detailed application has been made tob-CuZn (Belova and Murch, 1998). Fur-ther details can be found in the review byBakker (1984), see also the review byMurch and Belova (1998).

Interdiffusion in binary concentrated al-loys has been dealt with largely in Secs.3.2.3.2 and 3.3.1.8. Some recent calcula-tions (which use the “bond model” de-scribed above) have been performed byZhang et al. (1988) and Wang and Akbar(1993), see also the review by Murch andBelova (1998).

3.4.4 Diffusion in Ionic Crystals

In this section diffusion in ionic crystals(exemplified here by oxides) will be brieflydiscussed. A more detailed discussion hasbeen provided by Kofstad (1972) andSchmalzried (1995). Diffusion in otherionic crystals, especially alkali metal ha-lides and silver halides, has been reviewedby, for example, Fredericks (1975), Laskar(1990, 1992), Monty (1992).

3.4.4.1 Defects in Ionic Crystals

In order to discuss diffusion in ioniccrystals, we need first to discuss at an ele-mentary level the types of defects whicharise. Of almost overriding concern inionic crystals is the requirement of chargeneutrality.

Let us first consider stoichiometric crys-tals, say the oxide MO, and “intrinsic” de-fect production. The Schottky defect (actu-

ally a pair of defects) consists of a vacantanion site and a vacant cation site. It arisesas a result of thermal activation and notthrough interaction with the atmosphere.The Schottky defect generation is writtenas a chemical reaction, i.e.,

0 [ V≤M + VO•• (3-185)

where 0 refers to a perfect crystal, V≤M is avacant (V) metal (M) site, the primes referto effective negative charges (with respectto the perfect crystal), VO

•• is a vacant (V)oxgen (O) site and the dots refer to effec-tive positive charges. This is the Kröger-Vink defect notation (see, for example,Kofstad (1972)).

We can write an equilibrium constant Ks

for the reaction in the following form, validfor low defect concentrations:

Ks = [V≤M] [VO••] (3-186)

where the brackets [ ] indicate concentra-tions. Eq. (3-186) is usually called theSchottky product. Ks can be expressed inthe usual way

Ks = exp (– Gsf /kT ) (3-187)

where Gsf is the Gibbs energy of formation

of the Schottky defect, which can be parti-tioned into its enthalpy Hs

f and entropy Ssf

parts.The other type of defect occurring in the

stoichiometric ionic crystal is the Frenkeldefect. The Frenkel defect (actually a pairof defects) consists of a cation interstitialand cation vacancy or an anion interstitialand anion vacancy. In the latter case it hasalso been called an anti-Frenkel defect, al-though this nomenclature is now relativelyuncommon. Like the Schottky defect, theFrenkel defect is thermally activated. TheFrenkel defect generation is also written asa chemical reaction, for example for cat-ionic disorder:

MM [ M i•• + V≤M (3-188)

where MM is a metal atom (M) on a metalsite (M), M i

•• is an effectively doubly posi-tively charged metal ion on interstitial i siteand V≤M is an effectively doubly negativelycharged metal ion vacancy. We have as-sumed double charges here purely for illus-trative purposes.

The equilibrium constant for the Frenkeldefect reaction can be written as

KF = [M i••] [V≤M] (3-189)

provided that the defect concentration islow. This equation is often called the Fren-kel product. Again, the equilibrium con-stant KF can be expressed as exp (– Gf

F /kT )where Gf

F is the Gibbs energy of formationof the Frenkel defect, which again can bepartitioned into its enthalpy H f

F and en-tropy Sf

F parts.

3.4.4.2 Diffusion Theory in Ionic Crystals

In most mechanisms of diffusion exceptthe interstitial mechanism, an atom must “wait” for a defect to arrive at a nearestneighbor site before a jump is possible (seeSec. 3.3.4). Thus the jump frequency in-cludes a defect concentration term, e.g., cv,the vacancy concentration. Let us examinean example for diffusion involving theFrenkel defect. Although both an intersti-tial and a vacancy are formed, in oxidesone of them is likely to be much more mo-bile, i.e., to have a lower migration energy,than the other. (In the case of stoichiomet-ric UO2, for example, theoretical calcula-tion of migration energies suggests a muchlower migration energy for the oxygen va-cancy than for the interstitial (by eitherinterstitial or interstitialcy mechanisms)(Catlow, 1977).) At the stoichiometriccomposition,

[M i••] = [V≤M] (3-190)



so that the cation vacancy concentration isgiven by

[V≤M] = cv = exp (– GfF /2kT ) (3-191)

= exp(– H fF /2kT ) exp(S f

F /2k) (3-192)

The measured activation enthalpy for dif-fusion will be the migration enthalpy plushalf the Frenkel defect formation enthalpy.The reader might well ask how we knowwe are dealing with a Frenkel defect andnot a Schottky defect, and if we do know itis the Frenkel defect then how do we knowit is the vacancy mechanism that is operat-ing and not the interstitialcy mechanism?In general, we have to rely on independentinformation, principally computer calcula-tions of defect formation and migration en-thalpies, but structural information such asis provided by neutron diffraction and ther-modynamic information is also useful.

Another process of interest here is the in-trinsic “ionization” process whereby anelectron is promoted from the valence bandto the conduction band leaving behind ahole in the valence band. In the Kröger-Vink notation, we write for the intrinsicionization process (the electrons and holesmay be localized on the metal atoms)

0 [ e¢ + h• (3-193)

All ionic crystals are capable, in princi-ple, of becoming nonstoichiometric. Thelimit of nonstoichiometry is largely dic-tated by the ease with which the metal ioncan change its valence and the ability of thestructure to “absorb” defects without re-verting to some other structure and therebychanging phase. Nonstoichiometry can beachieved by either 1) an anion deficiency,which is accommodated by either anion va-cancies, e.g., UO2–x , or metal interstitials,e.g., Nb1+yO2 or 2) an anion excess, whichis accommodated by either anion intersti-tials, e.g., UO2+x or metal vacancies, e.g.,

Mn1–yO. As an example we will deal withan anion excess accommodated by metalvacancies.

The degree of nonstoichiometry andtherefore the defect concentration accom-panying it are functions of temperature andpartial pressure of the components. The de-fects produced in this way are sometimessaid to be “extrinsic”, but this terminologyis probably to be discouraged since theyare still strictly intrinsic to the material. Inan oxide it is usual, at the temperatures ofinterest, to consider only the partial pres-sure of oxygen since it is by far the morevolatile of two components. In carbides,where the temperatures of diffusion ofinterest are much higher, the metal partialpressure can be comparable to the carbonpartial pressure and either can be manipu-lated externally.

The simplest chemical reaction generat-ing nonstoichiometry is

–12

O2(g) [ O¥O + V¥

M (3-194)

where O¥O is a neutral (¥) oxygen ion on an

oxygen site and V¥M is a neutral metal va-

cancy. Physically, this corresponds to oxy-gen being adsorbed on the surface to formions and more lattice sites, thereby effec-tively making metal vacancies. These va-cancies diffuse in by cation–vacancy ex-change until the entire crystal is in equilib-rium with the atmosphere. Excess oxygennonstoichiometry corresponds also to oxi-dation of the metal ions. Here in this exam-ple, the holes produced are conceived to beassociated very closely with the metal va-cancies. The holes can be liberated, inwhich case the reaction can now be written

–12

O2(g) [ V¢M + O¥O + h• (3-195)

If the hole is then localized at a metal ion,we can consider this as the chemical equiv-alent of having M3+ ions formally presentin the sublattice of M2+ ions.

Assuming reaction Eq. (3-194) is thepreferred one, we find that the equilibriumconstant for this reaction is

(3-196)

thereby immediately showing that themetal vacancy concentration depends onthe partial pressure of oxygen in the fol-lowing way:

[V¥M] = K1 pO2

1/2 (3-197)

Accordingly, the metal vacancy concentra-tion is directly proportional to pO2

1/2.Assuming that the reaction in Eq. (3-

195) is the preferred one, we find that theequilibrium constant for this reaction is

(3-198)

The concentration of [O¥O] is essentially

constant and is usually absorbed into K2.The condition of electrical neutrality re-quires that

[h•] = [V¢M] (3-199)

thereby showing that the metal vacancyconcentration depends on the partial pres-sure of oxygen in the following way:

[V¢M] = K21/2 p2

1/4 (3-200)

Accordingly, the metal vacancy concentra-tion is directly proportional to pO2

1/4.Let us assume that the migration of

metal ions will principally be via vacanciesin the nonstoichiometric region, since theyare the predominant defect. Since the tracerdiffusion coefficient of the metal ions de-pends directly on vacancy concentration(see Sec. 3.3.4), then the tracer diffusioncoefficient will depend on oxygen partialpressure in the same way that either [V¥

M]or [V¢M] does. A measurement of the tracerdiffusion coefficient as a function of oxy-

Kp

2 1 22

=h•] [O ] [V ]O M

O

[/

× ′

Kp

1 1 22

=[V ]M

O

×

/

gen partial pressure will expose the actualcharge state of the vacancy, i.e., V¥

M or V¢M.In practice, this kind of differentiation

between charge states and even defecttypes does not often occur unambiguously.The result of plotting log D* versus log pO2

often does not show precisely one slopeand often there is also some curvature.These effects can sometimes be associatedwith contributions to the diffusion from thethree types of vacancy, V¥

M, V¢M, and V≤M,e.g., Co1–d O (Dieckmann, 1977). Probablythe most successful and convincing appli-cation of this mass-action law approach hasmade the defect types in Ni1–d O compat-ible with data from tracer diffusion, electri-cal conductivity, chemical diffusion, anddeviation from stoichiometry/partial pres-sure (Peterson, 1984).

At higher partial pressures of oxygen,which in some cases such as Co1–d O,Mn1–d O, and Fe1–d O can result in largedeviations from stoichiometry, we can log-ically expect that defect interactions couldplay an important role. First, the concentra-tions in the mass-action equations shouldbe replaced by activities. This so greatlycomplicates the analysis that progress hasbeen fairly slow. The retention of concen-trations rather than activities may be per-missible, however, because there is someevidence that non-ideal effects tend to can-cel in a log D/log pO2

plot (Murch, 1981).At high defect concentrations, defect clus-tering to form complex defects whichthemselves might move or act as sourcesand sinks for more mobile defects, is pos-sible. Two very well-known examples,identified by neutron scattering, are theKoch–Cohen clusters in Fe1–d O (Kochand Cohen, 1969) and the Willis cluster inUO2+x (Willis, 1978). Although attemptshave been made to incorporate such defectsinto mass-action equations, it is probablyfair to say that the number of adjustable pa-


3.5 Experimental Methods for Measuring Diffusion Coefficients 231

rameters thereby resulting and the lack ofany precise independent information on themobility or lifetime of the defect complexesmake any conclusions reached rather spec-ulative. New statistical mechanical ap-proaches have considerable promise, how-ever (see the review by Murch (1995)).

Somewhat analogous to changing the de-fect concentration, i.e., deviation from stoi-chiometry by changing the oxygen partialpressure, is the doping of ionic crystalswith ions of a different valence from thehost metal ion in order to produce defects.As an example, consider the solubility ofan oxide 2O3 in an oxide MO2. The im-plication in the stoichiometry 2O3 is thatthe ions have a valence of +3, comparedwith + 4 for M in the oxide MO2. Forcharge neutrality reasons the doped oxideadopts either oxygen vacancies, metal in-terstitials, or some electronic defect, thechoice being dictated by energetics. Let usconsider the first of these:

2O3 Æ VO•• + 2 ¢M + 3OO (3-201)

where ¢M is a ion on a M site. The sitefraction of oxygen vacant sites is not quitedirectly proportional to the dopant sitefraction because extra normal oxygen sitesare created in the process. The diffusioncoefficient of oxygen here is proportionalto the concentration of vacant oxygen sites.

For very large deviations from stoichiom-etry, or high dopant concentrations, thoseextrinsic defects greatly outnumber the in-trinsic defects. However, at low dopant con-centrations (or small deviations from stoi-chiometry), all the various mass-action lawsmust be combined. This complicates theanalysis, and space prevents us from dealingwith it here. For a more detailed discussionof this and the foregoing, we refer to Kof-stad (1972) and Schmalzried (1995).

Generally, cation diffusion in the oxidesof the transition metals (which show non-

stoichiometry (cation vacancies) in the cat-ion sublattice) is generally well understoodexcept at large deviations from stoichiome-try (see the review by Peterson (1984)).Other oxides such as MgO and Al2O3, al-though apparently simple, are less wellunderstood because of the low intrinsic de-fect population and the presence of extrin-sic defects coming from impurities (Subba-Rao, 1985). An understanding of oxygendiffusion in oxides has been hampered bythe lack of suitable radioisotopes, butmethods using 18O and secondary ion massspectrometry (SIMS) have rapidly changedthe situation (Rothman, 1990). Ionic con-ductivity (where possible) (see Sec. 3.3.2for the relation to diffusion) has tradition-ally been the measurement which has givenmuch more information on oxygen move-ment; see, for example, the review byNowick (1984).

3.5 Experimental Methods forMeasuring Diffusion Coefficients

In this section we briefly discuss themore frequently used methods of measur-ing diffusion coefficients.

3.5.1 Tracer Diffusion Methods

By far the most popular and, if per-formed carefully, the most reliable of theexperimental methods for determining “self” and impurity diffusion coefficientsis the thin-layer method. Here, a very thinlayer of radiotracer of the diffusant is de-posited at the surface of the sample. Thedeposition can be done by evaporation,electrochemical methods, decompositionof a salt, sputtering, etc. The diffusant ispermitted to diffuse for a certain time t athigh temperature, “high” being a relativeterm here. If the thickness of the layer de-

posited is much smaller than ÷---D t , then Eq.

(3-8) describes the time evolution of theconcentration–depth profile. After diffu-sion, the concentration–depth profile inthe sample is established by sectioning andcounting the radioactivity in each section.A number of techniques are available forsectioning. For thick sections ≥ 3 µm, me-chanical grinding is the standard method.Microtomes can be used for sections ofabout 1 µm, electrochemical methods for≥ 5 nm, and sputtering for ≥1 nm. The dif-fusion coefficient is obtained from theslope of the lnC (x, t) versus x2 plot (seeEq. (3-8)). Fig. 3-22 shows an exemplarytracer concentration profile of this type (Mundy et al., 1971).

A variant of the method is the “residualactivity” or “Gruzin (1952) method”.Rather than counting the activity in eachsection, the activity remaining in the sam-ple is determined. This requires an integra-tion of Eq. (3-8) and a knowledge of the ra-diation absorption characteristics of thematerial. At the limits of very soft or very

hard radiation the method can give com-parable accuracy to the counting of sec-tions.

A few elements do not have convenientradioisotopes. The diffusion part of the ex-periment is still performed in much thesame way but now with stable isotopes. Insome cases, e.g., 18O, the diffusant sourceis in the gas phase, although a thin sourceof oxide containing 18O can be deposited insome cases. After diffusion, nuclear reac-tion analysis of 18O can be used (see, forexample, the review by Lanford et al.(1984)), in order to establish the concentra-tion profile. Much more popular recently,especially for oxygen, is SIMS (see, for ex-ample, the reviews by Petuskey (1984),Kilner (1986) and Manning et al. (1996)).

Excellent detailed (and entertaining) ex-positions and critiques of all methodsavailable for measuring tracer diffusion co-efficients in solids have been provided byRothman (1984, 1990).

3.5.2 Chemical Diffusion Methods

The chemical or interdiffusion coeffi-cient can be determined in a variety ofways. For binary alloys, the traditionalmethod has been to bond together two sam-ples of different concentrations. Interdiffu-sion is then permitted to occur for a time tand at high temperature. The concentrationprofile can be established by sectioningand chemical analysis, but since about thelate 1960s the use of an electron micro-probe has been the usual procedure for ob-taining the concentration profile directly. Ifthe compositions of the starting samplesare relatively close, and the interdiffusioncoefficient is not highly dependent on com-position, then analysis of the profile withEq. (3-12) can lead directly to the interdif-fusion coefficient D at the average compo-sition. The more usual procedure is to use


Figure 3-22. A tracer concentration profile for self-diffusion in potassium at 35.5 °C (Mundy et al.,1971).

3.5 Experimental Methods for Measuring Diffusion Coefficients 233

the Boltzmann–Matano graphical integra-tion analysis (see Eqs. (3-13) and (3-14)) toobtain the interdiffusion coefficient and itscomposition dependence. If fine insolublewires (to act as a marker) are also incorpo-rated at the interface of the two samples,then the marker shift or Kirkendall shiftwith respect to the original interface can be measured. This shift, in association with the interdiffusion coefficient, can beused to determine the intrinsic diffusioncoefficients of both alloy components atthe composition of the marker (see Sec.3.2.2.4).

The rate of absorption or desorption ofmaterial from the sample can also be usedto determine the chemical diffusion coeffi-cient. This method is useful only where onecomponent is fairly volatile, such as oxy-gen in some nonstoichiometric oxides andhydrogen in metals. The surface composi-tion is normally assumed to be held con-stant. The concentration profile can be de-termined by sectioning or electron micro-probe analysis and can be analysed withthe aid of Eq. (3-9) to yield the interdiffu-sion coefficient. The Boltzmann–Matanoanalysis can be used if there is a composi-tion dependence.

The amount of material absorbed or de-sorbed from the couple, e.g., weightchange, can be used with Eq. (3-11) toyield the interdiffusion coefficient. Eqs. (3-9) and (3-11) refer to situations when ÷---

D tis very small compared with the geometryof the sample, i.e., diffusion into an infinitesample is assumed. When the sample mustbe assumed to be finite, e.g., fine particles,other, generally more complex, solutionsare required; see Crank (1975). A closelyrelated experiment can be performed wherethe gas phase is chemically in equilibriumwith the solid, but is dosed with a stableisotope. An example is 18O, a stable iso-tope of oxygen. The absorption by the solid

of 18O (exchanging with 16O) can be moni-tored in the gas phase with a mass spec-trometer in order to lead to the tracer diffu-sion coefficient of oxygen (see, for exam-ple, Auskern and Belle (1961)). A draw-back of methods which do not rely on sec-tioning is the necessary assumption ofrapid gas/surface reactions.

3.5.3 Diffusion Coefficientsby Indirect Methods

There are a number of methods for ob-taining diffusion coefficients that do notdepend on solutions of Fick’s Second Law.Unfortunately space does not permit a de-tailed discussion of these. We shall men-tion them here for completeness and directthe reader to the appropriate sources. Oftenthese methods extend the accessible tem-perature range for diffusion measurements.In all cases specific atomic models need tobe introduced in order to extract a diffusioncoefficient. Inasmuch as there are inevita-bly approximations in a model, the result-ing diffusion coefficients may not be as re-liable as those obtained from a concentra-tion profile. In some cases additional corre-lation information can be extracted, e.g.,nuclear methods. The diffusion coefficientsfound from these methods are essentially “self” diffusion coefficients but chemicaldiffusion coefficients for interstitial solutesare obtained from the Gorsky effect.

3.5.3.1 Relaxation Methods

In the relaxation methods net atomic mi-gration is due to external causes such asstress of magnetic field (Nowick and Berry,1972). The best known phenomena are (a)the Gorsky effect (Gorsky, 1935); for a re-view see Alefeld et al. (1970) and for an in-troduction see Borg and Dienes (1988); (b)the Snoek effect (Snoek, 1939); for an ex-

position see Wert (1970) and for an intro-duction see Borg and Dienes (1988); and(c) the Zener effect (Zener, 1947, 1951);for an introduction see Bocquet et al.(1996).

3.5.3.2 Nuclear Methods

A number of nuclear methods have be-come of increasing importance for deter-mining diffusion coefficients in solids. Thefirst of these is quasielastic neutron scat-tering (QNS), which has often been usedfor situations where the diffusion coeffi-cients are larger than about 10–11 m2 s–1

and the diffusing species exhibits a reason-ably high scattering cross-section. Most of the applications have been to hydrogen(protonic) in metals (see, for example, Janot et al., 1986). For a detailed introduc-tion to the subject see Lechner (1983) andZabel (1984). The second method is nu-clear magnetic resonance (NMR). NMR isespecially sensitive to interactions of thenuclear moments with fields produced bytheir local environments. Diffusion of a nu-clear moment can cause variations in thesefields and can significantly affect the ob-served resonance. In particular, diffusionaffects a number of relaxation times inNMR. In favorable cases diffusion coeffi-cients between 10–18 and 10–10 m2 s–1 areaccessible. For a detailed introduction tothe subject see Stokes (1984) and Heitjansand Schirmer (1998). We make specialmention of pulsed field gradient (PFG)NMR, which has been found to be espe-cially useful for studying anomalous diffu-sion (Kärger et al., 1998). Finally, we men-tion Mössbauer spectroscopy (MBS), whichshows considerable promise for under-standing diffusion processes in solids. Thegeneral requirement is that the diffusioncoefficient of Mössbauer active isotopeshould be larger than about 10–13 m2 s–1.

Most applications have been to solids in-corporating the rather ideal isotope 57Fe.See Mullen (1984) for a detailed discus-sion of MBS and Vogl and Feldwisch (1998)for several recent examples of applica-tions.

3.5.4 Surface Diffusion Methods

The techniques for measuring surfacediffusion are generally quite different fromsolid-state techniques. For short distance,microscopic or “intrinsic” diffusion (seeSec. 3.2.2.5), the method of choice is thefield ion microscope. Much elegant workhas been carried out with this technique byErlich and co-workers; see for example Er-lich and Stott (1980) and Erlich (1980).With this method single atoms can be im-aged and followed. Another method, thistime using the field electron microscope(Lifshin, 1992), correlates fluctuations ofan emission current from a very small areawith density fluctuations arising from surface diffusion in and out of the probearea (Chen and Gomer, 1979). Other meth-ods include quasi-elastic scattering of low energy He atoms (formally analogousto quasi-elastic neutron scattering) and re-laxation measurements, making use of dep-osition of the adsorbate in a non-equilib-rium configuration, followed by annealingwhich permits relaxation to equilibrium.Techniques useful for following the relaxa-tion process at this microscopic level in-clude pulsed molecular beam combinedwith fast scanning IR interferometry (Reutt-Robey et al., 1988) and work func-tion measurements (Schrammen and Hölzl,1983).

Many methods exist for long distance ormacrosopic diffusion. When the diffusingspecies is deposited as a source and the ap-propriate geometries are known then scan-ning for the concentration profile followed


3.7 References 235

by processing with the appropriate solutionto Fick’s Second Law (the Diffusion Equa-tion) Eq. (3-5) will give the mass transferdiffusion coefficient. Radioactive tracerscan certainly be used for this (Gjostein,1970). The concentration profile (in thecase of hetero-diffusion) can also be deter-mined using scanning SIMS, local XPS,scanning Auger, scanning EM and scan-ning STM (Bonzel, 1990). Of interest whenthe atoms are weakly adsorbed is laser induced thermal desorption (LITD) (Vis-wanathan et al., 1982). The field electronmicroscope can be used to image the diffu-sion front of adatoms migrating into a re-gion of clean surface (Gomer, 1958). Fi-nally, the “capillary method”, probably thebest known of the macroscopic methods,starts with a perturbed surface, usually in aperiodic way, say by sinusoidal grooving.There is now a driving force to minimizethe surface Gibbs energy. The time depen-dence of the decreasing amplitude of, say,the sinusoidal profile, can be processed togive the mass transfer diffusion coefficient;see Bonzel (1990).

The reader is referred to a number of thefine reviews in the area, e.g., Rhead(1989), Bonzel (1990), and the book editedby Vu Thien Binh (1983). The first of theserelates surface diffusion to a number oftechnologically important processes suchas thin film growth, sintering, and cataly-sis. The second is a comprehensive compi-lation of surface diffusion data on metalsand a detailed survey of experimentalmethods.

3.6 Acknowledgements

I thank my colleagues in diffusion fromwhom I have learnt so much. This workwas supported by the Australian ResearchCouncil.

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4 Statistical Theories of Phase Transitions

Kurt Binder

Institut für Physik, Johannes Gutenberg-Universität Mainz,Mainz, Federal Republic of Germany

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 2404.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.2 Phenomenological Concepts . . . . . . . . . . . . . . . . . . . . . . . . 2464.2.1 Order Parameters and the Landau Symmetry Classification . . . . . . . . . 2464.2.2 Second-Order Transitions and Concepts about Critical Phenomena

(Critical Exponents, Scaling Laws, etc.) . . . . . . . . . . . . . . . . . . . 2604.2.3 Second-Order Versus First-Order Transitions; Tricritical

and Other Multicritical Phenomena . . . . . . . . . . . . . . . . . . . . . 2694.2.4 Dynamics of Fluctuations at Phase Transitions . . . . . . . . . . . . . . . 2774.2.5 Effects of Surfaces and of Quenched Disorder on Phase Transitions:

a Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2794.3 Computational Methods Dealing with the Statistical Mechanics

of Phase Transitions and Phase Diagrams . . . . . . . . . . . . . . . . 2834.3.1 Models for Order–Disorder Phenomena in Alloys . . . . . . . . . . . . . 2834.3.2 Molecular Field Theory and its Generalization

(Cluster Variation Method, etc.) . . . . . . . . . . . . . . . . . . . . . . . 2884.3.3 Computer Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . 2934.4 Concepts About Metastability . . . . . . . . . . . . . . . . . . . . . . . 2984.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3034.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304



A prefactor of Friedel interactiona lattice spacingai nearest-neighbor distanceA, A¢ critical amplitude of specific heats B critical amplitude of the order parameterc concentrationC specific heatC phenomenological coefficientc lattice spacing in z-directionc cluster of geometric configurationDc concentration differenceC, C¢ critical amplitudes of ordering susceptibilitiesccoex concentration at coexistence curveci concentration of lattice site icijk… elastic constantsd dimensionalityD critical amplitude at the critical isothermd* marginal dimensionalityE electric fieldei random vectorsel (k, x) phonon polarization vectorF [F (x)] Helmholtz energy function in operator formf factor of order unityF Helmholtz energyDF* Helmholtz energy barrierF (j) Helmholtz energy functionalF scaling function of the Helmholtz energyf, f I, …, f IV, f coefficientsF0 background Helmholtz energy of disordered phasefcg coarse-grained Helmholtz energyfmn quadrupole moment tensorFreg background term of the Helmholtz energyg constant factor in the structure functionG reciprocal lattice vectorG (x) order parameter correlation functionG (x /x ) scaling function of the correlation functionG critical amplitude of the correlation functionG (x , x ) order parameter correlation functiongnc multispin correlation functionH ordering fieldH magnetic field HamiltonianH scaled ordering field

240 4 Statistical Theories of Phase Transitions


h, hc uniform magnetic field, critical valueHR random field (due to impurities)i number and index for different lattice sitesJ energy gained when two like atoms occupy neighboring sitesJ exchange interactionJ Fourier transform of the exchange interactionJij exchange interaction between spins at sites i and j (in cases where it is ran-

dom)Jm strength of the magnetic interactionJnn nearest-neighbor interactionJnnn next-nearest-neighbor interactionk wavevector of phononK1, K2 phenomenological coefficientskB Boltzmann constantkF Fermi wavenumberL coarse-graining length LagrangianM period (in Fig. 4-10)M magnetizationMl mass of an atommm sublattice magnetizationMs spontaneous magnetizationN degree of polymerizationna lattice sites in state ap pressureP polarizationp dielectric polarizationP (J) statistical distribution of random variable JP (Si) probability that a configuration Si occursp+ probability for spin upp– probability for spin downpb parameter in Landau equation (bicritical points)Q normal coordinateqa primitive vectors of the reciprocal latticeqmax value of q where c (q) is at a maximumq* nonzero wavenumberqEA ordering parameter for the Ising spin glassr phenomenological coefficient in the Landau expansionR effective range of interactionR ratio of Jnn and Jnnn

R* critical radius of a dropletRl

i site of atom l in unit cell ir, r ¢ coefficient in the Landau equationrc radius of polymer coilS entropy

S (q) structure factorS (q, w) scattering functionSi , Sj unit vectors in direction of the magnetic momentt timet reduced temperature 1– T/T0 (in Sec. 4.2.2)T0 temperature where r changes signT1 temperature where metastable ordered phase vanishesTb bicritical pointTc critical temperaturetcr reduced temperature at crossoverTL Lifshitz pointTl (x) transition temperature of superfluid 4HeTt temperature of tricritical pointu phenomenological coefficientu coefficient in Landau equationU enthalpyul displacement vectorUd surface of a d-dimensional spherev coefficient in the Landau equationv ( |x |) Friedel potentialV total volume of the thermodynamic systemV (xi – xj) pairwise interaction of two particlesVab interaction between atoms A and Bw coefficient in the Landau equationW action in classical mechanicsx, xn , xm , xa position vector and its componentsz dynamical critical exponentz coordinate (of position vector)z coordination numberZ partition function scaling variable

a1m, a2 notation for phases in phase diagramam magnetic phasean nonmagnetic phasea (x) short-range order parameter for metal alloysa, a ¢ critical exponents related to specific heatb exponent related to the order parameterG critical amplitude of the susceptibilityg, g ¢ critical exponents related to the susceptibilityG0 phenomenological rate factord critical exponent at the critical isothermdmn Kronecker symbole binding energy of a particle



eik strain tensorh exponent related to the decay of correlations at Tc

l wavelengthl label of the phonon branchL factor changing the length scalem chemical potentialm magnetic moment per spinDm chemical potential differencemcoex chemical potential at phase coexistencen exponent related to the correlation lengthx correlation lengthxi concentration difference (Eq. (4-126))r densityr (x) charge density distribution functiont characteristic timej phase shift in Friedel potentialF order parameterF (x) order parameter densityFi concentration difference between two sublatticesFms order parameter of metastable stateFs spinodal curvec (q) wavevector-dependent susceptibilitycel dielectric tensorcT response function of the order parametercT isothermal susceptibilitycT staggered susceptibilityy concentration difference between two sublatticesy I, II ordering parameter for D03 structureya amplitude of mass density wavesw characteristic frequency (for fluctuations)w scaling function

AF antiferromagneticcg coarse-grained (as index)coex coexistent (as index)CV cluster variation methodDAG dysprosium aluminum garnetESR electron spin resonancef.c.c. face-centered cubicLRO long-range orderMC Monte Carlo (method)MD molecular dynamicsMFA mean field approximationMFA molecular field approximation

ms metastable state (as index)NMR nuclear magnetic resonanceP paramagneticsc simple cubicSRO short-range order


4.1 Introduction 245

4.1 Introduction

This chapter deals with the general theo-retical concepts of phase transitions of ma-terials and their basis in statistical thermo-dynamics. This field has seen extensivescientific research over many decades, anda vast amount of literature exists, e.g., thestatistical theory of critical point phenom-ena fills a series of monographs that so farconsists of 17 volumes (Domb and Green,1972–1976; Domb and Lebowitz, 1983–1997). Introductory texts require wholebooks (Stanley, 1971; Ma, 1976; Patashin-skii and Pokrovskii, 1979; Yeomans, 1992),and so does a comprehensive account ofthe Landau theory of phase transitions (Tolédano and Tolédano, 1987).

This chapter therefore cannot give an ex-haustive description of the subject; insteadwhat is intended is a tutorial overview,which gives the flavor of the main ideas,methods, and results, with emphasis on theaspects which are particularly relevant formaterials science, and hence may provide auseful background for other chapters in thisbook. Hence this chapter will containhardly anything new for the specialist; forthe non-specialist it will give an initial or-ientation and a guide for further reading.To create a coherent and understandabletext, a necessarily arbitrary selection ofmaterial has been made which reflects theauthor’s interests and knowledge. Reviewscontaining complementary material havebeen written by De Fontaine (1979), whoemphasizes the configurational thermody-namics of solid solutions, and by Khacha-turyan (1983), who emphasizes the theoryof structural transformations in solids (which, from a different perspective, arediscussed by Bruce and Cowley, 1981).There are also many texts which either con-centrate on phase transformations of par-ticular types of materials, such as alloys

(Tsakalakos, 1984; Gonis and Stocks, 1989;Turchi and Gonis, 1994), magnetic materi-als (De Jongh and Miedema, 1974; Ausloosand Elliott, 1983), ferroelectrics (Blinc andZeks, 1974; Jona and Shirane, 1962), liquidcrystals (Pershan, 1988; De Gennes, 1974;Chandrasekhar, 1992), polymeric materials(Flory, 1953; De Gennes, 1979; Riste andSherrington, 1989; Binder, 1994; Baus etal., 1995), etc., or on particular types ofphase transitions, such as commensurate–incommensurate transitions (Blinc and Le-vanyuk, 1986; Fujimoto, 1997), multicriti-cal transitions (Pynn and Skjeltorp, 1983),first-order transitions (Binder, 1987a),martensitic transformations (Nishiyama,1979; Salje, 1990), glass transitions (Jäckle, 1986; Angell and Goldstein, 1986;Cusack, 1987; Zallen, 1983; Zarzycki,1991), percolation transitions (Stauffer andAharony, 1992), melting (Baus, 1987),wetting transitions (Dietrich, 1988; Sulli-van and Telo da Gama, 1985), metal– insu-lator transitions (Mott, 1974; Friedmannand Tunstall, 1978), etc. Other approachesfocus on particular ways of studying phasetransitions: from the random-phase approx-imation (Brout, 1965) and the effective-field theory (Smart, 1966) to advancedtechniques such as field-theoretical (Amit,1984) or real space renormalization (Burk-hardt and van Leeuwen, 1982), and com-puter simulation studies of phase transi-tions applying Monte Carlo methods (Binder, 1979, 1984a; Mouritsen 1984;Binder and Heermann, 1988; Binder andCiccotti, 1996) or the molecular dynamicstechniques (Ciccotti et al., 1987; Hockneyand Eastwood, 1988; Hoover, 1987; Binderand Ciccotti, 1996), etc. For particularmodels of statistical mechanics, exact de-scriptions of their phase transitions areavailable (Baxter, 1982). Of particularinterest also are the dynamics of criticalphenomena (Enz, 1979) and the dynamics

of first-order phase transitions (Gunton et al., 1983; Koch, 1984; Binder, 1984b,1989; Haasen et al., 1984; Zettlemoyer,1969; Abraham, 1974; Gunton and Droz,1983; Herrmann et al., 1992). The lattersubject is treated in two other chapters in this book (see the Chapters by Wagner et al. (2001) and Binder and Fratzl (2001))and will not be considered here.

In this chapter, we shall briefly discussthe statistical thermodynamics of phasetransitions on a phenomenological macro-scopic level (Sec. 4.2), i.e., the Landau the-ory of first- and second-order transitions,critical and multicritical phenomena, andalso the dynamics of fluctuations at phasetransitions, and the effects of quenched disorder in solids. The second part (Sec.4.3) is devoted to the complementary “mi-croscopic” approach, where we start from“model Hamiltonians” for the system underconsideration. The statistical mechanics ofsuch models naturally leads to the consider-ation of some computational methods, i.e.,methods based on the molecular field the-ory and its generalizations, and methodsbased on computer simulation techniques.Since in solid materials metastable phases

are ubiquitous, e.g., diamond is the meta-stable modification of solid carbon whereasgraphite is the stable phase, some com-ments about the statistical mechanics ofmetastability are made in Sec. 4.4.

4.2 Phenomenological Concepts

In this section, the main facts of the the-ory of phase transitions are summarized,and the appropriate terminology intro-duced. Rather than aiming at completeness,examples which illustrate the spirit of themain approaches will be discussed, includ-ing a discussion of critical phenomena andscaling laws.

4.2.1 Order Parameters and the LandauSymmetry Classification

Table 4-1 lists condensed matter systemsthat can exist in several phases, dependingon external thermodynamics parameterssuch as pressure, p, temperature, T, electricor magnetic fields (E, H). We assume thatan extensive thermodynamic variable canbe identified (i.e., one that is proportional


Table 4-1. Order parameters for phase transitions in various systems.

System Transition Order parameter

Liquid–gas Condensation/evaporation Density difference Dr = rl – rg

Binary liquid mixture Unmixing Composition difference Dc = c(2)coex – c(1)

coex

Nematic liquid Orientational ordering –12

·3 (cos2q) – 1ÒQuantum liquid Normal fluid a suprafluid ·yÒ (y : wavefunction)

Liquid–solid Melting/crystallization rG (G = reciprocal lattice vector)

Magnetic solid Ferromagnetic (Tc) Spontaneous magnetization MAntiferromagnetic (TN) Sublattice magnetization Ms

Solid binary mixture Unmixing Dc = c(2)coex – c(1)

coex

AB Sublattice ordering y = (DcII – DcI)/2Dielectric solid Ferroelectric (Tc) Polarization P

Antiferroelectric (TN) Sublattice polarization Ps

Molecular crystal Orientational ordering Ylm (J, j)

4.2 Phenomenological Concepts 247

to the volume of the system) which distin-guishes between these phases (examplesare also given in Table 4-1), called the “or-der parameter”. We shall denote the orderparameter as F, and the conjugate thermo-dynamic variable, the “ordering field”, asH. Using the thermodynamic potential, F,which has as “natural variables” a field, H(which is an “intensive” thermodynamicvariable, i.e., independent of the volume),and the temperature, T, we have

F = – (∂F/∂H )T (4-1)

the other derivative being the entropy,

S = – (∂F/∂T )H (4-2)

As examples, consider a ferromagnet, wherethe order parameter is the magnetization,M, or a ferroelectric, where the order pa-rameter is the dielectric polarization, P,

M = – (∂F/∂H)T (4-3a)

P = – (∂F/∂E)T (4-3b)

It is clear that such thermodynamic rela-tions can be written for any material, but aquantity qualifies as an “order parameter”when a particular value of the “orderingfield” exists where the order parameter ex-hibits a jump singularity between two dis-tinct values (Fig. 4-1). This means that forthese values of the ordering field a first-or-der phase transition occurs, where a firstderivative of the thermodynamic potentialF exhibits a singularity. At this transition,two phases can coexist; e.g., at the liq-uid–gas transition for a chemical potentialm = mcoex (T ), two phases with differentdensity coexist; and in a ferromagnet atzero magnetic field, phases with oppositesign of spontaneous magnetization can co-exist. Although the fluid-magnet analogy(Fig. 4-1) goes further, since the first-orderlines in the (m, T ) or (H, T ) plane in bothcases end in critical points which may even

be characterized by the same critical expo-nents (see below), there is also an impor-tant distinction that in the magnetic prob-lem, the Hamiltonian possesses a symme-try with respect to the change of sign of themagnetic field; reversing this sign and alsoreversing the sign of the magnetizationleave the Hamiltonian invariant. Becauseof this symmetry, the transition line mustoccur at H = 0. Conversely, if the system

Figure 4-1. The fluid-magnet analogy. On varyingthe chemical potential m at m (T )

coex, the density r jumpsfrom the value at the gas branch of the gas– liquid co-existence curve (rgas = r (1)

coex) to the value at the liquidbranch of the coexistence curve (rliquid = r (2)

coex) (topleft). Similarly, on varying the (internal) magneticfield H, the magnetization M jumps from the nega-tive value of the spontaneous magnetization (– Ms) toits positive value (top right). While this first-orderliquid–gas transition occurs at a curve mcoex(T ) inthe m – T-plane ending in a critical point (mc, Tc)where the transition is then of second order, the curvewhere phases with positive and negative spontaneousmagnetization can coexist simply is H = 0 (T < Tc)(middle). The order parameter (density differenceDr, or spontaneous magnetization Ms) vanishes ac-cording to a power law near Tc (bottom).

at H = 0 is in a monodomain state with either positive or negative spontaneousmagnetization, this symmetry is violated;this is described as “spontaneous symmetrybreaking”. No such obvious symmetry, onthe other hand, is identified for the liq-uid–gas transition of fluids. Consequently,the curve mcoex(T ) is a nontrivial functionin the m – T-plane, and no simple symmetryoperation acting on the gas phase atoms ex-ists that would transform this phase into aliquid, or vice versa. Similar lack of sym-metry between the phases is noted for un-mixing transitions in binary fluid or solidmixtures, where the order parameter is aconcentration difference, Dc = c(2)

coex – c(1)coex,

cf. Table 4-1, whereas solid binary mix-tures which exhibit sublattice ordering dopossess a symmetry. The order parameter inan alloy such as brass (b-CuZn) is the dif-ference between the relative concentrationof the two sublattices, y = (DcII – DcI)/2.However, the two sublattices physically arecompletely equivalent; therefore, the Ha-miltonian possesses a symmetry against theinterchange of the two sublattices, whichimplies that y changes sign, just as the(idealized!) ferromagnet does for H = 0(Fig. 4-1). In this example of an alloywhich undergoes an order–disorder transi-tion where the permutation symmetrybetween the two sublattices is spontane-ously broken, the “ordering field” conju-gate to the order parameter is a chemicalpotential difference between the two sub-lattices, and hence this ordering field is not directly obtainable in the laboratory.The situation is comparable to the case of simple antiferromagnets, the order pa-rameter being the “staggered magnetiza-tion” (= magnetization difference betweenthe sublattices), and the conjugate orderingfield would change sign from one sublat-tice to the other (“staggered field”). Al-though the action of such fields usually

cannot be measured directly, they never-theless provide a useful conceptual frame-work.

Another problem which obscures theanalogy between different phase transitionsis the fact that we do not always wish towork with the corresponding statistical en-sembles. For a liquid–gas transition, wecan control the chemical potential via thefluid pressure, and thus a grand-canonicalensemble description makes sense. How-ever, for the binary mixture (AB), the or-dering field would be a chemical potentialdifference Dm = mA – mB between the spe-cies. In an ensemble where Dm is the exter-nally controlled variable A atoms couldtransform into B, and vice versa. Experi-mentally, of course, we do not usually havemixtures in contact with a reservoir withwhich they can exchange particles freely,instead mixtures are kept at a fixed relativeconcentration c. (Such an equilibrium witha gas reservoir can only be realized forinterstitial alloys such as metal hydrides oroxides.) Thus the experiment is describedby a canonical ensemble description, with(T, c) being the independent variables, al-though the grand-canonical ensemble ofmixtures is still useful for analytical theo-ries and computer simulation. Now, as isobvious from Fig. 4-1, in a canonical en-semble description of a fluid, the first-order transition shows up as a two-phasecoexistence region. Here, at a given den-sity, r, with rgas < r < rliquid, the relativeamounts of the two coexisting phases aregiven by the lever rule if interfacial contri-butions to the thermodynamic potential canbe neglected. The same is true for a binarymixture with concentration c inside the two-phase coexistence region, c(1)

coex < c < c(2)coex .

These different statistical ensembles alsohave pronounced consequences on the dy-namic properties of the considered sys-tems: in binary mixtures at constant con-



centration, and also in fluids at constantdensity, the order parameter is a conservedquantity, whereas for a fluid at constantchemical potential, the order parameter isnot conserved. The latter situation occurs,for instance, for fluid–gas transitions inphysisorbed monolayers at surfaces, whichcan exchange molecules with the surround-ing gas phase which is in equilibrium withthis adsorbed layer.

An important distinction to which weturn next is the order of a phase transition.In the examples shown in the upper part ofFig. 4-1, a first derivative of the appropri-ate thermodynamic potential has a jumpsingularity and therefore such transitionsare called first-order transitions. However,if we cool a ferromagnet down from theparamagnetic phase in zero magnetic field,the spontaneous magnetization sets in con-tinuously at the critical temperature Tc

(lower part of Fig. 4-1). Similarly, on cool-ing the alloy b-CuZn from the state whereit is a disordered solid solution (in thebody-centered cubic phase), sublattice or-dering sets in continuously at the criticaltemperature (Tc = 741 K, see Als-Nielsen(1976)). Whereas the first derivatives ofthe thermodynamic potential at these con-tinuous phase transitions are smooth, thesecond derivatives are singular, and there-fore these transitions are also called sec-ond-order transitions. For example, in aferromagnet the isothermal susceptibilitycT and the specific heat typically havepower law singularities (Fig. 4-2)

(4-4)

(4-5)

C T F T

A T T T T

A T T T T

H H

T T

≡ − ∂ ∂

− >′ − <

⎧⎨⎩→

−

− ′

( / )

ˆ ( / ) ,ˆ ( / ) ,

2 20

1

1

=

c c

c c

=c

a

a

cg

g

T T

T T

F H

C T T T T

C T T T T

≡ − ∂ ∂

− >′ − <

⎧⎨⎩→

−

− ′

( / )ˆ ( / ) ,ˆ ( / ) ,

2 2

1

1=

c

c c

c c

In this context (see also Fig. 4-1), a, a¢, b,g, and g ¢ are critical exponents, while A, A¢,B, C, and C¢ are called critical amplitudes.Note that B and b refer to the spontaneousmagnetization, the order parameter (Fig. 4-1)

(4-6)

Behavior of the specific heat such as de-scribed by Eq. (4-5) immediately carriesover to systems other than ferromagnetssuch as antiferromagnets, the liquid–gassystem near its critical point, and brassnear its order–disorder transition; we mustremember, however, that H then means the

M B T TT T

s c=c→

−ˆ ( / )1 b

Figure 4-2. Schematic variation with temperature Tplotted for several quantities near a critical point Tc:specific heat CH (top), ordering “susceptibility” cT

(middle part), and correlation length x of order pa-rameter fluctuations (bottom). The power laws whichhold asymptotically in the close vicinity of Tc are in-dicated.

appropriate ordering field. In fact, this isalso true for Eq. (4-4), but then the physicalsignificance of cT changes. For a two-sub-lattice antiferromagnet, the ordering fieldis a “staggered field”, which changes signbetween the two sublattices, and hence isthermodynamically conjugate to the orderparameter of the antiferromagnet. Al-though such a field normally cannot be ap-plied in the laboratory, the second deriva-tive, cT (in this case it is called “staggeredsusceptibility”) is experimentally access-ible via diffuse magnetic neutron scatter-ing, as will be discussed below. Similarly,for the ordering alloy b-CuZn, H stands fora chemical potential difference between thetwo sublattices; the response function cT isagain physically meaningful and measuresthe peak intensity of the diffuse scatteringof X-rays or neutrons. (This scatteringpeak occurs at the superlattice Bragg spotcharacteristic of the sublattice orderingconsidered.) Finally, in the gas– liquidtransition considered in Fig. 4-1, H is thechemical potential and cT the isothermalcompressibility of the system.

As will be discussed in more detail inSec. 4.2.2, the divergences of second deriv-atives of the thermodynamic potential at acritical point (Eqs. (4-4), (4-5), Fig. 4-2)are linked to a diverging correlation lengthof order-parameter fluctuations (Fig. 4-2).Hence any discussion of phase transitionsmust start with a discussion of the order pa-rameter. The Landau theory (Tolédano andTolédano, 1987) which attempts to expandthe thermodynamic potential in powers ofthe order parameter, gives a first clue to thequestion of whether a transition is of sec-ond or first order (see Sec. 4.2.3).

We first identify the possible types of or-der parameters, since this will distinctly af-fect the nature of the Landau expansion. InEqs. (4-1) and (4-4) to (4-6) we havetreated the ordering field H and the order

parameter F as scalar quantities; althoughthis is correct for the gas–fluid transitionand for the unmixing transition of binarymixtures, the uniaxial ferro- or antiferro-magnets, uniaxial ferro- or antiferroelec-trics, etc., and for order–disorder transi-tions in alloys or mixed crystals (solid so-lutions), when only two sublatties need tobe considered, there are also cases wherethe order parameter must have a vector ortensor character. Obviously, for an iso-tropic ferromagnet or isotropic ferroelec-tric the order parameter in the three-dimen-sional space is a three-component vector(see Eq. (4-3)). It also makes sense to con-sider systems where a planar anisotropy ispresent, such that M or E (Eq. (4-3)) mustlie in a plane and hence a two-componentvector applies as an order parameter. How-ever, for describing antiferromagnetic or-dering and for order–disorder transitionswith many sublattices multicomponent or-der parameters are also needed, and thenumber of components of the order param-eter, the so-called “order parameter dimen-sionality”, is dictated by the complexity ofthe structure, and has nothing to do withthe spatial dimension. This is best under-stood by considering specific examples.Consider, for example, the ordering ofFe–Al alloys (Fig. 4-3): whereas in the dis-ordered A2 phase Fe and Al atoms are ran-domly distributed over the available latticesites, consistent with the considered con-centration (although there may be someshort-range order), in the ordered B2 phase(the FeAl structure) the bcc lattice is splitinto two inter-penetrating simple cubic (sc)sublattices, one taken preferentially by Fe,the other by Al atoms. This is the same(one-component) ordering as in b-CuZn.What is of interest here is the further “sym-metry breaking” which occurs for the D03

structure (realized in the Fe3Al phase):here four face-centered cubic (f.c.c.) sub-



lattices a, b, c, d must be distinguished. Al-though the concentrations on sublattices aand c are still equivalent, a further symme-try breaking occurs between the sublatticesb and d, such that only one sublattice (e.g.,

d) is preferentially occupied by Al, allother sublattices being preferentially occu-pied by iron. However, the role of the sub-lattices (a, c) and (b, d) can be inter-changed; hence a two-component order pa-rameter is needed to describe the structure,namely (see for example, Dünweg andBinder (1987))

(4-7)

where ci is the concentration of lattice site iand mm the sublattice “magnetization” inpseudo-spin language. As can example ofan ordering with m = 8 components, we canconsider the f.c.c. antiferromagnet MnO:as Fig. 4-4 shows, the magnetic structureconsists of an antiferromagnet arrangementof ferromagnetically ordered planes. If theordering were uniaxially anisotropic, we

yy

mm

Ia c b d

IIa c b d

=

=

m m m m

m m m m

m N ci

i

− + −− + + −

≡ −∑( / ) ( )1 2 1Œ

Figure 4-3. Body-centered cubic lattice showing theB2 structure (top) and D03 structure (bottom). Thetop part shows assignments of four sublattices, a, b,c, and d. In the A2 structure, the average concentra-tions of A and B atoms are identical on all four sub-lattices whereas in the B2 structure the concentra-tions at the b and d sublattices are the same (example:stoichiometric FeAl, with Fe on sublattices a and c;Al on sublattices b and d). In the D03 structure, theconcentrations at the a and c sublattice are still thesame, whereas the concentration at sublattice b dif-fers from the concentration at sublattice d (example:stoichiometric Fe3Al, with Al on sublattice b; allother sublattices taken by Fe). From Dünweg andBinder (1987).

B2

D03

Figure 4-4. Schematic view of the MnO structure,showing the decomposition of the fcc lattice into twoferromagnetic sublattices with opposite orientationof the magnetization (indicated by full and open cir-cles, respectively). Each sublattice is a stack of par-allel close-packed planes (atoms form a triangularlattice in each plane). Note that there are four equiv-alent ways in which these planes can be oriented inan fcc crystal.

would have m = 4 because of the four pos-sible ways of orienting the planes (andthere could be eight kinds of different do-mains coexisting in the system). Since forMnO the order parameter can take any or-ientation is a plane, it has two independentcomponents, and hence the total number oforder parameter components is m = 8 (Mu-kamel and Krinsky, 1976). An even largernumber of components is needed to de-scribe the ordering of solid 3He. In view ofthis, it is clear that even m Æ •, which isalso called the spherical model (Berlin andKac, 1952; Joyce, 1972) is a useful limit toconsider from the theoretical point of view.

Apart from this m-vector model for theorder parameter, Table 4-1 illustrates theneed to consider order parameters of tenso-rial character. This happens in molecularcrystals such as para-H2 (Fig. 4-5), N2, O2

and KCN, in addition to liquid crystals.Whereas the atomic degrees of freedomconsidered for ferro- or antiferromagneticorder are magnetic dipole moments, forferro- or antiferroelectric order electric di-pole moments, the degree of freedom of themolecules in Fig. 4-5 which now matters istheir electric quadrupole moment tensor:

(4-8)

where r (x) is the charge density distribu-tion function of a molecule, x = (x1, x2, x3),and dmn is the Kronecker symbol.

Proper identification of the order param-eter of a particular system therefore needs adetailed physical insight, and often is com-plicated because of coupling between dif-ferent degrees of freedom, e.g., a ferromag-netic material which is cubic in the para-magnetic phase may become tetragonal asthe spontaneous magnetization develops,owing to magnetostrictive couplings (for amore detailed discussion of this situation,

f x x xmn m nl

l mnr d= d=

∫ ∑−⎛

⎝⎜⎞

⎠⎟x x( )

13 1

32

see Grazhdankina (1969)). In this exampleit is clear that the spontaneous magnetiza-tion, M, is the “primary order parameter”whereas the tetragonal distortion (c/a –1),where c is the lattice spacing in the latticedirection where the magnetization direc-tion occurs and a is the lattice spacing perpendicular to it, is a “secondary orderparameter”. For purely structural phasetransitions where all considered degrees of freedom are atomic displacements, theproper distinction between primary andsecondary order parameters is much moresubtle (Tolédano and Tolédano, 1987).

We first formulate Landau’s theory forthe simplest case, a scalar order parameterdensity F (x). This density is assumed tobe small near the phase transition andslowly varying in space. It can be obtainedby averaging a microscopic variable over asuitable coarse-graining volume Ld in d-di-mensional space. For example, in an aniso-tropic ferromagnet the microscopic vari-able is the spin variable Fi = ±1 pointing inthe direction of the magnetic moment atlattice site i; in an ordering alloy such as b-CuZn the microscopic variable is the dif-ference in concentration of Cu between thetwo sublattices I and II in unit cell i, Fi =ci

II – ciI ; and in an unmixing alloy such as

Al–Zn, the microscopic variable is theconcentration difference between the localAl concentration at lattice site i (ci =1 if it


Figure 4-5. Schematic view of a lattice plane of apara-H2 crystal, indicating the orientational order ofthe ellipsoid H2 molecules.


is taken by Al and ci = 0 otherwise) and thecritical concentration, Fi = ci – ccrit . In allof these cases F (x) is considered to be anorder parameter field defined in continuumspace:

(4-9)

x being the center of gravity of the “vol-ume” Ld. The appropriate magnitude of thelinear dimension L of the coarse-grainingcell will be discussed later; obviously itmust be much larger than the lattice spac-ing in order for the continuum descriptionto make sense. Then a Helmholtz energyfunctional F [F (x)] is assumed:

where F0 is the background Helmholtz en-ergy of the disordered phase, and r, u, andR are phenomenological constants. (In fact,R can be interpreted as the effective rangeof interaction between the atomic degreesof freedom Fi , as will be seen below.) Ob-viously, Eq. (4-10) is the Taylor series-typeexpansion of F [F (x)] in powers of F (x)and —F (x), where just the lowest orderterms were kept. This makes sense if boththe coefficients u and R2 are positive con-stants at Tc, whereas the essential assump-tion which defines F as playing the role ofan order parameter of a second-order phasetransition is that the coefficient r changesits sign at the transition as the variable ofinterest (the temperature in the presentcase) is varied,

kB Tr = r¢ (T – Tc) (4-11)

Note also that in Eq. (4-10) we have as-sumed a symmetry in the problem against

1

12

14

12

0

2 4

2

k TF

Fk T

r u

Hk T d

R

B B

B

= (4-10)

d

[ ( )]

( ) ( )

( ) [ ( )]

F

F F

F F

x

x x x

x x

+ +⎧⎨⎩

− + ∇ ⎫⎬⎭

∫

F F( ) /x =i L

id

dL

Œ∑

the change of sign of the order parameterfor H = 0 and thus odd powers such asF3(x) do not occur; this is true for magnets(no direction of the magnetization is pre-ferred without magnetic field in a ferro-magnet) and for sublattice ordering of al-loys such as b-CuZn (since whether the Cuatoms preferentially occupy sublattices a, cin Fig. 4-3 or sublattices b, d is equivalent),but it is not true in general (e.g., third-orderterms do occur in the description of theCu3Au structure, or in the ordering of raregas monolayers adsorbed on graphite in the ÷–

3 structure, as will be discussed be-low).

In order to understand the meaning anduse of Eq. (4-10), consider first the fullyhomogeneous case, —F (x) ∫ 0, F (x) ∫F0;then F [F ] is the standard Helmholtz en-ergy function of thermodynamics, whichneeds to be minimized with respect to F inorder to determine the thermal equilibriumstate, and Ú dx = V the total volume of thesystem. Thus,

(4-12)

which is solved by

F0 = 0 , T > Tc

F0 = ± (– r/u)1/2 (4-13)= ± (r¢/kB u)1/2 (Tc /T – 1)1/2, T < Tc

Hence Eqs. (4-10) and (4-12) indeed yielda second-order transition as T is loweredthrough Tc. For T < Tc, a first-order transi-tion as a function of H occurs, since F0

jumps from (–r/u)1/2 to – (–r/u)1/2 as Hchanges sign. This behavior is exactly thatshown schematically in Fig. 4-1, withb =1/2, B= (r¢/kB u)1/2 and F0(H = 0) = MS.

If u < 0 in Eq. (4-10), however, we mustnot stop the expansion at fourth order butrather must include a term –1

6 vF6(x) (as-suming now v > 0). Whereas in the second-

10

0 0

0 03

k T VF

r uT HB =

= =∂∂

⎛⎝⎜

⎞⎠⎟

+F

F F

order case F (F) has two minima for T < Tc

which continuously merge as T Æ Tc andonly one minimum at F = 0 remains forT > Tc (Fig. 4-6a) F [F ] now has three min-ima for T0 < T < Tc, and the temperature T0

where r changes sign (r = r¢ (T – T0) now)differs from the phase transition tempera-ture Tc where the order parameter jumpsdiscontinuously from zero for T > Tc to(F0)Tc

= ± (3u/4v)1/2, see Fig. 4-6b.These results are found by analogy with

Eq. (4-12) from

(4-14)

which is solved by

(choosing the minus sign of the square rootwould yield the maxima rather than theminima in Fig. 4-6b). On the other hand,we know that [F – F (0)]/(V kBT ) = 0 in thedisordered phase, and Tc can be found (Fig.4-6b) from the requirement that the freeenergy of the ordered phase then is equal to

F02 22 2= − + −u u r/( ) ( / ) /v v v

1

0

0 0

0 02

04

k T VF

r u

T HB =

= + =

∂∂

⎛⎝⎜

⎞⎠⎟

+

F

F F F( )v

this value, i.e.,

(4-15)

With some simple algebra Eqs. (4-14) and(4-15) yield

Tc = T0 + 3u2/(32r¢v) (4-16)

and the “stability limit”, where the mini-mum describing the metastable orderedphase in the disordered phase above Tc dis-appears, is given by

T1 = T0 + u2/(8r¢v) (4-17)

The significance of such metastable statesas described by the Landau theory for first-order transitions and the associated stabil-ity limits T0 (for the disordered phase atT < Tc) and T1 will be discussed further inSec. 4.4.

The alternative mechanism by which afirst-order transition arises in the Landautheory with a scalar order parameter is the lack of symmetry of F against a signchange of F. Then we may add a term–13 wF3 to Eq. (4-10), with another pheno-

[ ( ) ( )]/( )F F V k T

r uT T

F

F F F

0

02

02

04

0

2 4 60

−

+⎛⎝

⎞⎠

B

== + =

c

v


Figure 4-6. Schematic variation of the Helmholtz energy in the Landau model at transitions of (a) second or-der and (b) first order as a function of the (scalar) order parameter F. Cases (a) and (b) assume a symmetryaround F = 0, whereas case (c) allows a cubic term.


menological coefficient, w. For u > 0, F (F)may have two minima (Fig. 4-6c); againthe transition occurs when the minima areequally deep. For r = r¢ (T – T0) this hap-pens when

Tc = T0 + 8w2/(81u r¢) (4-18)

the order parameter jumping there fromF0 = – 9r/w to F0 = 0. Again a stabilitylimit of ordered state in the disorderedphase occurs, i.e.

T1 = T0 + w2/(4u r¢) (4-19)

At this point, an important caveat should beemphasized: free energy curves involvingseveral minima and maxima as drawn inFig. 4-6 are used so often in the literaturethat many researchers believe these con-cepts to be essentially rigorous. However,general principles of thermodynamics indi-cate that in thermal equilibrium the ther-modynamic potentials are convex func-tions of their variables. In fact, F (F0)should thus be convex as a function of F0,which excludes multiple minima! For Fig.4-6a and b this means that for T < Tc instates with –F0 <F <F0 (where F0 is thesolution of Eqs. (4-13) or (4-14), respec-tively) the thermal equilibrium state is not a pure homogeneous phase: rather, theminimum free energy state is given by thedouble-tangent construction to F (F) andthis corresponds to a mixed phase state (the relative amounts of the coexisting phasesare given by the well known lever rule).Now it is standard practice, dating back tovan der Waals’ interpretation of this equa-tion of state for fluids, to interpret the partof F (F) in Fig. 4-6 which lies above theF (F) given by the double-tangent con-struction as a metastable state provided that cT = (∂2F/∂F2)T < 0, whereas stateswith cT < 0 are considered as intrinsicallyunstable states. As will be seen in Sec. 4.4,this notion is intrinsically a concept valid

only in mean-field theory, but lacks anyfundamental justification in statistical me-chanics. Schemes such as those shown inFig. 4-6 make sense for a local “coarse-grained free energy function” only (whichdepends on the length scale L introduced inEq. (4-9)), see Sec. 4.2.2, but not for theglobal free energy.

After this digression we return to thegeneralization of the Landau expansion,Eq. (4-10), the case where the order param-eter has vector or tensor character. Howcan we find which kinds of term appear inthe expansion?

Basically, there are two answers to thisquestion. A general method, which followsbelow, is a symmetry classification basedon group theory techniques (Landau andLifshitz, 1958; Tolédano and Tolédano,1987). A very straightforward alternativeapproach is possible if we consider a par-ticular model Hamiltonian (for a briefdiscussion of some of the most useful mod-els of statistical mechanics for studyingphase transitions, see Sec. 4.3.1). We canthen formulate a microscopic mean-fieldapproximation (MFA), such as the effec-tive field approximation of magnetism (Smart, 1966) or the Bragg-Williams ap-proximation for order–disorder phenomenain alloys (De Fontaine, 1979), where wethen expand the MFA Helmholtz energy di-rectly.

As an example of this approach, we con-sider a specific model of a ternary alloywhere each lattice site i may be taken by ei-ther an A, a B, or a C atom, assuming con-centrations cA = cB = cC =1/3, and assumingthat an energy J is achieved if two neigh-boring sites are taken by the same kind ofatom. This is a special case of the q-statePotts model (Potts, 1952; Wu, 1982), theHamiltonian being

(4-20)Potts = =− …⟨ ⟩∑ij

S S iJ S qi j

d , , , , 1 2

The states 1, 2, and 3 of the Potts spin standfor the three kinds of atomic species A, B,and C here and ·ij Ò denotes a sum overnearest neighbor pairs in the lattice. In theMFA, we construct the Helmholtz energyF = U – T S simply by expressing both en-thalpy U and entropy S in terms of the frac-tions na of lattice sites in states a. The en-tropy is simply the entropy of randomlymixing these species, and, using Stirlingsequation, this yields the standard expres-sion (see elementary textbooks on statisti-cal thermodynamics):

(4-21)

In the enthalpy term, MFA neglects correla-tions in the occupation probability of neigh-boring sites. Hence the probability of find-ing a nearest neighbor pair in a state is sim-ply n2

a , and in a lattice with coordinationnumber z there are z /2 pairs per site. Hence

(4-22)

and thus

(4-23)

We could directly minimize F with respect

to na , subject to the constraint

since each site should be occupied. In orderto make contact with the Landau expan-sion, however, we rather expand F in termsof the two order parameter componentsF1= n1 – 1/3 and F2 = n2 – 1/3 (note that allni =1/3 in the disordered phase). Hence werecognize that the model has a two-compo-nent order parameter and there is no sym-metry between Fi and –Fi . So cubic termsin the expansion of F are expected and dooccur, whereas for a properly defined orderparameter, there cannot be any linear term

aa

==

1

3

1∑ n ,

FV k T

z Jk T

n n nB B = =

= − +∑ ∑2 1

32

1

3

aa

aa aln

Uz J V

n==

− ∑2 1

32

aa

S V n n==

− ∑a

a a1

3

ln

in the expansion:

As expected, there is a temperature T0

(= z J/3kB) where the coefficient of thequadratic term changes sign.

Of course, for many phase transitions aspecific model description is not available,and even if a description in terms of amodel Hamiltonian is possible, for compli-cated models the approach analogous toEqs. (4-20) and (4-24) requires tedious cal-culation. Clearly the elegant but abstractLandau approach based on symmetry prin-ciples is preferable when constructing theLandau expansion. This approach startsfrom the observation that usually the disor-dered phase at high temperatures is more“symmetric” than the ordered phase(s) oc-curring at lower temperature. Recalling the example described in Fig. 4-3, referringto Fe–Al alloys: in the high temperatureA2 phase, all four sublattices a, b, c, and dare completely equivalent. This permuta-tion symmetry among sublattices is brokenin the B2 phase (FeAl structure) where theconcentration on sublattices a, c differsfrom the concentration on sublattices b, d.A further symmetry breaking occurs whenwe go from the B2 phase to the D03 phase(Fe3Al structure), where the concentrationon sublattice b differs from that on sublat-tice d. In such cases the appropriate struc-ture of the Landau expansion for F in termsof the order parameter F is found from theprinciple that F must be invariant againstall symmetry operations of the symmetrygroup G0 describing the disordered phase.In the ordered phase, some symmetry ele-

FV k T

z Jk T

z Jk T

B B

B

=

(4-2 )

− −

+ −⎛⎝⎜

⎞⎠⎟

+ +

+ + +…

63

3 13

92

4

12

22

1 2

12

2 1 22

ln

( )

( )

F F F F

F F F F



ments of G0 fall away (spontaneouslybroken symmetry); the remaining symme-try elements form a subgroup G of G0.Now the invariance of F must hold separ-ately for terms F k of any order k and thisrequirements fixes the character of theterms that may be present.

Rather than formulating this approachsystematically, which would require alengthy and very mathematical exposi-tion (Tolédano and Tolédano, 1987), werather illustrate it with a simple example.Suppose a cubic crystal exhibits a transi-tion from a para-electric to a ferroelectricphase, where a spontaneous polarizationP = (P1, …, Pn), n = 3, appears. F is thengiven as follows:

Whereas the quadratic term of a generaldielectric medium would involve the in-verse of the dielectric tensor,

this term is completely isotropic for cubiccrystals. Inversion symmetry requires in-variance against PÆ – P and hence nothird-order term occurs. The fourth-orderterm now contains the two “cubic invari-ants” Here we invoke

the principle that all terms allowed by sym-metry will actually occur. Now Eq. (4-25)leads, in the framework of the Landau the-ory, to a second-order transition if both

u > 0 and u + u¢ > 0 (4-26)

whereas otherwise we have a first-ordertransition (then terms of sixth order areneeded in Eq. (4-25) to ensure stability).

P Pii

ii

4 2 2and∑ ∑( ) .

ijij i jP P∑ −( ) ,cel

1

F xk T

Fk T

u P u P P

Rd

P

n

j

n

j

n

[ ( )]

( )

Px P

B Bel

= =

=

= d

(4-2 )

0 1 2

1

4

1

2 2

2

1

2

12

14

25

+ ⎡⎣⎢

+ + ′⎛

⎝⎜⎞

⎠⎟+…

+ ∇ +…⎤

⎦⎥

∫

∑ ∑

∑

−

<

c

mm

mm

mm

This approach also carries over to caseswhere the order parameter is a tensor. Forexample, for elastic phase transitions (Cowley, 1976; Folk et al., 1976) the orderparameter is the strain tensor eik. Apply-ing the summation convention (indices oc-curring twice in an expression are summedover), the Landau expansion is

F [eik (x)]

= F0 + Ú dx (–12 ciklm eik elm

+ –13 c(3)iklmrs eik elm ers

+ –14 c(4)iklmrsuv eik elm ers euv

+ … + gradient terms) (4-27)

Here the ciklm are elastic constants and c(3)

iklmrs and c(4)iklmrsuv analogous coefficients

of higher order (“anharmonic”) terms. Formost elastic transitions symmetry permitssome nonzero c(3)

iklmrs and hence leads tofirst-order transitions. Examples of suchsystems are the “martensitic transitions” inNb3Sn and V3Si; there the elastic distortionjumps at the transition from zero to a verysmall value (eik ≈ 10–4) and a Landau ex-pansion makes sense. Note that the Landautheory is not very useful quantitatively fortransitions which are very strongly first or-der, since in that case high-order terms in theTaylor expansion are not negligible. Moreabout martensitic phase transformations canbe found in the Chapter by Delaey (2001).

Just as in a ferromagnetic transition theinverse magnetic susceptibility c–1 van-ishes at Tc (Eq. (4-4)) and in a ferroelectrictransition the inverse dielectric susceptibil-ity cel

–1 vanishes at Tc (Eq. (4-25)), in a“ferroelastic” transition one of the elasticconstants ciklm vanishes at T0. For example,in KCN such a softening is observed for theelastic constant c44. Note that this materialhas already been mentioned as an exampleof a phase transition where the order pa-rameter is the electric quadrupole momenttensor (Eq. (4-8)), describing the orienta-

tion of the dumbbell-shaped CN–-ions: at the same time, it can be considered as an example for an elastic transition, wherethe order parameter is the strain tensor(emn (x))! Such an ambiguity is typical ofmany first-order transitions, because vari-ous dynamical variables (such as a localdielectric polarization P (x), a local quadru-pole moment fmn (x) and the strain emn (x)may occur simultaneously in a crystal andare coupled together. In a Landau expan-sion, such couplings are typically of the bi-quadratic energy–energy coupling type,i.e., various types of mixed fourth-orderterms occur, such as P2(x) emn em¢n¢ umn m¢n¢ .Although this is also true for second-ordertransitions, there the “primary order pa-rameter” is distinct from the fact that theassociate inverse response function cT

–1

vanishes at Tc. For the “secondary orderparameters” the corresponding inverse re-sponse function stays finite. With first-order transitions, the inverse responsefunction of the primary order parameterwould vanish at the hypothetical tempera-ture T0 (stability limit of the disorderedphase), which, however, cannot normallybe reached in a real experiment, see Sec.4.4. K+CN– is a good example of such acomplicated situation with three local or-der parameters coupled together, since theCN– ion also has a dipole moment. Afterthe first-order transition at T = 110 K fromthe cubic “plastic crystal” phase (wherethere is no long-range orientational order,the CN– dumbbells can rotate) to the tet-ragonally distorted, orientationally orderedphase, a second first-order transition oc-curs at T = 80 K to an antiferroelectricphase where the dipole moments order. Inthis system, only a microscopic theory (Michel and Naudts, 1977, 1978; De Raedtet al., 1981; Lynden-Bell and Michel, 1994)could clarify that the elastic interactionbetween the quadrupole moments fmn (x)

(mediated via acoustic phonons) is moreimportant than their direct electric quadru-pole–quadrupole interaction. The elasticcharacter of the phase transition is thereforean intrinsic phenomenon for this material.

We briefly discuss the appropriate con-struction of the order parameter (compo-nents) for order–disorder transitions. Fig.4-3 (and Eq. (4-7)) illustrated how we canvisualize the ordered structure in realspace, and apply suitable symmetry opera-tors of the point group. However, it is oftenmore convenient to carry through a corre-sponding discussion of the ordering in re-ciprocal space rather than in real space (re-member that the ordering shows up insuperlattice Bragg spots appearing in thereciprocal lattice in addition to the Braggspots of the disordered phase). For exam-ple, consider rare gas monolayers adsorbedon graphite: at low temperatures and pres-sures, the adatoms form a ÷–

3 structurecommensurate with the graphite lattice.This ÷–

3 structure can be viewed as a trian-gular lattice decomposed into three sublat-tices, such that the adatoms preferentiallyoccupy one sublattice. Mass density wavesare taken as an order parameter (Bak et al.,1979; Schick, 1981):

(4-28)

Here the qa are the three primitive vectorsassociated with the reciprocal lattice of therare gas monolayer (a being the latticespacing of the triangular lattice)

(4-29)

q

q

q

1

2

3

20

13

2 12

12 3

2 12

12 3

=

=

=

p

p

p

a

a

a

,

,

,

⎛⎝⎜

⎞⎠⎟

− −⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

r yyaa

a a

( ) [ exp( )

exp( )]

x q xq x

= i=1

3

∑ ⋅+ − ⋅−



and ya and y–a are the (complex!) orderparameter amplitudes. In constructing thefree energy expansion with the help of Eq.(4-28), note that the periodicity of theunderlying graphite lattice allows invariant“umklapp” terms (the phase factors ofthird-order terms add up to a reciprocal lat-tice vector of the graphite lattice). Keepingonly those terms in the Landau expansion,and which are nonzero in the ÷–

3 structure,with the real order parameter components(i = ÷––

–1)

(4-30)

the expression

is obtained. This is equivalent to the resultfor the three-state Potts model, Eq. (4-24)(Alexander, 1975), after the quadratic formis diagonalized. A three-dimensional ana-log of this order (where the planes exhibit-ing ÷–

3 structure are stacked together toform a hexagonal lattice) occurs in theintercalated compound C6Li (Guerard andHerold, 1975; Bak and Domany, 1979).

Similarly to the description of the den-sity modulation in the superstructure of adsorbed layers (in two dimensions) orinterstitial compounds (in three dimen-sions) in terms of mass density waves ofthe adsorbate (or interstitial, respectively),the superstructure ordering in binary alloyscan be described in terms of concentrationwaves (Khachaturyan, 1962, 1963, 1973,1983; De Fontaine, 1975, 1979). The sameconcept was used even earlier to describethe magnetic ordering of helimagnetic spin

FV k T

r

w u

B= (4-31)− +

+ − + +

12

13

314

12

22

13

1 22

12

22 2

( )

( ) ( )

F F

F F F F F

F

F

11

3

21

3

2 3

2 3

=

= i

=

=

aa a

aa a

y y

y y

∑

∑

−

−

−

−

( ) /( )

( ) /( )

structures in terms of „spin density waves“(Villain, 1959; Kaplan, 1959). We shalloutline the connection between this ap-proach and the MFA, generalizing the ap-proach of Eqs. (4-21)–(4-23) slightly, inSec. 4.3. Here we only mention that theseconcepts are closely related to the descrip-tion of structural transitions in solids,where the order parameter can often beconsidered as a “frozen phonon”, i.e., a dis-placement vector wave (Bruce and Cowley,1981). Note also that the approach of con-centration waves is not restricted to solids,but can also be used to describe meso-phases in fluid block-copolymer melts(Leibler, 1980; Fredrickson and Helfand,1987; Fredrickson and Binder, 1989; Bin-der, 1994), see Fig. 4-7, in liquid crystal-line polymers, etc.

As a final remark in this section we men-tion that not for all phase transitions in sol-ids there does exist a group–subgroup rela-tionship between the two groups G1 and G2

describing the symmetry of the phases co-existing at the transition. These phases can-not be distinguished by an order parameterwhich is zero in one phase and becomesnonzero in the other. Such transitions mustbe of first order. Examples of this situationare well known for structural phase transi-tions, e.g., the tetragonal–orthorhombictransition of BaTiO3 or the “reconstruc-tive” transition from calcite to aragonite(Guymont, 1981). For the so-called “non-disruptive transitions” (Guymont, 1981),the new structure can still be described inthe framework of the old structure (i.e., itssymmetry elements can be specified, theWyckoff positions can be located, etc.).Landau-type symmetry arguments stillyield information on the domain structuresarising in such phase transitions (Guymont,1978, 1981). The tetragonal–orthorhombictransition of BaTiO3 is considered to be anexample of such a non-disruptive transi-

tion, while “reconstructive transitions” andmartensitic transformations of the type ofthe f.c.c.–b.c.t. transition in Fe are “disrup-tive” (see the Chapter by Delaey (2001)).We shall not go into detail about such prob-lems here, however.

4.2.2 Second-Order Transitionsand Concepts about Critical Phenomena(Critical Exponents, Scaling Laws, etc.)

We now return to the case where at acritical point the order parameter of a sec-

ond-order transition vanishes continuously(Fig. 4-1). We consider the accompanyingcritical singularities (Fig. 4-2), in theframework of Landau theory (Eq. (4-10)).For a proper understanding of the criticalfluctuations of the order parameter, wemust no longer restrict the treatment to thehomogeneous case —F (x) ∫ 0 as was donein Eqs. (4-12) and (4-13); we now wish toconsider, for example, the response to aninhomogeneous, wavevector-dependent or-dering field H (x):

H (x) = Hq exp (i q · x) (4-32)

Although for a magnetic transition such afield cannot be directly applied in the la-boratory, the action of such fields can beindirectly probed by appropriate scatteringmeasurements, e.g., magnetic neutron scat-tering. This is because the scattering can be viewed as being due to the inhomogene-ous (dipolar) field that the magnetic mo-ment of the neutron exerts on the probe.Therefore, the scattering intensity for ascattering vector q is related to the wave-vector-dependent susceptibility (see Kittel(1967)). Similarly, by the scattering of X-rays the wavevector-dependent responsefunction to the sublattice ordering field canbe measured in ordering alloys such as b-CuZn (see Als-Nielsen (1976) for a discus-sion of such scattering experiments in vari-ous systems).

Hence in order to deal with Eq. (4-10)we now have to minimize a functionalrather than a simple function as was donein Eq. (4-12). In order to do so, we notethat the problem of minimizing Ú dx f (y,—y) in Eq. (4-10) is analogous to the prob-lem of minimizing the action in classicalmechanics, W = Ú dt (x, x ), where is theLagrangian and x = dx/dt, the velocity of a particle at point x (t = time). Just as inclassical mechanics, where this problem


Figure 4-7. (a) Chemical architecture of a diblockcopolymer. A diblock copolymer consists of a poly-merized sequence of A monomers (A-block) cova-lently attached to a similar sequence of B monomers.(b) The microphase separation transition occurswhen a compositionally disordered melt of copoly-mers, which are in random coil configurations (right)transforms to a spatially periodic, compositionallyinhomogeneous phase (left) on lowering the temper-ature. For nearly symmetric copolymers the orderedphase has the lamellar structure shown. Since thewavelength of the concentration wave here is of theorder of the coil gyration radius, which may be of theorder of 100 Å for high molecular weight, the order-ing occurs on a “mesoscopic” rather than micro-scopic scale (Å) and hence such phases are called“mesophases”. From Fredrickson and Binder (1989).


is solved in terms of the Euler–Lagrangeequation

we conclude here that

in order that the Helmholtz energy func-tional is a minimum. Eq. (4-10) thus yields

(4-33)

Using

F (x) = F0 + DF (x) = F0 + DFq exp(iq · x)

Eqs. (4-32) and (4-33) yield, on linearizingin DFq for small Hq , the wavevector-de-pendent susceptibility

From Eq. (4-34) we can simply read off the temperature dependence of the suscep-tibility cT (Eq. (4-4)) and the correlationlength x of order parameter fluctuations,which is generally expected to behave as

(4-35)

Above Tc, the terms 3u F02 + 5v F0

4 = 0,hence

c

x

T k Tr r T T

R rd R r d T T

= = (4-3 )

= =

B c

c

1 16

2 1 2

′ −

′ − −

( )

/( ) ( / ) ( ) /

x xx

n

n= c c

c c

ˆ ( / ) ,ˆ ( / ) ,

T T T T

T T T T

− >′ − <

⎧⎨⎩

−

− ′1

1

c

cx

( )q q

q≡

+ + +⎡

⎣⎢

⎤

⎦⎥

+

−

DF

F F

H

k Tr u

Rd

q

qT

=

= (4-3 )

B

13 5

14

02

04

22

1

2 2

v

r u

Rd

Hk T

F F F

F

( ) ( ) ( )

( )( )

x x x

xx

+ +

− ∇

3 5

22

v

=B

( / ) [ / ( )]∂ ∂ − ∇ ∂ ∂ ∇f fy y = 0

( / ) [ / ( / )]∂ ∂ − ∂ ∂ xt

x tdd

d d = 0

For T < Tc, using Eq. (4-13) the term 5vF04

near Tc is still negligible, and hence

Eqs. (4-36) and (4-37) simply imply thewell known Curie–Weiss law, i.e., the ex-ponents g, g¢, defined in Eq. (4-4), are, inthe framework of the Landau theory,

g = g¢ = 1 (4-38)

Comparing Eqs. (4-35) and (4-37) we alsoconclude that

n = n¢ = 1/2 (4-39)

The physical meaning of the correlationlength x is easily recognized if the wellknown fluctuation relationship

(4-40)

is expanded to second order in q as

which can be written as

Comparing Eqs. (4-34) and (4-42), we findthat in terms of the order parameter corre-lation function G (x),

(4-43)G( ) ( ) ( )x x= ⟨ ⟩ −F F F0 02

c ( ) [ ( ) ( ) ]

[ ( ) ( ) ]

[ ( ) ( ) ]

q x

x

x

x

x

x

=

(4-42)

B

10

12

0

0

02

22

02

02

k T

qd

x

∑

∑

∑

⟨ ⟩ −

× −⎧⎨⎩

⟨ ⟩ −

⟨ ⟩ − ⎫⎬⎭

F F F

F F F

F F F

c ( ) [ ( ) ( ) ]

( ) [ ( ) ( ) ]

q x

q x x

x

x

≈ ⎧⎨⎩

⟨ ⟩ −

− ⋅ ⟨ ⟩ − ⎫⎬⎭

∑

∑

10

12

0

02

202

k TB

(4-41)

F F F

F F F

c ( ) exp( )

[ ( ) ( ) ]

q q x

xx

= iB

1

0 02

k T∑ ⋅

× ⟨ ⟩ −F F F

c

x

T k Tr r T T

R rd R r d T T

= = (4-3 )

= =

B c

c

−′ −

− ′ − −

12

12

7

2 22 1 2

( )

/( ) ( / ) ( ) /

and cT and x2 are related to the zeroth andsecond moment of the correlation function:

(4-44)

(4-45)

Eq. (4-44) clearly shows that the critical di-vergence of cT occurs because the correla-tions G (x) become long ranged: whereasoff Tc the correlation function for large | x |decays exponentially,

ln G (x) Æ – | x | /x , | x | Æ • (4-46)

at Tc we have a power-law decay of the cor-relation function:

G (x) = G | x |–(d –2+h ), T = Tc (4-47)

The exponent describing this critical decayhas been defined such that

h = 0 (4-48)

in the Landau theory, whereas in generalh 0, as will be discussed below.

In order to make our collection of criticalexponents complete, we also consider thecritical isotherm

F0 = DH1/d , T = Tc (4-49)

Using Eq. (4-33) for a uniform field weconclude that

F0 = (kBTu)–1/3 H1/3 (4-50)

i.e., D = (kBTu)–1/3 and

d = 1/3 (4-51)

in the framework of the Landau theory. Finally, we turn to the specific heat, Eq. (4-5), which behaves rather pathologicallyin the Landau theory: for T > Tc and H = 0,Eq. (4-10) just implies F = F0, i.e., the specific heat associated with the orderingdescribed by F is identically zero above

x 2 212

=d

x G Gx x

x x∑ ∑( ) ( )

cT k TG=

B

1

xx∑ ( )

Tc. For T < Tc, we have instead, from Eqs.(4-10) and (4-13)

(4-52)

which implies C = r ¢2 (T /Tc) / (4 kB u) forT < Tc. This jump singularity of the specificheat at Tc instead of the power law in Eq.(4-5) is formally associated with vanishingcritical exponents:

a = a¢ = 0 (4-53)

The question must now be asked whetherthis description of critical phenomena isaccurate in terms of the Landau theory. Thefree energy functional F [F (x)] in Eq. (4-10) should be considered as an effectiveHamiltonian from which a partition func-tion Z can be obtained so that the trueHelmholtz energy becomes:

F = – kBT ln Z (4-54)

= – kBT ln Ú d [F (x)] exp – F [F (x)]/kBT

It is natural to expect that the main contri-bution to the functional integral comesfrom the region where the integrand is larg-est, i.e., the vicinity of the point whereF [F (x)]/kBT has its minimum. If we as-sume that the distribution over which theaverage in Eq. (4-54) is a delta function atthe value F0 yielding the minimum, i.e.,we assume that fluctuations of the orderparameter make a negligible contributionto the functional integral, then the mini-mum of F is equivalent to the minimum ofF (F). In general, however, this is not true,as the effects of fluctuations modify thecritical behavior drastically, and the Lan-dau theory does not hold for systems suchas b-CuZn.

A first hint of the conditions for whichthe Landau theory is valid can be obtained

F Fk T V

r u

ru

rT T

k u

− +⎛⎝

⎞⎠

− ≈ − ′−

002

02

22

2

2

2 4

41

4

B

c

B

=

=

F F

( / )



by considering the effect of fluctuationswithin the context of the Landau theory it-self. The criterion Ginzburg (1960) sug-gests is that the Landau theory is valid ifthe mean-square fluctuation of the orderparameter in a correlation volume is smallin comparison with the order parametersquare itself:

·[F (x) – F0]2ÒL= x O F02 (4-55)

Here L is the coarse-graining length intro-duced in Eq. (4-9) in order to transform themicroscopic lattice description of orderingphenomena in solids to a continuum de-scription. On the lattice level, the local or-der parameter Fi shows a rapid variationfrom one lattice site to the next, and themean-square fluctuation is very large. Forexample, for an Ising model of an aniso-tropic ferromagnet,

(4-56)

Where J is the exchange interaction, thesum ·i, j Ò extends once over all nearestneighbor pairs, and Si points in the direc-tion of the local magnetic moment at latticesite i, we have Fi = Si , i.e., Fi

2 ∫ 1. In thiscase ·[Fi – ·FiÒ]

2Ò = 1 – Ms2 is never small

in comparison with the square of the spon-taneous magnetization Ms

2 near Tc. Carry-ing out the local averaging defined in Eq.(4-9) reduces this local fluctuation in a“volume” Ld. This averaging should, inprinciple, be carried out over a length scaleL that is much larger than the lattice spac-ing a but much smaller than the correlationlength x,

a O L O x (4-57)

Whereas for a study of critical phenomenait is permissible to average out local effectson the scale of a lattice spacing, relevantspatial variations do occur on the scale ofx, as we have seen in Eqs. (4-34) and

Ising = =− − ±⟨ ⟩∑ ∑J S S H S Si j

i ji

i i,

, 1

(4-46). Replacing the right inequality inEq. (4-57) by the equality, L = x, as done inEq. (4-55), gives therefore the maximumpermissible choice for L, for which thefluctuations in Eq. (4-55) are smallest. Eventhen, however, it turns out that Eq. (4-55)typically is not fulfilled for T close to Tc. Inorder to see this, we write Eq. (4-55) interms of the local variable Fi and denotethe number of lattice sites i contained Ld for the choice L = x as N (t), wheret ∫1 – T /Tc. Then Eq. (4-55) becomes

(4-58)

where we have denoted the volume of sizeL = x centered at x as Vx (x). Making use ofthe translational invariance of correlationfunctions and ·FiÒ = Ms (t), the spontane-ous order parameter which is independentof i, Eq. (4-58) becomes

(4-59)

If the sum over correlations in Eq. (4-59)were to extend over all space, it would sim-ply be the “susceptibility” kBT c (t) (seeEq. (4-42)). Since the sum does containjust the volume region over which the Fisare strongly correlated with each other, thesum clearly is of the order of f kBT c (T ),where f <1 is a factor of order unity, whichshould have no critical (vanishing or diver-gent) temperature dependence as T ÆTc.Hence in the inequality Eq. (4-59) this fac-tor may also be omitted, and we conclude(Als-Nielsen and Laursen, 1980) that

c (t) O N (t) O Ms2(t) (4-60a)

Making use of N (t) = [x (t)/a]d, a being thelattice spacing, we obtain:

const. O [x (t)]d Ms2(t) c –1(t) (4-60b)

N t M t

N t M ti V

i i( ) [ ( )]

( ) ( )( )Œ x x

∑ ⟨ ⟩ −F F= s

s

02

2 2O

[ − ⟨ ⟩ ]∑ ∑i V

i ii V

iŒ Œx x( ) ( )x x

F F F2 2O

where in suitable units the constant on theleft-hand side of the equality is of orderunity. Using Eqs. (4-4), (4-6), and (4-35)we obtain (anticipating g = g¢ and n = n¢)

const. O t –n d+2b+g (4-61)

Inserting the Landau values for the criticalexponents b = 1/2, n = 1/2, g = 1 and know-ing the critical amplitude of x ~ R, the inter-action range, Eq. (4-60) implies

const. O (R/a)d t (d –4)/2 (4-62)

This condition for the validity of the Lan-dau theory for d < 4 always breaks down as t Æ 0 (T ÆTc). In fact, for d < 4 closeenough to Tc a regime occurs where fluctu-ations dominate the functional integral (Eq.(4-54)). The “crossover” from the mean-field regime (where the Landau descriptionis essentially appropriate) to the non-mean-field regime occurs at a reduced tempera-ture distance t = tcr , where Eq. (4-62) istreated as an equality: for d = 3,

tcr ~ (R/a)–6 (4-63)

Hence systems with a large but finite rangeof interaction behave in an essentially Lan-dau-like manner. An example of such a be-havior is the unmixing critical point inpolymer fluid mixtures with a high degreeof polymerization N : since each coil has aradius rc ~ N 1/2, the monomer density in-side the sphere taken by a polymer coil isonly of order r ~ N/rc

3 ~ N –1/2, which im-plies that each coil interacts with N 1/2

neighbor coils, and the effective interactionvolume (R/a)d in Eqs. (4-62) and (4-63)should be taken as N 1/2, i.e., tcr ~ N –1 (DeGennes, 1979; Binder 1984c, 1994).

At this point, we emphasize that usingN (t) = [x (t)/a]d in Eq. (4-60) is not valid insystems with long-range anisotropic inter-actions, such as uniaxial dipolar magnets.For a system such as LiTbF4 (Als-Nielsen

and Laursen, 1980), the essential part ofthe magnetic Hamiltonian is a magneto-static dipole–dipole interaction,

(4-64)

where m is the magnetic moment per spinand z the coordinate of x in the direction ofthe uniaxial anisotropy. Unlike the iso-tropic Ising Hamiltonian in Eq. (4-56), herethe sign of the interaction depends on thedirection in the lattice. Therefore, fluctua-tions display an essential anisotropy: in-stead of the isotropic result (Eq. (4-34)) ofthe well known Ornstein-Zernike type, the wavevector-dependent susceptibilityc (q) = S (q)/kBT (S (q) = structure factor)becomes anisotropic:

S (q) ~ [x –2 + q2 + g (qz /q)2]–1 (4-65a)

where qz is the z-component of q and g is aconstant. Whereas Eq. (4-32) implies thatS (q x =1) = S (q = 0)/2, the equation q x = 1which for Eq. (4-32) defines a sphere of ra-dius x –1 in q-space now defines an aniso-tropic surface. From

x –2 = q2 + g (qz /q)2 (4-65b)

we obtain an object having the shape of aflat disc, with radius 1/x in the qx – qy planebut maximum extension of order 1/x2 inthe qz direction. This object, defined byS (q) = –1

2 max S (q), can be interpreted asthe Fourier transform of the correlationvolume. This implies that the correlationvolume in real space is a long ellipsoid,with linear dimensions x in the qx , qy di-rection but with linear dimension x 2 in theqz direction. In d-dimensions, this argu-ment suggests for uniaxial dipolar systems:

N (t) = [x (t)/a]d+1 (4-66)

which yields in Eq. (4-60)

const. O t –n (d+1)+2b+g = t – (d–3)/2 (4-67)

= =− − ±≠∑m 2

2 2

53

1i j

i j iz

S S Sx

x| |,



where in the last equality, the Landau ex-ponents have been used. Comparing Eqs.(4-61) and (4-67) shows that d-dimensionaluniaxial dipolar systems somehow corre-spond to (d +1)-dimensional systems withisotropic short-range forces. Therefore themarginal dimension d*, above which theLandau theory predicts the values of criti-cal exponents correctly, is d* = 3 for uniax-ial dipolar systems, unlike the standard iso-tropic short-range case where d* = 4.

For elastic phase transitions in cubiccrystals where the combination of elasticconstants c11 – c12 softens as the criticalpoint is appproached, such as the systemPrAlO3, the structure factor S (q) has theform: (4-68)S (q) ~ [x –2 + q2 + A(qz /q)2 + B(q^/q)2]–1

where q = (qz , q^) and A and B are con-stants. In this case the correlation vol-ume defined from the condition S (q) =–12 max S (q) in real space is a flat diskwith radius x2 and diameter x, i.e., for elas-tic systems:

N (t) = [x (t)/a]d+2 (4-69)

In this case, the Ginzburg criterion, Eq. (4-60), yields

const. O t –n (d+2)+2b+g = t – (d–2)/2 (4-70)

instead of Eqs. (4-62) or (4-67), respec-tively. Since the marginal dimensionalityfor such systems is d* = 2, three-dimen-sional systems should be accurately de-scribed by the Landau theory, and this iswhat is found experimentally for PrAlO3.Examples for elastic phase transitionswhich are of second order are scarce, as isunderstandable from the symmetry consid-erations which lead to first-order transi-tions in most cases (Cowley, 1976; Folket al., 1976). However, for many first-orderferroelastic transitions the Landau descrip-tion also fits the experimental data very

well over a wide temperature range (Salje,1990).

For all other systems we do not expectthe Landau critical exponents to be an ac-curate description of the critical singular-ities. This is borne out well by the exact so-lution for the two-dimensional Ising model(Onsager, 1944; McCoy and Wu, 1973),where the critical exponents have the values

a = 0 (log), b = 1/8, g = 7/4,

d = 15, n = 1, h = 1/4 (4-71)

Obviously, these numbers are a long wayfrom the predictions of the Landau theory.Note that a = 0 in Eq. (4-71) has the mean-ing C ~ | log | t ||, unlike the jump singular-ity of the Landau theory.

Even in the marginal case of uniaxial di-polar ferromagnets (or uniaxial ferro-electrics; see, e.g., Binder et al. (1976) fora discussion), the Landau theory is notcompletely correct; it turns out (Larkin andKhmelnitskii, 1969; Aharony, 1976) thatthe power laws have the Landau form butare modified with logarithmic correctionfactors:

cT = G± t –1 | ln t |1/3, t Æ 0 (4-72a)

C = A± | ln t |1/3, t Æ 0 (4-72b)

Ms = B (– t)1/2 | ln (– t)|1/3, t Æ 0 (4-72c)

Ms|Tc= D H1/3 | lnH |1/3, H Æ 0 (4-72d)

Although for uniaxial ferroelectrics the ex-perimental evidence in favor of Eq. (4-72)is still scarce, convincing experimental ev-idence does exist for dipolar ferromagnetssuch as LiTbF4 (see, e.g., Als-Nielsen andLaursen (1980) for a review).

Eq. (4-72) results from including fluctu-ation contributions to the functional inte-gral in Eq. (4-54) systematically, which canbe done in a most powerful way be renor-malization group theory (Aharony, 1973,1976; Fisher, 1974; Wilson and Kogut,

1974; Ma, 1976; Amit, 1984; Yeomans,1992). A description of this theory is be-yond the scope of this chapter; we onlymention that it is this method which yieldshighly accurate approximations for the val-ues of critical exponents of three-dimen-sional systems. For example, for the uniax-ial magnets (as described by the Ising Ha-miltonian, Eq. (4-56), experimental exam-ples being the antiferomagnets MnF2 orFeF2) or the ordering alloy b-brass, the ex-ponents are predicted to be (LeGuillou andZinn-Justin, 1980):

a ≈ 0.110, b ≈ 0.325, g ≈ 1.240,d ≈ 4.82, n ≈ 0.63, h ≈ 0.032 (4-73)

Whereas within the Landau theory the “or-der parameter dimensionality” n does notmatter, as far as the values of the criticalexponents are concerned, n does matter ifwe go beyond the Landau theory. For ex-ample, for isotropic magnets as describedby the well known Heisenberg model ofmagnetism,

(4-74)

Si being a unit vector in the direction of themagnetic moment at lattice site i, we have(LeGuillou and Zinn-Justin, 1980):

a ≈ – 0.116, b ≈ 0.365, g ≈ 1.391,d ≈ 4.82, n ≈ 0.707, h ≈ 0.034 (4-75)

Again these results are in fair agreementwith available experimental data, such asthe isotropic antiferromagnet RbMnF3 (seeAls-Nielsen (1976) for a review of experi-ments on critical point phenomena). Notethat the negative value of a in Eq. (4-75)implies that the specific heat has a cusp offinite height at Tc.

The renormalization group theory alsoprovides a unifying framework and justifi-cation for two important concepts aboutcritical phenomena, namely “scaling” and

Heis = − ⋅ −≠∑ ∑i j

ij i j izJ H SS S m

“universality”. Again we wish to conveyonly the flavor of the idea to the reader,rather than to present a thorough discus-sion. Consider, for example, the decay ofthe correlation function of order parameterfluctuations at the critical point, Eq. (4-47):changing the length scale from r to r¢ = Lrwould change the prefactor G but leave thepower-law invariant. The physical inter-pretation of this fact is the “fractal struc-ture” of critical correlations (Mandelbrot,1982): just as a fractal geometric objectlooks the same on every length scale, thepattern of critical fluctuations looks thesame, irrespective of the scale. Slightly offTc, then, there should be only one relevantlength scale in the problem, the correlationlength x, which diverges as T ÆTc, and de-tails such as the precise behavior of thecorrelation functions for interatomic dis-tances should not matter. As a result, it isplausible that the correlation functionG (x, x ) should depend only on the ratio ofthe two lengths x and x, apart from a scalefactor:

G (x, x ) = Gx – (d–2+h) G (x /x ) (4-76)

Obviously, the condition G (0) =1 for the“scaling function” G (x /x ) ensures that Eq.(4-76) crosses over smoothly to Eq. (4-47)as T ÆTc, where x Æ∞.

From the “scaling hypothesis” in Eq. (4-76) we immediately derive a “scalinglaw” relating the exponents g, n and h. Us-ing Eq. (4-44) and denoting the surface of a d-dimensional unit sphere as Ud (Ud = 4pin d = 3) we obtain

c x

x

x

h

h h

Tx

d

d

k TG x

Gk T

G x

U Gk T

x G x x

U Gk T

G

= d

= d

= d (4-77)

B B c

B c

B c

1

1

0

2 1

0

∑ ∫

∫

∫

≈

−∞

− −∞

( )˜

( , )

ˆ˜ ( / )

ˆ˜ ( )

x



Since the integral over = x /x in Eq. (4-77)yields a constant, we conclude

cT ~ x (2–h) ~ | t |–n (2–h ) ~ | t |–g

cf. Eqs. (4-4) and (4-35). Thus the scalingrelation results in

g = n (2 – h) (4-78)

Eq. (4-76) indicates that G (x, x ) dependson the two variables x and x in a rather spe-cial form, namely, it is a homogeneous func-tion. A similar homogeneity assumption isalso true for the singular part of the free energy, (4-79)

F = Freg + t2–a F (H ), H = H G t –g–b/B

where the singular temperature dependencefor H = 0 is chosen to be compatible withEq. (4-5), Freg is a background term whichis analytic in both T and H even for T = Tc,while F (H ) is another “scaling function”.At this point, we present Eq. (4-79) as apostulate, but it should be emphasized thatboth Eqs. (4-76) and (4-79) can be justifiedfrom Eq. (4-54) by the renormalizationgroup approach.

Combining Eqs. (4-1) and (4-79) we ob-tain: (4-80)

Defining – (G /B) F ¢ (H ) ∫ B M (H ) with M (0) =1, we obtain:

F = Bt b M (H), b = 2 – a – g – b (4-81)

since for H= 0, Eq. (4-80) must reduce toEq. (4-6). Taking one more derivative wefind

cT = (∂F /∂H )T = G t –g M¢(H) (4-82)

hence Eq. (4-79) is compatible with Eq. (4-4), as it should be. The condition thatEq. (4-81) reduces to the critical isothermfor t Æ 0 requires that M (H Æ •) ~H b /(g+b ), in order that the powers of t

F G= =− ∂ ∂ − ′− − −( / )

ˆ

ˆ˜ ( ˜ )( )F H

Bt F HT

2 a g b

cancel out. On the other hand, this alsoyields

F|T=Tc~ H1/(1+g /b ) = H1/d ,

i.e., d = 1 + g /b (4-83)

The scaling assumptions also imply thatexponents above and below Tc are equal,i.e, a = a¢, g = g¢, and n = n¢. Now there isstill another scaling law which results fromconsidering all degrees of freedom inside acorrelation volume xd to be highly corre-lated with each other, while different corre-lation volumes can be considered as essen-tially independent: this argument suggeststhat the singular part of the Helmholtz en-ergy per degree of freedom can be writtenas Fsing≈ [1/Ns(t)] ¥ const., since indepen-dent degrees of freedom do not contributeany singular free energy. Since

Ns (t) ~ x d ~ | t |–dn

we conclude, by comparison with Eq. (4-79), that

dn = 2 – a (4-84)

Note that this so-called “hyperscaling rela-tion” with the Landau-theory exponents isonly true at the marginal dimension d* = 4,whereas the other scaling relationships ob-viously are fulfilled by Landau exponentsindependent of the system dimensionality.However, all scaling relationships (includ-ing Eq. (4-84)) are satisfied for the two-and three-dimensional Ising model (Eqs.(4-71) and (4-73)) and the three-dimen-sional Heisenberg model, Eq. (4-75).

Let us consider the two-dimensionalHeisenberg model. For this model the ef-fects of statistical fluctuations are so strongthat they destabilize the ordering alto-gether: no spontaneous ordering exists ind = 2 for any system with order parameterdimensionality n ≥ 2, and a critical pointoccurs only at zero temperature, Tc = 0, for

n > 2. Thus d = 2 is the “lower critical di-mensionality” d1 for n ≥ 2, whereas forn =1 we have d1=1: quasi-one-dimensionalorderings are always unstable. The casen = 2, d = 2 is very special: a phase transi-tion still occurs at a nonzero Tc, the so-called “Kosterlitz–Thouless” transition(Kosterlitz and Thouless, 1973); for T < Tc

we have Ms(T ) ∫ 0 while at the same timex (T ) = •, and the correlation functionshows an algebraic decay, Eq. (4-47), witha temperature-dependent exponent h. Atfirst sight, this behavior may appear fairlyesoteric, but in fact it is closely related to the “roughening transition” of crystalsurfaces (Weeks, 1980). Such rougheningtransitions have been observed for high-in-dex crystal faces of various metals (Sala-non et al., 1988). The power-law decay ofcorrelations, Eq. (4-47), can be related tothe behavior of the height–height correla-tion function of a crystal surface in therough state (Weeks, 1980). As is wellknown, the roughness or flatness of crystalsurfaces has a profound effect on their ad-sorption behavior, crystal growth kinetics,etc. (Müller-Krumbhaar, 1977).

The other important concept about criti-cal phenomena is “universality”: since theonly important length scale near a criticalpoint is provided by the correlation lengthwhich is much larger than all “micro-scopic” lengths such as lattice spacing andinteraction range, it is plausible that “de-tails” on the atomic scale do not matter, andsystems near a critical point behave in thesame way provided that they fall in thesame “universality class”. It turns out thatuniversality classes (for systems withshort-range interactions!) are determinedby both spatial dimensionality d and orderparameter dimensionality n, and in addi-tion, the symmetry properties of the prob-lem: e.g., Eq. (4-25) for n = 3 and u¢ = 0falls in the same class as the Heisenberg

model of magnetism, Eq. (4-74), since theferroelectric ordering would also be trulyisotropic. However, in the presence of anonvanishing “cubic anisotropy”, u¢≠ 0, ingeneral a different universality class re-sults. The effect of such higher order invar-iants in the Landau expansion are particu-larly drastic again in systems with reduceddimensionality; e.g., the model with n = 2but cubic anisotropy no longer exhibits aKosterlitz–Thouless transition, but rather a nonzero order parameter is stabilizedagain. Owing to the “marginal” characterof the cubic anisotropy in d = 2, however,the behavior is not Ising-like but rather thepathological case of a “universality class”with “nonuniversal” critical exponents oc-curs. The latter then do depend on “micro-scopic” details, such as the ratio of theinteraction strengths between nearest andnext nearest neighbors in the lattice (Krin-sky and Mukamel, 1977; Domany et al.,1978; Swendsen and Krinsky, 1979; Lan-dau and Binder, 1985). Again such prob-lems are not purely academic, but relevantfor order–disorder transitions in chemi-sorbed layers on metal surfaces, such as Oon W (110) (see Binder and Landau (1989)for a review of the theoretical modeling ofsuch systems).

One important consequence of univer-sality is that all gas–fluid critical points inthree-dimensional systems, all criticalpoints associated with unmixing of fluid orsolid binary mixtures, and all order–disor-der critical points involving a single-com-ponent order parameter (such as b-brass)belong to the same universality class as thethree-dimensional Ising model. All thesesystems not only have the same critical ex-ponents, but also the scaling functionsM (H ), G () etc. are all universal. Also,critical amplitude ratios are universal, such as C/C+, (Eq. (4-4), A/A¢ (Eq. (4-5)),or more complicated quantities such as



D G Bd –1. The theoretical calculation andexperimental estimation of such criticalamplitude ratios are discussed by Privmanet al. (1991).

An interesting problem concerns thecritical singularities associated with nonor-dering “fields”. These singularities occurbecause the ordering field F (x) consideredonly in Eqs. (4-10) and (4-54) couples toanother quantity: e.g., for an antiferromag-net in a magnetic field the order parameterF (the sublattice magnetization) couples tothe uniform magnetization M, to the latticeparameters, to the electron density, etc.Since the Hamiltonian is invariant againstinterchange of the sublattices which im-plies a sign change of F, this coupling isquadratic in F2. Whereas in mean fieldtheory the resulting critical behavior of thenonordering “field” is then proportional to·FÒ2 ~ t 2b, taking fluctuations into account,the actual critical behavior is proportionalto ·F2Ò ~ t (1–a), i.e., an energy-like singu-larity. Such energy-like singularities arepredicted for the electrical resistivity rel atvarious phase transitions (Fisher andLanger, 1968; Binder and Stauffer, 1976a),for the refractive index nr (Gehring andGehring, 1975; Gehring, 1977), etc. Look-ing for anomalies in the temperature deriv-ative drel (T )/dT, dnr (T )/dT etc., whichshould exhibit specific-heat-like singular-ities, is often a more convenient tool for lo-cating such phase transitions than measure-ments of the specific heat itself.

An important phenomenon occurs whena nonordering field which is coupled to theorder parameter in this way, such as themagnetization in an antiferromagnet in anexternal field, is held fixed. Suppose wefirst study the approach to criticality by let-ting the uniform magnetic field h tend to itscritical value hc(T ): we then have the lawF ~ [hc(T ) – h]b for the order parameterand M – Mc(T ) ~ [hc(T ) – h]1–a for the mag-

netization (due to the coupling F2(x) M (x)in the Hamiltonian, as mentioned above).However, if we now consider the variationof F with M, combining both laws we get(assuming a > 0)

F ~ [Mc (T ) – M ]b /(1–a) (4-85)

Considering now the phase transition in thespace of M, T as independent thermody-namic parameters, we then find that oncrossing the critical line Mc (T ) the expo-nent describing the vanishing of the orderparameter is b /(1–a) rather than b. Thiseffect generally occurs when the criticalline depends on extensive (rather than in-tensive) thermodynamic variables and iscalled “Fisher renormalization” (Fisher,1968). This is very common for order–dis-order phenomena in adsorbed layers atfixed coverage, or in alloys at fixed con-centration, for unmixing critical points internary mixtures, etc. Other exponents alsobecome “renormalized” similarly, e.g., g isreplaced by g /(1–a), etc. A detailed anal-ysis (Fisher, 1968) shows that no such“renormalization” of critical exponents oc-curs for systems with a fixed concentrationc if the slope of the critical line vanishes,dTc (c)/dc = 0, for the considered concen-tration. This is approximately true for b-brass, for instance.

4.2.3 Second-Order Versus First-OrderTransitions; Tricritical and Other Multicritical Phenomena

An important question for any phasetransition is deciding a priori whether itshould be a second- or first-order transi-tion; general principles are sought in an-swering this question, such that therewould be no need for specific experimentaldata.

In the Landau theory, general symme-try conditions exist which allow second-

order transitions (Lifshitz, 1942). We heremerely state them, without even attemptingto explain the group-theoretical language(see, e.g., Tolédano and Tolédano, 1987,for a thorough treatment):

(i) The order parameters F transform asa basis of a single irreducible repre-sentation X of the group G0 character-izing the symmetry properties of thedisordered phase.

(ii) The symmetric part of the representa-tion X3 should not contain the unitrepresentation.

(iii) If the antisymmetric part of X2 has arepresentation, the wavevector q asso-ciated with X is not determined bysymmetry. In this case q is expected to vary continuously in the orderedphase.

If these conditions are met, a transitioncan nevertheless be first order, because afourth-order term can be negative (see Sec.4.2.1). If they are not met, the transitionmust be first order, according to the Lan-dau rules.

The first of these rules essentially saysthat if in the ordered phase two quantitiesessentially independent of each other (notrelated by any symmetry operation, etc.)play the role of order parameter compo-nents F1, F2, there is no reason why in thequadratic term r1(T ) F1

2 + r2(T ) F22 of the

Landau expansion the coefficients r1(T ),r2(T ) should change their sign at the sametemperature. Thus, if F1, F2 are “primary”order parameter components, they shouldappear via a first-order transition.

The second condition essentially impliesthe absence of third-order terms (F3) in theLandau expansion. It turns out, however,that in d = 2 dimensions there are well-known counter-examples to these rules,namely the Potts model (Potts, 1952) withq = 3 and q = 4 (see Eq. (4-20)); as shown

by Baxter (1973), these models have sec-ond-order transitions. The critical expo-nents for these models are now believed to be known exactly, e.g., (q = 3) a =1/3,b =1/9, g =13/9 and (q = 4) a = 2/3,b =1/12, g = 7/6 (Den Nijs, 1979; Nienhuiset al., 1980). As mentioned in Sec. 4.2.1, anexperimental example for the three-statePotts model is the ÷–

3 superstructure of var-ious adsorbates on graphite. Convincingexperimental evidence for a specific heatdivergence of He4 on grafoil described bya =1/3 was presented by Bretz (1977). Thesystem O on Ru (001) in the p (2 ¥ 2) struc-ture (Piercy and Pfnür, 1987) falls into theclass of the four-state Potts models andagain the data are in reasonable agreementwith the predicted critical exponents.

In d = 3 dimensions, however, the three-states Potts model has a weak first-ordertransition (Blöte and Swendsen, 1979) sothat the Landau rule (ii) is not violated.This is of relevance for metallurgical sys-tems such as the CuAu ordering (Fig. 4-8)on the f.c.c. lattice, which, according toDomany et al. (1982), belongs to the classof the three-states Potts model, while theCu3Au structure belongs to the class of thefour-state Potts model. Another example ofthe three-state Potts model is the structuraltransition of SrTiO3 stressed in the [111]direction (Bruce and Aharony, 1975). Theorder–disorder transition for all systemsdescribed by these types of ordering are al-ways of first order, so there is no contradic-tion with this Landau rule, except for thetransition to a charge density wave state in2H-TaSe2 (Moncton et al., 1977) which ap-parently is second order although there is athird-order invariant (Bak and Mukamel,1979). It is possible, of course, that thetransition is very weakly first order so thatexperimentally it could not be distin-guished from second order. This remarkalso holds for other examples of apparent



violations of the Landau rules discussed byTolédano and Pascoli (1980), Tolédano(1981), and Tolédano and Tolédano (1987).

There are many transitions which theabove Landau symmetry criteria wouldpermit to be of second order but which areactually observed to be first order. Exam-ples are type I or type II antiferromagneticstructures, consisting of ferromagnetic(100) or (111) sheets, respectively, with an

alternating magnetic moment direction be-tween adjacent sheets (cf. Figs. 4-4 and 4-8a). Experimentally first-order transi-tions are known for FeO (Roth, 1958), TbP(Kötzler et al., 1979) (these systems haveorder parameter dimensionality n = 4), UO2

(Frazer et al., 1965; this is an example withn = 6), MnO (Bloch and Mauri, 1973), NiO(Kleemann et al., 1980) (examples withn = 8), etc. The standard phenomenologicalunderstanding of first-order transitions inthese materials invoked magnetostrictivecouplings (Bean and Rodbell, 1962) orcrystal field effects (in the case of UO2; seeBlume, 1966), which make the coefficientu in Eq. (4-10) negative and thus produce afree energy of the type shown in Fig. 4-6b.

It has been suggested that the first-ordercharacter of the phase transition in thesematerials is a fundamental property due to the large number n of order-parametercomponents and the symmetry of the Hamiltonian: renormalization group ex-pansions discussed by Mukamel et al.(1976a, b) and others suggested that allsuch transitions are “fluctuation-inducedfirst-order transitions”, i.e., all antiferro-magnets with n≥ 4 order parameters musthave first-order transitions. However, thereseem to be many counter-examples of cu-bic n ≥ 4 antiferromagnets with phase tran-sitions of apparently second order, such asBiTb (Kötzler, 1984), GdTb (n = 4; Hulli-ger and Siegrist, 1979), and GdSb (n = 8;McGuire et al., 1969). The reasons for this discrepancy between renormalizationgroup predictions and experiments are notclear.

There are various other cases where“fluctuation-induced first-order transitions”occur, as reviewed in detail by Binder(1987a). Here we mention only the exam-ple of the phase transition from disorderedblock copolymer melts to the lamellar mes-ophase (Fig. 4-7b). For this system the

Figure 4-8. Five crystallographic superstructures onthe face-centered cubic (fcc) lattice and their equiva-lent antiferromagnetic superstructures if in a binaryalloy A-atoms (full circles) are represented as “spinup” and B atoms (open circles) are represented as “spin down”. Case (a) refers to the CuAuI structure,(b) to the A2B2 structure, (c) to the A3B structure ofCu3Au type, (d) to the A3B structure of Al3Ti type,and (e) the A2B structure as it occurs in Ni2V orNi2Cr. In case (a) the labels 1, 2, 3, and 4 indicate thedecomposition of the fcc lattice into four interpene-trating simple cubic (sc) sublattices.

free-energy functional does not attain itsminimum at wavevector q = 0 in reciprocalspace but on a surface given by the equa-tion | q | = q* = 2 p /l, l being the wave-length of the lamellar pattern. This reflectsthe degeneracy that there is no preferred di-rection and hence the lamellae may beoriented in any direction. Owing to thelarge “phase space” in q-space where theinverse of the wavevector-dependent sus-ceptibility goes “soft” (c–1( | q | = q*) Æ 0as T Æ Tc), the mean-square amplitude ofthe local fluctuation of the order parameter·F2Ò – ·FÒ2 would diverge as T Æ Tc,which is physically impossible. It can beshown that this difficulty is avoided be-cause the strong local order parameter fluc-tuations turn u negative near Tc (Brazov-skii, 1975; Fredrickson and Helfand, 1987;Fredrickson and Binder, 1989), producinga free energy, as shown in Fig. 4-6b, for thecase of “symmetric” diblock copolymerswhere mean field theory would predict asecond-order transition. Experiments seemto confirm the first-order character of thetransition (Bates et al., 1988).

For certain systems, for which u > 0 andthus a second-order transition occurs, it ispossible by variation of a non-orderingfield to change the sign of the coefficient uat a particular point. This temperature Tt ,where u vanishes in Eq. (4-14), is called atricritical point. From Eq. (4-14) we thenimmediately find

(4-86)F0 = (– r /v)1/4 = (r ¢/v)1/4 (Tt /T – 1)1/4

Hence the tricritical order parameter expo-nent bt =1/4, while Eqs. (4-32) to (4-48) re-main unchanged and thus g t =1, n t =1/2,h t = 0. The critical isotherm, however, be-comes vF0

5 = H/kBT, i.e., d t = 5. From Eq.(4-15) we finally find

(4-87)F F

V k Tt t( ) ( ) /F0

1 203

− − −⎛⎝

⎞⎠B

=v

which implies a specific heat divergenceC ~ (– t)–1/2, i.e., a t =1/2. We note that thisset of exponents also satisfies the scalinglaws in Eqs. (4-78), (4-81), (4-83), and (4-84). Moreover, using this set of expo-nents in the Ginzburg criterion, Eqs. (4-58)to (4-61), we find that, with the Landau ex-ponents for a tricritical point, this condi-tion is marginally fulfilled, unlike the caseof ordinary critical points (Eq. (4-62)). Itturns out that d* = 3 is the marginal dimen-sion for tricritical points, and the mean-field power laws are modified by logarith-mic correction terms similar to those notedfor dipolar systems (Eq. (4-72)). Experi-mental examples for tricritical points in-clude strongly anisotropic antiferromagnetsin a uniform magnetic field (e.g., FeCl2,(Dillon et al., 1978), and dysprosium alu-minum garnet (DAG) (Giordano and Wolf,1975), see Fig. 4-9 for schematic phase diagrams), systems undergoing structuralphase transitions under suitable applied pres-sure, such as NH4Cl which has a tricriticalpoint at Tt = 250 K and pt = 128 bar (Yelon etal., 1974) or the ferroelectric KDP at Tt =113K and pt = 2.4 kbar (Schmidt, 1978), etc. Amodel system which has been particularlycarefully studied is He3–He4 mixtures: thetransition temperature Tl (x) of the normalfluid–superfluid He4 is depressed with in-creasing relative concentration x until, at atricritical point Tt= Tl(xt), the transition be-comes first order. This then implies a phaseseparation between a superfluid phase witha lower He3 content and a normal fluid He3-rich phase. Most common, of course, aretricritical phenomena in fluid binary mix-tures arising from the competition of gas–liquid transitions and fluid–fluid phaseseparations in these systems (Scott, 1987).

Just as the vanishing coefficient u in Eq.(4-10) leads to a multicritical point, the tri-critical point, another multicritical point isassociated with the vanishing of the coeffi-



cient of the [—F (x)]2 term; since this termwas also introduced as the lowest-orderterm of a systematic expansion, higher-or-der terms such as [—2F (x)]2 must then beincluded in Eq. (4-10). This problem ismost conveniently discussed in the frame-work of the wavevector-dependent suscep-tibility c (q), Eq. (4-34), which includingthe contribution resulting from a term [—2F (x)]2 can be written as (t0 =1– TL/T )

(4-88)

where K1 and K2 are phenomenological co-efficients. If K1 < 0, the divergence of c (q)does not occur for t0 = 0, q = 0 but at a valueqmax found from:

(4-89)

dd

=

=q

t K q K q

q K K

( )

( / )max/

0 12

24

1 21 2

0

2

+ +

⇒ −

c ( )ˆ

q =G0

0 12

24t K q K q+ +

The peak height of c (q) for |q | = qmax isthen described by

(4-90)

which implies that the actual critical pointoccurs at Tc = TL [1+ K1

2/(4K2)]. An exam-ple of such an ordering characterized by anonzero wavenumber q*= qmax is the meso-phase formation in block copolymer melts(Fig. 4-7) (see Leibler (1980)). Related phe-nomena also occur in crystals exhibiting“incommensurate superstructures”. Usu-ally c (q) is then not isotropic in q-space; inuniaxial systems, Eqs. (4-88) to (4-90) thenapply only if q is parallel to this preferreddirection. The superstructure described bythe wavelength l = 2p /qmax is determinedby the coefficients K1 and K2 and thus ingeneral it is not a simple multiple of the lat-tice spacing a, but (l /a) is an irrationalnumber. This is what is meant by the termincommensurate. Examples of such incom-mensurate structures are “helimagneticstructures” such as VF2, MnAu2, Eu, Hoand Dy (Tolédano and Tolédano, 1987).Note that in these systems the reason forthe incommensurate structure is not the existence of a negative coefficient in frontof the [—F (x)]2 term but the existence of

terms such as where

F+ = F 0 cos (p z /c), F – = F 0 sin (p z /c), cbeing the lattice spacing in the z-direction.Such terms involving linear terms in —F (x)are allowed by symmetry in certain spacegroups and are called “Lifshitz invariants”.Moving from lattice plane to lattice planein the z-direction in such a helimagneticspin structure, the spin vector (pointingperpendicular to z) describes a spiral struc-ture like a spiral staircase. Of course, suchincommensurate superstructures (alsocalled “modulated phases”) are not restrict-

FF FF FF FF+−

−+∂

∂− ∂

∂⎛⎝⎜

⎞⎠⎟z z,

c ( )ˆ

/( )maxqt K K

=G

0 12

24−

Figure 4-9. Schematic phase diagrams of antiferro-magnets with uniaxial anisotropy in an applied uni-form magnetic field H|| applied parallel to the easyaxis: Case (a) shows the case of weak anisotropy,case (b) the case of intermediate anisotropy, where inaddition to the antiferromagnetic ordering of the spincomponents in the direction of the easy axis, a spin-flop ordering of the transverse components also oc-curs. In case (a) both transitions T|| (H||), and T^(H||)are of second order and meet in a bicritical point. Forintermediate strength of the anisotropy the lineT^(H||) does not end at the bicritical point, but ratherin a critical end-point at the first-order transition line.Then a tricritical point also appears where the anti-ferromagnetic transition T|| (H||) becomes first order.For very strong anisotropy, the spin-flop phase disap-pears altogether.

ed to magnetic systems, and many exam-ples of dielectric incommensurate phaseshave been identified (e.g., NaNO2 (Tani-saki, 1961), Rb2ZnCl4, BaMnF4, thiourea[SC(NH2)2], see Tolédano and Tolédano(1987)). Whereas Rb2ZnCl4 and BaMnF4

also involve the existence of Lifshitz invar-iants, thiourea is an example of the casediscussed in Eqs. (4-88) to (4-90). Variousexamples of modulated superstructures ex-ist in metallic alloys; see Selke (1988,1989, 1992) and De Fontaine and Kulik(1985) for reviews of the pertinent theoryand experimental examples such as Al3Ti(Loiseau et al., 1985) and Cu3Pd (Broddinet al., 1986).

Whereas for TeTc the ordering of theincommensurate phase can be described interms of a sinusoidal variation of the localorder parameter density F (x), the nonlin-ear terms present in Eq. (4-10) at lowertemperatures imply that higher-order har-monics become increasingly important.Rather than a sinusoidal variation, thestructure is then better described in termsof a periodic pattern of domain walls (or“solitons”), and we talk about a “multi-soliton lattice” or “soliton staircase”. An-other important fact about modulatedphases is that the wavevector characteriz-ing the modulation period is not fixed atqmax, but varies with temperature or otherparameters of the problem. This is ex-pected from the third of the general Landaurules formulated at the beginning of Sec.4.2.3. Usually this variation of q stops atsome commensurate value where a “lock-in transition” of the incommensurate struc-tures occurs. In the incommensurate re-gime rational values of the modulation (“long-period superstructures”) may havean extended regime of stability, leading toan irregular staircase-like behavior of themodulation wavelength l as a function oftemperature (or other control parameters

such as pressure, magnetic field (for mag-netic structures), concentration (for alloys;see Fig. 4-10), etc.). Theoretically undercertain conditions even a staircase with aninfinite number (mostly extremely small!)of steps can be expected (“devil’s stair-case”, see Selke (1988, 1989, 1992) for adiscussion and further references). We alsoemphasize that the modulation need not in-volve only one direction in space (onewavevector), but can involve several wave-


Figure 4-10. Period M (in units of the lattice spacingin the modulation direction) for the alloy Al3–xTi1+x

as a function of the annealing temperature, as ob-tained from visual inspection of high-resolution elec-tron microscopic images. Each of the fifteen differentcommensurate superlattices observed is composed ofantiphase domains of length one or two (1-bands or2-bands, as indicated by sequences ·211Ò, ·21Ò etc.)based on the L12 structure. Very long annealing timeswere needed to produce these long-period super-structures supposedly at equilibrium. From Loiseauet al. (1985).


vectors. An example of the case of a two-rather than one-dimensional modulation isLiKSO4 (Pimenta et al., 1989).

Here we discuss only the special case of Eqs. (4-88)–(4-90) briefly where, bychanging external control parameters, it is possible to reach a multicritical pointwhere K1 vanishes, the so-called “Lifshitzpoint” (Hornreich et al., 1975). From Eq.(4-88) we then find that for T = Tc (= TL) wehave c (q) ~ q –4, and comparing this withthe result following from a Fourier trans-formation of Eq. (4-47), c (q)Tc

~ q –2+h , we conclude that hL = – 2 for the Lifshitzpoint, whereas for T > Tc we still havec (q = 0) = G0 t0

–1, i.e., gL = 1. Since forK1 = 0 Eq. (4-88) can be rewritten as

c (q) = G0 (1 – TL/T )–1/ [1 + x 4 q4]

x = [K2/(1 – TL/T )]1/4 (4-91)

we conclude that nL = 1/4 for the correla-tion length exponent at the Lifshitz point.

Also for the Lifshitz point we can askwhether the Landau description presentedhere is accurate, and whether or not statis-tical fluctuations modify the picture. Theresult is that the critical fluctuations arevery important. A marginal dimensionalityd* = 8 results in the case of an “isotropicLifshitz point”, where in

K1x (∂F /∂x)2 + K1y (∂F /∂y)2 + K1z (∂F /∂z)2

we have simultaneous vanishing of all co-efficients K1x , K1y , and K1z . Note that themore common case is the “uniaxial Lifshitzpoint”, where only K1z vanishes while K1x

and K1y remain nonzero. In this case themulticritical behavior is anisotropic, andwe must distinguish between the correla-tion length x|| describing the correlationfunction decay in the z-direction and thecorrelation length x^ in the other direc-tions. Eq. (4-91) then holds for x|| only,whereas Eq. (4-39) still holds for x^.

While experimental examples for Lif-shitz points are extremely scarce, e.g., thestructural transition in RbCaF3 under (100)stress (Buzaré et al., 1979), more commonmulticritical points are bicritical pointswhere two different second-order transi-tion lines meet. Consider, for example, thegeneralization of Eq. (4-10) to an n-compo-nent order-parameter field, but with differ-ent coefficient ri for each order-parametercomponent Fi :

For simplicity, the fourth-order term hasbeen taken as fully isotropic. If r1= r2 = …= rn , we would have the isotropic n-vectormodel discussed in Secs. 4.2.2 and 4.2.3.We now consider the case where some ofthese coefficients differ from each other,but a parameter p exists on which these co-efficients depend in addition to the temper-ature. The nature of the ordering will be determined by the term riFi

2, for which the coefficient ri changes sign at the high-est temperature. Suppose this is the casewhere i =1 for p < pb, then we have a one-component Ising-type transition at Tc1(p)given by r1(p, T ) = 0 (the other compo-nents Fi being “secondary order parame-ters” in this phase). If, however, for p > pb

the coefficient r2 (p, T ) = 0 at the highesttemperature Tc2 (p), it is the component F2

which drives the transition as the primaryorder parameter. The point p = pb, Tc1(p) =Tc2 (p) = Tb is then called a bicritical point.

An example of this behavior is found forweakly uniaxial antiferromagnets in a uni-

12

12

12

12 4

2

0

1 12

2 22

2 2 2

2

12 2

k TF

k T

r r

ru

Rd

n n

n

B B= (4-9 )

d

[ ( )]

( ) ( )

( ) [ ( )]

[( ) ( ) ]

FF

FF

x

x x x

x x

+ +⎧⎨⎩+…

+ +

+ ∇ +…+ ∇⎫⎬⎭

∫ F F

F

F F

form magnetic field H|| (Fig. 4-9). A modelHamiltonian for such a system can be writ-ten as

where we assume J < 0 (antiferromagne-tism) and for D > 0 the easy axis is the z-axis. For small H|| , we have a uniaxial anti-ferromagnetic structure and the order pa-rameter is the z-component of the stag-gered magnetization Ms

z. For strongerfields H|| , however, we have a transition to a spin-flop structure (two-component

=

(4-93)

− − ++ −

⟨ ⟩∑

∑i j

ix

jx

iy

jy

iz

jz

iiz

J S S S S

J S S H S,

( ) [ ]

1 D

||

order parameter, due to the perpendicu-lar components Ms

x and Msy of the stag-

gered magnetization). In the framework ofEq. (4-92), this would mean n = 3 andr2 ∫ r3≠ r1, with H|| being the parameter p.Both lines TN

|| (H||) and TN^ (H||) and the

first-order line between the two antiferro-magnetic structures join in a bicriticalpoint. A well known example of such a be-havior is GdAlO3 with Tb = 3.1242 K (Roh-rer, 1975). Another example, in our opin-ion, is the Fe–Al system where the ferro-paramagnetic critical line joins the second-order A2–B2 transition (Fig. 4-11). In themetallurgical literature, the magnetic tran-


Figure 4-11. The Fe–Al phase diagram, as obtained from a mean-field calculation by Semenovskaya (1974) (left), and experimentally by Swann et al. (1972) (right). The ferro-paramagnetic transition is shown by thedash-dotted line. The crystallographically disordered (A2) phase is denoted as an in the nonmagnetic and am inthe ferromagnetic state. The ordered FeAl phase having the B2 structure (Fig. 4-3) is denoted as a2, and the or-dered Fe3Al phase having the D03 structure (Fig. 4-3) is denoted as a1n or a1m, depending on whether it is non-magnetic or ferromagnetically ordered, respectively. Note that first-order transitions in this phase diagram areassociated with two-phase regions, since the abscissa variable is the density of an extensive variable, unlike theordinate variable in Fig. 4-9. From Semenovskaya (1974).


sition of Fe–Al alloys is often disregardedand then the multicritical point where theA2–B2 transitions change from second tofirst order is interpreted as a tricriticalpoint. Bicritical phenomena for structuralphase transitions are also known, such asSrTiO3 under stress in the [001] direction (Müller and Berlinger, 1975).

The structure of the phase diagram nearbicritical points has been analyzed in detailby renormalization group methods (Fisherand Nelson, 1974; Fisher, 1975a, b; Muka-mel et al., 1976a, b), experiment (Rohrerand Gerber, 1977) and Monte Carlo simu-lations (Landau and Binder, 1978). Apartfrom the fact that the exponents introducedso far take different values as T Æ Tb, thereis one additional exponent, the “crossoverexponent” F, describing the singular ap-proach of phase-transition lines towardsthe multicritical point. This behavior, andalso other more complicated phenomena(tetracritical points, tricritical Lifshitzpoints, etc.) are beyond the scope of thischapter. For an excellent introduction tothe field of multicritical phenomena we re-fer to Gebhardt and Krey (1979) and for acomprehensive review to Pynn and Skjel-torp (1983).

4.2.4 Dynamics of Fluctuationsat Phase Transitions

Second-order phase transitions alsoshow up in the “critical slowing down” ofthe critical fluctuations. In structural phasetransitions, we speak about “soft phononmodes”; in isotropic magnets magnonmodes soften as T approaches Tc from be-low; near the critical point of mixtures theinterdiffusion is slowed down; etc. Thiscritical behavior of the dynamics of fluctu-ations is again characterized by a dynamiccritical exponent z; we expect there to be

some characteristic time t which divergesas T Æ Tc:

t ~ x z ~ |1 – T /Tc |–n z (4-94)

Many of the concepts developed for staticcritical phenomena (scaling laws, univer-sality, etc.) can be carried over to dynamiccritical phenomena: “dynamic scaling”(Halperin and Hohenberg, 1967; Ferrell etal., 1967; Hohenberg and Halperin, 1977),“dynamic universality classes” (Halperinet al., 1974, 1976; Hohenberg and Halpe-rin, 1977), etc. However, the situation ismuch more complicated because the staticuniversality classes are split up in the dy-namic case, depending on the nature of theconservation laws, mode coupling terms inthe basic dynamic equations, etc. For ex-ample, we have emphasized that aniso-tropic magnets such as RbMnF3, orderingalloys such as CuZn, unmixing solid mix-tures such as Al– Zn, unmixing fluid mix-tures such as lutidine–water, and the gas–fluid critical point all belong to the sameIsing-like static universality class, but eachof these examples belongs to a different dy-namic universality class! Thus, in the an-isotropic antiferromagnet, no conservationlaw needs to be considered, whereas theconservation of concentration matters forall mixtures (where it means the order pa-rameter is conserved ) and for ordering al-loys (where the order parameter is not con-served but coupled to a conserved “nonor-dering density”). Whereas in solid mix-tures the local concentration relaxes simplyby diffusion, in fluid mixtures hydrody-namic flow effects matter, and also play arole at the liquid–gas critical point. In thelatter instance, conservation of energyneeds to be considered; it does not play arole for phase transitions in solid mixtures,of course, where the phonons act as a “heatbath” to the considered configurational de-grees of freedom.

We do not give a detailed account of crit-ical dynamics here, but discuss only thesimplest phenomenological approach (vanHove, 1954) for non-conserved order pa-rameters. We consider a deviation of the or-der parameter DF (x, t) from equilibrium atspace x and time t, and ask how this devia-tion decays back to equilibrium. The stan-dard assumption of irreversible thermody-namics is a relaxation assumption; we put

(4-95)

where G0 is a phenomenological rate factor.Using Eq. (4-10) and linearizing F (x, t) =F0 + DF (x, t) around F0 yields

Assuming that DF (x, t) is produced by a field H (x) = Hq exp (i q · x) which isswitched off at time t = 0, we obtain (seealso Sec. 4.2.2)

(4-97)

where Eq. (4-34) has been used. Since c (q)diverges for q = 0, T Æ Tc, the characteris-tic frequency w (q = 0) ∫ t –1 vanishes asw (q = 0) ~ cT

–1 ~ x –g /n = x –(2–h ), and bycomparison with Eq. (4-94) the result ofthe “conventional theory” is obtained:

z = 2 – h (4-98)

Although Eq. (4-98) suggests that there is ascaling relationship linking the dynamic tothe static exponents, this is not true in gen-eral if effects due to critical fluctuations aretaken into account. In fact, for noncon-served systems, z slightly exceeds 2 –h andextensive Monte Carlo studies were needed

DD

FF

Gq

qq q q

( )

( )exp [ ( ) ], ( ) / ( )

tt

0 0= =−w w c

∂∂

− +

− ∇⎫⎬⎭

tt r u t

Rd

t

D D

D

F G F F

F

( , ) ( ) ( , )

[ ( , )]

x x

x

=

(4-96)

0 02

22

3

∂∂

− ∂∂t

tDD

F GF

( , )( )

xx

= 0

to establish its precise value (Wanslebenand Landau, 1987).

It is important to realize that not all fluc-tuations slow down as a critical point is ap-proached, but only those associated withlong-wavelength order parameter varia-tions. This clearly expressed in terms of thedynamic scaling relationship of the charac-teristic frequency:

w (q) = qz w (), = qx (4-99)

where w () is a scaling function with theproperties w (•) = const. and w ( O1) ~ –z,and, hence, is consistent with Eq. (4-94).

Most experimental evidence on dynamiccritical phenomena comes from methodssuch as inelastic scattering of neutrons orlight, NMR or ESR spectroscopy, andultrasonic attenuation. Although the over-all agreement between theory and experi-ment is satisfactory, only a small fractionof the various theoretical predictions havebeen thoroughly tested so far. For reviews,see Hohenberg and Halperin (1977) andEnz (1979).

Finally, we also draw attention to nonlin-ear critical relaxation. Consider, for exam-ple, an ordering alloy which is held wellbelow Tc in the ordered phase and assumethat it is suddenly heated to T = Tc. The or-der parameter is then expected to decay to-wards zero with time in a singular fashion,F (t) ~ t –b /n z (Fisher and Racz, 1976). Sim-ilarly, if the alloy is quenched from a tem-perature T > Tc to T = Tc, the superstructurepeak appears and grows in a singular fash-ion with the time t after the quench,S (q, t) ~ tg /zn (Sadiq and Binder, 1984). Re-lated predictions also exist for the nonlin-ear relaxation of other quantities (orderingenergy, electrical resistivity) (Binder andStauffer, 1976a; Sadiq and Binder, 1984).

A very important topic is the dynamicsof first-order phase transitions, which willnot be discussed here because it is dis-



cussed elsewhere in this book (see Chap-ters by Binder and Fratzl (2001) and byWagner et al. (2001)).

4.2.5 Effects of Surfaces and of QuenchedDisorder on Phase Transitions:A Brief Overview

It a system contains some impurity at-oms which are mobile, we refer to “an-nealed disorder” (Brout, 1959). From thepoint of view of statistical thermodynam-ics, such mobile impurities act like addi-tional components constituting the systemunder consideration. For a second-ordertransition, small fractions of such addi-tional components have rather minor ef-fects: the transition point may be slightlyshifted relative to that of the pure material,and as the concentration of these impuritiesis normally strictly conserved, in principlethe so-called “Fisher renormalization” (Fisher, 1968) of critical exponents is ex-pected, as discussed in Sec. 4.2.2. Forsmall impurity contents, the region aroundthe transition point where this happens isextremely narrow, and hence this effect isnot important.

The effect of such impurities on first-or-der transitions is usually more important,e.g., for a pure one-component system, themelting transition occurs at one well-de-fined melting temperature, whereas in thepresence of mobile impurities this transi-tion splits up into two points, correspond-ing to the “solidus line” and “liquidus line”in the phase diagram of a two-componentsystem. This splitting for small impuritycontents is simply proportional to the con-centration of the impurities of a given type.In practice, where a material may containvarious impurities of different chemical na-tures and different concentrations whichoften are not known, these splittings of thetransition line in a high-dimensional pa-

rameter space of the corresponding multi-component system show up like a roundingof the first-order transition.

In solids, however, diffusion of impurityatoms is often negligibly small and such adisorder due to frozen-in, immobile impur-ities is called “quenched disorder” (Brout,1959). Other examples of quenched disor-der in solids are due to vacancies, in addi-tion to extended defects such as disloca-tions, grain boundaries, and external sur-faces. Quenched disorder has a drastic effecton phase transitions, as will be seen below.

Since the disorder in a system is usuallyassumed to be random (irregular arrange-ment of frozen-in impurities, for example),we wish to take an average of this randomdisorder. We denote this averaging over thedisorder variables, which we formally de-note here as the set xa, by the symbol[…]av in order to distinguish it from thethermal averaging ·…ÒT implied by statisti-cal mechanics. Thus the average free en-ergy which needs to be calculated in thepresence of quenched disorder is

whereas in the case of annealed disorder itwould be the partition function rather thanthe free energy which is averaged over thedisorder:

Eq. (4-101) again shows that annealed dis-order just means that we enlarge the spaceof microscopic variables which need to beincluded in the statistical mechanics treat-ment, whereas the nature of quenched aver-aging (Eq. (4-100)) is different.

F k T Z x

k T x k T

k T x k T

i

i

i

xi

ann B av

B B av

B B

= (4-1 )

= tr

= tr

−

− ( − )− ( − )

ln[ ]

ln [exp( , / )]

ln exp( , / )

,

a

a

aa

01

F

F

F

F

F k T Z x

k T x k Ti

i

= (4-1 )

= tr

B av

B Bav

−

− [ ( − )][ln ]

ln exp( , / )

a

a

00

FF

Experiments can be carried out with justone probe crystal and no averaging overmany samples as is implied by Eq. (4-100)needs to be performed. This is because wecan think about the parts of a macroscopicsample as subsystems over which such anaverage as written in Eq. (4-100) is actu-ally performed. This “self-averaging”property of macroscopically large systemsmeans that the average […]av over the setxa could also be omitted: it need only bekept for mathematical convenience.

We first discuss the physical effects ofrandomly quenched impurities (point-likedefects!) on phase transitions, assumingthe concentrations of these defects to bevery small. Various cases need to be distin-guished, depending on the nature of the local coupling between the defect and thelocal order parameter. This coupling maybe linear (“random field”), bilinear (“ran-dom bond”) or quadratic (“random aniso-tropy”). As an example, let us considermagnetic phase transitions described by then-vector model of magnetism, with an ex-change interaction depending on the dis-tance between lattice points ai and aj :

(4-102)

In Eq. (4-102) we have assumed that im-purities produce a random field Hi whichhas the properties

[Hi]av = 0 , [Hi2]av = H 2

R (4-103)

and Si is an n-component unit vector in thedirection of the local order parameter. Nowwe can show that for n ≥ 2 arbitrary smallamplitudes of the random field HR (whichphysically is equivalent to arbitrary smallimpurity concentration) can destroy uni-form long-range order (Imry and Ma,1975): the system is broken into an irregu-lar arrangement of domains. The mean size of the domains is larger as HR (or the

= − − ⋅ −≠∑ ∑i j

i j i ji

i izJ H S( )a a S S

impurity concentration, respectively) issmaller. Since no ideal long-range order isestablished at the critical point, this corre-sponds to a rounding of the phase transi-tion.

In the case n =1 and d = 3 dimensions,very weak random fields do not destroyuniform long-range order (Imbrie, 1984;Nattermann, 1998), although long-rangeorder is destroyed if HR exceeds a certainthreshold value H c

R (T ) which vanishes asT Æ Tc. But the random field drasticallychanges the nature of both static and dy-namic phenomena (Villain, 1985; Grin-stein, 1985). In d = 2 dimensions, arbitraryweak random fields destroy even Ising-like (n =1) order, producing domains ofsize xD with ln xD ~ (J/HR)2 (see Binder,1983a).

We now turn briefly to the physical real-ization of “random fields”. For symmetryreasons, no such defects are expected forferromagnets, but they may be induced in-directly by a uniform field acting on anantiferromagnet (Fishman and Aharony,1979; Belanger, 1998). Since antiferro-magnets in a field can be “translated” intomodels for order–disorder phenomena inalloys (the magnetization of the antiferro-magnet “translates” into the concentrationof the alloy, see Sec. 4.3.1) or (in d = 2 di-mensions) in adsorbed layers (the magnet-ization then “translates” into the coverageof the layer), random-field effects are im-portant for ordering alloys or order–disor-der transitions in monolayers, also. A niceexperimental example was provided byWiechert and Arlt (1993) who showed thatthe transition of CO monolayers on graph-ite near 5 K to a ferrielectric phase isrounded by small dilution with N2 mole-cules, in accord with the theory of random-field effects. Furthermore, random fieldsact on many structural phase transitionsowing to impurity effects: ions at low-sym-



metry, off-center positions may produceboth a random electric field and a randomstrain field. Also, symmetry generally per-mits random fields to act on tensorial order parameters in diluted molecular crystals(Harris and Meyer, 1985). For diluted mo-lecular crystals (such as N2 diluted with Ar,or KCN diluted with KBr, etc.) it is still un-clear whether the resulting “orientationalglass” phase is due to these random fieldsor to random bonds (for reviews, see Loidl(1989); Höchli et al. (1990); Binder andReger (1992); Binder (1998)).

The “random bond” Hamiltonian differsfrom Eq. (4-102) by the introduction of dis-order into the exchange terms rather thanthe “Zeeman energy”-type term,

(4-104)

Whereas in Eq. (4-102) we have assumedthe pairwise energy to be translationally in-variant (the exchange energy J dependsonly on the distance ai – aj between thespins but not on their lattice vectors ai andaj separately), we now assume Ji j to be arandom variable, e.g., distributed accord-ing to a Gaussian distribution

P (Ji j) µ exp [– (Ji j – J–

)2/2 (DJ)2] (4-105)

or according to a two-delta function distri-bution

P (Ji j) = pd (Ji j – J) + (1 – p) d (Ji j) (4-106)

If J–o DJ (Eq. (4-105)) or if 1– p O1 (Eq.

(4-106)), the ferromagnetic order occurringfor Eq. (4-104) if J > 0 or J

–> 0 is only

weakly disturbed, both for d = 3 and ford = 2 dimensions. Following an argumentpresented by Harris (1974), we can see thatthe critical behavior of the system remainsunaltered in the presence of such impuritiesprovided that the specific heat exponenta < 0, whereas a new type of critical behav-ior occurs for a ≥ 0 (see Grinstein (1985)

= − ⋅ −≠∑ ∑i j

ij i ji

izJ H SS S

for a review). Of course, the impurities willalways produce some shift of the criticaltemperature, which decreases as p in Eq.(4-106) decreases. When p becomessmaller than a critical threshold value pc,only finite clusters of spins are still mutu-ally connected by nonzero exchange bondsJ > 0, and long-range order is no longerpossible. The phase transition at T = 0 pro-duced by variation of p in Eq. (4-106) canhence be interpreted purely geometricallyin terms of the connectivity of finite clus-ters or an infinite “percolating” network ofspins (Stauffer and Aharony, 1992). Thistransition is again described by a com-pletely different set of exponents.

For the case where | J–

| O DJ in Eq. (4-105), a new type of ordering occurs,which is not possible in systems withoutquenched disorder; a transition occurs to astate without conventional ferro- or antifer-romagnetic long-range order but to a “spinglass phase” where the spins are frozen-inin a random direction (Binder and Young,1986; Young, 1998). The order parameterof the Ising spin glass was introduced byEdwards and Anderson (1975) as

qEA = [·SiÒ2T]av (4-107)

The nature of the phase transition in spinglasses and the properties of the orderedphase have been the subject of intense re-search (Binder and Young, 1986). Thisgreat interest in spin glasses can be under-stood because many different systems showspin glass behavior: transition metals witha small content x of magnetic impuritiessuch as Au1–xFex and Cu1–xMnx , dilutedinsulating magnets such as EuxSr1–xS, var-ious amorphous alloys, and also mixed dielectric materials such as mixtures ofRbH2PO4 and NH4H2PO4, where the spinrepresents an electric rather than magneticdipole moment. A related random orderingof quadrupole moments rather than dipole

moments if found in “quadrupolar glasses”(also called “orientational glasses”) such as K(CN)1–xBrx and (N2)xAr1–x , (Loidl,1989; Höchli et al., 1990; Binder andReger, 1992; Binder, 1998). However, it is still debated (Franz and Parisi, 1998)whether or not concepts appropriate for thedescription of spin glasses are also usefulfor the structural glass transition of under-cooled liquids (Jäckle, 1986).

Finally we mention systems with ran-dom anisotropic axes which can be mod-eled by the following Hamiltonian (Harriset al., 1973):

(4-108)

where the ei are vectors whose componentsare independent random variables with aGaussian distribution. This model is alsobelieved to exhibit destruction of long-range order due to break-up in domainssimilar to the random field systems. It isalso suggested that spin glass phases mayoccur in these systems. Again Eq. (4-108)is expected to be relevant, not only for dis-ordered magnetic materials but also fordielectrics where the spin represents elec-tric dipole moments, or displacement vec-tors of atoms at structural phase transitions,etc.

Very small fractions of quenched impur-ities which do not yet have an appreciableeffect on the static critical properties of asecond-order phase transition can alreadyaffect the critical dynamics drastically. Anexample of this behavior is the occurrenceof impurity-induced “central peaks” forstructural phase transitions in the scatteringfunction S (q, w) at frequency w = 0 in addi-tion to the (damped) soft phonon peaks(Halperin and Varma, 1976). In the abovesystems where the impurities disrupt theconventional ordering more drastically (such as random field systems and spin

= − − ⋅ − ⋅≠∑ ∑i j

i j i ji

i jJ ( ) ( )a a S S e S 2

glasses) the critical dynamics again have avery different character, and dynamicscharacteristic of thermally activated pro-cesses with a broad spectrum of relaxationtimes are often observed.

It is also interesting to discuss the effectof quenched impurities on first-order tran-sitions (Imry and Wortis, 1979). It is foundthat typically “precursor phenomena” nearthe first-order transition are induced, andthe latent heat associated with the transi-tion in the pure systems can be signifi-cantly reduced. It is also possible that suchimpurity effects may completely removethe latent heat discontinuity and lead to arounding of the phase transition.

Such rounding effects on phase transi-tions also occur when extended defectssuch as grain boundaries and surfaces areconsidered. These disrupt long-range orderbecause the system is then approximatelyhomogeneous and ideal (i.e., defect-free)only over a finite region in space. Whilethe description of the rounding and shiftingof phase transitions due to finite size ef-fects has been elaborated theoretically indetail (Privman, 1990), only a few casesexist where the theory has been tested ex-perimentally, such as the normal fluid–superfluid transitions of He4 confined topores, or the melting transition of oxygenmonolayers adsorbed on grafoil where thesubstrate is homogeneous over linear di-mension L of the order of 100 Å (see Marx,1989). While at critical points, a roundingand shifting of the transition normally setin when the linear dimension L and the cor-relation length x are comparable (Fisher,1971; Binder, 1987b), at first-order transi-tions the temperature region DT overwhich the transition is rounded and shiftedis inversely proportional to the volume ofthe system (Imry, 1980; Challa et al., 1986).

An important effect of extended defectssuch as grain boundaries and surfaces is


4.3 Computational Methods Dealing with Statistical Mechanics 283

that they often induce precursor effects tophase transitions, e.g., the “wetting transi-tion” (Dietrich, 1988; Sullivan and Telo daGama, 1985) where a fluid layer condens-ing at a surface is a precursor phenomenonto gas–fluid condensation in the three-dimensional bulk volume. Similarly “sur-face melting” and “grain-boundary melt-ing” phenomena can be interpreted as theintrusion of a precursor fluid layer at a sur-face (or grain boundary) of a crystal (DiTolla et al., 1996). Similarly, at surfaces ofordering alloys such as Cu3Au, the effectof the “missing neighbors” may destabilizethe ordering to the extent that “surface-in-duced disordering” occurs when the systemapproaches the transition temperature ofthe bulk (Lipowsky, 1984). Finally, wedraw attention to the fact that the local crit-ical behavior at surfaces differs signifi-cantly from the critical behavior in the bulk(Binder, 1983b). Such effects are outsidethe scope of this chapter.

4.3 Computational MethodsDealing with the StatisticalMechanics of Phase Transitionsand Phase Diagrams

While the phenomenological theories inSec. 4.2 yield a qualitative insight intophase transitions, the quantitative descrip-tion of real materials needs a more detailedanalysis. In this section, we complementthe phenomenological theory by a moremicroscopic approach. In the first step, theessential degrees of freedom for a particu-lar transition are identified and an appro-priate model is constructed. In the secondstep, the statistical mechanics of the modelare treated by suitable approximate or nu-merical methods.

Fig. 4-12 illustrates the modeling of or-der–disorder phase transitions in solids.

Transitions occur where the basic degree offreedom is the (thermally activated) diffu-sion process of atoms between various lat-tice sites. This happens for unmixing alloyssuch as Al–Zn or for ordering alloys suchas b-CuZn or the Cu–Au system. The mod-eling of such systems will be discussed inSec. 4.3.1. Many structural transitions areof a very different nature: we encounter pe-riodic lattice distortions where atomic dis-placements are comparable to those of lat-tice vibrations. Short wavelength distor-tions may give rise to “antiferrodistortive”and “antiferroelectric” ordering, as exem-plified by the perovskites SrTiO3, PbZrO3,etc. Long wavelength distortions corre-sponding to acoustic phonons give rise to“ferroelastic” ordering. Sec. 4.3.2 will dealwith the mean field treatment of such phasetransitions where the order parameter is aphonon normal coordinate, and Sec. 4.3.3is devoted to numerical methods going be-yond mean-field theory.

4.3.1 Models for Order–DisorderPhenomena in Alloys

Since the diffusive motion of atoms insubstitutional alloys (Fig. 4-12) is so muchslower than other degrees of freedom (suchas lattice vibrations), we may describe theconfigurational statistics of a substitutionalbinary alloy, to which we restrict attentionfor simplicity, by local occupation vari-ables ci. If lattice site i is taken by anatom of species B, ci =1, if it is taken by anatom of species A, ci = 0.

Neglecting the coupling between thesevariables and other degrees of freedom, theHamiltonian is then:

=

-

BB

AB

AA

0

2 1 4 1091 1

+ −

+ − −+ − − − +…

≠∑i j

i j i j

i j i j

i j i j

c c

c cc c

[ ( )

( ) ( ) ( )( ) ( ) ( )]

v

vv

x x

x xx x

where vAA, vAB, and vBB are interactionsbetween pairs of AA, AB, and BB atoms.In fact, terms involving three- and four-body interactions may also occur, but arenot considered here. Also, effects due tovacancies may easily be included but areneglected here.

As is well known, Eq. (4-109) can be re-duced to the Ising model, Eq. (4-56), by thetransformation Si = 1 – 2 ci = ± 1, apart froma constant term which is of no interest to ushere. The “exchange interaction” Jij

between spins i and j and “magnetic field”H in Eq. (4-56) are related to the interac-tion parameters of Eq. (4-109) by

J (xi – xj) ∫ Jij = [2vAB(xi – xj) (4-110)– vAA(xi – xj) – vBB(xi – xj)]/4

H i j i jj i

=

-

AA BB12

4 111

[ ( ) ( )]

( )

( )v vx x x x− − −

⎛

⎝⎜⎞

⎠⎟

−≠∑

Dm

where Dm is the chemical potential differ-ence between the two species.

The same mapping applies for the lat-tice-gas model of fluids, which at the sametime can be considered as a model of ad-sorbed layers on crystalline surface sub-strates (in d = 2 dimensions) (see Binderand Landau (1989) for a review) or as amodel of interstitial alloys, such as hydro-gen or light atoms such as C, N and O inmetals (Alefeld, 1969; Wagner and Horner,1974; Alefeld and Völkl, 1978). If weinterpret B in Eqs. (4-109)–(4-111) as anoccupied site and A as a vacant site, thenusually vAB = vAA= 0, such that

(4-112)

e being the binding energy which a particlefeels at lattice site i (for adsorbates at sur-faces, this is a binding to the substrate, anda similar enthalpy term is expected when

= 0 + − +≠∑ ∑i j

i j i j ii

c c cv ( )x x e


Figure 4-12. Degree of free-dom essential for the descrip-tion of various order–disor-der transitions in solids. Forfurther explanations, see text.


an interstitial is dissolved in a metal,whereas for modeling of an ordinary gas–liquid transition we would put e = 0). Therelationships analogous to Eqs. (4-110) and(4-111) are then:

(4-113)

Jij = – v (xi – xj)/4 (4-114)

where m is the chemical potential andv (xi – xj) the pairwise interaction of parti-cles at lattice sites xi and xj.

Of course, it is straigthforward to gener-alize this approach to binary (A–B) alloysincluding as a third option that a lattice siteis vacant (V) or to ternary alloys; thenmore complicated models result, e.g., thePotts (1952) model, Eq. (4-20), or themodel of Blume et al. (1971). Another gen-eralization occurs if one species (B) of a binary alloy is magnetic, e.g., Fe in Fe–Alalloys (see Dünweg and Binder (1987)). In the latter case, the Hamiltonian is (thetrue magnetic spin variable is now denotedas s i)

where Jm(xi – xj) denotes the strength ofthe magnetic interaction, which we haveassumed to be of the same type as used inthe Heisenberg model (Eq. (4-102)) withn = 3 and Hi = 0.

We will not go further into the classifica-tion of the various models here, but noteone general property of both models inEqs. (4-109) and (4-112) which becomesevident when mapping the Ising Hamilton-ian Eq. (4-56): for H = 0, this Hamiltonianis invariant against a change of sign of allspins. For H ≠ 0, it is invariant against the

= -

BB m

AB

AA

0 4 115

2 11 1

( )

[ ( ) ( ) ]

( ) ( )( ) ( ) ( )

+ − − − ⋅

+ − −+ − − − +…

≠∑i j

i j i j i j i j

i j i j

i j i j

c c J

c cc c

v

vv

x x x x

x xx x

ss ss

H i jj i

= − [ + + − ]≠∑( )/ ( )/( )

e m 2 4v x x

transformation Si, H Æ – Si , – H . Thisimplies a particle–hole symmetry of thephase diagram of the lattice gas model, or asymmetry against interchange of A and Bin the phase diagram of the binary alloymodel. Even in very ideal cases, whereboth partners A and B of a binary alloycrystallize in the same structure and havesimilar lattice spacings, the phase diagramis in reality not symmetric around the line·ciÒ = 1/2 (see the Cu–Au phase diagramreproduced in Fig. 4-13 as an experimentalexample). There are various conceivablereasons for this asymmetry of real phasediagrams: (i) the assumption of strictlypairwise potentials (Eq. (4-109)) fails, andterms such as cicjck , cicjckcl need to be in-cluded in the effective Hamiltonian; and(ii) the potentials vAA, vAB, and vBB in Eq.(4-109) are not strictly constant but depend

Figure 4-13. Partial phase diagram of copper–goldalloys in the temperature–concentration plane indi-cating the existence regions of the three orderedphases Cu3Au, CuAu, and CuAu3 (cf. Fig. 4-8a andc). These phases are separated from each other (andfrom the disordered phase occurring at higher tem-peratures) by two-phase coexistence regions. Theboundaries of these regions are indicated by full andbroken lines. Region II is a long-period modulatedversion of the simple CuAu structure occurring in re-gion I. Note that for strictly pairwise constant inter-actions (of arbitrary range!) in a model such as Eq.(4-109) the phase diagram should have mirror sym-metry around the line cAu = 50%. From Hansen(1958).

on the average concentration ·cÒ of the al-loy, since the effective lattice parameter aand the Fermi wavenumber kF and otherphysical characteristics entering the effec-tive potential of the alloy also depend on concentration ·cÒ. In fact, both reasonsprobably contribute to some extent in realalloys – the first-principles derivation ofmodel Hamiltonians such as Eq. (4-109)from the electronic structure theory for sol-ids is a challenging problem (Bieber andGautier 1984a,b; Zunger, 1994). Since theeffective interaction typically is oscillatoryin sign (Fig. 4-14) according to the Friedelform

v (| x |) = A cos (2 | kF | | x | + j)/| x |3 (4-116)

where A and j are constants, small changesin | kF | and/or the distances lead to rela-tively large changes in v (| x |) for distantneighbors. Whereas the interaction Eq. (4-116) results from the scattering of con-duction electrons at concentration inhomo-geneities, in other cases such as H in metalsthe effective interactions are due to theelastic distortions produced by these inter-stitials in their environment. The resultingelastic interaction is also of long range(Wagner and Horner, 1974).

Since the effective interactions thatshould be used in Eqs. (4-109) to (4-115)often cannot be predicted theoretically in areliable way, it may be desirable to extractthem from suitable experimental data. Formetallic alloys, such suitable experimentaldata are the Cowley (1950) short-range or-der parameters a (x) ∫ a (xi – xj) which de-scribe the normalized concentration corre-lation function, c ∫ ·ciÒ = (1 – M)/2, M ∫ ·SiÒ

(4-117)

which correspond to the normalized spincorrelation functions in the Ising spin rep-

a ( )( )

x xi ji j i jc c c

c c

S S M

M− ≡

⟨ ⟩ −−

⟨ ⟩ −−

2 2

21 1= resentation (Eq. (4-117) for a ferromagnet

is equivalent to the phenomenological Eq.(4-43)). In the disordered phase of an Isingspin system, it is straightforward to obtainthe wavevector-dependent response func-


Figure 4-14. (a) Interaction potential V lmn = – 2 J (x)(note that x = (l, m, n) –a

2, a being the lattice spacing) vs.

distance | x | (measured as ÷----l2 +

--m2-----

+ n2) for NiCr0.11

at T = 560 °C, as deduced from (b) short-range orderparameters a (x) measured by diffuse neutron scatter-ing. The circles show the results of the high-temper-ature approximation, Eq. (4-121), and crosses the re-sult of the inverse Monte Carlo method. Full andbroken curves represent a potential function of theFriedel form, Eq. (4-116), with different choices ofthe Fermi wavevector kF for ·100Ò and ·110Ò direc-tions, amplitude A and phase F being treated as fit-ting parameters. From Schweika and Hauboldt (1986).


tion c (q) describing the response to a wavevector-dependent field (cf. Eqs. (4-32),(4-34), and (4-40)), if we apply the molec-ular field approximation (see the next sub-section). The result is (Brout, 1965):

(4-118)

where J (q) is the Fourier transform of the“exchange interaction” J (x),

(4-119)

In Eq. (4-118), we have once again invokedthat there is a “fluctuation relation” (= static limit of the so-called “fluctuation-dissipation theorem”) relating c (q) to thestructure factor S (q) which is just theFourier transform of the correlations ap-pearing in Eq. (4-117):

(4-120)

Combining now Eqs. (4-118) and (4-120)we see that the reciprocal of this diffusescattering intensity S (q) in q-space is sim-ply related to the Fourier transformation ofthe interactions as

(4-121)

For the model defined in Eqs. (4-109) to(4-111), this expression even is exact in an expansion of 1/S (q) in a power series in1/T to leading order in 1/T, and is one ofthe standard tools for inferring informationon interactions in alloys from diffuse scat-tering data (Clapp and Moss, 1966, 1968;Moss and Clapp, 1968; Krivoglaz, 1969;Schweika, 1994). An example is shown in Fig. 4-14. Close to the order–disorderphase transition, we expect corrections tothe molecular field expression Eq. (4-121)due to the effects of statistical fluctuations

4 11

41

c cS

Jk T

c c( )( )

˜ ( )( )

− − −q

q=

B

S S S Mj i

i j i j( ) exp [ ( )] ( )( )

q q r r= i≠∑ ⋅ − ⟨ ⟩ − 2

˜ ( ) ( ) exp [ ( )]( )

J J ij i

j i jq x x q x x= i≠∑ − ⋅ −

S k TM

J k T M( ) ( )

[ ˜ ( )/ ] ( )q q

q= =B

B

c 1

1 1

2

2

−− −

(cf. Sec. 4.2.2); a more accurate procedureto deduce J (x) from experimental data ona (x) is the “inverse Monte Carlo method”(Gerold and Kern, 1986, 1987; Schweika,1989), but this method also relies on the as-sumption that an Ising model description aswritten in Eq. (4-109) is appropriate. Onthe level of the molecular field approxima-tion (for alloys this is usually referred to asthe Bragg–Williams (1934) approxima-tion) or the Bethe (1935) approximation, itis possible to avoid models of the type ofEq. (4-109) and include the configurationaldegrees of freedom in an electronic struc-ture calculation (Kittler and Falicov, 1978,1979). Although such an approach soundsvery attractive in principle, the results arenot so encouraging in practice, as shown inFig. 4-15. The results of this method forCu3Au are compared with Monte Carlo dataof a nearest neighbor Ising model (Binder,1980), with the cluster-variation treatmentof the same model (Golosov et al., 1973)and with experimental data (Keating andWarren, 1951; Moss, 1964; Orr, 1960).

The conclusion of this subsection is thatthe development of microscopic models forthe description of order–disorder phenom-ena in alloys is still an active area of re-search, and is a complicated matter, be-cause the validity of the models can only bejudged by comparing results drawn fromthe models with experimental data. How-ever, these results also depend on the ap-proximation involved in the statistical-me-chanical treatment of the models (e.g., thefull and broken curves in Fig. 4-15 refer tothe same nearest-neighbor Ising model,whereas the dash-dotted curve refers to adifferent model). In the next subsections,we consider various sophistications of thestatistical mechanics as applied to variousmodels for phase-transition phenomena.More details about all these problems canbe found in the Chapter by Inden (2001).

We have concentrated in this subsectionon alloys, but a similar discussion couldhave also been presented for order –disor-der phenomena in adsorbed monolayers(see for example, Binder and Landau (1981,1989), where the modeling of systems suchas H adsorbed on Pd(100) surfaces in terms

of lattice gas models is discussed), for mag-netic transitions (De Jongh and Miedema,1974), etc. Usually only the source of ex-perimental data used to extract the interac-tion parameters is different: e.g., for ferro-magnets such as EuS, rather than using dif-fuse magnetic neutron scattering in analogywith Eq. (4-118) it is more convenient toextract J (q) from the measurements of spinwave dispersion curves found from inelas-tic neutron scattering (Bohn et al., 1980).

4.3.2 Molecular Field Theory and its Generalization(Cluster Variation Method, etc.)

The molecular field approximation(MFA) is the simplest theory for the de-scription of phase transition in materials;despite its shortcomings, it still finds wide-spread application and has been describedin great detail in various textbooks (Brout,1965; Smart, 1966). Therefore we do nottreat the MFA in full detail here, but ratherindicate only the spirit of the approach.

We start with the Ising ferromagnet, Eq.(4-56). The exact Helmholtz energy can befound formally from the minimum of thefunctional (Morita, 1972)

(4-122)

where the sums extend over all configura-tions of the spins in the system and P (Si)is the probability that a configuration Sioccurs. Eq. (4-122) thus corresponds to thethermodynamic relation F= U–TS wherethe entropy S is written in its statisticalinterpretation. Minimizing Eq. (4-122) for-mally with respect to P yields the canonicaldistribution

Peq(Si) ~ exp[– Ising(Si)/kBT]

as desired.

= Ising

B

( ) ( )

( ) ln ( )

Si i

Si i

i

i

S P S

k T P S P S

= ±

= ±

∑

∑+1

1


Figure 4-15. (a) Order parameters for the A3B struc-ture (Fig. 4-8c) on the fcc lattice: Long-range orderparameter Y (LRO) and short-range order parameter–a1 for the nearest-neighbor distance (SRO) vs. tem-perature, according to the Monte Carlo method (Binder, 1980), the cluster variation (CV) method inthe tetrahedron approximation (Golosov et al., 1973),and the Kittler–Falicov (1978, 1979) theory. Data forCu3Au after Keating and Warren (1951) (LRO) andafter Moss (1964) (SRO). (b) Ordering energy DU(normalized to zero at Tc in the disordered state) vs.temperature for the fcc A3B alloy (top) and the fccAB alloy (bottom). Theoretical curves are from thesame sources as in (a); experimental data were takenfrom Orr (1960) and Orr et al. (1960).


The MFA can now be defined by factor-izing the probability P (Si) of a spin con-figuration of the whole lattice into a prod-uct of single-site probabilities pi which can take two values: p+ = (1 + M )/2 is theprobability that the spin at site i is up and p– = (1 –M )/2 is the probability that itis down, p+ – p– = M is the magnetization.Now the expression Jij Si Sj pi pj (cf. Eq. (4-56)) summed over the possible values p+ and p– simply yields Jij M2, and henceEq. (4-122) reduces to, using Eq. (4-119)

Minimizing MFA with respect to M nowyields the elementary self-consistencyequation:

(4-124)

As is well known, Eq. (4-124) implies asecond-order transition at Tc ∫ J (q = 0)/kB

with the same exponents as in the Landautheory.

Clearly, in factorizing P (Si) into aproduct of single-site probabilities andsolving only an effective single-site prob-lem, we have disregarded correlation in theprobabilities of different sites. A system-atic improvement is obtained if we approx-imate the probability of configurations notjust by single-point probabilities but by us-ing “cluster probabilities”. We considerprobabilities pnc(k, i) that a configuration kof the n spins in a cluster of geometric con-figuration c occurs (c may be a nearest-neighbor pair, or a triangle, tetrahedron, etc.).Note k = 1, … , 2n for Ising spins whereask = 1, … , qn for the q-state Potts model.

Mk T

J q M H= =B

tanh [ ˜ ( ) ]1

0 +

1 12

0 2

12

12

12

12

2

NJ q M H M

k TM M

M M

MFA

B

= = (4-1 3)˜ ( )

ln

ln

−

+ + +⎛⎝

⎞⎠

⎡⎣⎢

+ − −⎛⎝

⎞⎠⎤⎦⎥

These probabilities can be expressed interms of the multi-spin correlation func-tions gnc (i) ∫ ·Si Sj1

… SjnÒ, where the set ofvectors xj1

– xi, …, xjn – xi defines the n-point cluster of type c located at lattice sitei. The Helmholtz energy functional to beminimized in this cluster variation method(Kikuchi, 1951; Sanchez and De Fontaine,1980, 1982; Finel, 1994) is a more compli-cated approximation of Eq. (4-122) thanEq. (4-123). If the largest cluster consid-ered exceeds the interaction range, the en-ergy term in F= U– TS is treated exactly;unlike Eq. (4-123), the entropy is approxi-mated. We find

where the coefficients gnc are combinato-rial factors depending on the lattice geome-try and the clusters included in the approx-imation (Kikuchi, 1951), e.g., in the tetra-hedron approximation for the f.c.c. lattice,the sum over c in Eq. (4-125) includes the(nearest-neighbor) tetrahedron, the near-est-neighbor triangle, the nearest-neighborpair, and the single site.

Assuming the ordered structure to beknown, the symmetry operations of the as-sociated group can be applied to reduce the number of variational parameters in Eq. (4-125). In the MFA, there is a singlenon-linear self-consistent equation (Eq. (4-124)) or a set of equations involving theorder-parameter components if a problemmore complicated than the Ising ferromag-net is considered. In the CV method, amuch larger set of coupled non-linearequations involving the short-range orderparameters gnc (i) is obtained when we min-imize Eq. (4-125). Therefore, whereas thesimple MFA is still manageable for a wide

=

(4-1 5)

B

=

12

2

2

1

2

i jij

i n cnc

kn c

J i k T

p k i

j

n

∑∑ ∑∑

∑

+

×

g ,,

,

( )

( , )

r g

variety of systems (Brout, 1965; Smart,1966), the CV method is essentially re-stricted to Ising-type problems relevant forphase transitions in metallic alloys (DeFontaine, 1979; Finel, 1994). We discussthe merits of the various approaches in thenext section.

At this point, we return to the formula-tion of the MFA for the case of an arbitrarytype of ordering, rather than the simpletransition from paramagnetic to ferromag-netic considered in the above treatment(Eqs. (4-123) and (4-124)) of the Isingmodel (Eq. (4-56)). Hence Eq. (4-123) canobviously be generalized as follows (seealso De Fontaine, 1975):

where xi = ci – c = (M – Si)/2, and constantterms have been omitted. Expanding Eq.(4-126) in terms of xi yields, again omittinga constant term,

with the coefficients

f0¢¢(x ≠ 0) = – 4 J (xi – xj)

f0¢¢(x = 0) = kBT /[c (1 – c)] (4-128)

and

f0¢¢¢ = – kBT (2c – 1)/[c (1 – c)]2 (4-129)

f0IV= 2kBT [3c2– 3c + 1)]/[c (1 – c)]3

If we group the xis properly into the sub-lattices reflecting the (known or assumed)state, Eq. (4-127) essentially yields theLandau expansion in terms of the order-pa-rameter components, as discussed in Sec.4.2.1. Rather than defining sublattices a

D MFA

IV

= (4-1 7)12

2

13

14

0

03

04

i ji j i j

ii

ii

f

f f

,( )

! !

∑

∑ ∑

′′ −

+ ′′′ +

x x x x

x x

MFA

B

= (4-1 6)− −

+ + − −≠∑

∑

2 2

1 1i j

i j i j

ii i i i

J

k T c c c c

( )

[ ln ( ) ln ( )]

x x x x

priori, it is often convenient to introduceFourier transformations

(4-130)

which yield:

where G is a reciprocal lattice vector andf ¢¢(q) can be written as

f ¢¢(q) = – 4 J (q) + kBT/[c (1 – c)] (4-132)

Comparing Eqs. (4-121) and (4-132), werealize that the inverse structure factor (orinverse “susceptibility”, c–1 (q)) is simplyproportional to f ¢¢(q), as it should be, sinceX (q) and H (q) (apart from constants) arecanonically conjugate thermodynamic var-iables. Of special importance now are thepoints qc where J (q) has its maximum andcorrespondingly f ¢¢(q) has a minimum, be-cause for these wavevectors a sign changeof f ¢¢(q) in Eq. (4-131) occurs first (at thehighest temperature). Hence a stabilitylimit of the disordered phase is predicted as

kBTc = 4 J (qc) c (1 – c) (4-133)

If at this temperature a second-order transi-tion occurs, it should actually be describedby a “concentration wave” X (qc) as an or-der parameter (see also the discussion following Eq. (4-31)). In many cases themaxima of J (q) occur at special symme-try points of the Brillouin zone, and sym-metry considerations in reciprocal spaceare useful for discussing the resulting or-

D | | MFA

IV

=N

f X

Nf X X X

Nf X X X X

2

3

4

2

0

0

q

q q q

q q q q

q q

q q q

q q q G

q q q q

∑

∑

∑

′′

+ ′′′ ′ ′′

× + ′ + ′′ −

+ ′ ′′ ′′′

×

′ ′′

′ ′′ ′′′

˜ ( ) ( )

!( ) ( ) ( )

( )

!( ) ( ) ( ) ( )

(

, ,

, , ,

d

d qq q q q G+ ′ + ′′ + ′′′ − ) (4-1 1)3

XN i

ii( ) exp [ ]q q x= i

1 x∑ − ⋅



dered structures (De Fontaine, 1975, 1979;Khachaturyan, 1973, 1983).

This formulation of order–disorder tran-sitions in alloys is analogous to the treat-ment of structural transitions, for which amodel Hamiltonian similar to Eq. (4-127)can be written: rather than concentrationdeviations xi we now have displacementvectors ul (x) associated with a lattice vec-tor x for the l th atomic species in the unitcell. Just as it is useful to relate xi to theFourier transform X (q) of the concentra-tion deviation, it is useful to relate ul (x) tothe phonon normal coordinate Qk,l, de-fined as

(4-134)

where Ml is the mass of the atom of type lat site Ri

l in the i th unit cell, el (k, x) is aphonon polarization vector, l labels thephonon branch and k its wavevector (Fig.4-16). In this case, ·Qk0, l0

ÒT plays the role

u x k x e kk

kll

lN MQ( ) exp( ) ( , )

,,= i

1

lll∑ ⋅

of an order-parameter component for thetransition: in mean-field theory, the asso-ciate eigenfrequency vanishes at a temper-ature Tc (“soft phonon”). If this happens fora phonon with wavevector k0 at the Bril-louin zone edge, we have an antiferro-electric order, provided that the phonon ispolar, i.e., it produces a local dipole mo-ment. For non-polar phonons, such as forthe transition in SrTiO3 at Tc =106 K wherek0 = p (–12 , –1

2 , –12)/a (the soft phonon there

physically corresponds to an antiphase ro-tation of neighboring TiO6 octahedra, asindicated schematically in Fig. 4-12), thetransition leads to “antiferrodistortive” or-der. Long-wavelength distortions corre-sponding to optical phonons give rise to ferroelectric ordering (an example isPb5Ge3O11, where k0 = 0 and Tc = 450 K(Gebhardt and Krey, 1979)). There are alsoparaelectric–ferroelectric transitions offirst order, e.g., the cubic– tetragonal tran-sition of BaTiO3, such that no soft modeoccurs.

Just as the “macroscopic” ferroelectricordering can be associated with the normalcoordinate Qk0 = 0, l of the associate opticalphonon as a microscopic order parametercharacterizing the displacements on theatomic scale, the “macroscopic” ferroelas-tic orderings, where in the phenomenologi-cal theory, Eq. (4-27), a component of thestrain tensor emn is used as an order pa-rameter, can be related to acoustical pho-nons. Examples are the martensitic mate-rial In-25 at.% Tl where the combinationc11 – c12 of the elastic constants nearly sof-tens at Tc = 195 K, and LaP5O14 where c55

softens at Tc = 400 K and the structurechanges from orthorhombic to monoclinic(Gebhardt and Krey, 1979). Of course, thiscorrespondence between the microscopicdescription of displacements in crystals interms of phonons and the phenomenologi-cal macroscopic description in terms of po-

Figure 4-16. (a) Schematic temperature variation oforder parameter F = ·Qk0, l0

ÒT and square of the “soft-mode” frequency w2 (k0, l0) at a displacivestructural transition. (b) Schematic phonon spectrumof the solid at T > Tc. Note that either optical oracoustic phonons may go soft, and for optical pho-nons a softening may often occur at the boundary ofthe first Brillouin zone rather than at its center.

larization fields, strain fields, etc., is awell-known feature of condensed-mattertheory (Kittel, 1967).

As is well known, the Qk, l are definedsuch that the Hamiltonian of the crystal inthe quasi-harmonic approximation is diag-onalized (Born and Huang, 1954):

Eq. (4-135) corresponds precisely to Eqs.(4-127) and (4-131) if only the “harmonicterms” (i.e., quadratic terms in the expan-sion with respect to the xis) are retained.The vanishing of a soft mode, w2(k0, l0) ~(T – Tc), is again equivalent to the vanish-ing of a coefficient r ~ (T – Tc) of a quad-ratic term in a Landau expansion. Ofcourse, as in the Landau theory, higher-or-der “anharmonic” terms in Q are crucial forthe description of the ordered phase interms of a stable order parameter ·Qk0, l0

Òfor T < Tc.

These anharmonic terms thoroughlymodify the picture of the transition as ob-tained from MFA, as they lead to a cou-pling of the soft mode Qk0,l0

with othernoncritical modes. This coupling amongmodes gives rise to a damping of the softmode. In fact, under certain circumstanceseven an overdamped soft mode and the ap-pearance of a central peak are expected(Gebhardt and Krey, 1979; Bruce andCowley, 1981).

A treatment of the anharmonic higher-order terms of a Landau-like expansion in reciprocal space is cumbersome, as Eq.(4-131) demonstrates. Denoting the ampli-tude of the displacement vector producedby the soft mode Qk0, l0

in the unit cell i asFi, we may formulate a model for a one-

=

= (4-1 5)

U U

u u

U Q

i j l lil

jl

l i l j

02

02 2

12

12

3

+ ∂ ∂ ∂

×

+

′

′

′

∑

∑

, , , , ,

,,

[ /( ) ( ) ]

( ) ( )

( , )

a ba b

a b

llw l

x x

x x

kk

k| |

component structural transition as follows(Bruce and Cowley, 1981):

(4-136)

where r, u and C are phenomenological co-efficients. The last term corresponds to the harmonic interaction between displace-ments in neighboring lattice cells, whilethe anharmonicity has been restricted to the“single-site Hamiltonian” [rFi

2/2 + uFi4/4].

Note that Eq. (4-136) is fully equivalent tothe “Hamiltonian” D MFA in Eq. (4-127) iff0¢¢¢= 0 is chosen, since (Fi – Fj)

2 = Fi2 + Fj

2

– 2Fi Fj, and the coefficient C is thusequivalent to f0¢¢(xi – xj) where xi – xj is anearest-neighbor distance, a. On the otherhand, the “F4-model” in Eq. (4-136) canbe thought of as a lattice analog of the Helmholtz energy functional Eq. (4-10),putting (Fi – Fj)≈ a · —Fi (see Milchevet al. (1986) for a discussion).

An interesting distinction concerns thiseffective single-site Hamiltonian felt by theatoms undergoing the distortion. We haveassumed that the ordered structure isdoubly degenerate; the atoms below Tc cansit in the right or the left minimum of adouble-well potential. If the single-site po-tential above Tc is essentially of the sametype, and only the distribution of the atomsover the minima is more or less random,the transition is called “order–disordertype”. This occurs, for example, for hydro-gen-bonded ferroelectrics and is analogousto the sublattice ordering described abovefor alloys. On the other hand, if the single-site potential itself changes above Tc to asingle-well form, the transition is called“displacive”. Whereas it was often thoughtthat displacive structural transitions exhibitwell-defined soft phonons right up to theirtransition temperature Tc, it has now be-

F F F

F F

412

14

12

2 4

2

=i

i i

i ji j

r u

C

∑

∑

+⎡⎣⎢

⎤⎦⎥

+ −⟨ ⟩

( ),



come clear that all these structural transi-tions with a one-component order parame-ter acquire characteristics of order–disor-der transitions close to Tc, as expected fromthe “universality principle”, and thereforethe distinction between the character of astructural transition as being “order–disor-der” or “displacive” is not a sharp one (Bruce and Cowley, 1981).

It should also be noted that the softeningof w(k0, l0) near Tc does not mean that dis-placements ul (x) become very large. Infact, the mean square displacement of anatom at a structural transition is only ex-pected to have an energy-like singularity (Meißner and Binder, 1975)

·ul2ÒT – ·ul

2ÒTc~ (T /Tc – 1)1–a (4-137)

where a is the specific heat exponent. Thisis important because ·ul

2ÒT is easily de-duced experimentally from the Mössbauereffect, from the Debye–Waller factor de-scribing the temperature variation of Braggpeaks in X-ray or neutron scattering, etc.

We end this subsection with a commenton the theory of first-order structural tran-sitions. The common approach is to restrictthe analysis entirely to the framework ofthe quasi-harmonic approximation, inwhich the Helmholtz energy at volume Vand temperature T is written as

Therefore, if the effective potentials speci-fying the dynamical matrix ∂2U/[(∂xi

l)a∂xj

l)b] in Eq. (4-135) are known, the pho-non frequencies wV (k, l) for a given vol-ume and the free energy F (T,V) are ob-tained. Of course in this approach, knowl-edge of the structure of the material is assumed. First-order transitions between

F T V U T S

U V

k T k T

V

V

( , )

( ) ( , )

ln [ exp( ( , )/ )],

,

=

= (4-1 8)

B B

−

+

+ − −

∑

∑

012

3

1k

k

k

kl

l

w l

w l

different structures can be handled by per-forming this calculation for both phasesand identifying the temperature Tc wherethe free energy branches of the two phasescross. Since the quasi-harmonic theory is acalculation of the mean-field type, aspointed out above, first order transitionsalso show up via stability limits of thephases, where the soft modes vanish; thuswe are not locating Tc but rather tempera-tures T0 or T1 (cf. Fig. 4-6b), which are often not very far from the actual transi-tion temperature. This quasi-harmonic ap-proach to structural phase transitions hasbeen tried for many materials. Typical ex-amples include RbCaF3 (Boyer and Hardy,1981) and the systems CaF2 and SrF2

(Boyer, 1980, 1981a, b), which show phasetransitions to a superionic conducting state.

4.3.3 Computer Simulation Techniques

In a computer simulation, we consider afinite system (e.g., a cubic box of size L3

with periodic boundary conditions to avoidsurface effects) and obtain information onthe thermodynamic properties, correlationfunctions, etc. of the system (as specifiedby its model Hamiltonian) which is exact,apart from statistical errors. However, thisapproach is restricted to classical statisticalmechanics (including quasi-classical mod-els such as Ising or Potts models), althoughremarkable progress on application toquantum problems has been made (Kalos,1985; Kalos and Schmidt, 1984; De Raedtand Lagendijk, 1985; Suzuki, 1992; Ceper-ley, 1995). The principal approaches of thistype are the molecular dynamics (MD)technique and the Monte Carlo (MC) tech-nique (for reviews of MD, see Ciccottiet al., 1987; Hoover, 1987; Hockney andEastwood, 1988; Binder and Ciccotti,1996; of MC, see Binder, 1979, 1984a;Mouritsen, 1984; Binder and Heermann,

1988; Binder and Ciccotti 1996). In theMD method, we numerically integrate New-ton’s equation of motion which followsfrom the chosen Hamiltonian, assuming er-godic behavior, and the quantities of inter-est are obtained as time averages from thesimulation. This method requires that therelevant physical time scales involved inthe problem are not too different from eachother. The MD approach has often been ap-plied successfully to liquid–solid transi-tions. It would not be suitable to study or-der–disorder phenomena in solid alloys,since the time step in the MD method for asolid must be much less than a phonon fre-quency, and this time scale is orders ofmagnitude smaller than the time betweendiffusive hops of atoms to neighboring va-cant sites, which is the process relevant toequilibration of configurational degrees offreedom (Fig. 4-12).

For problems of the latter type, the MCmethod clearly is to be preferred. In theMC method, random numbers are used toconstruct a random walk through the con-figuration space of the model system. Us-ing the Hamiltonian, transition probabilitiesbetween configurations are adjusted suchthat configurations are visited according totheir proper statistical weight (Binder,1979, 1984a; Binder and Ciccotti, 1996).Again averages are obtained as “pseudo-time averages” along the trajectory of thesystem in phase space, the only differencefrom the MD method being that the tra-jectory is now stochastic rather than deter-ministic. Both methods have been exten-sively reviewed (see the references quotedabove); therefore, we do not give any de-tails here, but only briefly mention the dif-ficulties encountered when phase transi-tions are studied, and discuss a few typicalexamples of their application.

One principal difficulty is the finite-sizerounding and shifting of the transition. In

principle, this problem is well understood(Fisher 1971; Challa et al., 1986; Binder,1987b; Privman, 1990). In practice, thismakes it difficult to distinguish betweensecond-order and weakly first-order transi-tions. For example, extensive MD workwas necessary to obtain evidence that themelting transition of two-dimensional sol-ids with pure Lennard–Jones interaction isfirst order (Abraham, 1983, 1984; Bakkeret al., 1984), and that the suggested twocontinuous transitions involving the hexa-tic phase do not occur in these systems(Nelson and Halperin, 1979). However,this conclusion has been called into ques-tion by recent simulations for hard-diskfluids (Jaster, 1998) providing evidence forcontinuous two-dimensional melting.

Another difficulty is that the periodicboundary condition (for a chosen shape ofthe box) prefers certain structures of a solidand suppresses others which do not “fit”:this is particularly cumbersome for incom-mensurate modulated structures (Selke, 1988,1989; 1992) and for off-lattice systems,such as studies of the fluid–solid transitionor phase transitions between different lat-tice symmetries. For example, particles in-teracting with a screened Coulomb potential(this is a model for colloidal suspensions orcolloidal crystals (see Alexander et al.,1984)) may exhibit a fluid phase in addi-tion to an f.c.c. and a b.c.c. crystal, and thedetermination of a complete phase diagramis correspondingly difficult (Kremer et al.,1986, 1987; Robbins et al., 1988). The tra-ditional approach to dealing with such prob-lems is to repeat the calculation for differ-ent box shapes and compare the free ener-gies of the different phases. An interestingalternative method has been proposed byParrinello and Rahman (1980) and Parri-nello et al. (1983), who generalized the MDmethod by including the linear dimensionsof the box as separate dynamic variables.



Another severe problem is the occur-rence of metastability and hysteresis; thesystem may become trapped in a meta-stable state, the lifetime of which is longerthan the observation time of the simulation.The distinction of such long-lived meta-stable states from true equilibrium states isdifficult and may require computation ofthe Helmholtz energies of the phases inquestion.

These problems must be consideredwhen first-order transitions are studied bycomputer simulation so that suitable boxsizes and observation times are chosen andthe initial state is prepared accordingly.Then, employing sufficient effort in com-puting time, and performing a careful anal-ysis of all the possible pitfalls mentionedabove, very reliable and useful results canbe obtained which are superior in mostcases to any of the other methods that havebeen discussed so far (see Fig. 4-15 for acomparison in the case of f.c.c. alloys).Misjudgements of the problems mentionedabove have also led to erroneous con-clusions: in the f.c.c. Ising antiferromagnetwith nearest interaction J, the triple pointbetween the AB and A3B structures(Fig. 4-8a, c) was suggested to occur atT = 0 (Binder, 1980) while later a nonzerout low temperature was found, kBT /| J | ~ 1.0(Gahn, 1986; Diep et al., 1986; Kämmereret al., 1996). Unlike in the other methods,such problems can always be clarified bysubstantially increasing the computationaleffort and carrying out more detailed anal-yses of the simulation “data”.

A significant advantage of such a “com-puter experiment” is that interaction pa-rameters of a model can be varied system-atically. As an example, Fig. 4-17 showsthe temperature dependence of the long-range order parameters and some short-range order parameters of an f.c.c model ofan A3B alloy with nearest-neighbor repul-

sive (Jnn) and next-nearest-neighbor attrac-tive interaction Jnnn (Eq. (4-110)), for vari-ous choices of their ratio R = Jnnn/Jnn. Thecomparison with experimental data showsthat a reasonable description of Cu3Au isobtained for R = – 0.2, for T < Tc, whereasbetter experimental data for T >Tc areneeded before a more refined fit of effec-tive interaction parameters can be per-formed.

As an example showing that the MCmethod can deal with very complex phase

Figure 4-17. Temperature dependence of the long-range order parameter Y of the Cu3Au structure andof the nearest- and next-nearest-neighbor short-rangeorder parameter a1 and a2. Curves are Monte Carloresults of Binder (1986). Points show experimentaldata of Cowley (1950), Schwartz and Cohen (1965),Moss (1964), Bardham and Cohen (1976), and Keat-ing and Warren (1951). From Binder (1986).

diagrams, we consider the model in Eq. (4-115) for alloys with one magnetic com-ponent. However, in order to facilitatecomparison with the MFA and CV meth-ods, the magnetic interaction is chosen to beof Ising rather than Heisenberg type (Dün-weg and Binder, 1987). Fig. 4-18 showsthe resulting phase diagrams for the choiceof interaction parameters Jnnn/Jnn = 0.5 andJm/| Jnn | = 0.7. In this case, the MFA pre-dicts a wrong phase diagram “topology”(e.g., a direct transition between paramag-netic D03 and A2 phases never occurs,there is always a B2 phase in between), andit also grossly overestimates the transitiontemperatures. In contrast, the CV methodyields the phase diagram “topology” cor-rectly, and it overestimates the transitiontemperature by only a few per cent. Al-though the CV method does not alwaysperform so well (see Fig. 4-15, and Binder(1980) and Diep et al. (1986) for a discus-sion in the case of f.c.c. alloys), it is alwaysmuch superior to the simple MFA, which inmany cases fails dramatically.

One great advantage of the MC methodover the CV method is that it can also beapplied straightforwardly to models withcontinuous degrees of freedom, for whichthe CV method would be cumbersome towork out. As examples, Figs. 4-19 and 4-20 show the phase diagrams of the modelin Eqs. (4-93) and (4-136), namely the an-isotropic classical Heisenberg antiferro-magnet in a uniform field H|| along the easyaxis and the F4 model on the square lat-tice. Note that in the latter case the result of the MFA would be off-scale in this figure (e.g., Kc

–1 = 4 in the Ising limit). Fi-nally, Fig. 4-21 shows the phase diagram ofa model for EuxSr1–xS (Binder et al.,1979):

(4-139) = − ⋅≠∑1

2 i jij i j i jJ x x S S


Figure 4-18. Phase diagram of the body-centeredcubic binary alloy model with nearest and next near-est neighbor crystallographic interactions Jnn andJnnn (both being “antiferromagnetic” if Ising-modelterminology is used) and a ferromagnetic nearest-neighbor interaction Jm between one species, forJnnn /Jnn = 0.5, Jm/ |Jnn | = 0.7; c denotes the concentra-tion of the magnetic species. (a) Result of the MFA (Bragg–Williams approximation); (b) result of theCV method in the tetrahedron approximation, andpart (c) the MC result. The A2 phase is the crystallo-graphically disordered phase, the orderings of the B2and D03 phases are shown in Fig. 4-3. Two-phase co-existence regions are shaded. The magnetic orderingof the phases is indicated as para(-magnetic) andferro(-magnetic), respectively. From Dünweg andBinder (1987).


where Si are unit vectors in the direction ofthe Eu magnetic moment, and from spinwave measurements (Bohn et al., 1980) itis known that EuS essentially exhibitssuperexchange between nearest and next-

nearest neighbors only, further-neighborinteractions being negligibly small, withJnnn/ Jnn = – 1/2. Thus Fig. 4-21 exhibits aremarkable agreement between computersimulation and experiment (Maletta andFelsch, 1979); there are no adjustable pa-rameters involved whatsoever. Note thatanalytical methods for such problems withquenched disorder would be notoriouslydifficult to apply.

To conclude this subsection, we statethat computer simulation techniques are

Figure 4-19. Phase diagram of a uniaxial classicalHeisenberg antiferromagnet on the simple cubic lat-tice, as a function of temperature T and field H|| ap-plied in the direction of the easy axis. The anisotropyparameter D in the Hamiltonian Eq. (4-93) is chosenas D = 0.2. Both Monte Carlo results (crosses, circles,full curves) and the Landau theory fitted to the phasediagram in the region off the true bicritical point isshown (dash-dotted straight line). The Landau theorywould overestimate the location of the bicritical tem-perature Tb (Tb* > Tb), and fails to yield the singularumbilicus shape of the phase diagram lines Tc

^, Tc||

near the bicritical point. The broken straight lines de-note the appropriate choices of “scaling axes” for acrossover scaling analysis near Tb. The trianglesshow another phase diagram, namely when the fieldis oriented in the perpendicular direction to the easyaxis. The nature of the phases is denoted as AF (anti-ferromagnetic), SF (spin-flop) and P (paramagnetic).See Fig. 4-9a for a schematic explanation of thisphase diagram. From Landau and Binder (1978).

Figure 4-20. Critical line Kc–1 of the F4 model

(Eq. (4-136)) on the square lattice, shown in thespace of couplings K –1, L–1 defined as K = –rC/u,L = r (r + 4C)/(4u). Note that for L–1= 0, and K finite,the square Ising model results, the transition temper-ature of which is exactly known (Onsager, 1944; ar-row). () Monte Carlo simulation of Milchev et al.(1986); (+) MD results of Schneider and Stoll(1976); (, ¥) from a real space renormalizationgroup calculation of Burkhardt and Kinzel (1979).The disordered phase occurs above the critical line.From Milchev et al. (1986).

well suited for studying the phase diagramsof various systems, magnetic systems, me-tallic alloys, structural transitions, ad-sorbed layers at surfaces, etc. provided thata suitable model Hamiltonian is known.Apart from the phase diagram, detailed in-formation on long- and short-range order isalso accessible. Also, for certain problems,the kinetics of phase transitions can be studied, in particular for alloys where theMonte Carlo process can directly modelthe atomic jump processes on the crystallattice (Binder, 1979, 1984a; Kehr et al.,1989). Of course, there is also interest inclarifying the phase diagrams of materialsin their fluid states. Computer simulationscan now obtain accurate gas– liquid phasediagrams (Wilding, 1997), liquid binarymixtures (Wilding et al., 1998) and surfac-tants (Schmid, 1999).

4.4 Concepts About Metastability

Metastable phases are very common innature, and, for many practical purposes,not at all distinct from stable phases (re-member that diamond is only a metastablemodification of graphite!). Also, approxi-mate theories of first-order phase transi-tions easily yield Helmholtz energybranches that do not correspond to the ther-mal equilibrium states of minimumHelmholtz energy, and hence are com-monly interpreted as metastable or unstablestates (cf. Figs. 4-6 and 4-22). From the an-alog of Eq. (4-12) for H (0,

(4-140)

the limit of metastability where cT =(∂F /∂H )T diverges as follows:

(4-141)

Since in the metastable states

cT = (3 kBTu)–1 (F2 – F s2)–1

we see that cT Æ • as F ÆFs; in the(T,F ) plane, the spinodal curve F =Fs(T )plays the part of a line of critical points.

Similar behavior also occurs in manyother theories: for example, the van derWaals equation of state describing gas– liq-uid condensation exhibits an analogousloop of one-phase states in the two-phasecoexistence region.

Unfortunately, although the descriptionof metastability in the framework of theMFA seems straightforward, this is not soif more accurate methods of statistical ther-modynamics are used. A heuristic compu-

r uk T

r u

Hr r

u

T+

−

− −

31 1

0

3 3

23 3

2

0

F

F F

sB

s

c

= =

= =

=

c/ /

103

k T VF

r uH

k TTB B= =

∂∂

⎛⎝⎜

⎞⎠⎟

+ −F

F F


Figure 4-21. Ferromagnetic critical temperature ofthe model Eq. (4-139) of EuxSr1–xS vs. concentrationx of magnetic atoms. Circles denote experimentaldata due to Maletta and Felsch (1979), triangles theMC results for the diluted classical Heisenberg fccferromagnet with nearest (Jnn) and next-nearest (Jnnn) exchange, Jnnn = – –1

2Jnn. From Binder et al.

(1979).

4.4 Concepts About Metastability 299

tational approach, discussed in Sec. 4.3.2,is the CV method where we systematicallyimprove upon the MFA by taking more andmore short-range correlations into account,the larger the cluster is chosen. Computingthe equation of state for a nearest-neighborIsing ferromagnet in this way (Kikuchi,1967), it is found that the stable branch isnicely convergent, whereas the metastableloop becomes flatter and flatter the largerthe cluster, i.e., the critical field Hc (Eq. (4-141)) converges towards zero. An exactcalculation of the equation of state yieldsthe magnetization jump from F0 to –F0

as H changes from 0+ to 0– in Fig. 4-22, butdoes not yield any metastable state! This isnot surprising, however, because statisticalmechanics is constructed to yield informa-tion only on thermal equilibrium states.The partition function is dominated by thesystem configurations in the vicinity of theminimum of the Helmholtz energy. In amagnet with H < 0, states with negativemagnetization have lower Helmholtz en-ergy than those with positive magnetiza-tion. Hence the latter do not result from thepartition function in the thermodynamiclimit. Similar conclusions emerge from arigorous treatment of the gas–fluid transi-tion in systems with long-range interac-tions (Lebowitz and Penrose, 1966).

Two concepts emerged for the descrip-tion of metastable states (for a more de-tailed discussion see Binder (1987a)). Oneconcept (Binder, 1973) suggests the defini-tion of metastability be based on the con-sideration of kinetics: starting with the sys-tem in a state of stable equilibrium, the sys-tem is brought out of equilibrium by a sud-den change of external parameters (temper-ature, pressure, fields, etc.). For example,in the Ising-type ferromagnet of Fig. 4-22,at times t < 0 we may assume we have astate at H = 0+, F =F0 > 0: at t = 0, the fieldis switched to a negative value: then the

magnetization in equilibrium is negative,but in the course of the nonequilibrium re-laxation process from the initial state at F0

a metastable state may occur with Fms >0,although H < 0, which exists only over a finite lifetime tms . In order that Fms canclearly be identified in this dynamic pro-cess where the time-dependent order pa-rameter F (t) relaxes from F (t =0) = F0 tothe negative equilibrium value F (t Æ •), it is necessary that tms is much larger thanany intrinsic relaxation time. Then the timetms for the decay of the metastable state farexceeds the time needed for the system torelax from the initial state towards the met-astable state, and, in the “nonequilibriumrelaxation function” F (t) the metastablestate shows up as a long-lived flat partwhere F (t)≈Fms. The second concepttries to define a metastable state in theframework of equilibrium statistical me-chanics by constraining the phase space soas to forbid the two-phase configurations

Figure 4-22. Order parameter F vs. conjugate fieldH according to the phenomenological Landau theoryfor system at a temperature T less than the criticaltemperature Tc of a second-order phase transition(schematic). At H = 0, a first-order transition from F0

to –F0 occurs (thick straight line). The metastablebranches (dash-dotted) end at the “limit of meta-stability” or “spinodal point” (Fs , –Hs), respectively,and are characterized by a positive-order parametersusceptibility cT > 0, whereas for the unstable branch(broken curve) cT < 0.

that otherwise dominate the partition func-tion. Langer (1974) suggested that phaseseparation into two phases with order pa-rameters –F0 and +F0 coexisting at thefirst-order transition at H = 0 in Fig. 4-1and Fig. 4-22 is suppressed if we considera system (a) at fixed F between –F0 and+F0 but (b) constrain the system and di-vide it into cells of size Ld, and (c) requirethat the order parameter is not only glo-bally fixed at F but also inside each cell. If L is small enough, namely a O L O xcoex

(this is the same condition as noted for the construction of a continuum modelfrom the microscopic Hamiltonian, seeSec. 4.2.2, and Eqs. (4-9) and (4-57)),phase separation inside a cell cannot occur,and hence a coarse-grained Helmholtz en-ergy density fcg(F) of states with uniformorder parameter F is obtained, which hasexactly the double-well shape of the Lan-dau theory (Fig. 4-6a). Since fcg(F) thennecessarily depends on this coarse-grainedlength scale L, it is clear that the metastablestates of fcg(F) are not precisely those observed in experiments, unlike the dy-namic definition. While fcg(F) defines a “limit of metastability” or “spinodal curve”Fs where (∂2 fcg(F )/∂F2)T = cT

–1 vanishesand changes sign, this limit of metastabilityalso depends on L and hence is not relatedto any physical limit of metastability. In thecontext of spinodal decomposition of mix-tures, the problem of the extent to which aspinodal curve is relevant also arises and isdiscussed in detail in the chapter by Binderand Fratzl (2001).

In some systems where a mean-field de-scription is appropriate, the lifetime ofmetastable states can be extremely long,and then the mean-field concept of a limitof metastability may be very useful. Thisfact can again be understood in terms of the Ginzburg criterion concept as outlinedin Sec. 4.2.2. Since the spinodal curve

F =Fs(T ) acts like a line of critical points,as discussed above, we require (cf. Eq. (4-58))

·[F (x) – F]2ÒT,L O [F –Fs(T )]2 (4-142)

i.e., the mean-square fluctuations in a vol-ume Ld must be much smaller than thesquared distance of the order parameterfrom its value at criticality, which in thiscase is the value at the spinodal. Now, themaximum permissible value for L is thecorrelation length x in the metastable state,which becomes (4-143)

x = R/[6d u Fs(T )]1/2 [F – Fs (T )]–1/2

Eq. (4-143) exhibits the critical singularityof x at Fs(T ), corresponding to that of cT

noted above. With a little algebra we findfrom Eqs. (4-142) and (4-143) (Binder,1984c)

This condition can only be fulfilled if theinteraction range is very large. Even thenfor d = 3 this inequality is not fulfilled veryclose to the spinodal curve.

In order to elucidate the physical signifi-cance of this condition further and to dis-cuss the problem of the lifetime of meta-stable states, we briefly discuss the meanfield theory of nucleation phenomena. Inpractice, for solid materials, heterogeneousnucleation (nucleation at surfaces, grainboundaries, dislocations, etc.) will domi-nate; and this may restrict the existence ofmetastable states much more than expectedfrom Eq. (4-144). However, we considerhere only the idealized case that homoge-neous nucleation (formation of “droplets”of the stable phase due to statistical fluctu-ations) is the mechanism by which the met-astable state decays. Since there is no

1

4

2 2 2

2 2 6 2

6 4

O R T

R T T

R H H

d

d d d

d d

x −

− −

−

−−

−

[ ( )]

~ [ ( )] [ ( )]

~ ( )

( )/ ( ) /

( ) /

F FF F F

s

s s

c (4-1 4)


4.4 Concepts About Metastability 301

mechanism by which this process can besuppressed, metastable states cannot existin a region where strong homogeneous nu-cleation occurs. Therefore we estimate theintrinsic and ultimate limit of metastabilityby the condition that the Helmholtz energybarrier against homogeneous nucleationdecreases to the order of the thermal en-ergy kBT (Binder and Stauffer, 1976b).

As is well known, the nucleation barrierarises from competition between the favor-able) volume energy of a droplet of the newphase and the (unfavorable) surface Helm-holtz energy between the droplet and thesurrounding metastable background phase.In order to calculate this Helmholtz energybarrier in the framework of the Landau the-ory, we can still use Eqs. (4-10) and (4-33).Putting H (x) = 0 in Eq. (4-33) and consid-ering the situation that F (z Æ •) = F0,F (z Æ – •) = – F0 and solving for the con-centration profile F (z) (Cahn and Hilliard,1958), we can obtain the interface Helm-holtz energy associated with a planar (infi-nitely extended) interface perpendicular tothe z-direction (Fig. 4-23a). In the “classi-cal theory of nucleation” (Zettlemoyer,1969), this Helmholtz energy is then usedto estimate the Helmholtz energy barrier.However, this “classical theory of nuclea-tion” is expected to be reliable only formetastable states near the coexistencecurve, where the radius R* of a criticaldroplet (corresponding to a droplet Helm-holtz energy exactly at the Helmholtz en-ergy barrier DF*) is much larger than thewidth of the interfacial profile (which thenis also of the same order as the correlationlength xcoex at the coexistence curve, cf.Figs. 4-23a, b). Therefore it cannot be usedclose to the limit of metastability. Cahn andHilliard (1959) have extended the Landautheory to this problem, solving Eq. (4-33)for a spherical geometry, where only a radial variation of F (r) with radius r is

permitted, and a boundary conditionF (r Æ •) = Fms is imposed (Fig. 4-23b, c). Whereas for Fms near Fcoex = F0

this treatment agrees with the “classicaltheory of nucleation”, it differs signifi-cantly from it for F near Fs(T ): then thecritical droplet radius R* is of the same or-der as the (nearly divergent!) correlationlength x (Eq. (4-143)), and the profile isextremely flat, F (r) reaches in the dropletcenter only a value slightly below Fs ratherthan the other branch of the coexistencecurve. Calculating the Helmholtz energy

Figure 4-23. Order-parameter profile F (z) acrossan interface between two coexisting phases ±Fcoex,the interface being oriented perpendicular to the z-di-rection (a) and the radial order-parameter profile fora marginally stable droplet in a metastable statewhich is close to the coexistence curve (b) or close tothe spinodal curve (c). In (a) and (b) the intrinsic“thickness” of the interface is of the order of the cor-relation length at coexistence xcoex, whereas in (c) itis of the same order as the critical radius R*. FromBinder (1984c).

barrier DF*, we obtain for T near Tc

(Binder, 1984c; Klein and Unger, 1983)

(4-145)

whereas near the coexistence curve the re-sult is

(4-146)

where all prefactors of order unity are omit-ted. In a system with a large range R of in-teraction, the nucleation barrier is very highin the mean-field critical region, in whichR d (1 – T /Tc)

(4–d )/2 o1 (cf. Eq. (4-62)); thisfactor, which controls the Ginzburg criter-ion, also controls the scale of the nuclea-tion barrier as a prefactor (see Fig. 4-24).In this region the condition for the limit ofmetastability, DF*/kBT ≈ 1, is located veryclose to the mean-field spinodal. Then thedescription of nucleation phenomena interms of the diffuse droplets described byFig. 4-23c near the spinodal curve is mean-ingful (“spinodal nucleation”). On theother hand, for a system with short-rangeinteractions where R (measured in units of the lattice spacing in Eqs. (4-145) and(4-146)) is unity, the Helmholtz energybarrier becomes of order unity long beforethe spinodal curve is reached. The singu-larity at the spinodal then completely lacksany physical significance, as the meta-stable state decays to the stable phase longbefore the spinodal is reached.

It is instructive to compare the conditionDF*/kBT O1 with Eq. (4-144): this showsthat these conditions are essentially iden-tical! This is not surprising: the MFA is essentially correct as long as effects of statistical fluctuations are very small: the“heterophase fluctuations” (droplets of the

DFk T

RTT

dd d

*~

( )/ ( )

B c c

coex

coex1

4 2 1

−⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

− − −F F

F

DFk T

RTT

dd d

*~

( )/ ( )/

B c c

s

coex1

4 2 6 2

−⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

− −F FF

new phase) are also extremely rare, and theHelmholtz energy cost to form them shouldbe very high, as implied by Eqs. (4-145)and (4-146).

These concepts have been tested byMonte Carlo simulations on simple-cubicIsing models with various choices of therange of the interaction (Fig. 4-25, Heer-mann et al., 1982). For the case of nearest-neighbor interaction (each spin interactswith q = 6 neighbors) it is seen that cT

–1 dif-fers considerably from the MFA result, andthe range of fields over which metastablestates can be observed is not very large. Withan increase in the number of neighbors q,


Figure 4-24. Schematic plots of the Helmholtz en-ergy barrier for (a) the mean field critical region, i.e.,Rd(1– T/Tc)(4–d )/2 o1, and (b) the non-mean fieldcritical region i.e., Rd(1– T/Tc)(4–d )/2 O1. Note thatowing to large prefactors to the nucleation rate, theconstant of order unity where the gradual transitionfrom nucleation to spinodal decomposition occurs isabout 101 rather than 100. From Binder (1984c).

4.5 Discussion 303

this range increases, the value of cT–1 at

the field where nucleation becomes appre-ciable falls also, and the cT

–1 vs. H curvequickly converges towards the MFA pre-diction, except in the immediate neighbor-hood of the limit of metastability Hc.

It is expected that the ideas sketchedhere will carry over to more realistic sys-tems.

4.5 Discussion

There are many different kinds of phasetransitions in materials, and the fact that amaterial can exist in several phases may

have a strong influence on certain physicalproperties. The approach of statistical me-chanics tries to provide general conceptsfor dealing with such phenomena: classifi-cation methods are developed which alsotry to clarify which aspects of a phase transition are specific for a particular mate-rial and which are general (“universal”). Atthe same time, theoretical descriptions areavailable both on a phenomenologicallevel, where thermodynamic potentials areexpanded in terms of suitable order param-eters and the expansion coefficients are un-determined, and can only be adjusted to ex-perimental data, and on a microscopiclevel, where we start from a model Hamil-tonian which is treated either by molecularfield approximations or variants thereof orby computer simulation techniques.

This chapter has not given full details ofall these approaches, but rather tried togive a discussion which shows what thesemethods can achieve, and to give the readera guide to more detailed literature on thesubject. An attempt has been made to sum-marize the main ideas and concepts in thefield and to describe the general facts thathave been established, while actual materi-als and their phase transitions are men-tioned as illustrative examples only, withno attempt at completenesss being made.Although the statistical thermodynamics ofphase transitions has provided much physi-cal insight and the merits and limitations ofthe various theoretical approaches are nowwell understood generally, the detailedunderstanding of many materials is still ru-dimentary in many cases; often there arenot enough experimental data on the order-ing phenomenon in question, or the dataare not precise enough; a microscopicmodel Hamiltonian describing the interac-tion relevant for the considered ordering isoften not explicitly known, or is so com-plicated that a detailed theory based on

Figure 4-25. Inverse susceptibility of Ising ferro-magnets plotted against h = –H/kBT at T/Tc

MFA = 4/9for various ranges of the exchange interaction; eachspin interacts with equal strength with q neighbors.The full curve is the MFA, the broken curve a fit to adroplet model description. From Heermann et al.(1982).

such a model Hamiltonian does not exist.Many of the more sophisticated methods(CV method, computer simulation, etc.)are restricted in practice to relatively sim-ple models. Hence, there still remains a lotto be done to improve our understanding ofphase transitions.

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Alefeld, G. (1969), Phys. Stat. Sol. 32, 67.Alefeld, G., Völkl, J. (Eds.) (1978), Hydrogen in

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Alexander, S. (1975), Phys. Lett. 54 A, 353.Alexander, S., Chaikin, P. M., Grant, P., Morales, G.

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Als-Nielsen, J. (1976), in: Phase Transitions andCritical Phenomena, Vol. 5A: Domb, C., Green,M. S. (Eds.). New York: Academic, p. 88.

Als-Nielsen, J., Laursen, I. (1980), in: Ordering ofStrongly Fluctuating Condensed Matter Systems:Riste, T. (Ed.). New York: Plenum Press, p. 39.

Amit, D. J. (1984), Field Theory, the Renormaliza-tion Group and Critical Phenomena. Singapore:World Scientific.

Angell, C. A., Goldstein, M. (Eds.) (1986), DynamicAspects of Structure Change in Liquids andGlasses. New York: N.Y. Acad. of Sciences.

Ausloos, M., Elliott, R. J. (1983), Magnetic PhaseTransitions. Berlin: Springer Verlag.

Bak, P., Domany, E. (1979), Phys. Rev. B20, 2818.Bak, P., Mukamel, D. (1979), Phys. Rev. B19, 1604.Bak, P., Mukamel, D., Villain, J., Wentowska, K.

(1979), Phys. Rev. B19, 1610.Bakker, A. F., Bruin, C., Hilhorst, H. J. (1984), Phys.

Rev. Let. 52, 449.Bardham, P., Cohen, J. B. (1976), Acta Cryst. A32,

597.Bates, S. F., Rosedale, J. H., Fredrickson, G. H.,

Glinka, C. J. (1988), Phys. Rev. Lett. 61, 229.Baus, M. (1987), J. Stat. Phys. 48, 1129.Baus, M., Rull, L. F., Ryckaert, J. P. (Eds.) (1995),

Observation, Prediction and Simulation of PhaseTransitions in Complex Fluids. Dordrecht: KluwerAcad. Publ.

Baxter, R. J. (1973), J. Phys. C6, L445.

Baxter, R. J. (1982), Exactly Solved Models in Statis-tical Mechanics. London: Academic.

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Belanger, D. P. (1998), in: Spin Glasses and RandomFields: Young, A. P. (Ed.). Singapore: World Sci-entific, p. 251.

Berlin, T. H., Kac, M. (1952), Phys. Rev. 86, 821.Bethe, H. (1935), Proc. Roy. Soc. A150, 552.Bieber, A. Gautier, F. (1984a), J. Phys. Soc. Japan

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Statistical Physics. Berlin: Springer.Binder, K. (1980), Phys. Rev. Lett. 45, 811.Binder, K. (1983a), Z. Physik B50, 343.Binder, K. (1983b), in: Phase Transitions and Criti-

cal Phenomena, Vol. 8: Domb, C., Green, M. S.(Eds.). London: Academic Press, p. 1.

Binder, K. (1984a), Applications of the Monte CarloMethod in Statistical Physics. Berlin: SpringerVerlag.

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Binder, K. (1984c), Phys. Rev. A29, 341.Binder, K. (1986), in: Atomic Transport and Defects

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Binder, K. (1987a), Rep. Progr. Phys. 50, 783.Binder, K. (1987b), Ferroelectrics 73, 43.Binder, K. (1989), in: Alloy Phase Stability: Gonis,

A., Stocks, L. M. (Eds.). Dordrecht: Kluwer Acad.Publ., p. 232.

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Binder, K. Landau, D. P. (1981), Surface Sci. 108, 503.Binder, K., Landau, D. P. (1989), in: Molecule–Sur-

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5 Homogeneous Second-Phase Precipitation

Richard Wagner

Forschungszentrum Jülich GmbH, Jülich, Germany

Reinhard Kampmann

Institut für Werkstofforschung, GKSS-Forschungszentrum GmbH, Geesthacht, Germany

Peter W. Voorhees

Department of Materials Science and Engineering, Northwestern University, Evanston,Ill., USA

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3145.2 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 3155.2.1 General Course of an Isothermal Precipitation Reaction . . . . . . . . . . 3155.2.2 Thermodynamic Considerations – Metastability and Instability . . . . . . 3175.2.3 Decomposition Mechanisms: Nucleation and Growth versus Spinodal

Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3195.2.4 Thermodynamic Driving Forces for Phase Separation . . . . . . . . . . . 3225.3 Experimental Techniques for Studying Decomposition Kinetics . . . . 3265.3.1 Microanalytical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3265.3.1.1 Direct Imaging Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 3265.3.1.2 Scattering Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3285.3.2 Experimental Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3305.3.2.1 Influence of Quenching Rate on Kinetics . . . . . . . . . . . . . . . . . . 3305.3.2.2 Distinction of the Mode of Decomposition . . . . . . . . . . . . . . . . . 3325.4 Precipitate Morphologies . . . . . . . . . . . . . . . . . . . . . . . . . . 3345.4.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3345.4.2 Factors Controlling the Shapes and Morphologies of Precipitates . . . . . 3365.5 Early Stage Decomposition Kinetics . . . . . . . . . . . . . . . . . . . . 3395.5.1 Cluster-Kinetics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 3405.5.1.1 Classical Nucleation – Sharp Interface Model . . . . . . . . . . . . . . . . 3405.5.1.2 Time-Dependent Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . 3435.5.1.3 Experimental Assessment of Classical Nucleation Theory . . . . . . . . . 3455.5.1.4 Non-Classical Nucleation – Diffuse Interface Model . . . . . . . . . . . . 3475.5.1.5 Distinction Between Classical and Non-Classical Nucleation . . . . . . . . 3495.5.2 Diffusion-Controlled Growth of Nuclei from the Supersaturated Matrix . . 350


5.5.3 The Cluster-Dynamics Approach to Generalized Nucleation Theory . . . . 3525.5.4 Spinodal Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3565.5.5 The Philosophy of Defining of ‘Spinodol Alloy’ – Morphologies of

‘Spinodal Alloys’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3635.5.6 Monte Carlo Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3675.6 Coarsening of Precipitates . . . . . . . . . . . . . . . . . . . . . . . . . 3705.6.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3705.6.2 The LSW Theory of Coarsening . . . . . . . . . . . . . . . . . . . . . . . 3705.6.3 Extensions of the Coarsening Theory to Finite Precipitate Volume Fractions 3735.6.4 Other Approaches Towards Coarsening . . . . . . . . . . . . . . . . . . . 3775.6.5 Influence of Coherency Strains on the Mechanism and Kinetics of

Coarsening – Particle Splitting . . . . . . . . . . . . . . . . . . . . . . . 3775.7 Numerical Approaches Treating Nucleation, Growth and Coarsening as

Concomitant Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.7.1 General Remarks on the Interpretation of Experimental Kinetic Data

of Early Decomposition Stages . . . . . . . . . . . . . . . . . . . . . . . 3815.7.2 The Langer and Schwartz Theory (LS Model) and its Modification by

Kampmann and Wagner (MLS Model) . . . . . . . . . . . . . . . . . . . 3835.7.3 The Numerical Modell (N Model) of Kampmann and Wagner (KW) . . . . 3855.7.4 Decomposition of a Homogeneous Solid Solution . . . . . . . . . . . . . 3855.7.4.1 General Course of Decomposition . . . . . . . . . . . . . . . . . . . . . . 3855.7.4.2 Comparison Between the MLS Model and the N Model . . . . . . . . . . 3875.7.4.3 The Appearance and Experimental Identification of the Growth

and Coarsening Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3875.7.4.4 Extraction of the Interfacial Energy and the Diffusion Constant from

Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3895.7.5 Decomposition Kinetics in Alloys Pre-Decomposed During Quenching . . 3915.7.6 Influence of the Loss of Particle Coherency on the Precipitation Kinetics . 3925.7.7 Combined Cluster-Dynamic and Deterministic Description

of Decomposition Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 3945.8 Self-Similarity, Dynamical Scaling and Power-Law Approximations . . 3955.8.1 Dynamical Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3955.8.2 Power-Law Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 3985.9 Non-Isothermal Precipitation Reactions . . . . . . . . . . . . . . . . . 4015.10 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4025.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

310 5 Homogeneous Second-Phase Precipitation



a lattice parameterA solvent atomsai

j(c) activity of atomic species i in phase j at composition cB solute atomsc compositioncA atomic fraction of Ac0 initial compositioncr

mean compositionDc supersaturation or composition differenceca

e equilibrium composition of matrix phase ac¢a

e composition of matrix a in metastable equilibrium with coherent precipi-tates

cbe equilibrium composition of precipitate phase b

cp composition of incoherent particlesc¢p composition of coherent particlescij elastic constantcR composition of matrix at the matrix/particle interfaceC (i) equilibrium cluster distributionD diffusion constantD–

mean diameter of precipitated particlesE Young’s modulusf (R) size distribution function of precipitates with radii RF (c), f (c) Helmholtz energy or energy densityF¢ (c), f ¢(c) Helmholtz constraint energy or constraint energy densityDF* nucleation barrierDFel elastic free energyDFa/b interfacial free energyDFch (c) chemical driving forcefp precipitated volume fractionF (x) time-independent scaling functionG Gibbs energyG(| r – r0 |) two-point correlation function at spatial positions r, r0

H enthalpyi, i* number of atoms in a cluster or in a cluster of critical sizeJ s, J* steady state and time-dependent nucleation rateK aspect ratio of an ellipsoid of revolutionK* gradient energy coefficient in the CH spinodal theoryKR

LSW coarsening rate according to the LSW theoryk Boltzmann constantL ratio of elastic to interfacial energyM atomic mobilityNv number density of precipitatesnv number of atoms per unit volume

p pressureR radius of precipitatesR–

mean radius of precipitatesR* radius of a critical nucleus or a particle being in unstable equilibrium

with the matrixRg molar gas constantR (k); R (l) amplification factor in the CH spinodal theoryS entropyS(k) structure functionSm = S (km) maximum of structure functionT temperatureTA annealing temperatureTH homogenization temperaturet aging timeU internal energyV volumeVa, Vb molar volume of a or b phaseZ Zeldovich factor

a, a¢ matrix phaseb equilibrium precipitated phaseb¢ metastable precipitated phase; transition phaseg shear straind misfit parameterh = (1/a0) (∂a/∂c) atomic size factork , k scattering vector and its magnitudekm wavenumber of the maximum of the structure function S (k)l wavelengthm shear modulusmi

j chemical potential of component i in phase jn Poisson’s ratioW atomic volumesab specific interfacial energyt incubation period or scaled time in the MLS and KW models

AEM analytical transmission electron microscopyAFIM analytical field ion microscopy (atom probe field ion microscope)CH Cahn–Hilliard spinodal theoryCHC Cahn–Hilliard–Cook spinodal theoryCTEM conventional transmission electron microscopyEDX energy-dispersive X-ray analysisEELS electron energy loss spectroscopyFIM field ion microscopyHREM high resolution electron microscopyKW Kampmann–Wagner model



LBM Langer–Bar-On–Miller non-linear spinodal theoryLS Langer–Schwartz theoryLSW Lifshitz–Slyozov–Wagner theoryMCS Monte Carlo simulationMLS modified Langer–Schwartz modeln.g. nucleation and growthSAXS small angle X-ray scatteringSANS small angle neutron scatterings.d. spinodal decompositionTAP tomographic atom probe

5.1 Introduction

Many technologically important proper-ties of alloys, such as their mechanicalstrength and toughness, creep and corro-sion resistance, and magnetic and super-conducting properties, are essentially con-trolled by the presence of precipitated par-ticles of a second phase. This commonlyresults from the decomposition of a solidsolution during cooling. A fundamentalunderstanding of the thermodynamics, themechanism and the kinetics of precipita-tion reactions in metallic solids, leading toa well-defined microstructure, is thereforeof great interest in materials science.

As is reflected by the schematic binaryphase diagram of Fig. 5-1, for reasons ofentropy the single-phase state a of a solidsolution with composition c0 is thermody-namically stable only at elevated tempera-tures. At lower temperatures the free en-ergy of the system is lowered through un-mixing (‘decomposition’ or ‘phase separa-tion’) of a into two phases, a¢ and b.

In order to initiate a precipitation reac-tion, the alloy is first homogenized in thesingle-phase region at TH and then either

a) cooled down slowly into the two-phaseregion a¢ + b, or

b) quenched into brine prior to isothermalaging at a temperature TA within thetwo-phase region (Fig. 5-1).

In both cases, thermodynamic equilib-rium is reached if the supersaturation Dc,defined as

Dc (t) = c – cae (5-1)

becomes zero. (Here c (t) is the mean matrixcomposition at time t with c (t = 0) ∫ c0.)

For case a), which frequently prevailsduring industrial processing, the agingtemperature and the associated equilibriumsolubility limit ca

e (T ) decrease continu-ously. Equilibrium can only be reached ifthe cooling rate is sufficiently low within atemperature range where the diffusion ofthe solute atoms is still adequately high.The precipitated volume fraction ( fp) andthe dispersion of the particles of the secondphase can thus be controlled via the cool-ing rate.

Procedure b) is frequently used for stud-ies of decomposition kinetics under condi-tions which are easier to control and de-scribe theoretically (T = TA = const.; D =const.) than for case a). This leads to pre-cipitate microstructures whose volumefraction and particle dispersion depend onDc (T) and the aging time t.

Decomposition reactions involve diffu-sion of the atomic species via the vacancyand/or the interstitial mechanism. Hence,the precipitate microstructure proceed-ing towards thermodynamic equilibriumevolves as a function of both time and tem-perature. In practice, a metallurgist is oftenrequested to tailor an alloy with a specificprecipitation microstructure. For this pur-


Figure 5-1. Schematic phase diagram of a binary al-loy displaying a miscibility gap. Dashed lines showthe metastable coherent solvus line and a possiblemetastable intermetallic phase b¢. The long arrow in-dicates the quenching process.

5.2 General Considerations 315

pose it would be desirable to have a theo-retical concept, preferentially available asa computer algorithm, that allows suitableprocessing parameters to be derived for es-tablishing the specific microstructure ongrounds of the given thermodynamics ofthe alloy. This less empirical approach toalloy design, however, would require acomprehensive theory of precipitation ki-netics, which has not yet been developed toa satisfactory level, despite recent effortsand progress made in elucidating the kinet-ics of first-order phase transitions in solids.Furthermore, in materials science it is fre-quently desirable to predict the kinetic evo-lution of an originally optimized precipitatemicrostructure under service conditions,e.g., for high-temperature applications intwo-phase materials, where the precipitatedistribution might undergo changes be-cause of coarsening. Even though the ki-netics of coarsening are of great practi-cal importance, a completely satisfactorycoarsening theory has so far only been de-veloped in the limit of zero precipitatedvolume fraction (see Sec. 5.6). This limit,however, is never realized in technical al-loys, where the volume fraction of the mi-nor phase frequently exceeds 30%.

In the present chapter recent theoreticaland experimental studies on the kinetics ofphase separation in solids are reviewed fromthe point of view of the experimentalist.Special emphasis is placed on the questionsto what extent the theoretical results can beverified experimentally and to what extentthey might be of practical use to the physicalmetallurgist. We confine ourselves to deal-ing only with homogeneous and continuousphase separation mechanisms. Heterogene-ous nucleation at crystal defects, and dis-continuous precipitation reactions at mov-ing interfaces, as well as unmixing in solidsunder irradiation are treated separately, inthe chapter by Purdy and Bréchet (2001).

5.2 General Considerations

5.2.1 General Course of an IsothermalPrecipitation Reaction

As illustrated for a Ni–37 at.% Cu–8at.% Al alloy (Fig. 5-2 and 5-3), an isother-mal precipitation reaction (the kinetics ofwhich will be dealt with in more detail inSecs. 5.5, 5.6 and 5.7) is qualitatively char-acterized by an early stage during which anincreasing number density

Nv (t) = Ú f (R) dR (5-2)

of more or less spherical solute-rich clus-ters (‘particles’) with a size distribution f [R (t)] and a mean radius

(5-3)

are formed. In common with all homogene-ous precipitation reactions studied so far, inthe earliest stages of the reaction the parentphase a and the precipitate phase b share acommon crystal lattice, i.e., the two phasesare coherent. As inferred from the field ionmicrographs of Fig. 5.2a–c and from thequantitative data of Fig. 5-3, during theearly stages R

–increases somewhat and the

supersaturation, Dc, decreases slowly. Thissmall reduction in Dc, however, is sufficientto terminate the nucleation of new particles,as indicated by the maximum of Nv (t).

Beyond this maximum the precipitatenumber density decreases (Fig. 5-3) due tothe onset of the coarsening reaction, duringwhich the smaller particles redissolve, thusenabling the larger ones to grow (see Sec.5.6). During this coarsening process thesupersaturation Dc decreases asymptoti-cally towards zero.

As mentioned previously, within the mis-cibility gap, the solid solution a becomes un-stable and decomposes into the stable solidsolution a¢ and the precipitated phase b:a Æ a¢ + b

RR f R R

f R R= ∫

∫( )

( )

d

d

During the first step of the precipitationsequence a metastable precipitate (transi-tion phase b¢) is formed, frequently withlarge associated coherency strains and asmall interfacial energy sab. Often themetastable phase b¢ is an intermetalliccompound with a lower solute concentra-tion (c¢b

e ∫ c¢p) than the equilibrium precipi-tate (cp). Because of the large coherencystrains, the metastable (‘coherent’) solvusline (dashed line in Fig. 5-1) is shifted to-wards higher solute concentrations. Thisleads to a reduction of the supersaturation


Figure 5-2. a)–c) Neon field ion image of g¢-precipitates (bright images in dark matrix) in Ni–36.8 at.% Cu–8at.% Al aged for the given times at 580°C (Liu and Wagner, 1984). d) Three-dimensional distribution of Mg andZn atoms in a commercial Al–5.5 at.% Mg–1.35 at.% Zn alloy analyzed with the tomographic atom probe. Eachpoint represents an atom. The T ¢ precipitates are enriched in Mg and Zn (Bigot et al., 1997).

The terminal solute concentration of thesolvent (A)-rich matrix a¢ is given by theequilibrium solubility limit ca

e (T ); the ter-minal composition cp of the precipitatedphase is given by the solubility at TA at theB-rich side of the phase diagram (Fig. 5-1).Frequently, the interfacial energy sab(J/m2) between the matrix and the equilib-rium precipitate is rather high, particularlyif the width of the miscibility gap is large(see Sec. 5.7.4.4). In this case, the decom-position of a proceeds via the sequencea Æ a≤ + b¢ Æ a¢ + b

a) 2 min.; 2 R–

= 2,4 nm b) 180 min.; 2 R–

= 8 nm

c) 420 min.; 2 R–

= 11 nm d)


with respect to the (incoherent) equilib-rium solvus line (Fig. 5-1), and, hence, to areduction of the driving force for precipita-tion (see Sec. 5.2.4). Often only after ex-tended aging do the metastable phases a≤and b¢ transform further to the final equi-librium phases a¢ and b. The crystal struc-ture of the equilibrium phase b finally pre-cipitated is different from that of the parentphase, leading to either a coherent, a semi-coherent or a fully incoherent a¢/b interfaceboundary (Gleiter, 1983). The atomic struc-ture of the latter resembles that of a high-energy, high-angle grain boundary and, thus,is associated with a rather large interfacialenergy and small elastic strain energies.

The decomposition of Cu–Ti alloys withTi contents between ≈1 at.% and 5 at.%serves as an example of such a complex pre-cipitation sequence (Wagner et al., 1988):

a-Cu–(1 . . . 5 at.%) Ti

1st step T ≈ 350 °C

a≤-Cu–Ti (f.c.c. solid solution) +b¢-Cu4Ti (metastable, coherent, body-

centered-tetragonal structure,large coherency strains, smallinterfacial energy,sab= 0.067 J/m2)

2nd step extendedagingat T > 350 °C

a¢-Cu–Ti (f.c.c. solid solution) +b-Cu3Ti (stable, hexagonal structure,

incoherent, small strain, largeinterfacial energy,sab > 0.6 J/m2)

5.2.2 Thermodynamic Considerations –Metastability and Instability

Let us consider a binary alloy consistingof NA solvent atoms A and NB solute atomsB with NA + NB = N, or, in terms of atomicfractions, cA = NA/N and cB = NB/N, withcA + cB = 1. (The concentrations of atomsA and B are then given as cA nv and cB nv,where nv is the number of atoms per unitvolume.) As only one independent variableremains, we can refer to the composition ofthe alloy as c ∫ cB (0 ≤ c ≤ 1).

Decomposition of a supersaturated sin-gle-phase alloy into a two-phase state com-monly occurs at constant temperature Tand pressure p, and is thus prompted by apossible reduction in Gibbs energy (Gas-kell, 1983)

G = H – T S (5-4)

Figure 5-3. Time evolutionof the mean radius R

–, the

number density Nv, and the supersaturation Dc ofg¢-precipitates in Ni–36at.% Cu–9 at.% Al duringaging at 500°C (Liu andWagner, 1984).

with the enthalpy

H = U + pV (5-5)

Thermodynamic equilibrium is attainedwhen G has reached a minimum, i.e.,

dGT, p = dU + pdV – T dS = 0 (5-6)

For phase separation in solids the term p dVcan usually be neglected with respect to theothers in Eq. (5-6). Thus a good approxi-mation for G is given by the Helmholtz en-ergy

F = U – T S (5-7)

which will be used in the following sec-tions as the relevant thermodynamic func-tion. Equilibrium is achieved if F or thecorresponding Helmholtz energy density(F per unit volume or per mole) are mini-mized. Unmixing only takes place if thetransition from the single-phase state low-ers the Helmholtz energy, i.e., by conven-tion if DF < 0.

For a given temperature, volume and so-lute concentration of a heterogeneous bi-nary alloy, equilibrium between the twophases a and b can only be achieved if theconcentrations of A and B in the twophases have been established such that

(5-8)

In other words, at equilibrium the chemical

potentials of the

component i (either A or B) in the twophases are identical and the two phaseshave a common tangent to the associatedfree energy curves. This fact is illustratedin Fig. 5-4 for a supersaturated solid solu-tion of composition c0, which decomposesinto a B-depleted phases a¢ and a B-richphase b of composition ca

e and cbe , respec-

tively. cae and cb

e are fixed by the common

m mii

ii

Fn

Fn

αα

ββ

= ∂∂

= ∂∂

and

∂∂

⎛⎝⎜

⎞⎠⎟

= ∂∂

⎛

⎝⎜⎞

⎠⎟Fn

Fni T V n i T V nj j

α β

, , , ,

tangent to the Helmholtz energy curves ofthe a- and b-phases; thus cb

e – cae represents

the width of the miscibility gap at a giventemperature (cf. Fig. 5.4).

Fig. 5-5a shows schematically the phasediagram of a binary alloy with a two-phaseregion at lower temperatures for T = T1; theassociated Helmholtz energy versus com-position curve, F (c), is shown in Fig. 5-5b.In the single-phase field a, F initially de-creases with increasing solute concentra-tion due to the growing entropy of mixing(Eq. (5-7)). In the thermodynamicallyequilibrated two-phase region, F varies lin-early with c (bold straight line satisfyingthe equilibrium condition mi

a = mib).

The mean field theories (see Secs. 5.5.1and 5.5.4) dealing with the unmixing kinet-ics of solid solutions quenched into themiscibility gap are now based on the (ques-tionable) assumption that the quenched-insingle-phase states within the two-phaseregion can be described by a ‘constraint’Helmholtz energy F ′ (c) > F(c), e.g., as is


Figure 5-4. Helmholtz energy F as a function ofcomposition for a binary alloy with a miscibility gap.The changes in F and the resulting driving forces forunmixing are illustrated.


shown in Fig. 5-5b (bold dashed line).Based on this concept, the miscibility gapcan be subdivided into a metastable region,where F ′ (c) > F (c) and ∂2 F ′/∂c2 > 0, andan unstable region, for which F ′ (c) > F (c)but ∂2 F ′/∂c2 < 0. The unique spinodalcurve, which is defined as the locus of the

inflection points (∂2 F ′/∂c2 )T ∫ 0, separ-ates the two regions (Fig. 5-5a). The es-sence of the distinction between meta-stability and instability will be discussed inthe following section.

5.2.3 Decomposition Mechanisms:Nucleation and Growth versus SpinodalDecomposition

Experimentalists are often inclined todistinguish between two different kinds ofdecomposition reactions, depending onwhether i) the solid solution experiences ashallow quench (e.g., from points 0 to 1,Fig. 5-5a) into the metastable region or ii)whether it is quenched deeply into the un-stable region of the miscibility gap (e.g.,from points 0 to 2).

For case i, unmixing is initiated via theformation of energetically stable solute-rich clusters (‘nuclei’). As inferred fromFig.5-5 b) only thermal composition fluctu-ations with sufficiently large compositionalamplitudes ·c – c0Ò lower (DF < 0) theHelmholtz energy of the system and,hence, can lead to the formation of stablenuclei. According to the tangent construc-tion for the given composition c0 of the in-itial solid solution, the largest decrease inDF is obtained for a nucleus with composi-tion cp

nuc (which depends on c0) rather thanwith cp = cb

e. The latter is the compositionof the terminal second phase b coexistingin equilibrium with the a phase of compo-sition ca

e. The formation of stable nuclei vialocalized ‘heterophase’ thermal composi-tion fluctuations (Fig. 5-6 a) requires a nu-cleation barrier (typically larger than 5 kT )to be overcome (see Sec. 5.5.1) and is char-acterized by an incubation period. This de-fines the homogeneous solid solution atpoint 1 (Fig. 5-5 a) as being metastable,and the transformation as a nucleation andgrowth reaction.

Figure 5-5. a) Phase diagram of a binary model al-loy with constituents A and B. The two-phase regionis subdivided by the mean-field spinodal curve intometastable (hatched) and unstable regions (cross-hatched). b) Schematic Helmholtz energy versuscomposition curves at temperature T1. The bolddashed curve in the two-phase region shows the ‘constraint’ Helmholtz energy F ¢ (c) of the unstablesolid solution.

As will be outlined in Sec. 5.5.1, the de-cay of a metastable solid solution via nu-cleation and growth has frequently beendescribed in terms of cluster kinetics mod-els. The cluster kinetics approaches are es-sentially based on the Becker–Döring the-ory (Becker and Döring, 1935) of the dy-namics of solute cluster formation. There itis assumed that the non-equilibrium systemconsists of non-interacting solute-rich clus-ters of various size embedded in the matrix.

The time evolution of the cluster size dis-tribution and, hence, the dynamics of thedecay of the metastable alloy, are assumedto proceed via the condensation or evapo-ration of single solute atoms at each clus-ter.

In case ii, the non-equilibrium solid so-lution with an initial composition c0 > ca

s ,(Fig. 5-5 b) is unstable with respect to theformation of non-localized, spatially ex-tended thermal composition fluctuationswith small amplitudes. Hence, the unmix-ing reaction of an unstable solid solution,which is termed spinodal decomposition, isinitiated via the spontaneous formation and subsequent growth of coherent (‘homophase’) composition fluctuations(Fig. 5-6b).

The dynamic behavior of an unstable al-loy proceeding towards equilibrium has of-ten been theoretically approached in termsof spinodal theories (see Sec. 5.5.4 andChapter 6), amongst which the most well-known linear theory is due to Cahn (1966)and the most elaborate non-linear one isdue to Langer et al. (1975). As discussed inSec. 5.2.2, the spinodal theories are basedon the assumption that in each stage of thedecomposition reaction the Helmholtz en-ery of the non-equilibrium solid solution,which contains compositional fluctuations,can be defined. As in the cluster kineticsmodels, the driving force for phase separa-tion is again provided by lowering theHelmholtz energy of the alloy. If the formof the ‘constraint’ Helmholtz energy F¢ (c)of the non-uniform system is properly cho-sen, the composition profile with asso-ciated minimum Helmholtz energy can bedetermined (in principle!) at any instant ofthe phase transformation.

The existence of a unique spinodal curvewithin the framework of the mean fieldtheories (Cahn, 1966; Cook, 1970; Skripovand Skripov, 1979) has led to the idea (still


Figure 5-6. Spatial variation in solute distributionc (r = (x, y, z)) during a) a nucleation and growth re-action, and b) a continuous spinodal reaction at thebeginning (time t1) and towards the end (t3) of theunmixing reaction. The notation of the compositionsrefers to Fig. 5-5; c0

ms and c0us are the nominal com-

positions of the quenched-in metastable and unstablesolid solutions, respectively; R* is the critical radiusof the nuclei and l the wavelength of the compositionfluctuations. The direction of the solute flux is indi-cated by the arrows. After an extended reaction time(e.g., after t3), the transformation products are simi-lar and do not allow any conclusions to be drawnwith respect to the early decomposition mode.


widespread in the community of metallur-gists) that there is a discontinuity of themechanism and, in particular, of the de-composition kinetics at the boundarybetween the metastable and unstable re-gions. Therefore, many experiments havebeen carried out in order to determine thespinodal curve and to search for a kineticdistinction between metastable and un-stable states (see Sec. 5.5.5).

In reality, there is no need to develop dy-namical concepts which are confined to ei-ther the metastable nucleation and growthregime (case i) or to the unstable spinodalregime (case ii). In fact, the cluster-kineticsmodels and the spinodal theories can beseen as two different approaches used todescribe phase separation, the dynamics ofwhich are controlled by the same mecha-nism. i.e. diffusion of solvent and solute at-oms driven by the gradient of the chemicalpotential (Martin, 1978).

This fact is reflected in the attempts todevelop ‘unified theories’ which comprisespinodal decomposition as well as nuclea-tion and growth. Langer et al. (1975) triedto develop such a theory on the basis of anon-linear spinodal theory (see Sec. 5.5.4),whereas Binder and coworkers (Binder etal., 1978; Mirold and Binder, 1977) chosethe cluster kinetics approach by treating spi-nodal decomposition in the form of a gener-alized nucleation theory (see Sec. 5.5.3).These theories involve several assumptionswhose validity is difficult to assess a priori.The quality of all spinodal or cluster kineticsconcepts, however, can be scrutinized byMonte Carlo simulations of the unmixingkinetics of binary ‘model’ alloys. These arequenched into either a metastable or an un-stable state (see Sec. 5.5.6) and can be de-scribed in terms of an Ising model (Kalos etal., 1978; Penrose et al., 1978).

Although both the ‘unified theories’ andthe Monte Carlo simulations only provide

qualitative insight into early-stage unmix-ing behavior of a binary alloy, they have re-vealed that there is no discontinuity in thedecomposition kinetics to be expected dur-ing crossing of the mean-field spinodalcurve by either increasing the concentra-tion of the alloy and keeping the reactiontemperature (TA) constant or by loweringthe reaction temperature and keeping thecomposition (c0) constant. On the otherhand, the mean-field description is strictlyvalid only for systems with infinitely long-range interaction forces (Gunton, 1984)and, hence, in general does not apply tometallic alloys (polymer mixtures might beclose to the mean-field limit: Binder, 1983,1984; Izumitani and Hashimoto, 1985; seeSec. 5.5.4). Therefore, numerous experi-ments that have been designed by metallur-gists in order to determine a unique spino-dal curve simply by searching for drasticchanges in the dynamic behavior of an al-loy quenched into the vicinity of the mean-field spinodal must be considered withsome reservations.

In principle, the above-mentioned theo-ries and, in particular, the Monte Carlosimulations deal mainly with the dynamicevolution of a two-phase mixture in itsearly stages. They frequently do not ac-count for a further evolution of the precipi-tate or cluster size distribution with agingtime (i.e., coarsening; see Sec. 5.6) oncethe precipitated volume fraction is close toits equilibrium value. On the other hand,the time evolution of the precipitate micro-structure beyond its initial clustering stageshas been the subject of many experimentalstudies and is of major interest in practicalmetallurgy. For this purpose, numerical ap-proaches have been devised (Langer andSchwartz, 1980; Kampmann and Wagner,1984) which treat nucleation, growth andcoarsening as concomitant processes andthus allow the dynamic evolution of the

two-phase microstructure to be computedduring the entire course of a precipitationreaction (see Sec. 5.7).

As was pointed out in Sec 5.2.1, the re-action path of a supersaturated solid solu-tion can be rather complex, sometimes in-volving the formation of one or more inter-mediate non-equilibrium phases prior toreaching the equilibrium two-phase micro-structure. Unlike in the ‘early stagetheories’ mentioned above, these compli-cations, which are of practical relevance,can be taken into consideration in numeri-cal approaches. Even though they still con-tain a few shortcomings, numerical ap-proaches lead to a practical description ofthe kinetic course of a precipitation reac-tion which lies closest to reality.

There are several comprehensive reviewarticles and books dealing in a more gen-eral manner with the kinetics of first-orderphase transitions (Gunton and Droz, 1984;Gunton et al., 1983; Binder, 1987; Gunton,1984; Penrose and Lebowitz, 1979). Phaseseparation in solids (crystalline and amor-phous alloys, polymer blends, oxides andoxide glasses) via homogeneous nucleationand growth or via spinodal decompositionrepresents only one aspect among manyothers (Gunton and Droz, 1984). Apartfrom the above-mentioned numerical ap-proaches, the comprehensive articles byMartin (1978) and by de Fontaine (1982)cover most of the theoretical developmentsrelevant to the kinetics of (homogeneous)phase separation in metallic systems. Ageneral overview of the broad field of dif-fusive phase transformations in materialsscience including heterogeneous nuclea-tion and discontinuous precipitation notcovered in the present chapter, can befound in the article by Doherty (1983) or,as an introduction to this field, in the bookby Christian (1975), see also chapter byPurdy and Bréchet (2001).

5.2.4 Thermodynamic Driving Forcesfor Phase Separation

Even in the single-phase equilibriumstate, the mobility of the solvent and soluteatoms at elevated temperatures permits theformation of composition fluctuationswhich grow and decay again reversiblywith time. If the solid solution is quenchedinto the miscibility gap the two-phase mix-ture is the more stable state and, thus, someof these fluctuations grow irreversibly ow-ing to the associated reduction in Helm-holtz energy. The reduction in Helmholtzenergy during the transformation from theinitial to the final state provides the drivingforce DF. As we shall see in Sec. 5.5.1, it ispossible to calculate the formation rate andthe size of stable composition fluctuations(‘nuclei’ of the second phase) by means ofthe cluster kinetics approach once DF isknown. It should, however, be emphasizedat this point that for most alloys it is ratherdifficult to calculate DF with sufficient ac-curacy. This is seen as one of the majorhindrances to performing a quantitativecomparison between theory and experi-ment.

The driving force for precipitation ismade up of two different contributions:

i) the gain in chemical Helmholtz en-ergy, DFch < 0, associated with the forma-tion of a unit volume of the precipitatingphase b, and

ii) the expenditure of distortion Helm-holtz energy, DFel > 0, accounting for thecoherency strains which result from alikely variation of the lattice parameterwith the spatial composition fluctuations.

i) Chemical contribution, D Fch. Accord-ing to the tangent rule and referring to Fig.5-4, the chemical driving force for precipi-tation of the equilibrium b phase out of asolid solution with composition c0 is givenby the numerical value of xy––. Assuming the



precipitating phase b already has the finalbulk composition cb

e ∫ cp rather than cpnuc

(which is only a reasonable assumption ifthe supersaturation is not too large (Cahnand Hilliard, 1959a, b)), DFch has been de-rived for a unit volume of b phase with mo-lar volume Vb as (Aaronson et al., 1970 a,b):

(5-9)

where ai (c) is the activity of the solvent (i = A) or solute component (i = B) in theparent phase a for the given composition.

For most alloy systems the activity datarequired for a computation of DFch bymeans of Eq. (5-9) are not available. Inprinciple, they can be derived from a com-putation of the thermodynamic functionsby means of the CALPHAD method. Orig-inally this method was developed for thecalculation of phase diagrams by Kaufmanand coworkers (Kaufman and Bernstein,1970) on the basis of a few accuratelymeasured thermodynamic data to whichsuitable expressions for the thermodynam-ic functions had been fitted. For the sake ofa simplified mathematical description, thestable solid solutions of the particular alloysystem are frequently described in terms ofthe regular solution model, whereas thephase fields of the intermetallic com-pounds are approximated by line com-pounds. The thermodynamic functions ob-tained thereby are then used to reconstructthe phase diagram (or part of it). The de-gree of self-consistency between the recon-structed and the experimentally determinedphase diagram (or the agreement betweenthe measured thermodynamic data and thederived data) serves as a measure of the ac-curacy of the thermodynamic functions.

DFR T

V

ca c

a cc

a c

a c

chg

e B

Be

e A

Ae

=

× +⎡

⎣⎢

⎤

⎦⎥

–

ln( )

( )( – )ln

( )

( )

β

βα

βα

0 01

Hence, if, for example, F (c, T ) is knownfor the stable solid solution, this value canreadily be extrapolated into the adjacenttwo-phase region to yield F¢ (c) (see Fig.5-5) of the supersaturated homogeneoussolid solution or the related chemical po-tentials and activities. Of course, physi-cally this is only meaningful if we assume(as, in fact is done for the derivation of Eq.(5-9)) that the free energy of the non-equi-librium solid solution can be properly de-fined within the miscibility gap.

For many binary alloys and for some ter-nary alloys of technological significance,the thermodynamic functions have beenevaluated by means of the CALPHADmethod and are compiled in volumes of theCALPHAD series (Kaufman, 1977).

Hitherto, due to the lack of available ac-tivity data, the activities entering Eq. (5-9)were frequently replaced by concentra-tions, i.e.,

(5-10)

If b is almost pure B, then cbe ≈ 1 and the

chemical driving force is approximated as

(5-11)

For many alloys the condition cbe ≈ 1 is

not met. Nevertheless, it has been used forcomputation of DF. As discussed by Aa-ronson et al. (1970 a), the resulting errorcan be rather large. If the nucleating phaseis a solvent-rich intermetallic compound(e.g., Ni3Al or Cu4Ti), the computation ofDFch is particularly difficult, since up tonow the corresponding activities are nei-ther known nor furnished by the CAL-PHAD method, and Eqs. (5-10) and (5-11)are no longer valid.

DFR T

Vc

cch

ge≈ – ln

β α

0

DFR T

V

cc

cc

c

c

chg

ee

ee

=

× +⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

–

ln ( – ) ln–

–

β

βα

βα

0 011

1

ii) Reduction of the driving force by theelastic strain energy, D Fel. The formationof a composition fluctuation is associatedwith the expenditure of elastic strain Helm-holtz energy, DFel , if the solvent and the so-lute atoms have different atomic radii. Ac-cording to Cahn (1962), the Helmholtz en-ergy of a system containing homophasefluctuations is raised by

DFel = h2 Dc2 Y (5-12)

where h = (1/a0) (∂a/∂c) denotes the changein lattice parameter (a0 for the homogene-ous solid solution of composition c0) withcomposition, and Dc the composition am-plitude. Y is a combination of elastic con-stants and depends on the crystallographicdirection of the composition modulation. Itreduces to E/(1 – v), E and v being Young’smodulus and Poisson’s ratio, respectively,if the elastic anisotropy A ∫ 2 c44 + c12 –c11 is zero; otherwise, in order to minimizeDFel , the composition fluctuations are ex-pected to grow along the elastically soft di-rections, which for cubic crystals and A > 0are the ·100Ò directions.

In the context of heterophase fluctuations,the barrier against nucleation of the newphase is dominated by the matrix/ nucleusinterfacial energy (see Sec. 5.5.1). Since co-herent interfaces have a lower energy thanincoherent ones, a precipitate is usually co-herent (or at least semi-coherent) during theearly stages of nucleation and growth. Oftenits lattice parameter, ab, is slightly differentfrom that of the parent phase, aa. The result-ing misfit between the (unstrained) matrixand the (unstrained) precipitate:

(5-13)

can be accommodated by an elastic strain if both d and the particles are sufficientlysmall; this is commonly the case during theearly stages of nucleation and growth.

d =+

2a a

a aα β

α β

–

The problem of calculating DFel for co-herent inclusions is rather complex. It hasgenerally been treated within the frame-work of isotropic elastic theory for ellip-soidal precipitates with varying axial ra-tios, e.g. Eshelby (1957), and has been ex-tended to the anisotropic case (Lee andJohnson, 1982). Basically, these treatmentsshow that the strain energy depends on theparticular shape of the new phase. Only ifthe transformation strains are purely dilata-tional and if a and b have about the sameelastic constants will DFel become inde-pendent of the shape. It is then given as(Eshelby, 1957):

(5-14)

with g being the shear modulus.As is shown schematically in Fig. 5-7,

with respect to the Helmholtz energy of theincoherent equilibrium phase b, the Helm-holtz energy of the coherent phase b¢ israised by the elastic energy to match thetwo lattices. The driving force DF inc ∫ xy––

DFel = +⎛⎝

⎞⎠2

11

2g nn

d–


Figure 5-7. Helmholtz energy curves and associateddriving forces for precipitates of the coherent meta-stable (Fb¢) and incoherent equilibrium (Fb) phases.


is reduced to the value of DF coh ∫ uv–– , andthe solubility limit in the a phase increasesfrom ca

e to c¢ae.

Hence, if we refer to the resulting coher-ent phase diagram (cf. Fig. 5-1), the totaldriving force, DF , for coherent unmixing,implicitly already containing the elasticcontribution, is obtained by replacing, e.g.,in Eq. (5-11), the composition ca

e of the in-coherent phase by the composition c¢a

e ofthe coherent one:

(5-15)

As an example, for metastable iron-richf.c.c. precipitates (cb

e ≈ 99.9 at.%) withd = – 8 ¥ 10–3 formed at 500 °C in a super-saturated Cu–1.15 at.% Fe alloy (c0 = 1.15at.%, (ca

e ≈ 0,05 at.%; Kampmann and Wag-ner, 1986), Eq. (5-12) yields DFel≈0.13kJ/mol. This value is negligibly small withrespect to the rather large chemical drivingforce (≈20 kJ/mol). Since, however, DFch

decreases markedly with increasing tem-perature (or decreasing supersaturation),whereas DFel does not, coherent nucleationcommonly occurs at larger undercoolingsor supersaturations, whereas nucleation ofincoherent precipitates takes place atsmaller ones. This is in fact observed in theCu–Ti (see Sec. 5.2.1) and Al–Cu (Hornbo-gen, 1967) systems.

As will be shown in Sec. 5.5.1.1, the bar-rier DF* against formation of the newphase, and hence the nucleation rate, is notonly a function of the driving force but israther sensitive to the nucleus/matrix inter-face energy sab. Therefore, the phase nu-cleating first will not necessarily be theequilibrium phase with the lowest Helm-holtz energy but that with the lowest DF*,e.g. a coherent metastable phase with a low value of sab. This explains the likelyformation of a series of metastable phases(see Sec. 5.2.1) in various decomposing al-

Da

FR T

Vc

c=

′′– lng

eβ

0

loys, amongst which Al–Cu (c0 ≤ 2.5 at.%Cu) is probably the best known. There thestable phase (q-CuAl2) is incoherent, witha high associated sab (R 1 J/m2) (Hornbo-gen, 1967). This inhibits homogeneous nu-cleation and q is found to form only atsmall undercoolings, preferentially at grainboundaries (see Fig. 5-8). At larger under-coolings, a series of metastable copper-richprecipitates is formed in the order: GPIzones Æ GPII zones (q≤) Æ q¢ with differ-ent crystal structures (see Fig. 5-8). Gui-nier–Preston zones of type I (GPI) and typeII (GPII or q≤) are coherent and nucleatehomogeneously, whereas q¢ is semi-coher-ent and nucleates preferentially at disloca-tions.

Al–Mg–Si alloys represent an importantgroup of age-hardenable structural materi-als heavily used by industry in both castand wrought form. Hence, control and opti-mization of the precipitate microstructureduring cooling and heat treatment is of par-

Figure 5-8. Solubility limits of Cu in Al in the pres-ence of the metastable q≤ and q¢ phases (dashed) andthe stable q phase, as a function of temperature (afterHornbogen, 1967).

amount interest for optimum mechanicalproperties (Bratland et al., 1997). How-ever, due to its inherent complexity, theprecipitation sequence of these commercialalloys is still a matter of controversy (Duttaand Allen, 1991; Gupta and Lloyd, 1992;Edwards et al., 1998).

5.3 Experimental Techniques forStudying Decomposition Kinetics

5.3.1 Microanalytical Tools

In general, the course of a decomposi-tion reaction, including the early stages(during which composition fluctuationsand second-phase nuclei are formed, seeSec. 5.2.1) and the coarsening stages, can-not be followed continuously by any onemicroanalytical technique. The progress ofthe reaction is usually reconstructed fromthe microstructure that develops at variousstages of the phase transformation. Thus itis necessary to analyze the spatial exten-sion and the amplitude of compositionfluctuations of incipient second-phase par-ticles, as well as the morphology, numberdensity, size and chemical composition ofindividual precipitates at various stages ofthe phase transformation. For this purposemicroanalytical tools are required that arecapable of resolving very small (typically afew nm) solute clusters, and which allow(frequently simultaneously) an analysis oftheir chemical composition to be made.

The tools that, in principle, meet theserequirements can be subdivided into twogroups: direct imaging techniques andscattering techniques.

It is beyond the scope of this chapter todiscuss any one of the techniques belong-ing to either group in any detail. We shallonly briefly summarize the merits and theshortcomings of the various techniques

with respect to both the detection limit andthe spatial resolution of microanalysis.

5.3.1.1 Direct Imaging Techniques

Field ion microscopy (FIM) (Wagner,1982) as well as conventional (CTEM) (Hobbs et al., 1986) and high resolution (HREM) transmission electron microscopy(Smith, 1983) allow for direct imaging ofthe second-phase particles, provided thecontrast between precipitate and matrix issufficient.

In CTEM both the bright field and thedark field contrast of particles less than ≈ 5 nm in diameter are often either tooweak or too blurred for an accurate quanti-tative determination of the relevant structu-ral precipitate parameters. Hence, CTEMdoes not provide access to an experimentalinvestigation of the early stages of decom-position but remains a technique for study-ing the later stages. In contrast, HREM al-lows solute clusters of less than 1 nm diam-eter to be imaged, as was demonstrated forNi3Al precipitates in Ni–12 at.% (Si-QunXiao, 1989) and for silver-rich particles inAl–1 at.% Ag (Ernst and Haasen, 1988;Ernst et al., 1987). As is shown in Fig. 5-9for the latter alloy, the particles are notideally spherical but show some irregular-ities which, however, are small in compari-son with their overall dimensions. Hence,describing their shapes by spheres of radiusR is still a rather good approximation.

Prior to measuring the particle sizes di-rectly from the HREM micrographs, itmust be established via computer imagesimulations that there exists a one-to-onecorrespondence between the width of theprecipitate contrast (dark area in Fig. 5-9)and its true size. This was verified for theHREM imaging conditions used in Fig.5.9. The evolution of the size distributionin Al–1 at.% Ag with aging time at 413 K


5.3 Experimental Techniques for Studying Decomposition Kinetics 327

as derived from HREM micrographs isshown in Fig. 5-10.

The HREM lattice imaging techniquehas also been employed to determine thespacings of adjacent lattice planes in vari-ous alloys undergoing phase separation(Sinclair and Thomas, 1974; Gronsky etal., 1975). The smooth variations observedwere attributed to variations in lattice pa-rameter caused by composition modula-tions such as we would expect from spino-dal decomposition (e.g. Fig. 5-6 b at t1 ort2). Subsequent model calculations of high-resolution images, however, showed thatthe spatial modulation of the lattice fringesobserved on the HREM image can be sig-nificantly different from the spatial mod-ulation of the lattice plane spacings in thespecimen (Cockayne and Gronsky, 1981).In practice, it does not appear possible toderive reliable information on compositionmodulations from modulations of latticefringe spacings in HREM images.

The composition of the imaged particlescan, in principle, be obtained from analyti-cal electron microscopy (AEM), which isroutinely based on energy dispersive X-rayanalysis (EDX) or, less frequently, on elec-tron energy loss spectroscopy (EELS)(Williams and Carter, 1996). Due to beamspreading by electron beam/specimeninteraction, the spatial resolution for rou-tine EDX microanalysis is limited to R10 nm. This value is rather large and con-fines EDX microanalysis of unmixing al-loys to the later stages of precipitation. Themore interesting early stages of decom-position, however, where compositionchanges of either extended or localized so-lute fluctuations are expected to occur, arenot amenable to EDX microanalysis.

The spatial resolution of EELS may,with difficulty, reach 1 nm (Williams andCarter, 1996), but so far, EELS has notbeen employed for a systematic microanal-

Figure 5-9. HREM lattice fringe image of Al-1 at.%Ag aged for 92 h at 413 K with silver-rich precipi-tates (dark areas). For imaging the Al matrix a ratherlarge defocus ( 280 nm) had to be chosen, whichgives rise to the bright Fresnel fringes surroundingthe particles (reproduced by courtesy of F. Ernst (Ernst and Haasen, 1988)).

Figure 5-10. Time evolution of the particle size dis-tribution in Al–1 at.% Ag during aging at 413 K (af-ter Ernst and Haasen, 1988).

ysis of composition fluctuations duringearly-stage decomposition.

Analytical field ion microscopy (AFIM)(Wagner, 1982; Miller et al., 1996) can fa-vorably be applied for an analysis of ultra-fine solute clusters in the early stages ofdecomposition, as both the imaging resolu-tion of the FIM (see Fig. 5-2) and the spatialresolution of microanalysis of the integratedtime-of-flight spectrometer (‘atom probe’)are sufficient, i.e., <1 nm and ª 2 nm, re-spectively. (Although the atom probe de-tects single atoms from the probed volume,the requirement for the analysis to be sta-tistically significant confines the microana-lytical spatial resolution to ≥2 nm.) As isshown in Figs. 5-2a–c and 5-19c, both themorphologies of small particles and theirthree-dimensional arrangements can alsobe determined in the FIM, at least for pre-cipitates with sufficient contrast.

The volume sampled during an atomprobe FIM analysis is rather small (typi-cally about 200 nm3). Hence, in order toobtain statistically significant data con-cerning the average size (R) and the num-ber density (Nv) of the precipitates, the lat-ter ought to exceed 1023 m–3. Since afterthe early stages of precipitation Nv de-crease with time (see Sec. 5.2.1), AFIMshows its full potential as a microanalyticaltool in studies of the early stages of precip-itation, during which R is commonly smalland Nv sufficiently large (Haasen and Wag-ner, 1985).

With respect to analyses of the spatial ar-rangement and the composition of precipi-tated phases, the versatility of the AFIMwas considerably improved by the devel-opment of the tomographic atom probe(TAP) (Blavette et al., 1993, 1998; Milleret al., 1996). This instrument allows forthe three-dimensional reconstruction of asmall volume of the microscopically heter-ogeneous material on a sub-nanometric

scale (Fig. 5-2d). The spatial distributionof each chemical species can be directlyobserved with a spatial resolution betterthan 0.5 nm (Auger et al., 1995; Pareige etal., 1999; Al-Kassab et al., 1997). The vol-ume typically sampled is about 15 × 15× 100 nm3 and, thus, about a hundred timeslarger than the volume analyzed with theAFIM.

5.3.1.2 Scattering Techniques

The time evolution of the structure ofsupersaturated alloys, as well as of oxidesand polymer blends undergoing phase sep-aration, can be analyzed by means of smallangle scattering of X-rays (SAXS), neu-trons (SANS), and light. Light scattering isof course confined to transparent speci-mens, in which the domain sizes of theevolving second phase must be of the orderof ≈ 1 µm. It has been successfully appliedin studies of decomposition, e.g., of a poly-mer mixture of polyvinyl methyl ether (PVME) and polystyrene (PS) (Snyder andMeakin 1983a, b, 1985), and of variousglass-forming oxide systems (Goldstein,1965; Rindtone, 1975).

In principle, SAXS (Glatter, 1982) andSANS (Kostorz, 1979; Sequeira et al.,1995) provide access to a structural analy-sis of unmixing alloys, in both the earlystages where the composition fluctuationscan be small in spatial extension and in am-plitude, as well as in the later stages of de-composition where R and Nv may have at-tained values which are quite unfavorablefor a quantitative analysis by TEM or FIM.In addition, evaluation of the Laue scatter-ing allows the remaining supersaturation ofsolute atoms in the matrix to be deter-mined.

The metallurgist is left with the problemof extracting in a quantitative manner allthe structural data contained in scattering



curves such as are shown in Fig. 5-11. Thisis often not trivial (Glatter, 1982), in partic-ular for concentrated alloys (i.e., for mosttechnical two-phase alloys). Here the scat-tering curve reveals a maximum (at posi-tion km and with height Sm (km) in Fig.5-11) which results from an interparticleinterference of the scattered waves (Kos-torz, 1979). In general, the interparticleinterference function is not known, thusimpeding a straightforward quantitativeanalysis. Moreover, the composition andmorphology of the fluctuations are re-quired in order to perform such an analysis.For many alloys this information can onlybe obtained from AFIM or AEM. Thesecomplicating factors often demand theabove-mentioned microanalytical tools tobe employed jointly, rendering experimen-tal studies of decomposition rather difficultand tedious.

The small angle scattering intensity isproportional to the structure function S (k,t); k is the scattering vector with k ∫ |k | =4 p sin (q /2)/l; l is the wavelength of theX-rays or neutrons and q the scattering an-

gle. For binary alloys S (k, t) is the Fouriertransform of the two-point correlationfunction at time t (Langer, 1975):

(5-16)G (| r– r0| , t) = ·[c (r, t) – c0] [c (r0, t)– c0]Ò

The right-hand side in Eq. (5-16) denotesthe non-equilibrium average of the productof the composition amplitudes at two dif-ferent spatial positions, r and r0, in the al-loy with average composition c0.

S (k, t) contains all the structural infor-mation on the phase-separating system.Many theories and computer simulationsdealing with phase separation yield the ev-olution of S (k, t) and predict the shift ofkm (which is a measure of the averagespacing of solute clusters) and the growthof Sm (km) with time (Langer et al., 1975;Marro et al., 1975, 1977; Binder et al.,1978).

Hence, SAXS and SANS curves meas-ured after different aging times can be di-rectly compared with the predictions fromvarious theoretical kinetic concepts (seeSecs. 5.5.4, 5.5.6 and 5.8.1).

In ternary systems the situation is muchmore complicated. For a substitutional ter-nary alloy, there are three linearly indepen-dent pair correlation functions with threerelated partial structure functions (de Fon-taine, 1971, 1973), which linearly combineto the measured SAS intensity. An unam-biguous characterization of the kinetics ofphase separation in ternary substitutionalalloys requires the three partial structurefunctions to be determined separately. Thishas been attempted by employing the‘anomalous small angle X-ray scattering’technique for investigations of phase separ-ation in Al–Zn–Ag (Hoyt et al., 1987),Cu–Ni–Fe (Lyon and Simon, 1987, 1988)and Fe–Cr–Co (Simon and Lyon, 1989).

Unlike for X-rays, for thermal neutronsthe atomic nuclear scattering length is not

Figure 5-11. SANS curves of Cu-2.9 at.% Ti singlecrystals aged for the given times at 350°C. Note thateven the homogenized and quenched (‘hom’) speci-men yields SANS intensity, indicating that phaseseparation has occurred during quenching (Eckerlebeet al., 1986).

monotonically dependent on the atomicnumber (Kostorz, 1979). For this reasonuse of SANS is more universal and is fre-quently superior ro SAXS for decomposi-tion studies of binary alloys with mainlytransition metal constituents having similaratomic numbers. Consequently, SAXS hasonly been used extensively for studies ofthe unmixing kinetics in Al–Zn where thedifference (this, in fact, controls the con-trast in SAS) in atomic scattering lengthsfor X-rays is sufficiently large (Rundmanand Hilliard, 1967; Hennion et al., 1982;Forouhi and de Fontaine, 1987).

Furthermore, if the two phases differ notonly in composition but also in magnetiza-tion, both nuclear and magnetic SANScurves can be recorded (see Fig. 5-12).These two independently measurableSANS curves sometimes even allow thecomposition of the scattering centers to bedetermined, e.g., for Fe–Cu, where dia-magnetic copper-rich particles precipitatein the ferromagnetic a-Fe matrix (Kamp-mann and Wagner, 1986), or for phase-sep-arated amorphous Fe40Ni40P20 (Gerling etat., 1988).

Due to relatively simple calibration pro-cedures in SANS experiments, the scat-tered intensity which is expressed in termsof the coherent (nuclear or magnetic) scat-tering cross-section per unit volume,dS (k, t)/dW,can be measured in absoluteunits. This is directly related to the struc-ture function S (k, t) via (Hennion et al.,1982):

(5-17)

where b–

M and b–

p are the locally averaged(nuclear or magnetic) scattering lengths ofthe matrix and of the solute-rich clusters,respectively, and W

–is the mean atomic vol-

ume.

5.3.2 Experimental Problems

5.3.2.1 Influence of Quenching Rate on Kinetics

For studies on decomposition kineticsthe alloy is commonly homogenized in the single-phase region at TH (Fig. 5-1),quenched into brine and subsequently iso-thermally aged at TA. In order to capturethe initial stages of the decay of the super-saturated solid solution, both the quench-ing and the heating rate to TA are requiredto be sufficiently high in order to avoidphase separation prior to isothermal aging.This can be comfortably achieved in alloyswith small supersaturations. It is, however,often a problem (or even impossible) foralloys with large supersaturations and/orsmall interfacial energies, where the nucle-ation barriers are small (see Secs. 5.2.3 and5.5.1) and, hence, the nucleation rateslarge.

The driving force for unmixing in Al-1at.% Ag at 140°C is rather small(≈0.5 kJ/mol). Nevertheless, because of theextremely small interfacial energy of only

dd M p

SW W

( , ) ( – ) ( , )kk kkt b b S t= 12

2


Figure 5-12. Nuclear and magnetic SANS curves ofFe–1.4 at.% Cu aged for 200 h at 400°C. From theratio of magnetic and nuclear scattering intensities,the composition of the copper-rich clusters can be de-rived. Theoretical curves are shown as full lines(Kampmann and Wagner, 1986).


≈0.01 J/m2 (Le Goues et al., 1984c), thebarrier to nucleation is so low that precipi-tation commences instantaneously almostas soon as the solvus line is crossed. Thus,except for extremely high quenching rates,the non-equilibrium single-phase state can-not be frozen in and the as-quenched mi-crostructure already contains a large num-ber of small GP zones (Fig. 5-13).

The accessible quenching rates havebeen found to be insufficient for a suppres-sion of phase separation during quenchingof homogenized Cu–Ti alloys with Ti con-centrations exceeding ≈2.5 at.%. This isdiscernible from the SANS curve of the as-quenched specimen (Fig. 5-11) (see Sec.5.7.5).

Further problems result from the factthat the decomposition kinetics at TA arestrongly influenced by the concentration ofquenched-in excess vacancies. This de-pends strongly on the chosen homogeniza-tion temperature and quenching rate.

Fig. 5-14 shows SANS curves fromspecimens that were solution-treated at thegiven homogenization temperature TH.Following quenching, each specimen was

aged for 10 min at 350°C. The maximumin the SANS curves is found to be higher,and its position lower, the higher the ho-mogenization temperature. This indicatesthat phase separation in the early stagesprogresses the faster the higher the homog-enization temperature is chosen. Obviously,in Cu–Ti, the concentration of quenched-inexcess vacancies is correlated to TH andsignificantly influences the decomposition

Figure 5-13. HREM micrograph of GP zonesformed during quenching of Al–1 at.% Ag. Thesmallest GP zones have diameters of only 1 nm(reproduced by courtesy of P. Wilbrandt (Ernst et al.,1987)).

Figure 5-14. SANS curvesof Cu–2.9 at.% Ti quenchedfrom the given homogeniza-tion temperatures TH, and(apart from the bottomcurve) subsequently agedfor 10 min at 350°C (Ecker-lebe et al. 1986).

kinetics. This effect may be minimized ifthe alloy is first solution-treated at high TH

and subsequently equilibrated at a homog-enization temperature slightly above thesolvus temperature prior to quenching.

The inconsistency in the exact mode andkinetics of decomposition in Al–22 at.%Zn, investigated by different authors(Rundman and Hilliard, 1967; Gerold andMerz, 1967), has been explained by thepresence of different quenching rateswhich unavoidably lead to different statesof solute clustering in the as-quenchedspecimens (Agarwal and Herman, 1973;Bartel and Rundman, 1975).

The problems stated above make experi-ments devised for examination of the vari-ous spinodal theories particularly difficult.

5.3.2.2 Distinction of the Modeof Decomposition

The criterion that must be satisfied in or-der to distinguish spinodal decomposition(s.d.) from a nucleation and growth (n.g.)reaction is to prove by any microanalyticaltechnique that the amplitude of the compo-sition modulations of an alloy deeplyquenched into the miscibility gap increaseswith time during the initial stages of phaseseparation (see Sec. 5.2.2 and Fig. 5-6).Even apart from the problem outlined inthe preceding section, this is a rather diffi-cult task, since the cluster diameters or themodulation wavelengths in most of themore concentrated alloys investigated sofar have been found to range below the resolution limit of composition analysis,e.g. ≈2 nm for AFIM. Figure 5-15 revealsthat the compositions of the solute clustersin Ni–36 at.% Cu–9 at.% Al (Liu and Wagner, 1984) and in a hard-magneticFe–29 at.% Cr–24 at.% Co (Zhu et al.,1986) alloy have reached their equilibriumvalues and, hence, remain constant once

the clusters have attained sizes that are ac-cessible to chemical analysis by AFIM(3 nm and 1.8 nm, respectively). (In factboth ternary alloys may be regarded aspseudobinary. Thus the thermodynamicconsiderations of Sec. 5.2.2 still apply.)The observed features suggest the reactionin both alloys to be of the n.g. type. How-ever, it cannot be ruled out that these clus-


Figure 5-15. a) Composition and diameter D–

, of g¢-precipitates in Ni–36 at.% Cu–9 at.% Al as a func-tion of aging time at 580°C. The corresponding pre-cipitate microstructure is shown in Fig. 5-2a–c (Liuand Wagner, 1984). b) Composition and diameterr D

–

of the a1 (matrix) and a2 (precipitate ) phases inFe–29 at.% Cr–24 at.% Co as a function of agingtime at 640°C (Zhu et al., 1986).


ters result from a s.d. reaction which wasterminated after even shorter aging timesthan could be covered in these studies.

Fig. 5-16 shows the interconnected pre-cipitate microstructure of another hard-magnetic alloy, Fe–29 at.% Cr–14 at.%Co–21 at.% Al–0.15 at.% Zr, aged to yieldits optimum magnetic properties. In orderto characterize the decomposition processwith respect to the distribution of Crbetween the two phases, atom probe micro-analysis was performed.

The measured Cr concentration of thedarkly imaging precipitating a2-phase as afunction of aging time at T = 525 °C and T = 600°C is shown in Fig. 5-17a. Duringaging at 600°C for less than 5 min it in-creases continuously before it attains itsconstant equilibrium value. This behaviorindicates unequivocally a spinodal decom-position mechanism. At the lower agingtemperature of TA = 525°C, the Cr ampli-tude grows more slowly deeper within themiscibility gap due to the slower diffusion,but ultimately reaches higher equilibriumconcentrations in the a2-precipitates, thusshowing the spinodal behavior even moreclearly. The observed increase in Cr con-

centration in the a2-phase with decreasingaging temperature is due to the widening ofthe miscibility gap. As the Cr concentra-tion in the a2-phase increases, the propor-tions of Co and Fe are reduced in this phaseand are correspondingly increased in thea1-phase. Atom probe analysis yields aFe/Co ratio of 3 : 1 for both the a1- and a2-phases. This ratio corresponds to the tie-line in the ternary system along which

Figure 5-16. Field ion image of Fe–29 Cr–14 Co–2Al–0.15 Zr (at.%) aged into its optimum magneticstate (Zhu et al., 1986).

Figure 5-17. a) Evolution of the Cr concentration at525°C and 600°C in the Cr-rich a2-phase as a func-tion of aging time. b) Composition of the Fe-rich (a1)and Cr-rich (a2) phase after aging at 525°C for thegiven times (in minutes). The dashed line corre-sponds to the tie-line with a Fe/Co ratio of 3/1 (Zhuet al., 1986).

spinodal decomposition proceeds. Fig.5-17b shows the compositions of the iron-rich and chromium-rich phases for variousaging times at 525°C and the tie-line corre-sponding to an Fe/Co ratio of 3 : 1.

This study is one of the very rare ones onmetallic alloys to have identified a decom-position reaction unequivocally to be of thespinodal type.

Frequently, the occurrence of eitherquasi-periodically aligned precipitateswhich give rise to side-bands in X-ray orTEM diffraction patterns (Fig. 5-18) or of aprecipitated phase with a high degree ofinterconnectivity (e.g., Fig. 5-16) has beenemployed as a unique criterion by which todefine an alloy as spinodal. However, aswill be pointed out in Secs. 5.4.1 and 5.5.5,morphology alone cannot be used to unam-biguously distinguish spinodal decomposi-tion from a nucleation and growth reaction(Cahn and Charles, 1965).

5.4 Precipitate Morphologies

5.4.1 Experimental Results

There is a wide variety of differentshapes in individual precipitates and ofmorphologies in precipitate microstruc-tures. This is illustrated in Fig. 5-19 forthree different Cu–Ni-based alloys withternary additions of Al, Cr or Fe, the pre-cipitated volume fraction ( fp) of which israther large, e.g., fp (CuNiAl) ≈0.20, fp(CuNiCr) ≈0.19, and fp (CuNiFe) ≈0.37.

Aging of Ni–37 at.% Cu–8 at.% Al for167 h at 580 °C yields randomly distrib-uted spherical particles with R

– ≈11 nm anda number density of ≈3.6 ¥ 1022 m–3 (Fig.5-19a). In contrast, TEM of Cu–36 at.%Ni–4 at.% Cr aged for 240 h at 650 °C re-veals a modulated precipitate structurewith cuboidal particles aligned along the


Figure 5-18. TEM micrographs of Cu–48 at.% Ni–8at.% Fe aged for a) 8 h, b) 23 h, and c) 65 h at 500°C.Each insert shows two satellites around the bright(002) matrix reflections. The distance between thesatellites and the fundamental reflection is inverselyproportional to the wavelength of composition mod-ulations or to the precipitate spacing in modulatedstructures (reproduced by courtesy of R. P. Wahi (Wahi and Stajer, 1984)).

5.4 Precipitate Morphologies 335

three ·100Ò directions (Fig. 5-19 b). Re-placing Al or Cr by Fe leads to a mottled (‘sponge-like’) precipitate microstructurein Cu–48 at.% Ni–8 at.% Fe aged for 8 h at500 °C. The three-dimensional intercon-nectivity of the mottled structure becomesdiscernible by a reconstruction of a se-quence of FIM images which were taken atvarious distances underneath the originalsurface of the FIM specimen (Fig. 5-19 c).

Modulated structures have been pre-dicted (Cahn, 1965; Hilliard, 1970) toevolve from spinodal decomposition inelastically anisotropic cubic matrices, ow-ing to the tendency to minimize the cohe-rency strain energy (see Sec. 5.2.4, Eq. (5-12)), Furthermore, in material which iseither isotropic or for which the elastic en-ergy (see Eq. (5-12)) is negligibly small,spinodal decomposition is predicted togenerate an interconnected mottled precip-itate morphology. Computer simulations ofthe latter case, which were based on asuperposition of randomly oriented sinu-soidal composition fluctuations of fixedwavelength, random phase shifts and aGaussian distribution of amplitudes,

Figure 5-19. a) TEM dark field image of g¢-precipi-tates in Ni–37 at.% Cu–8 at.% Al aged for 167 h at580°C (Wagner et al. 1988). b) TEM bright field im-age of Cu–36 at.% Ni–4 at.% Cr aged for 240 h at650°C displaying particle alignment along the ·100Òmatrix directions (Wagner et al., 1988). c) FIM mi-crographs of Cu–48 at.% Ni–8 at.% Fe aged for 8 hat 500°C. Between each FIM micrograph 2 nm of thespecimen surface were removed (‘field evaporated’(Wagner, 1982)) in order to reveal the three-dimen-sional arrangement of the brightly imaged (Ni, Fe)-rich precipitated phase. In a the precipitates C, D, E,F, G are apparently isolated; after having removed2 nm, they have merged into one large extended par-ticle (b). At still greater depth (d; 6 nm) ‘particle’ Cappears again to be isolated from the others by thedarkly imaged matrix (reproduced by courtesy of W.Wagner (Piller et al., 1984)).

yielded an interconnectivity of the twoconjugate phases for volume fractionsranging from ≈0.15 to ≈0.85 (Cahn, 1965).The simulated morphological pattern (Fig.5-20) resembles the FIM images shown inFigs. 5-16 and 5-19c remarkably well.

This has made it quite tempting to definethese alloys and, more generally, all alloysdisplaying interconnected or quasi-periodi-cal morphologies, as spinodal alloys. Ingeneral, however, in more concentrated al-loys where the number density of clustersof the new phase is large and, hence, theintercluster spacing small, interconnectiv-ity and quasi-periodicity in the late-stagemicrostructure may result from othermechanisms. Examples of such mecha-nisms are coalescence of neighboring parti-cles (e.g., in the BaO–SiO2 system) or se-lective coarsening of elastically favorablyoriented particles (e.g., in Ni–Al (Ardelland Nicholson, 1966; Doi et al., 1988)).Hence, in essence, the distinction betweena spinodal reaction and nucleation andgrowth cannot be based solely on any spe-cific morphological features but requires a

complete study of the evolution of the newphase with aging time (see Sec. 5.5.5).

5.4.2 Factors Controlling the Shapesand Morphologies of Precipitates

In the case of homogeneous nucleation,the precipitating particles are commonlycoherent with the matrix. Their shape iscontrolled by the rather complex interplayof various factors, such as the magnitudeand anisotropy of the interfacial energy, thedifference in the elastic constants betweenmatrix and precipitate, and the crystalstructure of the latter (Khachaturyan,1983).

The complexity of the situation is illus-trated in Fig. 5-21a (Lee et al., 1977).There the anisotropic elastic strain energyDFel/d

2 of an ellipsoidal Ag-rich particlewith different orientational relationshipswith respect to a Cu or Al matrix is plottedas a function of the aspect ratio K ∫ T/R,where T and R denote the two axes of an el-lipsoid of revolution.

Assuming the interfacial energy to beisotropic, Fig. 5-21a reveals that the mini-mum strain energy is obtained for platelet-shaped particles (K1), the cubic direc-tions of which lie parallel to those of the Alor Cu matrices.

More generally, theory predicts (Lee etal., 1977) platelets to have the minimumand spheres to have the maximum strainenergy if the precipitated phase is elasti-cally softer than the matrix, regardless ofthe orientation relationship and elastic an-isotropy. For the reverse situation of a hardparticle embedded in a softer matrix, thesphere represents the minimum strain en-ergy shape.

However, the elastic constants of theprecipitating phase are often not available,rendering the computation of the shapewith minimum strain energy difficult. In


Figure 5-20. Computer-simulated cross-section of aspinodal structure in an isotropic solid displaying theinterconnectivity of the two incipient phases 1 and 2with equal volume fraction. (After Cahn, 1965).

5.4 Precipitate Morphologies 337

particular, this holds true if the nucleatingphase is a metastable transition phase.

In essence, however, the adopted shapeis determined by the balance between theinterfacial Helmholtz energy D Fa/b and theelastic Helmholtz energy DFel. D Fa/b isminimized for particles with spherical orfaceted shape (e.g., Figs. 5-9 and 5-19)whereas DFel, which is related to the parti-cle volume, commonly prefers a platelet-like morphology (e.g., Fig. 5-21a). Thus,during the early stages of precipitation,D Fa/b is the dominating term, whereasDFel prevails in the limit of large particles.

The nature of this elastic stress-inducedprecipitate–shape transition has been in-vestigated by Johnson and Cahn (1984) us-ing bifurcation theory. Assuming an elasti-cally isotropic inhomogeneous system andby restraining the morphology of the parti-cle to be ellipsoidal it was possible to ad-dress analytically the structure of the shapebifurcation. For example, in two dimen-

sions they found that at a critical size theshape of a circular precipitate will changeto an ellipse in a smooth manner as the sizeof the precipitate is increased; the shape bi-furcation is supercritical. A supercriticalbifurcation is analogous to the change inthe order parameter on passing through asecond-order phase transition. In contrast,in three dimensions, transcritical bifurca-tions (first-order transitions) are possible.Such a bifurcation implies that discontinu-ous changes in particle shape will occur asthe particle increases its size.

The work of Johnson and Cahn clearlyshowed the importance of particle shapebifurcations in understanding microstruc-tural evolution in systems with misfittingparticles. Their study, however, was re-stricted to ellipsoidal particle shapes. Thiswas remedied in the work of Thompson andVoorhees (1999) where the equilibriumshape of a particle in an elastically aniso-tropic homogeneous system with isotropic

Figure 5-21. a) Strain energy vs. aspect ratio of an Ag precipitate in Cu and Al matrices. The numbers 1, 4, 5and 7 designate different orientation relationships between matrix and precipitate. The lowest strain energy isobtained if the cubic directions of both particle and matrix are parallel to each other (curves 1), d is the linearmisfit. After Lee et al. (1977). b) The variation in a parameter, aR

2, as a function of L. aR2 is used to characterize

the symmetry of a given family of equilibrium shapes: it is zero if the shape is four-fold symmetric and non-zeroif it is two-fold symmetric (Thompson et al., 1994).

a) b)

interfacial energy was determined numeri-cally, see Fig. 5-21b. Here the equilibriumshapes were found as a function of the di-mensionless parameter L = d2 C44 l/sabwhere C44 is an elastic constant in a systemwith cubic elastic anisotropy, and l is thesize of a particle, e.g., l = 3V1/3/4p whereV is the volume of a precipitate, L is the ra-tio of the characteristic elastic to interfacialenergy. When the particle is small, interfa-cial energy is the dominant factor in settingthe particle shape and as L increases, theeffects of elastic stress become important.When L is small and the equilibrium shapehas a fourfold symmetry, a cube-like mor-phology with rounded corners and roundedsides will result. As L increases, at a criti-cal size there is a supercritical bifurcationto one of two plate-like shapes orientedalong the elastically soft directions. How-ever, due to the presence of interfacial en-ergy, the equilibrium shapes above the bi-furcation point are not plates for any finiteL. In the limit of infinite L the equilibriumshape is an infinitely long plate of zerothickness (Khachaturyan, 1983). The loca-tion of the bifurcation point is a function ofboth the elastic anisotropy and the differ-ence in elastic constants between thephases (Schmidt and Gross, 1997). Three-dimensional calculations have also beenperformed where the particles are not con-strained to be of a simple geometric shape(Mueller and Gross, 1998, Lee, 1997;Thompson and Voorhees, 1999).

Another qualitative difference betweenthe possible equilibrium shapes of a parti-cle in the presence of stress and that in theabsence of stress is that the morphology ofmisfitting particles can be non-convex. Inthe absence of stress, regardless of the typeof interfacial energy anisotropy, the equi-librium shape must be convex. In contrast,Thompson and Voorhees (1999) foundmetastable non-convex shapes in an elasti-

cally anisotropic system with a tetragonalmisfit, and Schmidt and Gross (1997)found strongly non-convex global energyminimizing, pincushion-like shapes whenthe particle is sufficiently soft in an elasti-cally anisotropic system. Similar pincush-ion-like precipitates have been observedexperimentally, but in a system with amuch smaller difference in elastic con-stants (Maheshwari and Ardell, 1992). Em-phasizing the difficulty in obtaining trueequilibrium shapes experimentally, Wangand Khachaturyan (1995) have suggested,however, that the shapes observed by Ma-heshwari and Ardell are kinetic in origin.

In an internally nitrided Fe 3 at.% Mo al-loy, (Fe, Mo)16N2-type precipitates werefound to nucleate as thin platelets with K ≈ 0.1 and with an undistored 100a-Fe

plane common to both the a-matrix and thenitrides. The observed morphology, as wellas the 100a-Fe habit planes, have beenaccurately predicted by applying macro-scopic linear elastic theory to tetragonalFe16N2 precipitates in ferritic iron, and as-suming the interfacial energy to be iso-tropic (Khachaturyan, 1983). The latter as-sumption, however, is debatable, since dur-ing aging at 600°C (Fe, Mo)16N2 plateletsonly grow markedly in diameter; theirthickness remains almost constant (Fig.5-22). This has been interpreted in terms ofthe large difference between the interfacialenergy of the habit plane (≈0.05 J/m2) andthat of the peripheral plane (≈0.3 J/m2)(Wagner and Brenner, 1978).

Under certain conditions, the elastic ormagnetic interaction of precipitates withexternal stress fields or magnetic fields al-lows for the generation of anisotropic,highly oriented precipitate microstructureswhich are sometimes of technological im-portance. If the transformation strain isnon-spherical, such as for Fe16N2 in ferriticFe–N, the particles may interact with an


5.5 Early Stage Decomposition Kinetics 339

externally applied stress field. The result-ing elastic interaction energy depends onthe particular orientation of the particlewith respect to this field. During aging, thiscauses a selective coarsening of favorablyoriented particles at the expense of the en-ergetically less favorably oriented ones.For instance, if the originally isotropicallydistributed Fe16N2 platelets undergo coars-ening in the presence of an external tensilestress applied along [001], a highlyoriented (anisotropic) precipitate structurewill result (Ferguson and Jack, 1984,1985), with the habit planes of all particlesbeing parallel to each other and perpendic-ular to [001]. In addition, these plateletsshow a strong uniaxial magnetic anisotropywith the direction of easy magnetizationparallel to the plate normal. This magneticproperty can also be used for completelyorienting the platelets during aging in anexternal magnetic field (Sauthoff andPitsch, 1987).

5.5 Early Stage DecompositionKinetics

As outlined in Sec. 5.2.3, the earlystages of unmixing of a solid solutionquenched into the miscibility gap are trig-gered by the growth and the decay of con-centration fluctuations. Basically, the ob-jective of any theory dealing with the ki-netics of early stage decomposition is theprediction of the particular shape, ampli-

Figure 5-22. FIM micrographs of (Fe, Mo)16N2

platelets (bright) in an a-iron matrix (dark). Duringcoarsening at 600°C for the given times, the aspectratio decreases from 0.1 (t = 0) to 0.04 (623 h),whereas the platelet thickness increases only from0.7 nm to 1.0 nm. The platelets intersect the sur-face of the semi-spherical field ion tip and thus ap-pear to be curved (Wagner and Brenner, 1978).

tude and spatial extension (or number of at-oms) of a solute fluctuation which becomescritical and, hence, stable against decay.Once formed, the critical fluctuations ren-der the supersaturated alloy unstable withrespect to further unmixing.

According to Sec. 5.2.4 and referring toFig. 5-6a, nucleation theories consider theformation rate of stable nuclei, the latterrepresenting spatially localized solute-richclusters (‘particles’ or ‘droplets’) withlarge concentration amplitudes. In this con-text, a distinction is made between classi-cal (Sec. 5.5.1.1) and non-classical nuclea-tion theory (Sec. 5.5.1.4) depending onhow the Helmholtz energy of the non-uni-form solid solution containing the clusterdistribution is evaluated. Once the Helm-holtz energy has been specified, the equi-librium distribution of heterophase fluctua-tions and the nucleation barrier (Sec. 5.2.3)can be calculated. The nucleation rate isthen obtained as the transfer rate at whichsmaller clusters attain the critical size.

A ‘composition wave’ picture ratherthan a ‘discrete droplet’ formalism is em-ployed in the spinodal theories (Sec. 5.5.4)which, as another approach, describe theearly-stage decomposition kinetics interms of the time evolution of the ampli-tude and the wavelength of certain stable ‘homophase’ fluctuations (Fig. 5-6b).

There are several papers dealing with theearly-stage unmixing kinetics. Russel(1980) and Aaronson and Russel (1982)consider both classical and non-classicalnucleation phenomena from a more metall-urgical point of view; the formulation of amicroscopic cluster theory of nucleationcan be found in the article by Binder andStauffer (1976). The comprehensive arti-cles by Martin (1978) and by Gunton andDroz (1984) disclose developments of bothnucleation and spinodal concepts not onlyto the theoretican but also to the materials

scientist. For more recent developments,see the chapter by Binder and Fratzl (2001).

5.5.1 Cluster-Kinetics Approach

5.5.1.1 Classical Nucleation –Sharp Interface Model

Suppose the homogenized solid solutionis quenched not too deeply into the meta-stable regime of the miscibility gap (e.g., topoint 1 in Fig. 5-5a). There it is isother-mally aged at a temperature sufficientlyhigh for solute diffusion. After a certaintime, a distribution of microclusters con-taining i atoms (i-mers) will form in thematrix.

Generally, classical nucleation is nowbased on both a static and a dynamic part.In the static part the changes of Helmholtzenergy associated with the formation of ani-mer and the cluster distribution f (i ) mustbe evaluated. In the dynamic part the kinet-ics of the decay of the solid solution whichnow is described by the given distributionof non-interacting microclusters, are calcu-lated in terms of the time evolution of f (i );ultimately this will furnish the formationrate of stable clusters, i.e. the nucleationrate.

Classical nucleation theory treats the so-lute fluctuations as droplets which were cutfrom the equilibrium precipitate phase band embedded into the a matrix; in thiscapillarity or droplet model the interfacebetween a and b is assumed to be sharp,e.g., Fig. 5-6a. In essence, the approxima-tion reduces the number of independentvariables which characterize a cluster, andwhich may vary during the nucleation pro-cess (e.g., the solute concentration, theatomic distribution within the cluster, itsshape, the composition profile across theinterface, etc.) and, hence, determine theHelmholtz energy of the system, to virtu-ally a single one. That is the number i of



atoms contained in the cluster or, in termsof a more macroscopic picture, the radius Rof the droplet; (4p /3) R3 = i Wb if Wb de-notes the atomic volume in the droplet.

Once the small nucleus containing only afew atoms is treated as a droplet of the newphase b having bulk properties (which, infact, is a rather debatable approximation),the interfacial Helmholtz energy and theHelmholtz energy are also considered to beentirely macroscopic in nature.

The formation of a coherent droplet withradius R which gives rise to some elasticcoherency strains, leads to a change ofHelmholtz energy,

(5-18)

where, according to Sec. 5.2.4, (D fch + D fel) is the driving force per unit volumeand sab the specific interfacial energy.

The first term in Eq. (5-18) which scaleswith R3 accounts for the gain of Helmholtzenergy on forming the droplet (i.e., it isnegative). The second term which scaleswith R2 has to be expanded on forming theinterphase boundary and hence is a positivecontribution to DF (R).

Fig. 5-23 shows the dependence of thetwo contributions in Eq. (5-18) on thedroplet radius. The resulting droplet forma-tion energy DF (R) passes through a maxi-mum at R ∫ R* or i ∫ i* with

(5-19)

Accordingly, clusters with R = R* are inunstable equilibrium with the solid solu-tion, i.e., the Helmholtz energy of thesystem is lower if it contains clusters withsizes below or beyond R* or i*. Therefore,only clusters with radii that exceed the ra-

Rf f

if f

*– ( )

*– ( )

=+

=+

⎡

⎣⎢

⎤

⎦⎥

2

43

2 3

s

p s

ab

b

ab

D D

D D

chem el

chem el

or

W

D D D abF R f f R R( ) ( )= + +ch el43

43 2p p s

dius R* of the critical nucleus are predictedto grow continuously. This requires a fluc-tuation to become a critical nucleus, first toovercome the activation barrier for nuclea-tion, or the nucleation energy,

(5-20)

As outlined in Sec. 5.2.4, D fchem de-creases strongly with decreasing supersatu-ration, or, for a given concentration, withincreasing aging temperature (Fig. 5-1).Correspondingly, since sab and D fel areonly weakly dependent on temperature,both R* and DF* increase strongly withdecreasing supersaturation. Hence, thedroplet model ought to approximate the en-ergetics of an unmixing alloy better thesmaller the supersaturation.

Once we have specified the free energyof formation of a droplet in terms of its sizeR or i (Eq. 5-18)), the Helmholtz energy ofthe system containing a number of f (i, t) ofnon-interacting clusters of size i per unitvolume at time t is given as (Frenkel, 1939;

D DD D

ab3

F R Ff f

( *) *( )

≡ =+

163 2p s

chem el

Figure 5-23. Schematic representation of Helmholtzenergy changes associated with cluster formation asa function of cluster radius (R) or number i of atomsin the cluster. DF 0 is the Helmholtz energy of the homogeneous solid solution, Z the Zeldovich factor.

Russell, 1980):

(5-21)

The entropy of mixing Smix arises fromdistributing the clusters on the available N0

lattice sites per unit volume of the crystal.The kinetics of early-stage unmixing are

governed by the change in the cluster sizedistribution function f (i, t) with agingtime. Microscopically, this may occur viadifferent processes.

Volmer and Weber (1926), Becker andDöring (1935) and Zeldovich (1943), theworks which most nucleation theories arebased on, assume the transitions betweensize classes in an assembly of non-interact-ing droplets to occur via the condensationor evaporation of single solute atoms.Hence, since only a transition between sizeclasses i and i + 1 is allowed, its flux, Ji isgiven as (Russell, 1980):

Ji = b (i ) f (i, t) – a (i +1) f (i +1, t) (5-22)

where b (i) is the condensation rate anda (i + 1) the evoporation rate of a singleatom to a cluster of size i or from a clusterof size (i + 1), respectively.

In equilibrium, the fluxes Ji must vanishand f (i, t) becomes identical with the equi-librium cluster size distribution C (i ) forwhich F in Eq. (5-21) attains a minimum:

C (i) = N0 exp [–DF (i )/kT] (5-23)

It is worth noting that in this case thenumber of critical nuclei C* ∫ C (i ∫ i*) isproportional to exp (–DF*/kT ). Then, ac-cording to Eq. (5-22), the condensationrate and evaporation rate are related to eachother via

(5-24)

In the conventional nucleation theory itis now assumed that the evaporation rate

a b( ) ( )( )

( )i i

C iC i

+ =+

11

F F i f i t T S= ∑D ( (1

n

mix) , ) –

a (i + 1) derived for the equilibrium situa-tion (‘principle of detailed balance’) is stillvalid for the non-equilibrated systemwhere f (i, t) ≠ C (i ) and Ji ≠ 0. Under suchan assumption, which becomes reasonablewhen i-mers are able to relax internallybetween atomic condensation or evapora-tion, the flux of clusters between sizeclasses i and i + 1 is obtained from Eqs. (5-22) and (5-23) (Russell, 1980):

(5-25)

In the earliest theory on nucleation (Volmer and Weber; VW, 1926) it is as-sumed that f (i, t) ∫ 0 for clusters withi > i* (or R > R*), and that clusters with R> R* decay artificially into monomers,thus keeping the matrix supersaturationabout constant. The resulting quasi-steady-state distribution of cluster sizes is ob-tained from Eq. (5-23) and shown in Fig.5-24). In this theory the steady-state nucle-ation rate J s

V–W is obtained as the productof the number C* of critical nuclei and therate b at which a single solute atom im-pinges on the critical nucleus rendering itsupercritical, i.e.:

J sVW = b C* = b N0 exp (–DF*/kT ) (5-26)

One of the shortcomings of the VW the-ory is that supercritical droplets with i > i* are assumed not to belong to thecluster size distribution. This was clearedup by the theory of Becker and Döring(1935). In their theory (BD) the non-equi-librium steady-state distribution of smallclusters with i i* is identical with that ofVW but unlike in the latter theory, clusterswith i* i ic (ic is a somewhat arbitrarilychosen cut-off size) are considered to be-long to the size distribution (Fig. 5-24).

J i C if i tC i

f i tC i

i C if i t C i

i

i = ++

⎡⎣⎢

⎤⎦⎥

≈ ∂∂

⎡⎣⎢

⎤⎦⎥

b

b

( ) ( )( , )( )

–( , )( )

– ( ) ( )( , ) / ( )

11



Hence, according to Eq. (5-22) a decayor dissolution of supercritical droplets withi > i* becomes a likely process in the BDtheory and is accounted for by the Zeldo-vich factor Z. With the assumption that therate b ∫ b* at which a solute atom impingeson a critical droplet is proportional to itssurface area, integration of the cluster fluxequation (Eq. (5-25)) yields the steady-state nucleation rate J S

BD of the BD theoryas (Russell, 1980):

J sBD = Z b* N0 exp (–DF*/kT ) (5-27)

with

(5-28)

According to Eq. (5-27), nucleation is athermally activated process with an activa-tion energy identical to that (DF*) offorming a critical nucleus of size i* or R*.Furthermore, like the thermodynamicmodel that yields the number of critical nuclei proportional to exp (–DF*/kT ) (Eq.(5-23)), the kinetic treatment predicts thesteady-state nucleation rate also to be pro-portional to the same exponential factor.

ZkT

F

ii i

= ∂∂

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

–

*

/

12

2

2

1 2

p∆

The Zeldovich factor Z is of the order of1/20 to 1/40; graphically, its reciprocalvalue corresponds approximately to thewidth of the potential barrier, DF (R) orDF (i ), at a distance kT below the maxi-mum (Fig. 5-23).

5.5.1.2 Time-Dependent NucleationRate

It must be pointed out that the abovegiven steady-state nucleation theories donot provide any information on the mo-mentary cluster size distribution or on thenucleation rate prior to reaching steady-state conditions, i.e., the time-dependentnucleation rate J*(t). Commonly the latteris considered in terms of the steady-statenucleation rate J s and an incubation periodt via:

J* (t) = J s exp (–t /t) (5-29)

Steady-state will be achieved once theclusters have attained sizes for which theprobability of redissolution is negligiblysmall. Referring to Fig. 5-23 and recallingthe physical meaning of the Zeldovich fac-tor Z, this will be the case for clusters withi > i* +1/2 Z (Feder et al., 1966). As thegradient of DF (i ) within the region 1/Z israther small, the clusters will move acrossthis region predominantly by random walkwith the jump frequency b*. The time

(5-30)

to cover the distance 1/Z by random walk isidentified with the incubation period.

So far it has been assumed that duringnucleation the supersaturation and, hence,the driving force remains unchanged. At itsbest this may be valid for extremely lownucleation rates: in this case DF*, Z and tmay be considered time-independent quan-tities. Russell (1980) proposed the nuclea-

tb

= 12 * Z

Figure 5-24. Quasi-stationary cluster size distribu-tions of the Volmer–Weber (V–W) and Becker–Döring (B–D) theory.

tion process to terminate itself understeady-state conditions if the incubationtime t is shorter than some critical time tc.If, however, t > tc, phase separation will becompleted without steady-state ever havingbeen achieved. This situation, which in factis met in most decomposing alloys studiedup to now (see Sec. 5.7.4.1), is referred toas catastrophic nucleation; under suchconditions DF*, Z and t become time de-pendent. Assuming that nucleation ceasesonce the diffusion fields and around pre-cipitates with radius R and composition cb

e

embedded in a matrix with initial composi-tion c0 begin to overlap (Fig. 5-6 a), andfurthermore that nucleation becomes un-likely within a region R/e (e ≈1/10) aroundthe particle where the solute content hasdecreased below (1 – e) c0, the critical timetc is estimated to be:

(5-31)

where D is the solute diffusion coefficient.For a quantitative assessment of classi-

cal nucleation theory, the atomic impinge-ment rate b* in Eq. (5-27) must be known.For spherical nuclei it was evaluated to be(Russell, 1970):

(5-32)

where a is the lattice parameter. b* is pro-portional to the nucleus surface, as was as-sumed in the original Becker–Döring the-ory.

According to Sec. 5.4.2 the nucleusshape with minimum energy may deviatefrom the spherical one due to different en-ergies of the interfaces (some may be co-herent, some semi- or incoherent, Sec.5.2.1) bounding the nucleus in differentcrystallographic directions. It has beenshown (Chan et al., 1978) that in this case

b p*

*= 4 2

04

R Dc

a

tc

c D Jc

e

s2=⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

e6

0

3

3

1 5

1

2

1β

( )

/

the expressions for DF*, Z, b* and t givenabove for spherical clusters, have merely tobe multiplied with numerical factors butotherwise remain unchanged. The magni-tude of these numerical factors depends onthe particular equilibrium shape of the nu-cleus. Ignoring volume strain energy, theequilibrium shape of the nucleus can be de-termined by means of the Wulff construc-tion on a three-dimensional sab-plot (Her-ring, 1953). The latter is obtained from aradial plot of different vectors in every di-rection e.g., vv1, vv2, vv3 in Fig. 5-25), thelength of which is proportional to the en-ergy of the interface perpendicular to theparticular vector. The surface connectingthe tips of all vectors represents the polarsab-plot, wherein the cusps indicate inter-faces with good atomic matching , i.e., lowinterfacial energies. Subsequently planes


Figure 5-25. Polar sab-plot for particles with aniso-tropic interfacial energies sab. The normals drawn atthe tip of each vector vv are the Wulff planes, the innerenvelope of which gives the equilibrium shape of theprecipitate. If the sab-plot displays deep cusps (e.g.,for vv1, vv3, and vv5) the Wulff construction yields a fac-eted polyhedron.


are drawn (the so-called Wulff planes) per-pendicular to the vectors vv1, vv2, vv3, etc.which intersect the sab-plot (Fig. 5-25).The inner envelope of the Wulff planesyields the equilibrium shape of the nucleuswith facets at the cusps of the sab-plot(Martin and Doherty, 1976).

LeGoues et al. (1982) employed a near-est-neighbor interaction, regular solutionmodel for a computation of the sab-plot asa function of T/Tc (Tc is the critical temper-ature) within the miscibility gap of an f.c.c.solid solution. At low temperatures theyfound the nucleus shape to be fully facetedby 100 and 111 planes whereas at 0.5Tc and near-zero supersaturation, the nu-cleus shape can be rather well approxi-mated by a sphere. They furthermoreshowed that beyond T ≈0.4 Tc the parame-ters DF*, Z, t, and b* which enter the time-dependent (classical) nucleation rate (Eqs.(5-27) and (5-29)) need no longer be cor-rected for deviations from spherical shape.

Recently the question of to what extentthe energy of the coherent interphase boun-dary depends on its crystallographic orien-tation and on temperature has been read-dressed by employing the cluster variationmethod (CVM) with effective cluster inter-action parameters (Sluiter and Kawazoe,1996; Asta, 1996). The latter are specificfor the particular two-phase alloy systemunder consideration. They can be derivedfrom the results of ab initio total energycalculations as was demonstrated by com-puting the thermodynamic properties ofinterphase boundaries between disorderedf.c.c. matrices and ordered (L12 type) pre-cipitates in Al–Li (Sluiter and Kawazoe,1996) and Al–Sc (Asta et al., 1998) as wellas disordered GP zones (cf. Fig. 5-13) inAl–Ag alloys (Asta and Hoyt, 1999).Between Tc and T = 0.5 Tc the CVM-calcu-lated energies for the cube 100 and octa-hedral 111 orientations of the flat inter-

face between Al-rich matrix and Ag-richGP zones are found to be equal within 5%,and to increase from zero to about30 mJ/m2 if the temperature is loweredfrom Tc to ~ 0.5 Tc. This result is compat-ible with the spherical particle morphologyresolved in the HREM micrographs (Figs.5-9 and 5-13).

5.5.1.3 Experimental Assessment ofClassical Nucleation Theory

A quantitative assessment of classicalnucleation theory in solids is inherentlydifficult and thus has prompted only a fewstudies. First of all, the range of supercool-ing (or supersaturation) has to be chosensuch that nucleation is homogeneous andthe nucleation rates are neither unmeasure-ably slow nor beyond the limits of catas-trophic nucleation. Secondly, the quench-ing rate must be sufficiently high in orderto avoid phase separation during thequench but also sufficiently low in order toavoid excess vacancies to be quenched in.

Servi and Turnbull (1966) studied in awell-designed experiment homogeneousnucleation kinetics of coherent Co-richprecipitates in Cu–1 … 2.7 wt.% Co al-loys. By using electrical resistivity meas-urements, they could determine rather ac-curately the precipitated volume fraction.Assuming that the growth is diffusion-con-trolled (Sec. 5.5.2), from the latter the par-ticle density at the end of the precipitationprocess could be derived as a function ofaging temperature and composition. Thethus indirectly obtained number density ofCo-rich particles, which was later corrobo-rated by CTEM studies, agreed within oneorder of magnitude with the value pre-dicted by classical nucleation theory if theinterfacial energy was taken as ≈0.19 J/m2;this value was derived from discrete latticecalculations (Shiflet et al., 1981).

The validity of classical nucleation the-ory, as proven by the Servi–Turnbull studyon Cu–Co alloys, was later challenged byLeGoues and Aaronson (1984). Theyargued that the supersaturation employedin the Servi–Turnbull investigations wasprobably too high to avoid phase separa-tion during quenching and also probablytoo high to avoid concomitant coarseningduring the precipitation reaction. Employ-ing a discrete lattice point model, which in-corporates coherency strain energy, Le-Goues and Aaronson first evaluated the ‘window’ of temperatures (DT ≈50°C) andcompositions (0.5 to 1.0 at.% Co) at whichhomogeneous nucleation kinetics would beneither too sluggish nor too fast, and atwhich no interference with coarseningwould be expected. Prior to CTEM analy-ses, the isothermally nucleated particleshad to be subjected to diffusion-controlledgrowth in order to increase their radius be-yond a certain size (R ≈5 nm) at which theCo-rich particles became easily discerniblein the CTEM. The experimental resultswere interpreted in terms of classical nu-cleation theory (Eq. (5-29)). The agree-ment between the experimentally obtainednucleation rates and the theoretically pre-dicted ones was again rather good, thusproviding further support for the validity ofclassical nucleation theory. Furthermore,as is shown in Fig. 5-26, for smaller super-saturations the nucleation energies DF*and the critical radii R* as evaluated fromclassical nucleation theory are almost iden-tical to the corresponding quantities calcu-lated from either the non-classical nuclea-tion theory (cf. Sec. 5.5.1.4, Eq. (5-37)) or adiscrete lattice point theory. Hence, at leastfor smaller supersaturation, the classicaltheory predicts the nucleation rates about aswell as the two more sophisticated theories.

In another attempt to assess classical nu-cleation theory, Kirkwood and coworkers

(Kirkwood, 1970; West and Kirkwood,1976; Hirata and Kirkwood, 1977) studiedearly-stage precipitation of g ¢-Ni3Al inNi–Al alloys, also using CTEM. Theyfound the g ¢-Ni3Al precipitate number den-sity to decrease instantaneously upon ag-ing, which is indicative of an extremelyfast nucleation process and the observation


Figure 5-26. Nucleation barrier DF* (a) and criticalradius R* (b) as a function of supersaturation at T =0.25 Tc according to classical theory (cl. th.), non-classical Cahn–Hilliard continuum model (C–H) anddiscrete lattice model (DLM). R* is determined suchthat it corresponds to (c0 + cc)/2 where cc is the com-position of the nucleus at its center. (After Le Goueset al., 1984a).


of a coarsening process rather than a nucle-ation event. By making certain assumptionsthey attempted to infer the steady-state nu-cleation rate from the particle density ob-served during coarsening. The observedlack of agreement between the experimen-tally derived nucleation rates and the theo-retically predicted ones may be seen to re-sult from these experimental difficulties.

5.5.1.4 Non-Classical Nucleation –Diffuse Interface Model

In the droplet model that classical nucle-ation is based on, it is assumed that thecomposition of the nucleus is more or lessconstant throughout its volume and that itsinterface is sharp. This made it possible totake the change of volume Helmholtz en-ergy and the interfacial Helmholtz energyseparately into account (Eq. (5-18)). Innon-classical nucleation theory, developedby Cahn and Hilliard (1959a, b), the inho-mogeneous solid solution in its metastablestate is considered to contain homophasefluctuations with diffuse interfaces and acomposition which varies with positionthroughout the cluster (Fig. 5-6 b). Hence,unlike in the droplet model a critical fluctu-ation has now to be characterized by atleast two parameters, its spatial extensionor wavelength l and its spatial compositionvariation. The necessary Helmholtz energyformalism, which no longer treats volumeenergy and surface energy separately, waselaborated by Cahn and Hilliard (1958).They wrote the Helmholtz energy changeassociated with the transfer from the homo-geneous system with composition c0,

(5-33a)

to that of the inhomogeneous system with

(5-33b)F f c K c VV

= + ∇∫[ ( ) * ( ) ]2 d

F f c VV

0 0= ∫ ( ) d

as:

(5-33c)

The basic idea behind these expressions isto subdivide the solid with volume V intomany small volume elements dV, a proce-dure which is often referred to as ‘coarsegraining’ (Langer, 1971). The free energyof each volume element is taken as f (c) dV,f (c) is considered to be the Helmholtz en-ergy per unit volume of the bulk materialwith composition c which is equal to themean composition c (r) within dV locatedat position r. In essence, coarse graining re-quires the number of atoms within eachvolume element to be sufficiently largesuch that c (r), as well as f (c), can be spec-ified as continuous functions. On the otherhand, the number of atoms within dV mustbe small enough in order to avoid phaseseparation within the individual volumeelement. For the non-uniform system withcomposition fluctuations, Cahn and Hilli-ard (1958, 1959a, b) assumed that the localHelmholtz energy contains further termsdepending on the composition gradient (—c). They finally showed that the non-uni-form environment of atoms in a composi-tion gradient may be accounted for by add-ing to the local Helmholtz energy a singlegradient energy term which is proportionalto (—c)2. Hence, the resulting Helmholtzenergy of a volume element dV is ex-pressed as f (c) + K* (—c)2 where the con-stant K* denotes the gradient energy coeffi-cient. Summing up all contributions fromthe various volume elements yields Eq.(5-33c).

The continuum model is base on the as-sumption that the Helmholtz energy f (c)varies smoothly with composition, i.e. thewavelength of the fluctuations must belarge compared with the interatomic spac-

DF F F

f c f c K c VV

=

= + ∇∫( – )

[ ( ) – ( ) * ( ) ]

0

02 d

ing. Furthermore, for a continuum modelits is required that the two evolving phasesare fully coherent with each other and thushave the same crystal structure with similarlattice parameters. If the lattice parameterchanges with composition, which is com-monly the case in crystalline solids, the re-quirement for coherency leads to cohe-rency strains which according to Eq. (5-12)are accounted for by an elastic energy term

fel (c) = h2 Y (c – c0)2 (5-34)

Combining Eqs. (5-34) and (5-33c) yields

(5-35)

This expression may be interpreted in sim-ilar terms as for classical nucleation [Eq.(5-18)]: neglecting the elastic energy term,the positive contribution of the gradient en-ergy as a barrier to nucleation acts like thesurface Helmholtz energy in the dropletmodel and is finally overcome by the gainin chemical Helmholtz energy once thecomposition difference between the fluctu-ation and the homogeneous matrix has be-come sufficiently large.

Assuming isotropy, the composition pro-file c (r) of a spherical fluctuation (r is theradial distance from the fluctuation center)is obtained from a numerical integration of(Cahn and Hilliard, 1959a,b):

(5-36)

with the boundary conditions dc/dr = 0 atthe nucleus center (r ≈ 0) and far away fromit (r Æ •) where c ∫ c0. The critical nu-cleus is then determined as the fluctuationwhich, like a critical droplet, is in unstableequilibrium (Eq. (5-35)) with the matrix.Its composition is established such that the nucleation barrier (e.g., de Fontaine,

2 42

20

Kc

r

Kr

cr

fc

fc

c c

**

–d

d

dd

+ = ∂∂

∂∂

DF f c f c K c

Y c c VV

= + ∇

+

∫ [ ( ) – ( ) * ( )

( – ) ]

02

20

2h d

1982),

(5-37)

attains a minimum. Like in classical nucle-ation, D f * is the vertical distance from thetangent at c0 to the Helmholtz energy curveat c* (cf. Fig. 5-5b). Suppose D F* isknown; the nucleation rate is then obtainedfrom Eq. (5-27) or (5-29). Provided theconstraint free energy is known, the com-position profile of a critical nucleus in asolid solution with composition c0 and gra-dient energy coefficient K* (K* is of the or-der 10 to 100 J m–1 (at. fract.)–2) can be ob-tained from integrating Eq. (5-36). Fig.5-27 shows the composition profiles of nu-clei in supersaturated solid solutions withdifferent solute concentrations c0 quenchedto T/Tc = 0.25, where Tc is the critical tem-perature (LeGoues et al., 1984b); these cal-culations are based on the regular solutionmodel for f (c) (cf. Sec. 5.2.4). For smallsupersaturation (c0 = 2 ¥10–3) the compo-sition within the nucleus is constant andcorresponds to cb

e of the equilibrium b-phase. The interface is almost sharp, simi-lar to that assumed in the classical dropletmodel. Correspondingly, the free energy offormation of the nucleus (DF*) and its size(R*) as evaluated from either classical ornon-classical theory are almost identical(Fig. 5-26) and tend to infinity as c0 ap-proaches ca

e . With increasing supersatura-tion the composition profile becomes in-creasingly diffuse and the solute concentra-tion at the nucleus center decreases (Fig.5-27); this situation is no longer accountedfor by the droplet model. Furthermore, asthe initial composition (c0) approaches thespinodal one (cs

a), unlike the dropletmodel, non-classical theory predicts DF*to go to zero (Fig. 5-26a), and the spatial

∆ ∆

∆

F c r F

f c Kcr

r r

[ *( )] *

( *) *

≡

= + ⎛⎝

⎞⎠

⎡

⎣⎢

⎤

⎦⎥

∞

∫42

0

2p dd

d



extent of the critical fluctuations (or R*) toincrease again; at c0 ∫ cs

a, R* finally be-comes infinite (Fig. 5-26 b) suggesting adiscontinuity of the decomposition mecha-nism at the spinodal line (cf. Sec. 5.2.3).

This apparent discontinuity at the spino-dal results from the assumption that the de-composition path is controlled by the mini-mum height of the nucleation barrier, i.e.,by DF*, which close to the spinodal is inthe order of a few kT (Fig. 5-26a) and,hence, rather low. As pointed out by deFontaine (1969), in a system close to thespinodal many fluctuations exist, the spa-tial extent of which is much smaller thanthat of the critical fluctuation; neverthe-

less, their formation energies are onlyslightly higher than DF*. In this case, theprobability of the alloy decaying via theformation of those ‘short-wavelength’fluctuations requiring a slightly higher ac-tivation energy is much higher than for adecay via the formation of critical ‘long-wavelength’ fluctuations.

5.5.1.5 Distinction Between Classicaland Non-Classical Nucleation

For a practical application of non-classi-cal nucleation theory to experimental stud-ies of unmixing, the total constraint freeenergy f¢ (c) rather far away from equilib-rium, as well as K*, have to be known inorder to calculate the composition profile[Eq. (5-36)] and the nucleation energy [Eq.(5-37)]. This is the case only for simple al-loy systems and, hitherto, has inhibited theapplication of the non-classical theory toeither scientifically or industrially impor-tant alloy systems. Furthermore, so far ithas been assumed that a nucleus is in equi-librium with the infinite matrix and that thenuclei do not interact during the nucleationstage. This may be a valid approximationas long as the supersaturation is small, i.e.,where according to Figs. 5-26 and 5-27classical nucleation theory applies. It oughtto become, however, a poor approximationfor more concentrated alloys where non-classical nucleation applies. In this case thenucleation rate is high (then the steady-state nucleation regime will not be ob-served) and, hence, the nucleus densitylarge. These complications in non-classicalnucleation make it desirable to provide theexperimentalist with a criterion for the ap-plicability of classical nucleation theorywhich is much easier to handle. Cahn andHilliard (1959a,b) suggest classical theorywill apply if the width l of the (diffuse)interface is considerably smaller than the

Figure 5-27. Composition profile of a spherical nu-cleus (center at r = 0) in four different metastablesolid solutions with composition c0 quenched to T =0.25 Tc into the miscibility gap. The continuous linewas calculated from the Cahn–Hilliard continuummodel (Eq. (5-36)), the crosses represent calcula-tions based on the discrete lattice point model (afterLeGoues et al., 1984a). The computations are basedon a regular solution model with the solubility limitca

e = 0.37 ¥ 10–3 and the spinodal composition cas =

6.41 ¥ 10–2, a is the lattice parameter of an assumedf.c.c. lattice and l the interfacial width.

core of the nucleus where the compositionis about constant (Fig. 5-27).

If we assume that a perceptible nuclea-tion rate is obtained for DF*≤ 25 kT (inthe literature, DF*≈ 60 kT is frequentlystated for the maximum barrier at whichnucleation becomes measurable; this value,however, is much too high for homogene-ous nucleation to occur in solids) and em-ploys Eqs. (5-19) and (5-20), classical nu-cleation theory may be applied if (Cahnand Hilliard, 1959a, b):

(5-38)

For Cu–Ti (sab = 0.067 J/m2; T = 623 K),Fe–Cu (sab = 0.25 J/m2; T = 773 K), andCu–Co (sab = 0.17 J/m2; T = 893 K) thismeans that the interfacial width has to besmaller than ≈ 0.56, 0.33, and 0.42 nm, re-spectively, and, hence, has to be rathersharp. So far it has not been possible tomeasure the composition profile across anucleus/matrix interface. However, analyt-ical field ion microscopy of Cu–1.9 at.% Tiaged at 350°C for 150 s revealed the Ticoncentration of particles with radii of only≈1 nm to decrease from 20 at.% (corre-sponding to Cu4Ti, cf. Sec. 5.2.1) to that ofthe matrix within one to two atomic (111)-layers (von Alvensleben und Wagner,1984). For the chosen aging conditions nu-cleation was evaluated to terminate withinabout 60 s (Kampmann and Wagner, 1984),thus after aging for 150 s the analyzed par-ticles must have already experienced somegrowth beyond the original size of the nu-cleus. Nevertheless, these results providerather good evidence that during nucleationthe Cu4Ti clusters may be considered drop-lets with sharp interfaces rather than spa-tially extended (long-wavelength) fluctua-tions with diffuse interfaces, even thoughDF* is only about 10 kT.

lkT

2 4

751

ps ab

Although some caution is advisable intreating these tiny particles (critical nucleiin solid-state transformation are tens tohundreds of atoms in size) in terms of acontinuum theory1, and assigning them amacroscopic surface and thermodynamicbulk properties, classical nucleation theoryseems to be appropriate for an estimationof the nucleation rate. This conclusion willbe corroborated in Sec. 5.7.4.

5.5.2 Diffusion-Controlled Growthof Nuclei from the Supersaturated Matrix

Suppose that after nucleation the stablenucleus is embedded in a still supersatu-rated matrix. As is illustrated in Fig. 5-28,the particle will then be surrounded by aconcentration gradient which provides thedriving force for solute diffusion, and thusgives rise to its growth.

The growth rate can be controlled eitherby the rate at which atoms are supplied tothe particle/matrix interface by diffusion orby the rate at which they cross the interface(Tien et al., 1973). It may be rationalizedthat for small particles the interface reac-tion is likely to be the rate-controlling stepsince the diffusion distances are rathershort; once the particles have grown to acertain size, the matrix will be depletedfrom solute atoms and the associated re-duction of the driving force makes diffu-sion likely to be the slower and, thus, rate-controlling step (Shewmon, 1965). Thetransition from one step to the other de-pends upon the relative magnitudes of so-lute diffusion and interface mobility.

After termination of nucleation it is com-monly assumed that the mobility of theinterface is sufficiently high in order to al-


1 Many FIM studies of two-phase alloys (e.g., Cu–1at.% Fe) revealed the shapes of even tiny clusterswith as little as about twenty atoms to be alreadycompact rather than ramified (Wagner, 1982).


low the solute concentration cR at thecurved interface to achieve local equilib-rium (Doherty, 1983). In this case diffusionis the rate-controlling step for the growthof stable coherent nuclei in a homogeneousprecipitation reaction. An analytical solu-tion of the adequate field equation,

(5-39)

requires certain approximations (Zener,1949). These were critically examined andcompared by Aaron et al. (1970) for thediffusion-controlled growth of sphericaland platelet-shaped particles. Regardless ofthe particular approximation used, both theradius R of a spherical precipitate and thehalf-thickness T of a platelet grow with timeaccording to a parabolic growth law as:

R = li (D t)1/2 (5-40)

and

T = lj (D t)1/2 (5-41)

respectively. The rate constants li and lj in-crease with increasing supersaturation, or,more specifically, with an increase in thefactor

(5-42)kc t cc c

*( ) –

–= 2 R

p R

D c r tc r t

t∇ = ∂

∂2 ( , )

( , )

in a manner which, in particular for largervalues of k*, depends upon the approxima-tion assumed (Aaron et al., 1970). For pre-cipitating intermetallic compounds, k* isone or two orders of magnitude larger (e.g.,≈ 0.3 for g¢-Ni3Al in Ni–Al and ≈ 0.2 forb¢-Cu4Ti in Cu–Ti) than for precipitatingphases with cp ≈ 1 such as Fe–Cu, Cu–Feand Cu–Co.

For small supersaturations, c (r, t) in Eq.(5-39) may be approximated as being time-independent. In this case, an isolated spher-ical particle with radius R surrounded bythe concentration field illustrated in Fig.5-28 will grow at a rate

(5-43)

where D (assumed to be independent ofcomposition) is the volume diffusion coef-ficient in the matrix. According to theGibbs–Thomson equation, the compositioncR of the matrix phase at a curved interfaceis different from that of a flat interface, thelatter being in equilibrium with the equilib-rium solute concentration ca

e of the a-ma-trix phase, and varies with the precipitateradius as (Martin and Doherty, 1976)

(5-44)

with ga (cR) and ga (cae ) being the activity

coefficients of the solute atoms in the a-phase at the concentration cR and ca

e , re-spectively. If the solid solution shows reg-ular solution behavior (ga then becomes in-dependent of c), Eq. (5-44) yields the well-known Gibbs–Thomson equation:

(5-45)

or for larger radii in its linearized version,

(5-46)c R cV

R T RRe

g( ) = +

⎛

⎝⎜⎞

⎠⎟a

ab b12 1s

c R cV

R T RRe

g( ) exp=

⎛

⎝⎜⎞

⎠⎟a

ab b2 1s

ln( ) –

–– ln

( )

( )

c R

c

c

c c R T

V

Rc

cR

e

e

pe

g

Re

a

a

a

ab b a

a a=

⎛

⎝⎜⎞

⎠⎟1 2 s g

g

dd

R

p R

Rt

Rc t cc c

DR

≡ =v ( )( ) –

–

Figure 5-28. Schematic concentration field in thematrix surrounding a nucleus with radius R and com-position cp.

By assuming a monodispersive particledistribution and cR≈ c (t Æ •) = ca

e (or c¢ae),

and c (t) ≈ c0, integration of Eq. (5-43) yields

(5-47)

It should, however, be pointed out that, sofar, there is little experimental evidence forthe existence of a precipitation regime dur-ing which particle growth strictly followsEq. (5-47) (Kampmann and Wagner, 1984).In Sec. 5.7.4.3 the reason for this lack ofexperimental evidence will be provided, aswell as a guideline to the design of experi-ments that allow a verification of the exis-tence of diffusion-controlled particlegrowth with the predicted kinetics to bemade. Most quantitative experimental ob-servations of growth rates which yieldedgood agreement with the diffusion-con-trolled models outlined above, were con-fined to large particles, frequently withsizes in the micrometer range (Aaronson etal., 1970a, b; Doherty, 1982).

5.5.3 The Cluster-Dynamics Approachto Generalized Nucleation Theory

As has been pointed out in Sec. 5.5.1.1,the cluster-dynamics approach chosen byBecker and Döring (1935) is based on theassumption that the equilibrium cluster dis-tribution function can be specified by Eqs.(5-23) and (5-18). Even though the result-ing predictions of classical nucleation the-ory ought to become valid asymptoticallyfor large droplets at small supersaturations,there remain some inherent deficits (Binder, 1980; Gunton et al., 1983).

For instance, for small clusters the sur-face energy entering Eq. (5-18) shouldcontain size-dependent corrections to themacroscopic equilibrium energy of a flatinterface. Furthermore, the separation ofthe droplet formation Helmholtz energy

R tc c

c cD t( )

–

–( )

/

/=⎛

⎝⎜⎞

⎠⎟2 0

1 2

1 2a

a

e

pe

into a bulk and a surface term appears ratherdebatable. To avoid some of these deficits,there have been several attempts to developmore accurate descriptions of clusters, tak-ing into account their different sizes andshapes (e.g., Binder and Stauffer, 1976;Stauffer, 1979), and to derive an equilibriumdistribution of clusters which is more realis-tic than the one given by Eq. (5-23). As inthe classical nucleation theory, the latter isan important quantity for a computation ofthe cluster formation rate in terms of themore recently elaborated theories of clusterdynamics which, in essence, are extensionsof the Becker–Döring theory (Binder andStauffer, 1976; Penrose and Lebowitz,1979; cf. Gunton et al., 1983 and the chapterby Binder and Fratzl (2001) for a generaldiscussion of the various developments).

Unlike Becker and Döring, however,who confined the microscopic mechanismof cluster growth or shrinkage to the con-densation or evaporation of monomers(Sec. 5.5.1.1), in their generalized nuclea-tion theory Binder and coworkers considerthe time evolution of the cluster size distri-bution function f (i, t) more generally interms of a cluster coagulation or clustersplitting mechanism, i.e., in a single clusterreaction step i-mers (i = 2, 3, …) are alsoallowed to be added or separated from anexisting cluster (e.g., Binder, 1977; Miroldand Binder, 1977). Thus, the related kineticequation for the time evolution of f (i, t) (Binder, 1977),

(5-48)

contains four different terms. The first ac-counts for an increase of i-mers due to

ddt

f i t f i i t

f i t

f i t f i i t f i t f i t

i i ii

i ii

i

i i ii

i

i ii

( , ) ( , )

– ( , )

( , ) ( – , ) – ( , ) ( , )

,

,

–

– ,

–

,

= + ′

+

× ′ ′ ′

+ ′ ′′ =

∞

′′=

′ ′′=

′′=

∞

∑

∑ ∑

∑

a

a b

b

1

1

1

1

1

1

12

12



splitting reactions (i + i¢) Æ (i, i¢) whichare assumed to be proportional to the mo-mentary number f (i + i¢, t) of clusters ofsize (i + i¢), where the proportionality co-efficient ai + i¢, i¢ is the rate constant. Thesecond term describes the decrease of i-mers because of splitting reactions i Æ(i – i¢, i¢); as a reverse process, coagulationof clusters with sizes (i – i¢) and i¢ contrib-utes to a further increase of i-mers (thirdterm), some of which are lost again by coag-ulation reactions between i-mers and i¢-mers to yield (i + i¢)-mers (fourth term). Evidently, for i¢ = 1 the monomer evapora-tion and condensation process, assumed tobe the rate controlling step in the Becker–Döring theory, is contained in Eq. (5-48).

If, again as in classical nucleation (Sec.5.5.1.1, Eq. (5-24)), detailed balance con-ditions are assumed to apply between split-ting and coagulation, then the rates a and bin Eq. (5-48) can be replaced by a singlereaction rate W, e.g.,

(5-49)W (i, i¢) ∫ ai + i¢, i¢ C (i + i¢) = bi, i¢ C (i ) C (i ¢)

Here, C (i ) again denotes the cluster con-centration which is in thermal equilibriumwith the metastable matrix.

Numerical integration of Eq. (5-48) withEq. (5-49) provides the time evolution ofthe cluster concentration. This requires,however, a knowledge of the initial clusterdistribution f (i, t = 0), the reaction ratessuch as W (i, i¢), and of C (i). These quan-tities are not commonly available for alloysundergoing phase separation; this has sofar impeded a quantitative comparisonbetween the predictions of the generalizednucleation theory and experimental resultson nucleation kinetics.

However, the solution of the kineticequations with plausible assumptions forthe missing quantities (Mirold and Binder,1977, see also the chapter by Binder andFratzl (2001) has provided a profound,

though basically qualitative, insight intothe dynamics of cluster formation andgrowth. As is shown in Fig. 5-29, the in-itially (t = 0) monotonically decreasingcluster size distribution attains for interme-diate times a minimum at a size that corre-sponds to about that of the critical nucleusin classical nucleation theory.

The broad maximum of f (i, t) (which,in fact, does not appear in the correspond-ing curve of the Becker–Döring nucleationtheory (cf. Fig. 5-24), and its shift to largersizes with increasing time is due to thoseparticles which have nucleated at earliertimes and, hence, have already experiencedgrowth. At t Æ • equilibrium is reached.In this case large clusters (iÆ •) of the precipitated equilibrium phase with com-position cb

e are embedded in a matrix (with composition ca

e) which then still con-tains small clusters with the equilibriumsize distribution shown in Fig. 5-29.

Figure 5-29. Cluster size distribution at differentaging times (arbitrary units) as obtained from a nu-merical solution of Eqs. (5-48) and (5-49). (AfterBinder and Stauffer, 1976.)

The qualitative features of the f (i, t)-curves emerging from the generalized nu-cleation theory and displayed in Fig. 5-29are corroborated by computer simulations(cf. Sec. 5.5.6). They are indicative of theoccurrence of nucleation and growth asconcomitant processes during the earlystages of unmixing. For most solids under-going phase separation, therefore, it is notpossible to investigate experimentally nu-cleation and growth as individual processesproceeding subsequently on the time scale.Apart from well designed experiments, it isthus usually impossible to verify experi-mentally the kinetics predicted by classicalor non-classical nucleation theory (Secs.5.5.1.1 to 5.5.1.3) or predicted by growththeories (Sec. 5.5.2). This will be substan-tiated in more detail in Sec. 5.7.4.

The generalizing character of the nucle-ation theory of Binder and coworkers isfounded on the fact that, in the sense ofSec. 5.2.3, it comprises nucleation in themetastable regime as well as the transitionto spinodal decomposition in the unstableregime of the two-phase region. Conse-quently, the artificial divergency of boththe critical radius R* of the nucleus and thewavelength l* of a critical fluctuation, in-herent to the Cahn–Hilliard theories of non-classical nucleation (Sec. 5.5.1.4) and spi-nodal decomposition (Sec. 5.5.4), is nolonger discernible on approaching the spi-nodal line either from the metastable (‘nucleation’) or from the unstable (‘spinodal’) regime of the two-phase re-gion (Fig. 5-30). On crossing the spinodalcurves the size of a critical cluster de-creases steadily until it becomes compar-able to the correlation length of typicalthermal fluctuations (Binder et al., 1978).Thus, no discontinuity of the mechanism orof the decomposition kinetics is expectedto occur on crossing the border between themetastable and unstable regions.

From the experimental point of view it israther difficult and tedious to obtain statis-tically significant information from directimaging techniques (FIM, HREM) on thetime evolution of the cluster size distribu-tion during the earliest stages of unmixing.Attempts were made by Si-Qun Xiao andHaasen (1991) who employed HREM tobinary Ni–Al alloys. This information,however, is implicitly contained in thestructure function S (k, t) from small angle


Figure 5-30. a) Schematic phase diagram of a bi-nary alloy (components A and B) with a symmetricalmiscibility gap.b) Variation of the critical radius R* of a nucleus andof the wavelength lc of a critical fluctuation withcomposition of the alloy at T0 according to the non-classical Cahn–Hilliard nucleation theory (C–H non-cl.) and the Cahn–Hilliard spinodal theory (C–Hspin.). Approaching the spinodal composition ca

s

from either the metastable or the instable regioncauses R* and, respectively, to diverge. In contrast,the generalized nucleation theory of Binder and co-workers yields the size of the critical cluster to de-crease steadily until it becomes comparable with thecorrelation length of typical thermal fluctuations.There is no peculiarity on crossing the spinodalcurve.


scattering techniques, e.g., SAXS or SANS(Sec. 5.3.1.2). Binder et al. (1978) com-puted the time evolution of S (k, t) ongrounds of their cluster dynamics model.For this purpose, rather crude approxima-tions and simple assumptions had to bemade concerning the originally complexshape of the size distribution function ofthe clusters, their shapes and their compo-sition profiles. Hence, the resulting S (k, t)is only qualitative in nature and cannot beused for a quantitative comparison with ex-perimentally obtained structure functions.

As is shown in Fig. 5-31a, S (k, t) is pre-dicted to develop a peak the height Sm ∫S (km) of which increases with time due tothe increasing cluster volume fraction,while the peak position is shifted towardssmaller values of k, reflecting the coarsen-ing of the cluster size distribution withtime. Fig. 5-31b shows the variation ofboth the peak height Sm (k ∫ km) and thepeak position km with time. On a double-logarithmic plot both curves display somcurvature, though the shift of km may bereasonably well represented as a power law,

km (t) µ t–1/6 (5-50)

during the ‘aging period’ covered in Fig.5-31a and 5-31b.

Even though it is not feasible to con-vert the scaled times shown in Fig. 5-31aand 5-31b to real times, experimentalSANS curves (Fig. 5-32a) and the relatedSm (t) µ dSm (t)/dW (cf. Eq. 5-17)) andkm (t)-curves (Fig. 5-32 b) from Cu–2.9at.% Ti homogenized at TH, quenched andisothermally aged at 350°C (Eckerlebe etal., 1986) display the characteristic fea-tures qualitatively predicted by the gener-alized nucleation theory.

Kampmann et al. (1992) developed a ge-neric cluster-dynamics model that can beused for the quantitative interpretation ofexperimental kinetic data.

Unlike previous models, in this model adistinction can be made between the clus-tering phase being a disordered solid solu-tion, e.g., Cu–Co with a positive pair ex-change energy, or an ordered phase, e.g.,

Figure 5-31. a) Time evolution of the structure func-tion S (k, t), and b) of its peak height Sm and peak po-sition km as predicted by the generalized nucleationtheory of Binder and coworkers for a 3-dimensionalIsing model (c0 = 0.1, T/Tc = 0.6). After Binder et al.(1978), see also the chapter by Binder and Fratzl(2001).

Ni3Al in the Ni–Al system with a negativepair exchange energy (Staron and Kamp-mann, 2000a, b). In the latter case the con-tinuation of the atomic order must be ac-counted for when computing the growthprobability of the long-range orderedNi3Al cluster, the growth of which is as-sumed to proceed via diffusional incorpo-ration of a solute monomer (cf. Sec. 5.7.7).


Figure 5-32. a) Time evolu-tion of the structure function (SANS experiments) of aCu–2.9 at.% Ti single crystalaged at 350 °C for the giventimes. b) Time evolution of thepeak height Sm µ dS m/dW andthe peak position km. (AfterEckerlebe et al., 1986.)

5.5.4 Spinodal Theories

As has been pointed out in Sec. 5.2.3 theevolution of non-localized, spatially ex-tended solute-enriched fluctuations intostable second phase particles is treated interms of spinodal theories. The generalconcepts of the spinodal theories, includingmore recent extensions, are comprehen-sively discussed by Binder and Fratzl(2001) in Chapter 6 of this volume. We willtherefore restrict ourselves to a summary ofthe essential predictions of those theorieswhich may be examined by experimentalstudies. Although often not justified, in ma-terials science some of these predictions arefrequently employed as sufficient criteria todistinguish a spinodal reaction from an un-mixing reaction via nucleation and growth.

In their continuum model for spinodaldecomposition, which is based on the freeenergy formalism of a non-uniform binaryalloy outlined in Sec. 5.5.1.4 (Eq. (5-35)),Cahn and Hilliard (1959a, b) derived thefollowing (linearized) diffusion equation:

(5-51)∂

∂= ∇ ∂

∂⎛

⎝⎜+ ⎞

⎠⎟⎡

⎣

⎢⎢

× ∇ ⎤⎦⎥

c tt

Mn

f

cY

c t K c t

c

( , )

( , ) – * ( , )

r

r r

v

22

22

4

0

2

2

h


for the time dependence of the compositionc (r, t) at position r. As the number of at-oms per unit volume, nv, accounts for thefact that the derivative has to be taken withrespect to the concentration of componentB, M is the atomic mobility and is related tothe interdiffusion coefficient D via

(5-52)

The other symbols in Eq. (5-51) were de-fined in Sec. 5.5.1.4.

As M is always positive, D takes the signof ∂2 f /∂c2 and so is negative inside the spi-nodal regime (Sec. 5.2.2), giving rise to an‘uphill diffusion’ flux of solute atoms (cf.Fig. 5-6 b).

The linearized version of the more gen-eral nonlinear diffusion equation holds ifthe amplitude c (r) – c0 of the compositionfluctuation is rather small and both M and(∂2 f /∂c2)|c0

are independent of composi-tion. These approximations inherently con-fine the Cahn–Hilliard theory (CH theory)to the earliest stages of phase separation.

By expanding the atomic distributionc (r, t) into a Fourier series, Eq. (5-51) canbe written as

(5-53)

with the amplitude

(5-54)

of a Fourier component with wavenumberk = 2p /l if l denotes the wavelength of thecomposition fluctuation. Eq. (5-51) has asolution for every A (k, t) with

(5-55)

where A (k, 0) is the initial amplitude ofthe Fourier component with wavenumberk . The so-called amplification factor R (k)

A t A R t( , ) ( , ) exp [ ( ) ]kk kk kk= 0

A c c i( ) [ ( ) – ] exp ( )kk kk= ∫ r r r0 d

∂∂

=At

R A( )

( ) ( )kk kk kk

M D nf

cc

≡ ∂∂

˜v

2

2

0

is defined as: (5-56)

With reference to Eqs. (5-16), (5-54) and(5-55), the structure function S (k, t) or therelated small angle scattering intensity (Eq.(5-17)) at time t is obtained as

(5-57)

where S (k, 0) denotes the equal-timestructure function of the system equili-brated at the homogenization temperature.

Inside the spinodal region (∂2 f /∂c2)|c0is

negative. Thus for a given negative valueof the latter, R (k) becomes positive for allwavenumbers k satisfying

(5-58 a)

or, correspondingly, for all fluctuationwavelengths l larger than lc with

(5-58 b)

Long wavelength fluctuations with l > lc

thus will be amplified exponentiallywhereas short wavelength fluctuations willdecay with aging time. For lm ∫ 2 lc, R (l)attains a maximum giving rise to fastestgrowth of fluctuations with wavelength lm.On approaching the coherent spinodalcurve where the denominator in Eq.(5-58b) becomes zero (cf. Sec. 5.5.3), thecritical wavelength lc diverges in a similarmanner as does the critical radius R* (Fig.5-30). Thus both the linear theory on spino-dal decomposition and the non-classicalnucleation theory (Sec. 5.5.1.4) predictabrupt changes in the decomposition kinet-ics on crossing the spinodal line from ei-ther side. This has prompted several ex-perimental studies to determine the spino-

l p hc2

2

224 2

0

= ∂∂

⎛⎝⎜

⎞

⎠⎟K

f

cY

c

* –

k k h22

22

0

2 2< = ∂∂

⎛⎝⎜

⎞

⎠⎟c

2 f

cY K

c

– *

S t S R t( , ) ( , ) exp [ ( ) ]kk kk kk= 0 2

RMn

f

cY K

c

( ) – *kk = ∂∂

⎡

⎣⎢ + +

⎤

⎦⎥⎥v

2

22 2 2

0

2 2h k k

dal curve on grounds of a kinetic distinctionbetween metastable and unstable states.

In terms of the time evolution of the struc-ture function [Eq. (5-57); now assumed to beisotropic] or the small angle scattering inten-sity, S (k, t) should increase exponentially fork < kc with a peak at the time-independentposition km=kc /2. Furthermore, all S (k)

curves taken at various times should cross ata common cross-over point, at kc.

Most SAXS or SANS studies of binaryalloys, oxides or glasses that were deeplyquenched into the miscibility gap failed tocorroborate the predictions of the linear-ized CH theory. This is exemplified byCu–2.9 at.% Ti isothermally aged at350°C. As illustrated in Fig. 5-32b, thepeak position of the structure function,which is frequently identified with themean cluster spacing l

–(l– ≈ 2p/km=lm), is

not found to be time-independent but israther shifted towards smaller values of kowing to spontaneous coarsening of theclustering system. Furthermore, no com-mon cross-over point at any kc (Fig. 5-32a)exists nor does S (k, t) grow exponentiallyfor any value of k (Fig. 5-33).

According to Eqs. (5-56) and (5-57), aplot of (5-59)

versus k2 should yield a straight line with R (k) ∫ 0 at kc, whereas a pronounced cur-vature at larger values of k has been ob-served experimentally in alloys (e.g., Fig.5-34) and glass systems (Neilson, 1969).Cook (1970) attributed this curvature torandom thermal composition fluctuationswhich were not accounted for in the origi-nal CH theory. Even with the incorporationof the thermal fluctuations into the linear-ized theory, at its best the resultingCahn–Hilliard–Cook (CHC) spinodal the-ory is seen (Langer, 1973; Gunton, 1984)to be valid only for the earliest stages ofdecomposition in systems for which therange of the interaction force is consider-ably larger than the nearest-neighbor dis-tance, i.e., in systems which are almostmean-field-like in nature. This is not thecase for metallic alloys, oxides and glasses,but is for polymer blends. In fact, results

Rt

S t S( ) / ( / ) ln [ ( , ) / ( , )]k k k k k2 21 2 0= ∂∂


Figure 5-33. Time evolution of dS /dW for Cu–2.9at.% Ti aged at 350°C for various constant wave-numbers k (Eckerlebe et al., 1986).

Figure 5-34. Variation of R (k, t)/k2 with k2 as de-termined for two different time intervals (Eckerlebeet al., 1986).


from kinetic studies of decomposition invarious polymer mixtures, some of whichare cited in Table 5-1 (for reviews, cf. Gun-ton et al., 1983; Binder and Fratzl (2001),Chapter 6 of this volume), were found to beconsistent with the predictions of the line-arized spinodal CHC theory (e.g., Okadaand Han, 1986). In contrast, apart from aSANS study on the kinetics of phase separ-ation in Mn–33 at.% Cu (Gaulin et al.,1987), none of the scattering experimentson metallic alloys or oxides which are listedin Table 5-1 and which probably had beenquenched into the unstable region of themiscibility gap, corroborated the predic-tions of the CHC theory. SANS curvestaken from Mn–33 at.% Cu during short-term aging (between 65 s and 210 s at450°C) could be fitted to the CHC structurefunction with km being time-independentduring this period. Thus, it was concludedthat phase separation in Mn-33 at.% Cu is ofthe spinodal type and the kinetics follow theCHC predictions (Gaulin et al., 1987). TheS (k) curve of the as-quenched state alreadyrevealed a peak, in fact, at a smaller value ofkm than for the aged samples. This is indic-ative of the fact that phase separation oc-curred during the quench owing to a finitequench rate. In this case, as has been exten-sively discussed by Hoyt et al. (1989), km

may indeed become initially time-indepen-dent, thus pretending an experimental con-firmation of the CHC linear approximation.

In the framework of the mean-field theo-ries the spinodal curve is well defined asthe locus where ∂2 f /∂c2 vanishes (Sec.5.2.2). It can then be easily evaluated, e.g.,from an extrapolation of the thermody-namic data which are known for the single-phase solid solution, into the (‘mean-field’)spinodal regime (Hilliard, 1970). Since,however, neither metallic alloys nor glassor oxide mixtures behave as mean-field-like systems, no uniquely defined spinodal

curve exists for these solid mixtures (cf.Binder and Fratzl (2001), Sec. 6.2.5, thisvolume). Thus in general, it is currently notfeasible to predict on theoretical groundswhether a metallic, glass or oxide solidmixture has been quenched into the meta-stable or unstable region of the miscibilitygap. In this context, it remains debatablewhether each of the metallic, glass or oxidemixtures that were studied by means ofSAXS or SANS, and which have been re-ported to undergo spinodal decomposition(Table 5-1), were truly quenched into andaged within the spinodal region of thephase diagram. Nevertheless, extensive ex-perimental SAXS or SANS studies, in par-ticular on Al–Zn alloys and Fe–Cr (Table5-1), did not reveal any evidence for a dras-tic change in the time evolution of the S (k, t)curves with a change in the initial supersat-uration. For the Al–Zn systems in particu-lar, it was shown (Simon et al., 1984) thatthe clustering rate increases rapidly withincreasing supersaturation, i.e. both km andSm are larger the deeper the quench.

Extensions of the Cahn–Hilliard theoryof spinodal decomposition to ternary ormulticomponent systems have been elab-orated by de Fontaine (1972, 1973) andMoral and Calm (1971). However, to ourknowledge the kinetic predictions fromthese extensions have not yet been com-pared with experimental results, probablybecause of difficulties in determining thepartial structure functions of ternary alloys(cf. Sec. 5.3.1.2).

As has been pointed out in Sec. 5.2.3,the formulation of a ‘unified theory’ com-prising both nucleation and growth as wellas spinodal decomposition can also be at-tacked on grounds of a spinodal theory.This has been attempted in the statisticalmodel of Langer, Bar-On and Miller(1975), which takes into account thermalfluctuations and nonlinear terms which are


Table 5-1. Some small angle scattering experiments – sometimes jointly employed with other microanalyticaltools for studying the morphology – primarily designed for an investigation of spinodal decomposition. If nototherwise stated the concentrations are in at.% (N.d.: not determined.)

Decomposing solid Experimental Morphology of two- Authorstechnique phase microstructure

Metallic alloysAl–4 wt.% Cu SAXS Modulated structure Naudon et al. (1976)

Al–22% Zn SAXS Gerold and Merz (1967)

Al–22% Zn SAXS, TEM Modulated structure Agarwal and Herman (1973)

Al–20.7 … 49.1% Zn SAXS Bonfiglioli and Guinier (1966)

Al–5.3 … 6.8% Zn SANS GP-zones Hennion et al. (1982)

Al–5.3 … 12.1% Zn SANS, TEM Guyot and Simon (1981)Simon et al. (1984)

Al–22% Zn–0.1 Mg SAXS, TEM Interconnected spherical Forouhi and de Fontaineclusters at early times; (1987)

regularly spaced plateletsat later stages

Al–12 … 32 Zn SAXS (synchro- Hoyt et al. (1987, 1989)tron radiation)

Al–2.4% Zn–1.3 Mg SANS Blaschko et al. (1982, 1983)

Al–3.8 … 9% Li with SANS N.d. Pike et al. (1989)additions of Cu, Mn

Au-60% Pt SANS Singhal et al. (1978)

Fe-28% Cr–10% Co SANS, TEM, AFIM ‘Sponge-like’ structure Miller et al. (1984)

Fe–34% Cr SANS N.d. Katano and Iizumi (1984)

Fe–30 … 50% Cr SANS N.d. Ujihara and Osamura (2000)

Fe–29.5% Cr–12.5% Co Anomalous SAXS N.d. Simon and Lyon (1989)(synchrotron source)

Fe–52% Cr SANS N.d. La Salle and Schwartz (1984)

Fe–20 … 60% Cr SANS N.d. Furusaka et al. (1986)

Cu–2.9% Ti SANS, TEM Modulated structure Eckerlebe et al. (1986)

CuNiFe SANS Modulated structure Aalders et al. (1984)

CuNiFe SANS, TEM, AFIM Mottled structure Wagner et al. (1984)

CuNiFe Anomalous SAXS N.d. Lyon and Simon (1987)(synchrotron source)

Cu–2% Co SANS N.d. Steiner et al. (1983)

Mn–25 … 52% Cu SANS (analysis of N.d. Vintaikin et al. (1979)integrated intensity)

Mn–33% Cu SANS N.d. Gaulin et al. (1987)

Ni–13% Al SANS Anisotropic clustering Beddoe et al. (1984)

Ni–12.5% Si SANS, SAXS, TEM Modulated structure Polat et al. (1986, 1989)(‘side bands’)

Ni–11.5% Ti SANS, TEM Modulated structures Cerri et al. (1987, 1990)


neglected in the CH theory. The LBM non-linear theory of spinodal decompositionalso accounts for the later stages of phaseseparation, though not for the coarseningregime. If coherency strains are neglectedit approximates the kinetic evolution of thestructure function as

(5-60)

dd v

v

S tt

Mn

Kf

cS A t

Mn

k T

c

( , )–

* ( ) * ( )

k k

k k

k

=

× + ∂∂

⎛⎝⎜

⎞⎠⎟

+⎡

⎣⎢⎢

⎤

⎦⎥⎥

+

2

2

2

22

2

2

0

With certain approximations, nonlinear ef-fects and thermal fluctuations are con-tained in the term A* (t) and in the lastterm, respectively. Setting A* (t) ∫ 0 repro-duces the corresponding equation of mo-tion of the CHC theory, and, if the fluctua-tion term is also omitted, Eq. (5-57) of theoriginal linear CH theory is regained.Langer et al. (1975) proposed a computa-tional technique for solving Eq. (5-60).This required several approximations to bemade, e.g., on f (c) and on M, which inher-ently leaves the LBM approach with somefundamental shortcomings. Nevertheless,with respect to the features displayed in the

Table 5-1. Cont.


Glasses, oxides a

B2O3–(15 wt.% PbO SAXS Craievich (1975)–5 wt.% Al2O3) Acuña and Craievich (1979)

(quasi-binary system) Craievich and Olivieri (1981)

B2O3–(27 wt.% PbO SAXS Craievich et al. (1986)–9 wt.% Al2O3) (synchrotron source)

Vycor glass SANS ‘Sponge-like’ structure Wiltzius et al. (1987)consisting of a SiO2-rich

and a B2O3-alkalioxide-rich phase

SiO2–13 mole% Na2O SAXS, TEM ‘Sponge-like’ structure Neilson (1969)

TiO2–(20–80 mole%) SnO2 SAXS, TEM, X-ray Lamellar modulations Park et al. (1976)diffraction (‘side along [001] (Fig. 5-38)band’ analysis)

Polymer mixtures

Critical mixture of per- Light scattering Wiltzius et al. (1988)deuterated and protonated Bates and Wiltzius (1989)

1,4-polybutadiene

polybutadiene and Izumitani and Hashimoto (1985)styrene–butadiene

copolymer mixtures

polystyrene–polyvinyl- Light scattering Sato and Han (1988)methylether (PS–PVME) Sato et al. (1989)

Snyder et al. (1983a, b)Hashimoto et al. (1986a, b)

Okada and Han (1986)

a For a comprehensive survey on oxides and glasses up to 1978, see Jantzen and Herman (1978).


Figure 5-35. a) Phase diagram of a binary alloy witha symmetric miscibility gap centered at csym.b) Time evolution of the structure function of an al-loy with c0 = csym quenched and aged at TA. The in-sert shows the distribution function of compositionconfigurations at two different times. At t = 80 theevolving two-phase structure with compositions c1

and c2 or y1 and y2, respectively, with

already becomes discernible. All units are dimen-sionless.c) LBM predictions on the variation of R (k, t)/k2

with k2 as determined from b) according to Eq.(5-59). Qualitatively there is good agreement withthe LBM predictions and experimental results, e.g.,displayed in Fig. 5-34. (After Langer et al., 1975.)

yc c

c cy

c c

c c1

12

2= =–

–

–

–sym

esym

syme

symand

a b

Figure 5-36. Time evolutionof the structure function (SANS intensity) of Fe–40at.% Cr aged at 515°C for thegiven times. Full lines arecalculated from the LBM the-ory with three free fitting pa-rameters. (After Furusaka etal. 1986.)


time evolution of S (k, t), the predictions ofthe LBM theory (Fig. 5-35) are in rathergood agreement with both experimentalstudies (e.g., Fig. 5-32 a and 5-34) andMonte Carlo studies (Sec. 5.5.6, Fig. 5-41).

In their study of phase separation inFe–(20, 30, 40 and 60) at.% Cr alloys, Fu-rusaka et al. (1986) compared the experi-mental S (k, t) curves with Eq. (5-60) ofthe LBM theory. For shorter aging times at515°C, the experimental data points takenfrom Fe–40 at.% Cr could well be fitted byEq. (5-60) (Fig. 5-36) if kT/K*, MK*, andA*/K* were employed as three independentfitting parameters. On the grounds of theagreement with the spinodal theory ofLBM, and additionally, as this particularalloy was evaluated to lie within the (mean-field) spinodal region, it has beenconcluded that Fe–40 at.% Cr is a ‘spino-dal alloy’.

Based on a similar LBM analysis ofSANS data from Fe–30 … 50 at.% Cr, Uji-hara and Osamura (2000) corroborated thisconclusion recently.

The nonlinear theory of spinodal decom-position developed by Langer (1971) forbinary alloys has also been extended to ter-nary substitutional systems (Hoyt, 1989)by deriving the time-dependent behavior ofthe three linearly independent partial struc-ture functions (cf. Sec. 5.3.1.2). An experi-mental examination of the kinetic predic-tions, however, is still lacking.

5.5.5 The Philosophy of Defininga ‘Spinodal Alloy’ –Morphologies of ‘Spinodal Alloys’

Referring to the previous section it israther difficult or even impossible to assesson a thermodynamic basis whether an al-loy, glass or oxide system was trulyquenched into and aged within the spinodal

region of the miscibility gap. Furthermore,the various spinodal theories are difficult tohandle and many parameters in the result-ing kinetic equations are often not avail-able for most solid mixtures. Hence, likethe non-classical nucleation theory, thevarious elegant theories that describe thekinetics of phase separation of a ‘spinodalalloy’ are of little use for the practical met-allurgist.

Because of these problems, materialsscientists have up to now employed mor-phological criteria for the definition of a‘spinodal alloy’. These are simply relatedto the predictions of the linear spinodal CHtheory concerning the morphological evo-lution of a solid mixture undergoing spino-dal decomposition.

Most crystalline solid solutions show avariation of lattice parameter with compo-sition leading to coherency strains. The as-sociated strain energy fel = h2 Y (c–c0)2

(Eq. (5-34)), which is accounted for in Eq.(5-51), reduces the driving force for phaseseparation. This effect shifts the locus ofthe original chemical (mean field) spinodalto lower temperatures, yielding the coher-ent spinodal curve (Eq. (5-58)). If the pa-rameter Y (cf. Sec. 5.2.4), which is a com-bination of various elastic constants, de-pends on the crystallographic direction, fel

also becomes anisotropic. Therefore, thelocus of the coherent spinodal may alsovary with the crystallographic direction.This becomes particularly discernible forthe tetragonal TiO2–SnO2 oxide system (Park et al., 1975, 1976), where fel attains aminimum for composition waves along[001] and is larger for waves along [100]and [010]. Hence, the coherent spinodalsplits up into a [001], a ·101Ò, and a ·100Òbranch (Fig. 5-37). As a consequence theSnO2-rich modulations form preferentiallyalong [001], giving rise to a lamellar struc-ture at later aging stages (Fig. 5-38). For

most cubic metallic systems the elastic an-isotropy parameter A (Sec. 5.2.4 ii) is posi-tive, rendering the minimum fel along theelastically soft ·100Ò directions. Accordingto the CH theory the growth rate will thusbe highest along the three ·100Ò directions,giving rise to the frequently observed mod-ulated precipitate microstructures (Table5-2, Figs. 5-18, 5-19b). In isotropic materi-als such as polymers, glasses, or, for in-stance, Fe–Cr and Fe–Cr–Co alloys, themodulations do not grow along any prefe-rential directions. The resulting two-phasemicrostructure is of the ‘sponge-like’ typeand sometimes referred to as a ‘mottled’ or‘interconnected’ precipitate microstructure(e.g., Fig. 5-16, 5-19 c, 5-20). Based onthese considerations, two-phase alloys dis-playing either lamellar, modulated orinterconnected precipitate microstructuresare commonly termed ‘spinodal alloys’ bymetallurgists. Furthermore, the morphol-ogy of a two-phase material is sometimesemployed to derive the locus of the spino-dal. This is illustrated in Fig. 5-39 for theCr-rich ferrite phase of a cast duplex stain-

less steel that undergoes phase separationduring tempering between 350°C and450°C with an associated embrittlement(Auger et al., 1989). Increasing the Cr con-tent of the ferritic Fe–Ni–Cr solid solutionleads to the formation of Cr-rich a¢-precip-itates during tempering. As the a¢-phase isdiscernible as individual particles byCTEM, it is concluded that phase separa-tion occurred via nucleation and growth. Afurther increase in the Cr content, to about25 wt.%, yielded a ‘sponge-like’ micro-structure after tempering below 400°C.This was attributed to a spinodal mecha-nism. Thus the spinodal is drawn as the linethat separates the two morphologies (Fig.5-39).

It must be pointed out, however, thatinterconnected or modulated structuresrepresent two-phase microstructures in thelater stages of the reaction (e.g., Fig. 5-18).Even though they are widely believed toresult from spinodal decomposition in thesense of the CH theory, their formationmight be of a rather different origin. For in-stance, a strong elastic interaction of a high


Figure 5-37. The phase diagram and the spinodalcurves for compositionwaves along [001], ·101Ò,and ·100Ò directions forelastically anisotropic te-tragonal TiO2–SnO2. Thespinodals were calculatedon the basis of the regularsolution model. (FromPark et al., 1976.)


number density of individual nuclei, eachof which is surrounded by a solute-de-pleted zone in which no further nucleationcan occur, may also lead to regularly ar-ranged precipitates, i.e., modulated struc-tures (Ardell et al., 1966; Doi et al., 1984,1988; Doi and Miyazaki, 1986). On theother hand, initially interconnected micro-structures in Cu–Ni–Fe alloys were foundto break up into isolated plates (Piller et al.,1984). The later-stage microstructure isthus neither sufficient to draw any conclu-sions on the early-stage decompositionmode nor for a definition of a ‘spinodalalloy’. We are therefore still left with thequestion as to what really is a spinodal al-loy? To answer this question unequiv-ocally, we must verify by any microanalyt-ical technique that the amplitude of thecomposition waves increases graduallywith time until the evolving second phasehas finally reached its equilibrium compo-sition. Such an experimental verification isa difficult task and, to our knowledge, was

only shown for decomposing Fe–Co–Cr (Figs. 5-16 and 5-17), AlNiCo permanentmagnetic materials (Hütten and Haasen,1986) and Fe–Cr (Brenner et al., 1984) bymeans of AFIM, and for a phase separatingpolystyrene–polyvinylmethylether poly-mer mixture employing nuclear magneticresonance methods (Nishi et al., 1975).

As has ben outlined in Sec. 5.2.3, re-gardless of whether phase separation oc-cours via nucleation and growth or via spi-nodal decomposition, the underlying mi-croscopic mechanism is diffusion of thesolvent and solute atoms. In this sensethere is thus no need to distinguish betweenthe two different decomposition modes andthe term ‘spinodal alloy’ is simply seman-tic in nature. This is also reflected in thevarious attempts to develop ‘unifiedtheories’ comprising either mode.

From the practical point of view the mi-crostructure of virtually all technical two-phase alloys corresponds to that of the laterstages. Hence, the practical metallurgistworries little about the initial stages of un-mixing but is instead interested in predictingthe growth and coarsening behavior of pre-cipitate microstructures in the later stages.This will be the subject of Secs. 5-6 and 5-7.

Figure 5-38. CTEM micrograph of equimolarTiO2–SnO2 displaying a lamellar structure consist-ing of alternating TiO2- and SnO2-rich layers formedafter aging at 900 °C for 60 min. From Park et al.(1976).

Figure 5-39. Variation of the phase boundaries inthe Fe–Ni–Cr ferrite phase with chromium content asderived from microstructural observations. (AfterAuger et al., 1989.)


Table 5-2. TEM and AFIM studies on phase separating solids that revealed ‘spinodal precipitatemicrostructures’. If not otherwise stated the concentrations are in at.%.


Metallic alloysAl–4 wt.% Cu TEM (‘side-bands’) Modulated structure Rioja and Laughlin (1977)

Al–(2.4–3%) Li HREM, Radmilovic et al. (1989)X-ray diffraction

Cu3–4MnxAl TEM (‘side-bands’) Modulated structure; Bouchard and Thomas (1975)at later stages

Cu2MnAl + Cu3Al-plates

Cu–(1.5–5.2 wt.%) Ti TEM (‘satellite Modulated structure Laughlin and Cahn (1975)analysis’)

Cu–2.7% Ti AFIM Modulated structure Biehl and Wagner (1982)

Ni–12% Ti AFIM Modulated structure Grüne (1988)

Ni-29% Cu–21% Pd TEM Modulated structure Murata and Iwama (1981)

Ni-base superalloys AFIM Short wavelength Bouchon et al. (1990)(≈2.5 nm) Cr fluctuation

in g-phase

Nimonic 80 A TEM (‘side-bands’) Modulated structure Wood et al. (1979)

Fe–25% Be AFIM Modulated structure Miller et al. (1984, 1986)

CuNiFe AFIM Interconnected percolated Piller et al. (1984)structure (cf. Fig. 5-19c)

CuNiFe TEM (‘side-bands’) Modulated structure Wahi and Stager (1984)(cf. Fig. 5-18) Livak and Thomas (1974)

Co–10% Ti FIM Modulated structure Davies and Ralph (1972)

Co–3 wt.% Ti TEM (‘side-bands’) Modulated structure Singh et al. (1980)–1 … 2 wt.% Fe

Mn–30 wt.% Cu TEM, magnetic Interconnected Yin et al. (2000)susceptibility structure

Steels

Cast Duplex stainless TEM, AFIM After decomposition Auger et al. (1989)original ferritic

Fe–Cr–Ni shows‘sponge-like’ structure

Ferrous martensiteFe–15 wt.% Ni–1 wt.% C TEM Tweed structure Taylor (1985)

Fe-25 wt.% Ni TEM Tweed structure Taylor (1985)–0.4 wt.% C

Fe–13 … 20 wt.% Mn X-ray diffraction Modulated structure at Miyazaki et al. (1982)(‘side-bands’), TEM early times; isolated

Fe2Mo particles at laterstages

Amorphous alloysTi50Be40Zr10 AFIM Wavy composition profile Grüne et al. (1985)


5.5.6 Monte Carlo Studies

The basic features of early stage decom-position as a stochastic process, without ar-tificial distinction between nucleation,growth and coarsening regimes, have beenextensively studied in model alloys bymeans of Monte Carlo (MC) computer sim-ulations, mainly by Lebowitz, Kalos andcoworkers (Lebowitz and Kalos, 1976;Binder et al., 1979; Marro et al., 1975,1977; Penrose et al., 1978; Lebowitz et al.,1982; see also Binder and Fratzl, 2001;Chapter 6 of this volume). The binary modelalloys are usually described in terms of athree-dimensional Ising model with pairwisenearest-neighbor interactions on a simplerigid cubic lattice the sites of which are oc-cupied by either A or B atoms, and a phasediagram which displays a symmetrical mis-cibility gap centered at 50 at.% solute con-centration (Fig. 5-40). The microscopic dy-namics of this system have commonly beendescribed by the Kawasaki model (Kawa-saki, 1972). There a nearest-neighbor pairof lattice sites is chosen at random, then theatoms on those sites may be interchangedwith a probability that depends on the ener-gies of the configuration before and afterthe exchange in such a way that detailedbalancing holds (Penrose, 1978). Monte

Carlo techniques are then employed tocarry out this stochastic process.

The Kawasaki dynamics employed forMC simulations are far from being repre-sentative of a real binary alloy system as

Table 5-2. Cont.


Glasses, ocides

TiO2–60 mole% SnO2 TEM Tetragonal system with Stubican and Schultz (1970)lamellar modulations

along [001]

TiO2–50 mole% SnO2 HREM Tetragonal system with Horiuchi et al. (1984)lamellar modulations

along [001]

SiC–(50–75 mole%) AlN TEM (‘satellites’) Modulated structure Kuo and Virkar (1987)

Figure 5-40. Phase diagram of the 3-dimensional Is-ing model approximating a binary model alloy. Interms of the mean field theory (Sec. 5.2.2) ‘alloys’#1 to # 4 are quenched into the metastable regime, ‘alloys’ # 5 to # 7 beyond the classical spinodal lineare quenched into the unstable region of the phase di-agram. (After Lebowitz et al., 1982.)

the atomic exchange is assumed to occurdirectly rather than indirectly via the va-cancy mechanism. Nevertheless, MC simu-lations based on Kawasaki dynamics on anatomic scale yielded the time evolution ofthe cluster configuration and the structurefunction S (k, t) of model alloys quenchedinto any region below the solubility line(Fig. 5-40) without worrying about com-plicating factors, such as an insufficientquenching rate, excess vacancies, or latticedefects, which we commonly face when in-vestigating real alloys (Sec. 5.3.2). Fur-thermore, MC simulations have allowed acritical examination of the various theoret-ical approaches in terms of cluster dynam-ics models or spinodal models to be made.For practical computational limitations,however, the maximum size of the modelalloy has commonly been restricted toabout 50 ¥ 50 ¥ 50 lattice sites. Because ofthe size limitation, in general MC kineticexperiments can only cover the earlierstages of a precipitation reaction in alloyswhere the supersaturation is sufficientlyhigh for the formation of a large numberdensity of clusters or nuclei, and where thecluster sizes are still considerably smallerthan the linear dimension of the modelsystem. This is frequently not the case inreal alloys.

In essence, the results from MC simula-tions based on Kawasaki dynamics have re-covered the predictions both from the gen-eralized nucleation theory of Binder andcoworkers (Sec. 5.5.3) and from the non-linear spinodal LBM approach. In particu-lar, the time evolution of the structure func-tion in the kinetic Ising model displaysqualitatively the same features as the corre-sponding ones obtained from the latter the-ories (cf. Figs. 5-31 and 5-35b) or from ex-periment (e.g., Fig. 5.32a).

For instance, S (k, t) of ‘alloy’ # 4 (Fig.5-41b), which according to Fig. 5-40 lies

close to the spinodal line within the meta-stable regime, evolves similarly to that of‘alloy’ # 5 quenched into the center of thespinodal region (Fig. 5-41c). Thus, inagreement with the generalized nucleationtheory and the LBM spinodal approach, butunlike the predictions of linearized CH the-ory of spinodal decomposition, MC simu-lations again reveal i) no evidence for anyabrupt change of the decomposition kinet-ics on crossing the spinodal curve, ii) nocommon cross-over point of the S (k)curves taken after different aging times, iii)no exponential growth of the scattering in-tensity for a certain time-independent wavevector in any time regime, and, further-more, iv) the peak position of S (k, t) at km

is not found to be time-independent but isshifted towards smaller values of k indicat-ing the immediate growth of clusters.

Frequently, the MC data for the time ev-olution of the peak position (km) and thepeak height (Sm) of the structure functionhave been fitted to simple power laws (cf.Sec. 5.8.2), such as km (t) µ t –a and Sm(t)d tb ; a and b were estimated to range from0.16 to 0.25 and 0.41 to 0.74, respectively,depending on the initial supersaturation ofthe ‘alloy’ (Marro et al., 1975, 1977; Sur etal., 1977). Lebowitz et al. (1982) pointedout, however, that due to the finite (small)size of the system it is difficult to extractfrom computer simulations precise and re-liable information about the analyticalform of km (t), for example and that it ispossible to fit the same MC data with otherfunctional forms than the power laws givenabove. (In Sec. 5.8.2 we will show, in fact,that apart from the late stages of coarsen-ing, it is usually not feasible to interpret ex-perimental kinetic data over an extendedaging period in terms of a power-law be-havior with a time-independent exponent.)

The Kawasaki dynamics are based on anunrealistic exchange mechanism between



neighboring atoms. In order to account forthe more realistic case of atomic diffusionbeing based on the vacancy mechanism,the Ising model was more recently ex-tended to binary alloys comprising vacan-cies (Yaldram and Binder, 1991 a,b; Fratzl

and Penrose, 1994). Comparing the growthrates of clusters in the Ising model of atwo-dimensional model alloy showed thatthe asymptotic regime of cluster coarsen-ing (R

–~ t1/3; cf. Sec. 5.6.2) is approached

much faster with vacancy dynamics than

Figure 5-41. Time evolution of the structure func-tion at T/Tc = 0.59 as obtained from MC computersimulations. With reference to Fig. 5-40:a) for ‘alloy’ #1 with c0 = 0.05 (after Lebowitz et al.,1982),b) for ‘alloy’ #4 with c0 = 0.2 (after Sur et al., 1977)andc) for ‘alloy’ #5 with c0 = 0.5 (after Marro et al.,1975). The given times are in units of a Monte Carlostep, i.e., the average time interval between two at-tempts at exchanging the occupancy of a specificsite. The numerical results for S (k, t) at the discretevalues of k were connected by straight lines.

with Kawasaki dynamics, in particular atlarger supersaturations; the cluster shapesare found not to depend on the particulardynamics model (Fratzl and Penrose,1994).

In a first attempt to interpret experimen-tal data with kinetic data obtained fromMC simulations, Soisson et al. (1996) sim-ulated phase separation in a Fe–1.34 at.%Cu alloy. The alloy was modeled as a rigidb.c.c. crystal with 2 ¥ L3 lattice sites (L var-ying from 32 to 64) containing one singlevacancy; the energetics and the parametersof the atomistic kinetic model were ad-justed to the thermodynamic and diffusiondata of the Fe–Cu system. The MC resultsconfirmed earlier results from magneticSANS studies (Beaven et al., 1986) that theprecipitating clusters are pure copper; fur-thermore the time evolution of the precipi-tated volume fraction agreed quite wellwith the one experimentally determinedfrom resistivity measurements (Lê et al.,1992). In the later stages dynamical scalingbehavior was shown to hold, indicating thatthe cluster pattern remains similar as agingtime increases (cf. Sec. 5.8.1).

5.6 Coarsening of Precipitates

5.6.1 General Remarks

For most two-phase alloys, the simplemodel of diffusional growth of isolatednon-interacting particles with uniform size,on which Eq. (5-47) is based, frequentlydoes not give a realistic description of thefurther dynamic evolution of the precipi-tate microstructure beyond its nucleationstage. In reality, towards the end of the nu-cleation period a more or less broad parti-cle size distribution f (R) is established(Fig. 5-10 and 5-29). According to theGibbs–Thomson equation (5-45), the solu-

bility cR (R) in the presence of small parti-cles with a large ratio of surface area tovolume is larger than that for larger ones.With reference to Eq. (5-43), this leads to asize-dependent growth rate, which is posi-tive for larger particles with c > cR and neg-ative for smaller ones with c < cR. Thegrowth rate becomes zero for particles withc = cR which are in unstable equilibriumwith the matrix. Their radius R* is derivedfrom Eq. (5-46) as (5-61)

Hence, driven by the release of excessinterfacial energy, larger precipitates willgrow at the expense of smaller ones whichdissolve again given rise to a change in theprecipitate size distribution. This process,which is commonly referred to as coarsen-ing or Ostwald ripening2, frequently re-duces the precipitate number density form≈1025 m–3 to less than 1019 m–3 in typicaltwo-phase alloys during aging (cf. Fig.5-3). Usually, the coarsening process isconsidered to be confined to the lateststages of a precipitation reaction. However,as will be shown in Sec. 5-7, coarseningmay accompany the growth process out-lined in Sec. 5.5.2, or may even start whilethe system is still in its nucleation period,depending on the initial supersaturation ofthe solid solution.

5.6.2 The LSW Theory of Coarsening

In essence, the coarsening of randomlydispersed second-phase particles is a multi-particle diffusion problem which is diffi-cult to handle theoretically. In their classicLSW coarsening theory, Lifshitz and Slyo-

RV

R T c cK

c c*

ln ( / ) ln ( / )≈ ≡ ′

2 1 1ssab b

aab

age e


2 In its original meaning, Ostwald ripening is con-fined to a coarsening reaction where the second-phase particles act as the only sinks or sources of so-lute atoms.

5.6 Coarsening of Precipitates 371

zov (1961), and Wagner (1961) calculatedthe time evolution of f (R, t) which satis-fies the continuity equation:

(5-62)

On the basis of the continuity equation andof Eq. (5-43), the time evolution of themean particle radius R

–(t) and the precipi-

tate number density Nv (t) are derived. Cer-tain approximations, however, had to bemade in order to solve the equations of mo-tion analytically:

a) Both terminal phases a and b are dilutesolutions; their thermodynamics can bedescribed by a dilute solution and thelinearized version (Eq. 5-46) of theGibbs–Thomson equation may be used.

b) The precipitated volume fraction fp =(4p /3) R

–3 Nv is close to zero. In such adilute system interparticle diffusionalinteractions such as those occurring inmore concentrated alloys (cf. Sec.5.6.3) can be neglected, and a particleonly interacts with the infinite matrix.

c) fp ≈ const., i.e., the decomposition isclose to completion with the supersatu-ration Dc ≈ 0. This inherently confinesthe LSW theory to the late stages of aprecipitation reaction.

With these assumptions the LSW theoryyields in the asymptotic limit Dc (t) Æ 0temporal power laws

R–3(t) = KR

LSW t (5-63a)

Nv (t) = KNLSW t –1 (5-63b)

Dc (t) = KCLSW t –1/3 (5-63c)

for the time evolution of the average parti-cle radius, the particle density and thesupersaturation, respectively, and a particlesize distribution fLSW (R/R

–), the shape of

which is time invariant under the scaling ofthe average particle size R

–(Fig. 5-42). It

∂∂

+ ∂∂

⎡⎣⎢

⎤⎦⎥

=ft R

fRt

dd

0

must be emphasized that these predictionshold only in the limit t approaching infin-ity, since the particle size distribution thatis present at the beginning of coarseningcan be quite different from the time-invari-ant form.

Extensions of the LSW theory to binarysystems with non-zero solubilities and non-ideal solution thermodynamics, and ade-quate modification of the Gibbs–Thomsonequation, have reproduced the temporalpower laws (Eq. 5-63). The correspondingrate constants for this more realistic casewere derived as (Schmitz and Haasen,1992; Calderon et al., 1994):

(5-64a)

(5-64b)

(5-64c)Kc c

DKC

e e

R=9

42 3( – ) /b a

Kf

K fNp

R=

3

41

3p

KD V

c c GR e e=

′′8

9 2b ab

b

s

a a( – )

Figure 5-42. CTEM analyses of the normalized g¢-particle size distribution in Ni–8.74 wt.% Ti after ag-ing at 692°C for the given times. For comparison, theshape-invariant distribution function fLSW of theLSW theory is included. (After Ardell, 1970.)

Ga≤ is the second derivative of the molarHelmholtz energy of the a phase; f3 is thethird moment of the time-independentscaled particle radius distribution function f (R/R

–) which in the zero volume fraction

limit (i.e. f = fLSW; see Sec. 5.6.3) is 1.129.For an ideal solution and the dilute solu-

tion limit (i.e. cae 1, cb

e ≈1) the rate con-stant KR (Eq. 5-64a) adopts the form

(5-65)

of the original LSW theory (Calderon etal., 1994).

Numerous experimental studies on awide variety of two-phase alloys attemptedto examine the coarsening kinetics pre-dicted by the LSW theory. Regardless ofthe particular alloy system and the micro-analytical technique employed (e.g.,CTEM on Ni–Al, Ni–Ti: Ardell, 1967,1968, 1970; and on Ni–Si: Cho and Ardell,1997; AFIM on Ni–Al: Wendt and Haasen,1983; SANS on Fe–Cu: Kampmann andWagner, 1986), these studies frequently re-vealed the experimental size distributionfunction to be considerably broader thanfLSW (R/R

–) (Fig. 5-42), whereas a plot of R

–3

or Nv–1 vs. t yielded more or less straight

lines (due to limited statistics, the errorbars are usually rather large). From theslopes of these LWS plots the product sab Dcan be derived with ca

e usually taken fromthe known phase diagram. Frequently Dcwas also measured and plotted versus t–1/3

(Eq. (5-63c)) in order to determine D/s 2ab

(e.g., Ardell, 1967, 1968, 1995; Wendt and Haasen, 1983). Thus apparently abso-lute values for both sab and D have beendetermined from values of sab D andD/s 2

ab. However, the rate constants KRLSW

and KCLSW in Eq. (5-63) were derived

with the assumption c): Dc≈0 or fp= const.Once this condition is fulfilled, it is no

KD V c

R TRLSW

e

g=

8

9b a abs

longer feasible to follow minor changes ofDc with time quantitatively by any of theexperimental techniques frequently em-ployed, such as CTEM, AFIM, SAXS orSANS (Sec. 5.3.1); in the asymptotic limitDc Æ0, it even appears difficult to measureDc (t) in alloys containing a ferromagneticphase with magnetic techniques (Ardell,1967, 1968), though these direct methodsare certainly more sensitive than CTEM,AFIM or SAS. On the other hand, in thoseearlier decomposition stages, where Dc (t)is experimentally accessible, an LSW anal-ysis cannot be performed, and, in particu-lar, the rate constant KR

LSW (Eq. (5-65)) isno longer valid (see Sec. 5.7.4.3). Thus, inpractice, an LSW analysis based on Eqs.(5-63) and (5-65), at its best, can be per-formed only in dilute systems and yieldsonly a value for the product Dsab. This factwas ignored in most LSW analyses basedon the independent measurement of bothR–

(t1/3) and Dc (t –1/3); hence, the values ofD and sab derived from this type of analy-sis must be regarded with some reserva-tion. However, frequently the interfacialenergy was derived exclusively from Eq.(5-65) under the assumption that the effec-tive diffusivity D is identical with that ob-tained from an extrapolation of availablehigh-temperature data to the aging temper-ature, thus neglecting the influence ofquenched-in vacancies (Sec. 5.3.2.1). (InSec. 5.7.4.4 two different methods will bediscussed which allow a separate deriva-tion of sab and D to be made from experi-mental data, one method without evenknowing Dc (t).)

For non-ideal solid solutions, sab mustbe derived from Eq. (5-64a). Apart fromknowing D, this requires detailed informa-tion on the thermodynamics of the a matrixphase in order to derive Ga≤ . For many bi-nary alloys the necessary thermodynamicfunctions have become available via the



CALPHAD method (Sec. 5.2.4). If notavailable, they must be derived from moreor less adequate solution models, with theregular solution model being the less so-phisticated one. For the case of a binarymodel alloy with a miscibility gap and reg-ular solution behavior, Calderon et al.(1994) demonstrated that depending on ag-ing temperature, sab may be significantlydifferent if assessed from the correct formof the rate constant for a regular solution or from the original LSW form, i.e. Eq. (5-65).

5.6.3 Extensions of the Coarsening Theory to Finite Precipitate VolumeFractions

There have been many efforts to extendthe LSW theory to the more realistic caseof a finite precipitated volume fraction andto investigate its influence on the shape ofthe distribution function and the coarsen-ing kinetics (Ardell, 1972). The centralchallenge is to determine the effects ofinterparticle diffusional interactions on thegrowth rate of a particle of a given size.Theories accounting for these diffusionalinteractions fall into two broad categories:effective medium theories, and statisticaltheories that are based upon a solution tothe diffusion field in this multiparticlesystem. All theories are in agreement thatthe presence of a non-zero volume fractiondoes not change the exponents of the tem-poral power laws given by LSW (Eq. (5-63)) or the existence of a time-indepen-dent scaled particle size distribution. Thepresence of a non-zero volume fraction,however, does change the amplitudes of thetemporal power laws and the shape of thescaled distribution functions. The rate con-stant KR

LSW increases with the volume frac-tion (Fig. 5-43) and the time-independentdistribution function becomes broader and

more symmetric as the volume fraction in-creases (Fig. 5-44). The reason for the in-crease in the rate constant is clear: as thevolume fraction increases, shrinking parti-cles move closer to growing particles andthus the concentration gradients are larger,and the rate of growth and shrinkage in-creases.

The effective medium theories deter-mine the growth or shrinkage rate of a par-ticle of a given size using a medium that isconstructed, presumably, to give the statis-tically averaged growth rate of particles ofa certain size. Examples of such theoriesare due to Tsumuraya and Miyata (1983),Brailsford and Wynblatt (1979) and, morerecently, Marsh and Glicksman (1996).While the effective medium employed byBrailsford and Wynblatt has been shownanalytically to be consistent with a solutionto the multiparticle diffusion equation, this

Figure 5-43. The rate constant K, relative to that ofLSW, KLSW, as a function of the volume fraction fP.Shown on the figure are the predictions of Brailsfordand Wynblatt (BW), Marsh and Glicksman (MG),Marqusee and Ross (MR), Tokoyama and Kawasaki(TK) and the simulations of Akaiwa and Voorhees(AV) where denote the use of the monopole ap-proximation of the diffusion field and denotemonopole and dipole approximations to the diffusionfield.

has not been done with the other theories.This, then, is the major disadvantage ofthese theories. A significant advantage,however, is that they are simple to applyand make predictions over a wide range ofvolume fractions, even as high as 0.95. Inaddition, both the Brailsford and Wynblattand Marsh and Glicksman theories corre-spond reasonably well with the first-princi-ples statistical theories for volume frac-tions below 0.3. Above 0.3, these are theonly theories that make predictions on therate constants and particle size distribu-tions.

The statistical theories are based upon asolution to the multiparticle diffusion prob-lem. The particles are assumed to be spher-ical and the concentration in the matrix atthe interface of each particle is given by theGibbs–Thompson equation, (5-45). Forvolume fractions below 0.1 the solution tothe diffusion equation is represented as amonopole source or sink of solute in thecenter of each particle (Weins and Cahn,1973). At higher volume fractions dipolar

terms must be included (Akaiwa and Voor-hees, 1994). Given, then, a spatial distribu-tion of particles and a particle size distribu-tion, it is possible to determine the coarsen-ing rate of each particle. The more chal-lenging step is to determine from this infor-mation the statistically averaged growthrate of a particle of a given size. This canbe done analytically (Marqusee and Ross,1984; Tokuyama and Kawasaki, 1984; Yaoet al., 1993), or numerically by placing alarge number of particles in a box and de-termining their coarsening rate (Voorheesand Glicksman, 1984; Akaiwa and Voo-rhees, 1994; Mandyam et al., 1998). Al-though all of these theories begin with thesame solution to the diffusion equation,none of the predictions for the rate con-stants and particle size distributions are inagreement, illustrating the difficulty in per-forming the statistical averaging.

Nevertheless, a number of qualitative ef-fects of a finite volume fraction becomeclear after examining the predictions ofthese theories. (a) The growth rate of a par-ticle of a given size is a function of its sur-rounding particles. For example, if a parti-cle of size R1 is surrounded by particles ofradii R < R1, this particle will grow, but ifthis particle is surrounded by particles withradii R > R1, it will shrink. Thus, unlike inthe LSW theory the growth rate of a parti-cle is not solely a function of its size. (b)The local diffusional interactions give riseto spatial correlations between particlesthat are not random. For example, the aver-age interparticle separation for a systemundergoing coarsening is larger than thatfor a random spatial distribution. This isbecause the probability of finding a smalland large particle almost touching, whichcan occur in a system with a random spatialdistribution, is low, as the strong diffu-sional interactions that occur when a largeand a small particle are located close to


Figure 5-44. Steady-state (time-invariant) precipi-tate size distributions at various volume fractions.For comparison the corresponding LSW distributionfor zero volume fraction is also shown. (After To-kuyama and Kawasaki, 1984.)


each other causes this small particle to dis-appear. At larger interparticle separationdistances, however, large particles tend tobe surrounded by small particles as theyare feeding solute to the growing large par-ticles (Akaiwa and Voorhees, 1994). Thus,there are both spatial and particle size cor-relations (Marder, 1987). Information onthe spatial correlations between particles isneeded to determine the structure functionthat is measured using small angle scatter-ing. The agreement between the predic-tions of Akaiwa and Voorhees (1994) forthe structure function and the experimen-tally measured structure function in Al–Lialloys is reasonable (Che et al., 1997). (c)Another effect of interparticle diffusionalinteractions is that the center of mass of theparticles is not fixed, but moves in a man-ner consistent with the concentration gra-dients in the system. This particle motionhas been determined theoretically (Mar-der,1987; Akaiwa and Voorhees, 1994), andhas been observed in experiments on coars-ening in transparent solid–liquid mixtures(Voorhees and Schaefer, 1987).

Experiments in which the coarsening ki-netics of solid particles in a liquid (Fig. 5-45) have been measured show clearlythat the coarsening rate increases with vol-ume fraction (Hardy and Voorhees, 1988,and references therein). In two-phase solidsystems, however, there are reports that therate constant is independent of the volumefraction (Cho and Ardell, 1997). The rea-son for these contradictory results may bethe elastic stress that is present in the two-phase solid systems. Although the pre-dicted particle size distributions of the non-zero volume fraction theories are broaderthan those of LSW, nearly all experimen-tally measured particle size distributionsare broader than the predictions of the non-zero volume fraction theories. Until re-cently there have been no experiments per-

formed in a system wherein the materialsparameters, such as the interfacial energyand diffusion coefficient, are known (inde-pendent of a coarsening experiment) andwhere this system satisfies all the assump-tions of theory. Therefore, it is unclear ifthe disagreement between theory and ex-periments is due to an artifact of the systememployed in the experiments or a defect inthe theories. Recent experiments that em-ploy a system in which the materials pa-rameters are known and satisfy all the as-sumptions of theory have been performedusing solid Sn-rich particles in a Pb-richeutectic liquid (Alkemper et al., 1999).There was no convective motion of the par-ticles because the experiments were per-formed in the microgravity environment ofthe Space Shuttle. These experiments showthat there can be very long transients asso-ciated with the coarsening process. In par-ticular, during the course of the experimentthe initially broad particle size distributionevolved slowly towards that predicted bytheory. Even with a factor of four change inthe average particle size the distributionnever reached the steady-state distributionpredicted by theory. This may be the reasonwhy the experimentally measured particlesize distributions (measured using othersystems) rarely agree with theory. Never-theless, the evolution of the distributionwas found to correspond quite well withthe predictions of two theories for transientOstwald ripening.

Coarsening of two-phase alloys withmodulated microstructures consisting oflarge volume fractions ( fp ≥ 30%) of iso-lated coherently misfitting particles, e.g.,Cu–Ni–Fe and Cu–Ni–Cr (Fig. 5.19),Cu–Ni–Si (Yoshida et al., 1987), or mod-ulated Co–Cu (Miyazaki et al., 1986), wasreported to be rather sluggish. The time ex-ponent a of the coarsening rate, defined byR– µ t a, of these alloys was consistently


Figure 5-45. Evolution of Sn-rich particles with fp = 0.64 in a liquid Pb–Sn matrix during coarsening at 185°C.The top row is at constant magnification and illustrates a typical coarsening process. For scaling the absolutesize of the microstructure, in the bottom row the magnification has been multiplied by the ratio of the averageintercept length L

–at time t to L

–at t = 75 min. (From Hardy and Voorhees, 1988.)


found to be less than 1/3, the value ex-pected from conventional coarsening the-ory on the basis of Ostwald ripening. Fur-thermore, it was shown that coarsening of a·100Ò modulated structure in less concen-trated Co–Cu alloys (≤ 20 at.% Cu; fp ≈20%) still follows the t1/3 kinetics, whereasthe coarsening rate becomes extremelysmall in a Co–50 at.% Cu alloy ( fp ≈50%),as indicated by the small time exponent a < 1/50 (Miyazaki et al., 1986).

As was pointed out in Sec. 5.5.5 “mot-tled structures” or “interconnected micro-structures” can result from spinodal de-composition of alloys with vanishing cohe-rency strains. Unlike for modulated struc-tures, elastic interactions can therefore beruled out as a likely reason for the stabilityof interconnected microstructures againstcoarsening. Nevertheless, the coarseningrate of interconnected phases is frequentlyalso found not to follow the t1/3 kinetics.Experimental studies on the coarsening ki-netics of interconnected phases in hardmagnet materials such as Fe–Cr–Co (Zhuet al., 1986), Al–Ni–Co (Hütten and Haa-sen, 1986; Katano and Iizumi, 1982) in-stead yielded time exponents a between 1/4and 1/10. Evidently, an adequate theoreti-cal description of the coarsening kinetics ofboth modulated and mottled microstruc-tures has to go beyond the mere modelingof diffusional interaction between an en-semble of isolated spherical particles.

5.6.4 Other Approaches TowardsCoarsening

In addition to the LSW-type coarseningmechanism, which is based on the evopora-tion and condensation of single atoms fromdissolving and growing precipitates,Binder and Heermann (1985) also con-siderred a cluster-diffusion-coagulationmechanism likely to become operative dur-

ing the intermediate stages of coarsening.Depending on the specific (local) micro-scopic diffusional mechanism which is as-sumed to contribute to the shift of the cen-ter of gravity of the particles and theirlikely coagulation, the time exponents a forthe related coarsening rate were evaluatedbetween a = 1/6 and a = 1/4 and hence aresmaller than predicted by the LSW theory(a = 1/3).

According to Fig. 5-32b the position ofthe SANS peak intensity at km varies inproportion to t –0.25. This might, in fact, beinterpreted in terms of a cluster-diffusion-coagulation mechanism being dominantprior to LSW-type coarsening. In Sec.5.8.2 it will be shown, however, that evenif single atom evaporation or condensationin the LSW sense is assumed to be the onlymicroscopic mechanism contributing toparticle growth, the time exponent a al-ready displays a strong time dependence.Depending on the initial supersaturation ofthe alloy, a (t) may then vary between al-most zero and 0.5 during the course of a pre-cipitation reaction. The question of whethera even reaches its asymptotic value 1/3 de-pends on whether the experiment spans asufficiently long range of aging times.

From the experimental point of view itthus does not appear feasible to decidemerely on grounds of the measured timeexponent whether the cluster-diffusion-coagulation mechanism influences thegrowth or coarsening kinetics of a precipi-tate microstructure.

5.6.5 Influence of Coherency Strains on the Mechanism and Kinetics of Coarsening – Particle Splitting

The LSW theory, as well as the exten-sions to finite volume fractions, assumethat the coarsening process is driven en-tirely by the associated release of interfa-

cial energy. In the presence of elasticstresses, however, such as those induced bycoherency strains, the coarsening processcan be driven by the release of both theinterfacial and elastic energy. The elasticenergy can be decomposed into an elasticself-energy, the elastic energy of an iso-lated particle in an infinite medium, and aninteraction energy that is due to the pres-ence of other particles in the system. Theelastic self-energy is a strong function ofthe shape of a particle and is responsiblefor the shape bifurcations mentioned inSec. 5.4.2. Moreover, the elastic interac-

tion energy is also a function of the shapeof the particles, the importance of whichhas been emphasized by Onuki and Nishi-mori (1991).

In the presence of misfit strains, inter-particle elastic interactions can give rise topronounced spatial correlations betweenthe precipitates. Ardell et al. (1966) pro-posed that the elastic interactions betweeng¢–Ni3Al particles are responsible for thealignment of the particles along the elasti-cally soft directions of the crystal duringcoarsening. The resulting structure ofmany nickel-based alloys reveals that the


Figure 5-46. High-magnification scanningelectron micrograph of concave g¢-precipi-tates in an aged Ni–23.4 Co–4.7Cr–4 Al– 4.3Ti (wt.%) superalloy prior to splitting.(Reproduced by courtesy of D. Y. Yoon(Yoo et al., 1995).)

Figure 5-47. As in Fig. 5-46, showing asplit g¢-precipitate.


g¢-type precipitates are rather uniformlydistributed (Fig. 5-19 b), which is some-times reminiscent of modulated structures.These strong spatial correlations developby two basic mechanisms. The first is thatparticles aligned as stringers along theelastically soft directions will grow at theexpense of particles not so aligned (Ardellet al., 1966). The second is that alignmentwill occur by particles migrating throughthe matrix (Voorhees and Johnson, 1988).This migration is a result of the elasticinteractions inducing a non-uniform inter-facial concentration and, hence, non-uni-form concentration gradients along the par-ticle interface.

It has been observed (Miyazaki et al.,1982; Doi et al., 1984; Kaufmann et al.,1989) that during coarsening, individualg¢-type precipitates can split into groups oftwo or eight smaller cuboidal particles. Us-ing a deep etching technique and a scan-ning electron microscope, the recent experi-ments of Yoo et al.(1995) have illustratedclearly the three-dimensional morphologyof the particles undergoing the splittingprocess (Figs. 5-46 and 5-47). Such a split-ting process clearly cannot be driven by areduction in interfacial energy and thus thecause of the splitting has been ascribed to adecrease in the elastic energy. Although theelastic self-energy does not change onsplitting, assuming that the particle mor-phologies are the same before and aftersplitting, if the elastic interaction energy isnegative, e.g., as is the case for particlesaligned along the elastically soft directionsof an elastically anisotropic crystal, the to-tal elastic energy will decrease upon split-ting (Khachaturyan and Airapetyan, 1974;Khachaturyan et al., 1988; Johnson andCahn, 1984; Miyazaki et al., 1982). How-ever, constraining the morphology of theparticles to be invariant upon splitting ne-glects the effects of the elastic interaction

on the morphology of the particles andhence on the magnitude of the elastic inter-action energy itself. In fact, two-dimen-sional calculations (Thompson et al., 1994)wherein the morphology of a particle wasnot constrained found that a misfittingfourfold symmetric particle in an elasti-cally anisotropic homogeneous crystal wasstable with respect to interfacial perturba-tions at least up to L = 26 (cf. Sec. 5.4.2).Thus the conclusion reached on the basis ofelastic energy considerations that splittingis possible appears to be due to the assump-tion that the morphology of the precipitateis invariant upon splitting. Miyazaki et al.(1982) proposed that the splitting processbegins with the formation of the matrixphase in the middle of the precipitate. Thisidea was verified through the diffuse inter-face calculations of Wang et al. (1991).However, the experimental results of Yooet al. (1995) confirmed the hypothesis ofKaufmann et al. (1989) that the splittingprocess is initiated via the amplification ofperturbations along the precipitate–matrixinterface and not by the formation of thematrix phase in the center of the particle.Thus, the cause of the splitting remains, atthis point, unexplained. However, thestability of a cuboidal shaped particle inthree dimensions has not been examined,and the effects of elastic inhomogeneity arestill to be explored fully. Recent work byLee (2000) has shown that splitting is pos-sible for certain differences in elastic con-stants between particle and matrix.

In addition to inducing particle migra-tion and selective coarsening, elastic inter-actions may alter the kinetics of the coars-ening process. Calculations employingfixed (spherical or circular) particle mor-phologies have shown that in a two-particlesystem inverse coarsening is possible,wherein a small particle will grow at theexpense of a large particle (Johnson,1984;

Miyazaki et al.,1986; Johnson et al., 1990).However, similar calculations of the evolu-tion of two precipitates in an elastically an-isotropic homogeneous system wherein themorphology is not fixed have shown thatinverse coarsening does not occur (Su andVoorhees, 1996). The difference betweenthese two results is due to the strong shapedependence of the elastic interaction en-ergy; the particle morphology must be al-lowed to change in a manner that is consis-tent with the elastic and diffusion fields inthe system. Inverse coarsening can still oc-cur, however, when more than two particlesinteract elastically (Su and Voorhees, 1996;Wang et al., 1992). If, however, L is suffi-ciently large and the system is both elasti-cally anisotropic and inhomogeneous,Schmidt et al. (1998) have shown that twoarbitrarily shaped particles can indeed bestable with respect to coarsening. Thesetwo results imply that it is essential not toconstrain the morphology of the precipi-tates when computing the evolution of themicrostructure and that elastic inhomoge-neity can be an important factor in deter-mining the stability of the system with re-spect to coarsening. Paris et al. (1995) havealso emphasized the importance of elasticinhomogeneity in determining the stabilityof a system with respect to coarsening.This implies that determining the evolutionof a system during coarsening with a largenumber, sufficient to provide accurate sta-tistical information, of arbitrarily shapedparticles is a very challenging problem.Nevertheless, some attempts have beenmade. For example, Onuki and Nishimori(1991) have found through diffuse inter-face calculations that when the precipitatesare softer than the matrix, stabilization ofthe system with respect to coarsening maybe possible. Although the size of thesystem examined is small, the exponent ofthe temporal power law for the average

particle size is time dependent and, at thevery least, much less than the classicalvalue of 1/3 predicted by the LSW theory.In contrast, Ising model simulations (Fratzland Penrose, 1996) and diffuse interfacecalculations (Nishimori and Onuki, 1990)in an elastically anisotropic homogeneoussystem do not show stabilization with re-spect to coarsening, again indicating theimportance of elastic inhomogeneity. Byassuming that the systems are statisticallyinvariant, Leo et al. (1990) have shown thatthe exponent for the average size scale ofthe precipitates can be altered by the pres-ence of elastic stress, in this case attaining


Figure 5-48. Variation of the mean radius R–

of g¢-particles with aging time in two different Ni–Cu–Sialloys with fp = 0.18 (Ni–47.4 Cu–5Si) and fp = 0.5(Ni–35.1 Cu–9.8 Si), numbers indicate at.%. (AfterMiyazaki and Doi, 1989.)

Figure 5-49. Variation of the standard deviation s ofthe g¢-particle size distribution with aging time forthe same alloys as in Fig. 5-48. (After Miyazaki andDoi, 1989.)

5.7 Numerical Approaches to Concomitant Processes 381

a value of 1/2. This result does not contra-dict Onuki’s, as the onset of stabilizationclearly violates the statistically time invar-iant assumption of Leo et al.

There have been many investigations ofthe kinetics of coarsening in elasticallystressed solids. Experiments performed inNi–Cu–Si alloys by Yoshida et al. (1987),and Miyazaki and Doi (1989) show a cleardeparture from the classical t1/3 predictionsfor the average particle radius with thegrowth rate of the average particle radiusand the width of the particle size distributiondecreasing in time (Figs. 5-48 and 5-49). Asimilar departure from the t1/3 function hasalso been observed in the Ti–Mo system byLangmayr et al. (1994). In contrast, inNi–Al (Ardell, 1990), Ni–Mo–Al (Fähr-mann et al., 1995) and Ni–Si (Cho and Ar-dell, 1997) alloys the exponent for the aver-age particle size appears to be 1/3. The rateconstant, however, appears to be indepen-dent of the volume fraction of precipitate.Although it has been speculated that elasticinteractions are responsible for this result,this has not been confirmed due to the diffi-culty of performing calculations in systemsthat are sufficiently large to yield accuratestatistical information.

5.7 Numerical ApproachesTreating Nucleation, Growth and Coarsening as ConcomitantProcesses

5.7.1 General Remarks on the Interpretation of Experimental KineticData of Early Decomposition Stages

In this section we turn our attention to decomposition studies of alloys with sufficiently high nucleation barriers(DF*/kT t7) decomposing via nuclea-

tion, growth and coarsening. In contrast tothe considerations of the preceding sec-tions we now ask how experimental kineticdata can be interpreted if they refer to veryearly decomposition stages which includenucleation, growth and coarsening as con-comitant rather than consecutive processeson the time scale. In attempting such inter-pretations we have to recall that these earlydecomposition stages especially are char-acterized by dramatic changes in supersat-uration and, thus, in the driving forces fornucleation and growth processes. Further-more, during these early stages the size dis-tribution function evolves and is subjectedto drastic alterations with rather short ag-ing periods where nucleation, growth andcoarsening must be seen as competing andoverlapping processes. Evidently theserather complicated phenomena are notproperly accounted for by splitting thecourse of decomposition into a nucleationregime, a growth regime, and a coarseningregime. Moreover, the kinetic theories de-veloped for either regime (Secs. 5.5.1, 5.5.2and 5.6, respectively) are based on ideal-ized assumptions which are frequently notexpected to be close to reality. This holdstrue for the classical nucleation theories ofVolmer and Weber and Becker and Döring(Sec. 5.5.1). In these theories the supersat-uration is assumed to be constant. This mayonly be fulfilled – if at all – during the ear-liest nucleation stage. Furthermore, thesetheories are based on artificial assumptionsof the cluster size distribution in the vicin-ity of the critical radius (Fig. 5-24), whichare not consistent with the fact that duringthe nucleation process, many growing pre-cipitates slightly larger than the criticalsize are formed. The theory of diffusionalgrowth by Zener and Ham (Sec. 5.5.2) de-scribes only the time evolution of precipi-tates with uniform size. However, towardsthe end of nucleation as well as at the be-

ginning of the LSW coarsening regime wecertainly have to deal with a polydispersedprecipitate microstructure. Thus, even ifnucleation and LSW coarsening were to be well separated on the time scale, inbetween these two regimes the Zener–Hamtheory cannot be expected to correctly pre-dict the measured growth kinetics quantita-tively. Finally, the coarsening theories ofLifshitz, Slyozov and Wagner are based onthe linearized version of the Gibbs–Thom-son equation (Eq. (5-46)) and on the as-sumption that the supersaturation is closeto zero. These restrictions also hold for themore recent theories (Sec. 5.6.3) whichtake into account finite precipitated vol-ume fractions and overlapping concentra-tion profiles between precipitates.

Thus, any theory for the kinetics of pre-cipitation that can be employed either for amore realistic interpretation of experimen-tal data or for a prediction of the dynamicevolution of a second-phase microstructureunder elevated temperature service condi-tions, has to treat nucleation, growth andcoarsening as concomitant processes. Thiswas accounted for in the cluster dynamicstheories of Binder and coworkers (Secs.5.5.3, 5.6.4 and Binder and Fratzl (2001),Chapter 6 of this volume). However, aswas pointed out by these authors (Binder etal., 1978; Mirold and Binder, 1977), thetheory developed yields a reasonable qual-itative prediction of the features of experi-mentally observable quantities but does notattempt their quantitative interpretation.

A further decomposition theory treatingnucleation, growth, and coarsening as con-comitant processes was developed byLanger and Schwartz (LS model; Langerand Schwartz, 1980). This theory was for-mulated for describing droplet formationand growth in near-critical fluids. It waslater modified by Wendt and Haasen(1983) and further improved by Kamp-

mann and Wagner (MLS model; 1984) insuch a way that it could be applied for thedescription of the kinetics of precipitateformation and growth in metastable alloysof rather high degrees of supersaturation.The MLS model is still based on the sameassumptions as the original LS theory. Inparticular, the explicit form of the size dis-tribution is not accounted for and the long-time coarsening behavior is assumed tomatch the LSW results, i.e., is described byEqs. (5-63a, 5-65). A priori, it is not pos-sible to foresee the influence of these as-sumptions on the precipitation kinetics.

Therefore, Kampmann and Wagner(KW; 1984) have devised an algorithm thataccurately describes the entire course ofprecipitation within the framework of clas-sical nucleation and growth theories. Thisalgorithm is accurate in the sense that, un-like the LS and the MLS theories, it con-tains no simplifying assumption; in partic-ular, in this algorithm, termed the Numeri-cal model (N model), the time evolution ofthe size distribution is computed withoutany approximations. From a comparison ofthe N model with experimental data it ispossible to determine some crucial precipi-tation parameters of the particular alloysystem as well as to scrutinize the existingnucleation and growth theories with re-spect to their applicability to decomposi-tion reactions in real materials. Further-more, the N model allows an evaluation tobe made of how realistic the various ap-proximations are which occur in both theLWS theory and the MLS model.

In this section we briefly introduce theMLS and the N models. An attempt ismade to demonstrate their ability to de-scribe the entire course of the decomposi-tion reaction; we will also show how someessential parameters of the decomposingalloy, such as the diffusion constant and thespecific interfacial energy, can be evalu-



ated by fitting the theoretically predictedcurves R

–(t) and Nv (t) to the corresponding

experimental data. Furthermore, on thegrounds of the N model, it is possible to ex-amine whether certain time intervals existduring the course of a precipitation reac-tion in which the kinetic evolution is pre-dicted by either the growth or the coarsen-ing theory with sufficient accuracy.

5.7.2 The Langer and Schwartz Theory(LS Model) and its Modification by Kampmann and Wagner (MLSModel)

In the LS theory it is assumed that thesystem contains NLS droplets per unit vol-ume of uniform size R

–LS. In order to ac-

count for coarsening, the continuous distri-bution function f (R, t) and the number ofparticles of critical size, i.e., f (R*, t) dR*,must be known. However, in the LS theorythis is not the case. LS, therefore, intro-duced an apparent density fa (R*, t) (Fig. 5-50) which is given as:

(5-66)

fa (R*, t) is thus proportional to NLS and in-versely proportional to the width of f (R, t).The constant b = 0.317 is chosen in such away that for t Æ • the coarsening ratedR

– 3LS/dt is identical to KR

LSW (Eq. (5-65)).Unlike the LSW theory, where R

–=R*, in

the LS theory only particles with R > R*arecounted as belonging to f (R, t), i.e.

(5-67)

Thus, R–

LS >R* at all stages of decomposi-tion, keeping fa (R*, t) in Eq. (5-66) finite.Due to nucleation at a rate J and dissolu-tion, NLS changes with time according to:

(5-68)d

ddd

LSa

Nt

J f R tRt

= – ( *, )*

RN

f R t R RR

LSLS

d=∞

∫1

( , )*

f R t Nb

R RLSa LS( *, )

– *=

The growth of particles of mean size R–

LS isgiven as:

(5-69)

The term v (R–

LS ) is given by Eq. (5-43) andaccounts for the growth rate of the particlesin the supersaturated matrix. The secondterm accounts for the change in the truedistribution function caused by the dissolu-tion of fa (R*, t) dR* particles with radiibetween R* and R* + dR*. The third termdescribes the change in f (R, t) caused bythe nucleation of particles which must beslightly larger than those of critical size,

dd

dd

LSLS

LSa

LS

LSLS

Rt

R

R Rf R t

NRt

NJ R t R R R

=

+

+ +

v ( )

( – *)( *, ) *

[ * ( )] ( * * – )1 d

Figure 5-50. Relationship between the ‘true’ contin-uous size distribution function f (R, t) yielding themean radius R

–and the related parameters of the LS

model. LS assumed a monodispersive distribution,fLS, of particles with radius R

–LS. fa (R*, t) dR* is the

apparent number density of particles with radiibetween R* and R* + dR*. In the LS theory, only par-ticles in the hatched region belong to the precipitatedphase.

i.e., R = R* + dR* and dR* O R*. Togetherwith the continuity equation3,

(5-70)

and Eq. (5-66), Eqs. (5-68) and (5-69) arethe rate equations which describe the entirecourse of precipitation in the LS model.After proper scaling, these equations werenumerically integrated in conjunction withsteady state nucleation theory (Eq. (5-27)).

The LS model is based on the assumptionthat the equilibrium solubility of smallclusters can be determined from the linear-ized version of the Gibbs–Thomson equa-tion, i.e. Eq. (5-46). This linearization,however, generally does not hold for smallclusters in metallic alloys (cf. Sec. 5.6.2).This becomes immediately evident for theCu–1.9 at.% Ti system isothermally agedat 350°C. At t = 0, R* ist 0.48 nm or 0.13nm, depending on whether R* is computed(sab = 0.067 J/m2, see Sec. 5.7.4.4) fromthe non-linearized or from the linearizedversion of Eq. (5-46). This example clearlydemonstrates that Eq. (5-46) must be usedin its non-linearized version, particularyfor systems with large values of sab and/orlarge supersaturations.

The MLS model is based on the non-line-arized Gibbs–Thomson equation (Eq. (5-45)). Thus, the growth rate in the MLSmodel is:

(5-71)

In order to write the rate equations (5-68)to (5-70) in a properly scaled version, weintroduce the following parameters:

dd

LSLS

pe

LS

e m

g LS

Rt

Rc c

DR

c cV

R T R

= =

×⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

v

ab

( )–

– exp

1

2 1

a

as

( – ) ( – )c c R N c cp LS LS43

30

p =

(5-72)

Apart from RN, all parameters are dimen-sionless; unlike in both the original LSmodel and the study of Wendt and Haasen(1983), in the MLS model RN rather thanthe correlation length is used as the scalinglength. By straigthforward scaling of Eqs.(5-27), (5-61) and (5-71) we obtain thescaled version of the equations of motion(5-68) and (5-69). From these, the numberdensity n of particles is eliminated by vir-tue of the scaled continuity equation

(5-73)

We finally obtain

(5-74)

(5-75)

Eqs. (5-74) and (5-75) are the basic equa-tions of the MLS model and are numeri-cally integrated with s0 = 0.1 J/m2; valuesof dr*/r* (x0) used by Kampmann andWagner (1984) ranged from 0.05 to 0.2.

In the LS theory only the steady-statenucleation rate has been used; in contrast,

dd

dd

p

p

rt x x

xt

x rx r

rxx x x

dr r

+

=

+ +⎛⎝⎜

⎞⎠⎟

b k

k

Jk

s

s

s

(ln )

–[ – exp ( / )]

–

–˜

ln* –

2

3

0

11

1

1

dd

dd

p

rt

rx x r x x x

xt

r x xx x

+ +⎛⎝⎜

⎞⎠⎟

=

31 1

3

0

4

0

– ln – ln

– ˜ –

–

b kk

J

s

s

n = 113

0

rx xx

––p

RVR T

k c c

c c c c R R

R R R R

n N RD

Rt J J

nN

t

Nm

g

e

ep p

eLS N

N N

LS NN2

dd

dd

= = =

= = =

= =

= = =

2

43

0

3

s s s x

x x r

r dr dp t

t

s0

ab 0 0 a

a a

; / ; / ;

/ ; / ; / ;

* * / ; * * / ;

; ; ˜


3 Note that particles with R = R* are not contained inEq. (5-70); we therefore call these particles ‘apparent’.


in Eqs. (5-74) and (5-75) of the MLSmodel, KW employed the time-dependentnucleation rate J* (Eq. (5-29)), which in-volves the incubation time. In the scaledversion the latter (tw) was evaluated from:

(5-76)

When deriving this equation, it was as-sumed that the minimum time (tw, min) for aparticle to reach the critical size r* is givenby Eq. (5-47). However, for subcritical nu-clei cR is considerably larger than ce

a, andthe probability of their redissolution israther large; hence, Eq. (5-47) overesti-mates the growth rate significantly. Thisfact is counterbalanced by the introductionof the parameter cw. For each particular al-loy system, cw is determined in such a waythat after a period tw the first particles be-come ‘observable’ with a reasonable numberdensity. For all alloy systems investigated sofar, cw ranges from 1.4 to 3.5. Hence, be-cause of the stochastic nature of clustergrowth and dissolution, on average it takesabout two to twelve times longer for a clus-ter to attain a size R >R* beyond which itsfurther growth may be evaluated in a deter-ministic manner on the basis of Eq. (5-47).

5.7.3 The Numerical Model (N Model)of Kampmann and Wagner (KW)

Unlike the MLS model, in the N modelthe time evolution of f (R, t) – or f (r, t) ifwe stick to the same nomenclature – iscomputed. For this purpose f (r, t) is sub-divided into intervals [rj + 1, rj] with|rj – rj + 1| /rj 1 and nj particles in the j-thinterval. In contrast to the MLS model, dis-solving particles with r < r* are assumedto belong to the precipitating phase, i.e.

n nn

njj

j

j jj

j

= == =∑ ∑and

1 1

0 01r r

t xxx

r tw c c( )–

–* ,min= ≅1

2

1

12 2 2p

w w w

(5-77)

The continuity equation in the N modelthen reads:

(5-78)

Thus the continuous time evolution off (r, t) is split into a sequence of individualdecomposition steps; these step are chosenin such a way that within each correspond-ing time interval Dti the changes of all ra-dii rj, (ti) and of the supersaturation x (ti)remain sufficiently small. Then, both thenucleation and the growth rates can be con-sidered as being constant during Dti andthe change of f (r, t) and of x can be reli-ably computed. The N model of KW con-tains some algorithms which ensure arather high accuracy of the numerical cal-culation; it amounts to ≈ 0.5% in the caseof the growth rate of r.

5.7.4 Decomposition of a Homogeneous Solid Solution

5.7.4.1 General Course of Decomposition

KW discussed the decomposition reac-tion of an ideally quenched alloy with ca

e =0.22 at.%, cp = 0.20 at.%, c0 = 1.9 at.%and sab = 0.067 J/m2. Just on the basis ofthese input data, the decomposition reac-tion can be calculated within the frame-work of the MLS and N models. Fig. 5-51shows the predictions from both the MLS(full lines) and the N models (discrete sym-bols) for the time evolution of the radii r– (t) and r* (t), the number density n (t),the supersaturation x (t), and the nuclea-tion rate J (t), The chosen values of x0 =8.7, ca

e , s and D correspond to those forCu–1.9 at.% Ti aged at 350°C (see Secs.5.7.4.3 and 5.7.4.4). Surprisingly, bothmodels yield qualitatively similar results.

( – ) –x r x xp 11

30

0

j

j

j jn=∑ =

j j j= + +with 112

r r r( )

During the early nucleation period (t Ù 1)both J and n increase, whereas x and,hence, r* remain roughly constant. At thisstage, the N model gives a rather narrowsize distribution function f (r, t) (Fig.5-52). Since cR is still close to c–≈ c0, the

growth rate of nucleated precipitates is alsoclose to zero, i.e., r– remains about con-stant. After (t Û 1, those precipitates nucle-ated first become considerably larger thanr* and f (r, t) becomes much broader. Thisis the beginning of the growth period


Figure 5-51. Evolution of various scaled precipitation parameters with scaled aging time t according to both theN model and the MLS model. The chosen values of x0, sab correspond to those for Cu–1.9 at.% Ti aged at 350°C.

Figure 5-52. Evolution of the scaled size distribution function in Cu-1.9 at.% Ti with aging times as computedwith the N model. For comparison, the distribution functions fLSW (r, t) of the LSW theory with the known values of Nv (t) and R

–(t) are also shown. D = 2.5 × 10–15 cm2/sec.


which is characterized by (i) the highestgrowth rate ever observed during thecourse of precipitation; (ii) a ratio r–/r*which becomes significantly larger than 1;(iii) the maximum number density (nmax,Nv,max) of particles which remains aboutconstant; (iv) a stronger decrease of x; and,consequently, (v) a decrease of J from itsmaximum value; in this particular case, Jnever reaches its steady-state value. At theend of the growth regime (t ≈ 15) thesupersaturation has dropped significantlyand the growth rate becomes small. Thiseffect causes r* to converge towards r– andmakes dn/dt < 0 (Eq. (5-68)). During thesubsequent transition period (t ≈ 15), thegrowth rate of r– is primarily controlled bythe dissolution of particles with r < r* andonly to a lesser extent by the uptake of so-lute atoms from the matrix, the supersatu-ration of which is still about 20%. Duringthe transition period, the true distributionfunction f (r, t) continuously approachesthe one predicted by LSW (Fig. 5-52). Atthis stage, however, fLSW is still a ratherpoor approximation for f (r, t). This simplyreflects the influence of the linearization ofthe Gibbs–Thomson equation (Eq. (5-46))on which the LSW theory is based. Fort Æ •, dr–3/dt approaches asymptotically a constant value, i.e., the reaction is withinthe asymptotic limit of coarsening wherethe supersaturation is almost zero. At thisstage f (r, t) is well approximated by fLSW

with only minor deviations for small parti-cle radii.

5.7.4.2 Comparison Between the MLSModel and the N Model

The precipitation reaction starts with anidentical nucleation rate and identical par-ticle sizes in both the MLS and the N mod-els. Therefore, both models are expected toyield identical results which, in fact, is ob-

served in Fig. 5-51. During the latergrowth period, dissolution of particles withr < r* commences. At this stage, the MLSmodel only counts particles with r > r* asbelonging to the second phase (Eq. (5-67)),and, furthermore, assumes fa (r = r*) to beproportional to 1/(r– – r*) (Eq. (5-66)). Atthis stage, this is a rather poor approxima-tion because the N model yields a steepslope for f (r Æ r*, t) with a rather smalldensity n (r = r*). These facts make theMLS model predict considerably larger rateconstants for the decrease of the particlenumber density and, hence, for the growthof r– at the end of the growth period (t ≈ 15).

Depending on the particular choice of b(Eq. (5-66)), the coarsening rate of theMLS model approaches asymptotically thevalue KR

LSW (Eq. (5-65)) from the LSW the-ory. In this asymptotic limit the mean radiifrom both models and, hence, their coars-ening rates become identical. At this stagethe N model yields r– Ó r* as predicted byLSW, whereas the MLS model yields r– –r* = const., i.e., R

–LS> R*, as required by

Eq. (5-67).We can conclude that the MLS model,

which requires much less computing timethan the N model, provides a good surveyof the general course of precipitation.However, due to the simplifying assump-tions made, it does not predict the precipi-tation kinetics with the same accuracy asthe N model. This is particularly evidentfor those precipitation stages where theshape of f (r, t) is extreme, as, for in-stance, during the later growth stages in theexample discussed above.

5.7.4.3 The Appearance andExperimental Identificationof the Growth and Coarsening Stages

In the following, the results from theMLS and the N models are compared with


Figure 5-53. Variation of R–

,R*, R

–LS and of the relative

supersaturation with agingtime for Cu–1.9 at.% Ti ascomputed with the N modeland the MLS model for thegiven set of thermodynamicsdata; also shown are the ex-perimental data for R

–(t) from

von Alvensleben and Wagner(1984).

Figure 5-54. Variation of R2 and R*2 with time during the growth regime for Cu–1.9 at.% Ti. During the periodmarked by the two arrows the kinetics follow the power-law R

– t1/2.b) Variation of the coarsening rate dR

– 3/dt with aging time; KRLSW is the value predicted by the LSW theory,

Eq. (5-65).


experimental data obtained from a FIMatom probe study of early-stage precipita-tion in Cu–1.9 at.% Ti at 350°C (von Al-vensleben and Wagner, 1984).

Fig. 5-53 shows the time evolution of R*and or R

–in physical units. Again these ex-

perimental data points are well describedby the N model. Fig. 5-53 also reveals thatthe experimental data cover neither theearly nucleation period nor the growth re-gime, but rather start (t = 2.5 min) at theend of the latter region. From Fig. 5-54a,where R

–2 is plotted as a function of t, it isrecognized that during the period t ≈0.7 min to 1.2 min, R

–2 varies linearly with tas suggested by Eq. (5-40). Since this timeperiod extends only over 0.5 min, KW con-cluded that for Cu–1.9 at.% Ti the timewindow is too short for revealing the R– t1/2 kinetics experimentally; in fact, thesame holds true for many other alloysystems analyzed by KW using the Nmodel. Furthermore, from Fig. 5-54a theslope of the straight line has been evaluatedto be 0.88 nm2/min, whereas the corre-sponding growth rate from Eq. (5-47) iscomputed (ca

e = 0.22 at.%, cp = 20 at.%, D = 2.5 ¥ 10–15 cm2/sec) to be 2.5 nm2/min. This result clearly demonstrates thatno growth regime exists which is ade-quately accounted for by Eq. (5-47). Inother words, the idealizations made in thederivation of Eq. (5-47) do not approxi-mate the true situation in Cu–1.9 at.% Ti.However, if c (t) and cR in Eq. (5-43) arereplaced by their mean values respectively,during the period for which R

– t1/2 holds,then Eqs. (5-40 and 5-42) with li = k*yield a value for the growth rate (0.104 nm2/min) which is only 16% higher than thetrue value.

In Fig. 5-54b the rate constant dR–3/dt for

coarsening is plotted versus t. It is evidentthat for Cu–1.9 at.% Ti the rate constantKR

LSW= 1.2 ¥ 10–24 cm3/sec from the LSW

theory (Eq. (5-65)) is only reached for ag-ing times beyond ≈104min! At this stage R

–

has already grown to ≈6.4 nm. From thisresult, which reflects the influence of thelinearization of Eq. (5-46), KW inferredthat the LSW theory predicts the correctcoarsening rate once

(5-79)

If this relation holds, f (R, t) is almost iden-tical to fLSW (Fig. 5-52) and c/ca

e ≈ 1.According to the results from the N

model, during the early coarsening stagesthe precipitation kinetics deviate signifi-cantly from those predicted by the LSWtheory (e.g., Fig. 5-54b). Thus the widelyused LSW analyses resulting in a determi-nation of D and sab on the basis of Eq. (5-63a) and Eq. (5-63c) should not be ap-plied to early coarsening stages duringwhich the relation does not hold.

5.7.4.4 Extraction of the InterfacialEnergy and the Diffusion Constant from Experimental Data

KW determined sab and D separately byfitting R

–(t) and Nv (t) as obtained from the

N model to the corresponding experimentaldata. Fig. 5-55 shows the variation of Nv

and J* with aging time as computed withthe N and the MLS models, together withthe corresponding experimental data. Thepeak number density Nv, max of particles ina precipitation reaction is essentially gov-erned by the value of DF* via the nuclea-tion rate equation (5-27). Since DF*s 3

ab(Eq. (5-20)), Nv, max depends sensitively onthe value of the interfacial energy sab. Thevery strong dependence of Nv, max on sab isclearly revealed by Fig. 5-55, showing agood fit of Nv (t) for sab = 0.067 J/m2 andonly poor agreement for sab = 0.071 J/m2.On the other hand, a variation in D mani-

RV

R T13

m

g

s ab

fests itself in a parallel shift of the entirecurve on the time scale. Thus, a fit of R

–(t)

and of Nv (t) as obtained from the N modelto the experimental curves allows both saband D to be determined quite accurately.

From a variety of different two-phase al-loys, the available kinetic data have beeninterpreted in terms of the N model. Table5-3 presents the interfacial energies sab to-gether with the width of the (coherent) mis-


Figure 5-55. Variation of Nv and J* with aging time for Cu–1.9 at.% Ti as computed with the N model and theMLS model for sab = 0.067 J/m2; for this value of sab the computed Nv (t) curve agrees well with the experi-mental data; poor agreement is obtained for sab = 0.071 J/m2.

Table 5-3. Correlation between the width of the miscibility gap and the coherent interfacial energies sab forvarious two-phase alloys as determined from a fit of the N model to experimental kinetic data.

Alloy Aging Composition/type Width of coherent Coherent interfacialtemperature of precipitates miscibility gap energy

at.% °C at.% sab J/m2

Ni–14 Al 1 550, 500 g ¢-Ni3Al ≈ 15 ≈ 0.016Ni–26 Cu–9 Al 2 550, 500 g ¢-(Cu, Ni)3Al ≈ 20 ≈ 0.052

540, 500 Not determined ≈ 20 ≈ 0.050580, 500 Not determined ≈ 20 ≈ 0.052

Cu–1.9 Ti 3 350, 500 b¢-Cu4Ti ≈ 20 ≈ 0.067Cu–2.7 Ti 4 350, 500 b¢-Cu4Ti ≈ 20 ≈ 0.067Cu–1.5 Co 5 500, 500 >95 at.% Co ≈ 95 ≈ 0.171Fe–1.4 Cu 6 400, 500 >98 at.% Cu ≈ 100 ≈ 0.250

Fe–0.64 Cu 6 400, 500 >98 at.% Cu ≈ 100 ≈ 0.250

1 Wendt and Haasen (1983) (AFIM)2 Liu and Wagner (1984) (AFIM)3 Alvensleben and Wagner (1984) (AFIM, CTEM)4 Eckerlebe et al. (1986) (SANS)5 Gust (1986), unpublished (magnetic measurements)6 Kampmann and Wagner (1986) (SANS)


cibility gap for each given alloy. It is evi-dent that there is a pronounced correlationbetween sab and the compositional width,i.e., the broader the gap, the larger is sab.This is consistent with various theoreticalpredictions on the interfacial energy (seeLee and Aaronson, 1980, for a comprehen-sive discussion of this aspect).

5.7.5 Decomposition Kinetics in AlloysPre-Decomposed During Quenching

The versatility of the N model is furtherillustrated by its ability to predict the pre-cipitation kinetics in alloys which have ex-perienced some phase separation duringquenching. This is exemplified for Cu–2.9at.% Ti, the decomposition reaction ofwhich was studied by Kampmann et al.(1987) by means of SANS. They found thatthe cooling rate of their specimen was notsufficient to suppress the formation ofCu4Ti precipitates during the quench; infact, as is shown in Fig. 5-56, the soluteconcentration decreased from c0 = 2.9 at.%to 2.2 at.% Ti during the quench. With fur-ther aging, the supersaturation decreasedcontinuously through the formation of ad-ditional clusters and the growth of existing

ones. After aging for ≈100 min at 350°Cthe metastable b¢-solvus line is nearlyreached at c¢a

e≈0.22 at.% Ti (cf. Sec. 5.2.1).The experimentally determined kinetic

behavior of the precipitate number density,of their mean radius, and of the supersatu-ration are displayed in Fig. 5-57a–c andcompared with the predictions of the Nmodel. For the computations, the resultfrom the SANS evaluation was taken intoaccount, which yielded the homogenizedsample to already contain ≈2 ¥ 1025 clus-ters/m3 with R

– ≈ 0.7 nm. These could im-mediately grow by further depleting thematrix from solute atoms. Moreover, at t = 0 the supersaturation was still largeenough for nucleating new clusters withsmaller radii at a nucleation rate J* (Fig.5-57a). Thus, during the first minutes ofaging, the cluster number density in-creased. At this stage, the alloy contained asort of bimodal cluster distribution: thelarger ones formed at a smaller supersatu-ration during quenching, and the smallerones resulted from nucleation at 350°C.Due to both nucleation of new clusters andgrowth of pre-existing ones, the supersatu-ration and, hence, the nucleation rate de-creased rapidly; after aging for ≈3 min, nu-

Figure 5-56. Decrease in the solute con-centration in the matrix with aging time, asdetermined from Laue scattering (· ), fromthe integrated intensity (¥), and the Guinierapproximation () of SANS curves.

cleation is virtually terminated. After about10 min the critical radius R*, which is cor-related with the momentary supersatura-tion, reaches the mean value R

–of the glo-

bal size distribution. At this instant, R* hasgrown beyond the mean radius of thesmaller, freshly nucleated clusters; thesenow redissolve leading to a further de-crease in Nv. Now the size distribution isagain governed by the larger precipitatesformed during quenching. After about500 min, Nv decreases with t as expectedfrom the LSW theory.

With regard to the accuracy of both theSANS experiment and, in particular, theSANS data evaluation, the agreementbetween the experimental kinetic data andthose from the N model is rather good foran interfacial energy sab = 0.067 J/m2.This value is identical to that determinedfor the less concentrated Cu–1.9 at.% Ti al-loy (Sec. 5.7.4.4). In the early stages (t 10 min) the experimental R

–is considerably

larger than the theoretical one. This simplyreflects the fact that the scattering power ofa particle is weighted with R6; hence, fort 10 min, essentially the radius of onlythe larger particles within the bimodal dis-tribution was determined. The diffusion coefficient (D = 3 ¥ 10–16 cm2/sec) wasfound to be about one order of magnitudesmaller than in the Cu-1.9 at.% Ti alloy(Sec. 5.7.4.4). It was not possible to decidewhether this difference in the D values re-flects the concentration dependence of D,or whether it was caused by differences inthe homogenization temperatures of thetwo alloys (Cu–2.9 at.% Ti: TH = 780°C;Cu–1.9 at.% Ti: TH = 910°C).

5.7.6 Influence of the Loss of Particle Coherency on the Precipitation Kinetics

In many two-phase systems the particleslose coherency once they have grown be-


Figure 5-57. Time evolution of (a) the cluster num-ber density Nv and the nucleation rate J*, (b) of theirmean radius R

–and of the critical radius R*, and (c) of

the supersaturations. Experimental SANS results:discrete symbols; computational results: full lines.


yond a certain size RT . The associated in-crease in the interfacial energy and de-crease in the solubility limit (cf. Fig. 5-1)leads to an enhanced driving force and,hence, to accelerated kinetics for furthercoarsening of the incoherent microstruc-ture with respect to the coherent one. Thiseffect can also be accounted for by the Nmodel. This is shown in Fig. 5-58 forFe–1.38 at.% Cu aged at 500°C. As in-ferred from CTEM, the Cu-rich particlestransform at RT ≈ 2.8 nm from the meta-stable b.c.c. structure into the f.c.c. equilib-rium structure. The associated loss of cohe-rency occurs at particle number densitieswell beyond the maximum number density Nv, max (Fig. 5-58). Thus, following theprocedure outlined in Sec. 5.7.4.4, the co-herent interfacial energy could be deter-mined by fitting the N model to the experi-mental (SANS) kinetic data of the still co-herent system. The value sab ≈ 0.27 J/m2

obtained is considerably smaller than thecorresponding value s inc

ab ≈ 0.50 J/m2

which was derived from a thermodynami-cal analysis of the Fe–Cu system (Kamp-mann and Wagner, 1986). As is shown inFig. 5-58, the loss of coherency in factleads to a momentary acceleration of thegrowth kinetics. It is, however, not suffi-cient to bridge the discrepancy between theexperimental kinetic data and the theoreti-cally predicted ones displayed in Fig. 5-58.As the predicted kinetics are much moresluggish than the experimentally deter-mined ones, we might speculate that ne-glecting particle interaction accounts forthe observed discrepancy. Inspection ofFig. 5-43, however, reveals that a consider-ation of finite volume effects in Fe–1.38at.% Cu with fp ≈ 1% would increase thecoarsening rate only by less than a factor1.3 whereas a factor of ≈10 is required tomatch the results from the N model and theSANS experiments at the later stages of

Figure 5-58. Kinetic evolution of the precipitatednumber density (top) and of the mean radius (bottom)as predicted by the N model for Fe–1.38 at.% Cu.The solid points refer to experimental data derivedfrom nuclear and magnetic SANS experiments. Thedashed lines show the kinetic evolution without con-sideration of the b.c.c. Æ f.c.c. transformation of thecopper-rich particles.

precipitation. At present, it is unclearwhether the various theories dealing withfinite volume effects (Sec. 5.6.3) are stillinsufficient or whether some heterogenousprecipitation at lattice defects accounts forthe observed discrepancy.

At first glance the experimental kineticdata for R

–(t) and Nv (t) in Fig. 5-58 might

be seen as being amenable to an LSW anal-ysis in terms of Eq. (5-63). Analyses of theSANS data for t 103 min, however,showed the width of the particle size distri-bution to be much broader (standard devia-tion: s ≈ 0.31) than expected from the LSW theory or its modification (s ≈ 0.23).Furthermore, the measured supersaturationwas still far from being close to zero.Hence, the conditions for an LSW analysisof the experimental data are not at all ful-filled.

5.7.7 Combined Cluster-Dynamic and Deterministic Description of Decomposition Kinetics

There remain two main shortcomings ofthe N model outlined in Sec. 5.7.3. Firstly,the stochastic nature of the nucleation pro-cess is solely accounted for by using thetime- and concentration-dependent nuclea-tion rate (i.e., Eq. (5-29)), whereas thegrowth (and shrinkage!) is computed in adeterministic manner on the basis of Eq. (5-43). Thus, the stochastic nature of early-stage growth which, in particular, becomeseffective in systems with small nucleationenergies, is not adequately accounted for inthe N model.

Secondly, as in all previous theories, thematrix concentration cR at the cluster–ma-trix interface was also calculated on the ba-sis of the Gibbs–Thomson equation, Eq.(5-46). This relation, which describes anequilibrium between the cluster size R andthe matrix concentration c, is generally not


adequate for describing the growth (R >R*)or dissolution (R < R*) of clusters.

To overcome these shortcomings, Kamp-mann et al. (Kampmann et al., 1992; Staronand Kampmann, 2000 a, b) extended the Nmodel by a cluster-dynamics (C–D) simu-lation of the kinetics during the nucleationand growth stages (cf. Sec. 5.5.3). To savecomputation time, in the later growth andcoarsening stages where the stochastic pro-cess is no longer relevant, the cluster-dy-namics approach is linked to the determin-istic description of the original N model.As in the latter, the input parameters enter-ing the extended C–D model are c0, ca

e , sand D with variations in D leading only toshifts of the computed curves (e.g., R

–(t) or

J (t)) parallel to the time scale. Therefore,this C–D simulation can be used in as ver-satile a manner as the original N model for the interpretation of experimental data (Staron and Kampmann, 2000b).

The cluster-dynamics algorithm was de-vised such that its corresponding Helm-holtz energy functional approximatesclosely that of the regular solution model.As proved by CALPHAD studies, this is a rather good approximation for Cu–Co.Fig. 5-59 shows the computed time evolu-tion of the cluster size distribution duringthe nucleation period and the nucleationrate with input parameters correspondingto a Cu–0.8 at.% Co model alloy aged at500°C. At t = 5 sec all clusters are still be-low the critical size4 and, thus, Fig. 5-59aprovides insight into the incubation process.

The nucleation rate J SB–D calculated ac-

cording to Eq. (5-27) with c ∫ c0 exceedsthe maximum computed one (Fig. 5-59b)by more than a factor of 1000. This dis-crepancy cannot be attributed to incubation

4 In the C–D approach the critical radius is definedas that size for which the probabilities of clustergrowth and shrinkage are identical.

5.8 Self-Similarity, Dynamical Scaling and Power-Law Approximations 395

effects; it can only be resolved if – insteadof Eq. (5-27) – a more adequate equation isused for calculating the Becker–Döring nu-cleation rate:

(5-80)

The modified exponent takes into accountthat no Helmholtz energy DF* (1) isneeded for forming monomers (which arealready dissolved in the matrix during thesolution treatment).

Accordingly the number of lattice points(N0) in Eq. (5-28) must be replaced by thenumber of monomers n1 (t), and for a calcu-lation of DF* the concentration ofmonomers c1 (according to Fig. 5-60, c1 =0.7 at.% rather than the nominal concentra-tion of c0 = 0.8 at.%) must be used. Thefactor 1⁄2 in Eq. (5-80) takes into accountthat monomers serve both as initial clusters(containing i = 1 solute atom) capable ofgrowing, and also as reaction species beingable to initiate the growth of “monomerclusters”. As shown in Fig. 5-59b, withthese physically justified modifications,excellent agreement is obtained betweenthe maximum nucleation rate (Jmax) as ob-tained from the cluster-dynamics model,and J S

B–D (Eq. (5-80)).

5.8 Self-Similarity,Dynamical Scaling and Power-Law Approximations

5.8.1 Dynamical Scaling

According to the LSW theory and its ex-tensions to finite volume fractions (Sec.5.6), the distribution of relative particlesizes R /R

–evolves during extended aging

(t Æ •) towards an asymptotic, time-invar-iant form, the particular shape of which de-

ˆ * ( )

exp – [ *( ) – *( )]/

J Z n t

F c F kT

B–DS =

×

12

1

1

1

b

D D

Figure 5-59. a) Evolution of the cluster size distri-bution during the nucleation stage for a Cu–0.8 at.%Co model alloy aged at T = 500°C, with pair ex-change parameter W = 6.24, ca

e = 0.2 at.%, DCuCo=1¥10–19 m2/s.b) Time evolution of the nucleation rate J (t) (solidline) according to the cluster-dynamics calculation.JS

B–D is based on Eq. (5-27), J SB–D is the modified

Becker–Döring nucleation rate (Eq. (5-80)). (AfterKampmann et al., 1992.)

pends on the precipitated volume fraction(Fig. 5-44). The final time invariance of f (R /R

–) reflects the fact that once the pre-

cipitated volume fraction has reached itsequilibrium value, consecutive configura-tions of the precipitate microstructure aregeometrically similar in a statistical sense,i.e., all consecutive configurations are sta-tistically uniform on a scale that is consid-erably larger than some characteristiclength such as the mean particle size R

–or

the mean center-to-center distance L–

=Nv

–1/3. The self-similarity of the microstruc-tural evolution has found its expression inthe dynamical scaling of the structure func-tion S (k, t) (Binder and Stauffer, 1974;Binder et al., 1978). Furukawa (1981) pro-posed S (k, t) to satisfy (after some tran-sient time t0) a scaling law of the form:

S (k, t) = l3 (t) F [k l (t)]; t t0 (5-81)

where F [k l (t)] ∫ F (x) is the time-inde-pendent scaling function. As the scalingparameter, l (t) denotes some characteristiclength and contains exclusively the timedependence of S (k, t).

Strong theoretical support for the valid-ity of the scaling hypothesis, Eq. (5-81),during the later stages of decomposition

was first provided by Monte Carlo simula-tions of the time evolution of binary modelalloys (Sec. 5.5.6). From these studies itwas concluded that there is a small thoughsystematic dependence of F (x) on the in-itial supersaturation, at least for small x(Lebowitz et al., 1982); for large values ofx, the scaling function appears to be uni-versal in that it becomes independent oftemperature and precipitated volume frac-tion and even of the investigated material (Fratzl et al., 1983). By analogy to thePorod law of small-angle scattering, in thisregime F (x) decays in proportion to x–4.

In experiments designed to test the valid-ity of the scaling behavior, l (t) is com-monly related to either the radius of gyra-tion RG, the mean particle radius, or to ei-ther km

–1 (t) or k1–1 (t) if km and k1 denote

the maximum and the first moment of S (k,t), respectively. The scaling function F (x)is then simply obtained, for instance, byplotting km

3 S (k, t) versus k /km. If scalingholds, F (k /km) is time independent 5. After


Figure 5-60. Time evolution of thenumber density (nv), the mean (dy-namical) critical radius (R*), themean radius of supercritical clus-ters (R

–); the conventionally defined

supersaturation x – 1 = c/cea – 1 is

compared with the supersaturationx mono– 1 = c1/ce

a – 1 of monomers.For t ≤ 100 min the computationwas performed cluster-dynamicallyand afterwards continued determin-istically on the basis of the Nmodel of Sec. 5.7.3. (After Kamp-mann et al., 1992.)

5 In order to test whether experimental data satisfythe scaling law, Fratzl et al. (1983) have proposed adirect method by which the evaluation of RG, R

–, km

or k1 can be avoided and by means of which F (x)can be determined graphically.


some initial transient time this, in fact, wasobserved in the glass systems B2O3–PbO–Al2O3 (Craievich et al., 1981, 1986) and inseveral binary alloys such as Mn–Cu (Fig.5-61), Al–Zn (e.g., Simon et al., 1984;Hoyt and de Fontaine, 1989) or Ni–Si (Po-lat et al., 1989). Dynamical scaling behav-ior was also found in some ternary alloyssuch as Al–Zn–Mg (Blaschko and Fratzl,1983). For Cu–Ni–Fe, which was studiedby means of anomalous SAXS (Lyon andSimon, 1987), the scaling behavior isfound to be obeyed only by the partialstructure functions, indicating that thissystem does not behave like a pseudo-bi-nary system. For Fe–Cr the results are con-troversial. In contrast to Katano and Iizumi(1984) and Furusaka et al. (1986), La Salleand Schwartz (1984) reported that dynami-cal scaling does not hold. The decomposi-tion kinetics in Fe–Cr at about 500°C arefairly sluggish and it may well be that evenafter the longest chosen aging time (100 h)the system had not yet reached the scalingregion where the microstructure displaysself-similarity.

In principle, the structure function S (k,t) contains all the information on the vari-ous structural parameters of a decomposingsolid such as f (R, t), Nv (t), R

–(t), morphol-

ogy, etc. which, for instance, may controlits mechanical properties. In practice, how-ever, commonly only R

–and Nv and neither

f (R, t) nor the morphology can be extractedfrom experimental data. This stems mainlyfrom a lack of knowledge of the interparti-cle interference function, which containsthe spatial correlations of the precipitatemicrostructure and which manifests itselfin the appearance of a maximum in the S(k, t) curves of less dilute systems. Further-more, both the limited range of k overwhich S (k, t) can be measured and thelarge background in conventional small-angle scattering experiments often render

the quantitative extraction of informationon the precipitate microstructure rather dif-ficult.

From a practical point of view, scalinganalyses are sometimes seen to allow allinformation contained in S (k, t) to be deci-phered. Provided the explicit form of F (x)could be predicted on grounds of a first-

Figure 5-61. Time dependence of the scaling func-tion F (k/km) = k3

m S (k, t) for Mn–33 at.% Cu at450°C. For later times (t > 5115 s) F(k/km) becomestime independent and, hence, dynamic scaling holds(top). The structure functions taken at earlier times(965, 1602, 2239, 2886 and 3532 s) do not yet dis-play scaling behavior (bottom). (After Gaulin et al.,1987.)

principles theory, it could be comparedwith experimental data and employed for acomparison of S (k, t) curves taken fromdifferent materials. Up to now, however,this is not feasible. For this reason variousphenomenological theories for F (x) havebeen conceived (e.g., Furukawa, 1981;Hennion et al., 1982) amongst which themodel of Rikvold and Gunton (1982) maybe regarded as the one that is most conven-ient for a comparison with experimentaldata as it contains the precipitated volumefraction as the only parameter. This modelassumes the two-phase microstructure toconsist of a ‘gas’ of spherical second phaseparticles (with an identical scattering formfactor) each of which is surrounded by azone depleted of solute atoms. With simpleapproximations on the probability distribu-tion for pairs of particles with certain inter-particle spacings, the explicit analyticalform for F (x) was derived. However, dueto the various assumptions invoked, theRikvold–Gunton model is restricted tosmaller precipitated volume fractions. Inspite of its simplicity, fair agreement wasreported between the theoretical F (x) andthe scaling functions obtained from com-puter simulations and from scaling analy-ses of S (k, t) curves taken from decom-posed Al–Zn and Al–Ag–Zn alloys (Simonet al., 1984). In constrast, in studies ofAl–Zn (Forouhi and de Fontaine, 1987)and of Ni–Si (Chen et al., 1988) the pre-dicted F (x) was found to be much broaderthan the experimental ones; scaling analy-ses for borate glasses also could not verifythe theoretically predicted form of F (x)(Craievich et al., 1986). This may stemfrom the inadequate assumptions on thechosen probability distribution in whichlong-range correlations are neglectedand/or from precipitate morphologies de-viating from spheres, e.g., platelets inAl–Zn.

In contrast to Al–Zn, the small misfitstrains between the Al-rich matrix and thespherical d¢-Al3Li precipitates renders theAl–Li system ideal for an accurate test ofscaling in the later stages of coarsening.Using SAXS, Che et al. (1997) found scal-ing behavior to be obeyed in the coarseningregime of each of the Al–Li alloys with dif-ferent volume fractions ranging from 0.18to 0.23. The breadth of the scaled structurefunction, measured by the full width at halfmaximum, versus the equilibrium volumefraction, agreed well with the boundary in-tegral-based computer simulation ofAkaiwa and Voorhees (1994) (cf. Sec.5.6.3).

As a concluding remark to this section itis probably fair to state that scaling analy-ses currently can neither furnish the practi-cal metallurgist with more information onthe precipitate microstructure nor on its dy-namic evolution than has been possible byconventional analyses of S (k, t) curvesprior to the emergence of the scaling hy-potheses. It is felt that precise informationon the size distribution, the morphology,and the spatial arrangement of precipitatesin a two-phase microstructure becomesmore readily available from studies em-ploying direct imaging techniques, e.g.,CTEM or AFIM, in particular, as dynami-cal scaling only holds in the later stages ofaging where the precipitate microstructure,in general, can be easily imaged and re-solved by these techniques. Furthermore,once dynamical scaling is satisfied, thesystem is close to the asymptotic limitwhere the LSW theory or its extensionsmay be applied for a prediction of its dy-namic evolution during further aging.

5.8.2 Power-Law Approximations

So far not explicit assumption has beenmade about the time dependence of the



chosen characteristic length, e.g., km–1 (t),

entering the scaling law, Eq. (5-81). As theself-similarity of a precipitate microstruc-ture and, hence, the scaling law is impli-citly contained in the LSW theory of coars-ening, the region of validity of dynamicalscaling coincides with the LSW regime ofcoarsening. km

–1 (t) is thus expected to showthe simple power-law behavior,

km–1 (t) t a (5-82)

with a = 1/3. Accordingly, if scaling holds,the maximum of the structure functionmust evolve in time as

Sm (t) t b (5-83)

with b = 3a (Eq. (5-81)). Such a power-lawbehavior was frequently corroborated byscattering experiments on materials thatwere aged in the scaling region, and also bycomputer simulations (e.g., Lebowitz et al.,1982), which indicated that scaling wouldhold.

The more recent theoretical develop-ments on the kinetics of phase separationhave predicted various other values for theexponent a. On the basis of their cluster–diffusion–coagulation model (Sec. 5.6.4),for intermediate times Binder and cowork-ers predict a = 1/6 and a = 1/5 or 1/4 forlow and intermediate temperatures, respec-tively (Binder and Stauffer, 1974; Binder1977; Binder et al., 1978). As is shown inFig. 5-31b, approximation of Sm (t), whichdisplays some curvature, by a power-law(Eq. (5-83)) yields b = 0.7 rather than 0.48as implied by scaling. The LBM theory ofspinodal decomposition, which accountsfor some coarsening at earlier stages (Sec.5.5.4), yields a = 0.21. As outlined in Sec.5.5.6, these values agree quite well withthe corresponding values (a = 0.16 to 0.25and b = 0.41 to 0.71, depending on thesupersaturation and ‘aging temperature’)obtained from fitting power laws to the

corresponding data from computer simula-tions.

Inspired by the theoretical predictions,many scattering experiments on alloyswere interpreted in terms of power-law ap-proximations. Frequently the existence oftwo well-defined kinetic regimes with dis-tinct values of a has been reported (e.g.,Fig. 5-32b). At earlier times, a rangesbetween Ù0.1 and ≈0.2, whereas at laterstages a and b are found close to the values1/3 and 1 predicted by the LSW theory.Sometimes this has been taken as evidence(e.g., Katano and Iizumi, 1984) for the firstregime to be dominated by the cluster–diffusion–coagulation mechanism (Sec.5.6.4), whereas in the second regime, theevolution proceeds according to the LSWmechanism via the evaporation and con-densation of single solute atoms.

However, the interpretation of and Sm (t)and km (t) in terms of two distinct kineticregimes, each of which is well described bya power law, seems rather debatable. Acloser examination of the Sm (t) and km

curves (e.g., Fig. 5-32b) always revealssome curvature prior to reaching the scal-ing region. This clearly shows that the ex-ponents a and b are time dependent; thus,apart from the LSW regime, a power-lawapproximation must be seen as a ratherpoor description of the dynamic evolutionof a decomposing solid and commonlydoes not disclose the specific growthmechanism dominating at a certain agingregime. This becomes particularly evidentby employing the Numerical Model (Sec.5.7.3) for a derivation of the exponent a (t)= ∂ log R

–/∂ log t. As the N model com-

prises nucleation, growth and coarseningas concomitant processes on the basis ofjust one growth mechanism – single-atomevaporation or condensation in the LSWsense – a plot of a (t) versus t allows acloser examination to be made of the valid-

ity of power-law approximations at any in-stant. This is shown in Fig. 5-62 for Cu–1.9at.% Ti, the experimental kinetic data ofwhich are well described by the N model(cf. Figs. 5-53 and 5-55). Towards the endof the nucleation regime, a (t) increases

sharply from ≈0.15 to its maximum value a= 0.56 which is indicative of a diffusion-controlled growth of the particles. The du-


Figure 5-62. Variation ofthe time exponent a with ag-ing time as evaluated forCu–1.9 at.% Ti by means ofthe N model.

6 Accounting for modified boundary conditions, thecluster-dynamics approach even yields a maximumvalue a = 0.7 (Kampmann et al., 1992).

Figure 5-63. Dynamical scaling of the structure function of Cu-2.9 at.% Ti beyond t = 250 min; SANS results

5.9 Non-Isothermal Precipitation Reactions 401

ration of the growth regime where R–

evolves according to the parabolic powerlaw R

– t1/2 is rather short for this alloy(cf. Sec. 5.7.4.3). At the end of the growthregime, where the particle number densityhas reached its maximum value (Fig.5-55), a (t) drops within about 250 s to avalue of less than 0.1. In the subsequenttransition regime at intermediate times,a (t) increases continuously and approachesonly slowly the LSW-coarsening regimewhere dynamical scaling holds. Evidently,there is no time regime between the appear-ance of the growth regime and the LSW re-gion where power-law behavior is ob-served. On the other hand, it may be in-ferred from Fig. 5-62 that the kinetic evo-lution of R

–during intermediate aging

stages might be artificially interpreted interms of power laws if the time windowcovered by the experiment was too short;in this case any exponent between 0.15 and0.33 may be derived. Thus, if the time win-dow for the kinetic experiment is not prop-erly chosen, a ≈ 0.2 may be obtained, butwithout the cluster–diffusion–coagulationmechanism being operative. As has beenpointed out in Sec. 5.6.4.3, with increasingsupersaturation, the growth regime withR– µ t 1/2 disappears completely and a (t)takes values only between zero and 1/3.Furthermore, the transition period (wherea< 1/3) becomes shorter. In this case, LSWcoarsening and dynamical scaling are ob-served after rather short aging times. This isillustrated in Fig. 5-63 for Cu–2.9 at.% Tiwhich satisfies dynamical scaling after ag-ing for ≈250 min at 350°C, whereas for theless concentrated Cu–1.9 at.% Ti alloy scal-ing only holds after about 5 ¥ 104 min.

5.9 Non-Isothermal PrecipitationReactions

For many age-hardening commercial al-loys, the microstructure is established dur-ing continuous cooling from the processingtemperature, i.e., the solidification or hot-working temperature in cast or wrought al-loys, respectively, or during cooling afterlaser surface treatments or welding (Brat-land et al., 1997, and references therein).The resulting non-isothermal transforma-tion usually involves nucleation andgrowth of the precipitates as concomitantprocesses. Modeling of non-isothermaltransformation rates thus ought to accountfor the independent variations of the nucle-ation and the growth rates with temperatureand with the temperature-dependent solu-bilities in the matrix and the precipitatedphase. Furthermore, heterogeneous nuclea-tion at lattice defects often must also betaken into account, as usually the drivingforce for homogeneous nucleation be-comes sufficient only once the alloy hasbeen cooled sufficiently deep into the mis-cibility gap.

Whereas previous models, which havebeen based on isokinetic behavior and theadditivity concept, are limited to a descrip-tion of only diffusional growth (or dissolu-tion during reheating) (e.g., Shercliff and Ashby, 1990a, b; Myhr and Grong,1991a,b; Onsoien et al., 1999), Grong andMyhr (2000) considered the non-isother-mal precipitation reaction as a coupled nu-cleation and growth process in terms of anumerical solution and two analytical mod-els. As validated through a comparisonwith the numerical results, both analyticalmodels, i.e., a simplified state variable so-lution and a solution based on the Avramiequation, yielded an adequate descriptionof the overall non-isothermal transforma-tion behavior comprising the variation of

the nucleation and growth rates as well asthe precipitated volume fraction with tem-perature and cooling rate.


The authors would like to thank Prof. R.Bormann and Dr H. Mertins for their criti-cal comments on this chapter and Mrs E.Schröder, A. Conrad-Wienands and B.Feldmann for their kind endurance and as-sistance in preparing the manuscript. Thesupport of this work by the Deutsche For-schungsgemeinschaft (Leibniz-Programm)is also gratefully acknowledged.

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6 Spinodal Decomposition

Kurt Binder

Institut für Physik, Johannes Gutenberg-Universität Mainz,Mainz, Federal Republic of Germany

Peter Fratzl

Erich-Schmid-Institut der Österreichischen Akademie der Wissenschaftenund Montan-Universität Leoben, Leoben, Austria

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4146.2 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4166.2.1 Phenomenological Thermodynamics of Binary Mixtures

and the Basic Ideas of Phase Separation Kinetics . . . . . . . . . . . . . . 4166.2.2 The Cahn–Hilliard–Cook Nonlinear Diffusion Equation . . . . . . . . . . 4206.2.3 Linearized Theory of Spinodal Decomposition . . . . . . . . . . . . . . . 4216.2.4 Spinodal Decomposition of Polymer Mixtures . . . . . . . . . . . . . . . 4266.2.5 Significance of the Spinodal Curve . . . . . . . . . . . . . . . . . . . . . 4286.2.6 Towards a Nonlinear Theory of Spinodal Decomposition

in Solids and Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4376.2.7 Effects of Finite Quench Rate . . . . . . . . . . . . . . . . . . . . . . . . 4416.2.8 Interconnected Precipitated Structure Versus Isolated Droplets,

and the Percolation Transition . . . . . . . . . . . . . . . . . . . . . . . . 4426.2.9 Coarsening and Late Stage Scaling . . . . . . . . . . . . . . . . . . . . . 4456.2.10 Effects of Coherent Elastic Misfit . . . . . . . . . . . . . . . . . . . . . . 4476.3 Survey of Experimental Results . . . . . . . . . . . . . . . . . . . . . . 4506.3.1 Metallic Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.3.2 Glasses, Ceramics, and Other Solid Materials . . . . . . . . . . . . . . . . 4546.3.3 Fluid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4556.3.4 Polymer Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4576.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4606.4.1 Systems Near a Tricritical Point . . . . . . . . . . . . . . . . . . . . . . . 4606.4.2 Spontaneous Growth of Ordered Domains out of Initially Disordered

Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4626.4.3 Phase Separation in Reduced Geometry and near Surfaces . . . . . . . . . 4676.4.4 Effects of Quenched Impurities; Vacancies; Electrical Resistivity

of Metallic Alloys Undergoing Phase Changes . . . . . . . . . . . . . . . 4686.4.5 Further Related Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . 470


6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4736.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

410 6 Spinodal Decomposition



a lattice spacingA, B, … symbols for chemical elementsa (t) structural relaxation variableA, B symbols for Landau expansion coefficients or other constantsa0, …, a3 constantsB critical amplitude of the order parameterc (x), ci local concentration, concentration of site iC critical amplitude of the scattering functioncs

B, csp(T ) concentration at the spinodal curveccoex concentration at the coexistence curvec– average concentrationccrit critical concentrationd spatial dimensionalityD0 diffusion constantD2 damping coefficient of second soundDt tracer diffusion coefficiente electron chargeeij elastic strain tensore0

ij elastic misfit tensorEact activation energyF thermodynamic free energyD free energy functionDF* nucleation free-energy barrierF*MF mean-field result for the nucleation barrierf0 parameter or universal constant of order unityfcg free energy density (coarse-grained)fD Debye functiong cooling rate Hamiltonianä Planck constantJ concentration current densityJ interaction strength of the Ising modelK elastic modulusk wavevector, scattering vector in diffraction experimentsk modulus of kkB Boltzmann constantkc critical wavenumber in Cahn’s theorykF radius of Fermi spherekm wavenumber of maximal growthL length scale of coarse-graining celll number of atoms within a cluster Liouville operatorM mobility

meff effective mass of an electronN chain length of a polymerN number of atoms per cm3

nL concentration of a cluster containing L atomsPL(M) magnetization distribution functionq rescaled wavenumberQB superstructure Bragg spotr range of effective interactionr radius of the grain·R2Ò mean-square end-to-end distance of a polymer chain·R2

gyrÒ gyration radius of a polymer chainR (k) rate factorr0 phenomenological coefficientS (k, t) equal-time structure factors (x, t) entropy densityS rescaled structure factorT temperaturet timeT0 temperature at which a quench is started at t =0Tab Oseen tensor (component)Tc critical temperatureTi , Tf initial, final temperaturetij elastic stress tensorTt tricritical pointu phenomenological coefficient of the Landau expansionu2 second sound velocityV volumev phenomenological coefficient of the Landau expansionw elastic energy densityW (l, l¢) cluster reaction matrixWA, WB time constant of element A, Bx Lifshitz–Slyozov exponentx position vectorx, y, z spatial coordinatesXA, XB degree of polymerization of a polymer chain of type A, B scaling variableZ-A%, Z-B% atomic numbers of constituents A, B

a phenomenological coupling constantb critical exponent of the order parameterg critical exponent of the structure factorg phenomenological coefficientG, g rate factorsh viscosityha change of lattice constant with alloy composition



hT random forceQ temperature at which behavior of polymer in solution is idealk factor describing screening of Coulomb interactionl wavelengthlc critical wavelength in Cahn’s theorylijmn elastic stiffness tensorL generalized Onsager coefficientm chemical potential differencem elastic modulusn critical exponent of the correlation lengthnP Poisson coefficient in elasticity theoryx interfacial width; correlation lengthx critical amplitude of the correlation lengthxcoex correlation length at the coexistence curver1, r2, … probability distributionss subunit length of a polymersij elastic stress tensort rescaled coordinate (time)t time constantj pair interaction energyF volume fractionFc volume fraction at the critical pointc Flory–Huggins parametercs Flory–Huggins parameter at the spinodal curvecn

–1 phenomenological coefficienty complex order parameter

ABV element-A–element-B–vacancy modelCHC Cahn–Hilliard–Cook (method)ESA European Space AssociationLBM Langer–Baron–Miller (method)LSW Lifshitz–Slyozov–Wagner (method)m (index) maximumMF mean fieldp (index) polarizationV vacancy (also as index)

6.1 Introduction

This chapter deals with the dynamics ofphase changes in materials, which arecaused by transferring the material into aninitial state that is not thermodynamicallystable (e.g. by rapid cooling (“quenching”)or, occasionally, rapid heating; in fluids thesystem may also be prepared by a rapidpressure change). It is generally believed(Gunton et al., 1983; Binder, 1984a, 1987a,1989; Kostorz, 1991; Wagner and Kamp-mann, 2000) that in homogeneous materi-als such phase changes may be initiated bytwo different types of mechanism, corre-sponding to two different types of statisti-cal fluctuations, namely “homophase fluc-tuations” and “heterophase fluctuations”.Fig. 6-1 illustrates these fluctuations qual-itatively for the example of a binary mix-ture, where the variable to consider is a lo-cal concentration variable c (x). By “local”we do not mean the scale of a lattice site i(in a crystalline solid), since then the asso-ciated concentration variable ci could take

only two values: ci =1, if the site is takenby a B atom, and ci = 0, if it is taken by anA atom in an AB mixture. Instead here weare interested in an averaged concentrationfield obtained by “coarse graining” over acell of size L3 (or L2 if we consider a two-dimensional layer):

(6-1)

where x is the center of gravity of the cellover which ci is averaged. This intermedi-ate length scale L must be large in compar-ison with the lattice spacing, since only ifthe cell contains many lattice sites is a con-tinuum description as anticipated in Fig.6-1 warranted. On the other hand, L mustbe small in comparison with the lengthscales of the statistical fluctuations that wewish to consider, i.e., the wavelength l ofthe concentration wave in Fig. 6-1a, or thewidth x of the interface between a dropletof the new phase and its environment inFig. 6-1b. As we shall see, the existence ofthis intermediate length scale L severely re-

c L ci

i( )x =cell

− ∑3

Œ


Figure 6-1. Schematic diagram of unstable thermodynamic fluctuations in the two-phase regime of a binarymixture AB at a concentration cB (a) in the unstable regime inside two branches cs

B of the spinodal curve and (b)in the metastable regime between the spinodal curve cs

B and the coexistence curve c(1)coex. The local concentration

c (r) at a point r = (x, y, z) in space is schematically plotted against the spatial coordinate x at some time t afterthe quench. In case (a), the concentration variation at three distinct times t1, t2, and t3 is indicated. The diame-ter of the critical droplet, whose cross-section is shown in case (b), is denoted by 2R*, and the width of the inter-facial region by x. Note that the concentration profile of the droplet reaches the other branch of the coexistencecurve c(2)

coex in the droplet center only for weak “supersaturations” of the mixture, where cB – c(1)coex O cs

B – cB andR* o x ; for the sake of clarity, the figure is therefore not drawn to scale (Binder, 1981).


stricts the quantitative validity of the theo-retical concepts developed in the presentchapter (see also Binder, 2000). This chap-ter will exclusively consider the mecha-nism of Fig. 6-1a, where in a thermody-namically unstable initial state long-wave-length delocalized small-amplitude statis-tical fluctuations grow spontaneously inamplitude as the time after the quench in-creases. For a binary mixture, this mecha-nism is called spinodal decomposition(Cahn, 1961, 1965, 1966, 1968); the rele-vant fluctuations can then be considered asa wavepacket of “concentration waves”,and one such wave is shown in Fig. 6-1a.As will be discussed below, this mecha-nism should not occur inside the wholetwo-phase coexistence region of the phasediagram of the mixture, but rather only in-side a smaller region, the boundary ofwhich is given by the spinodal curve (Cahnand Hilliard, 1958, 1959). Between the spi-nodal curve cs

B and the coexistence curvec(1)

coex (or in between the other branch of thespinodal and c(2)

coex the mechanism of drop-let nucleation and growth is involved (Fig.6-1a). This latter mechanism is discussedin the chapter by Wagner et al., 2001).Apart from these phase transformationmechanisms triggered by spontaneous ther-mal fluctuations, heterogeneous nucleationprocesses must also be considered (Zettle-moyer, 1969); near inhomogeneities insolids such as grain boundaries, disloca-tions, external surfaces, or point defectssuch as substitutional or interstitial impur-ities or clusters thereof, microdomains ofthe new phase may already be formed be-fore the quench, or at least their formationafter the quench is greatly facilitated. Gen-eral theoretical statements about theseheterogeneous mechanisms, however, arehardly possible without a detailed discus-sion of the nature of such defects. Suchproblems will not be discussed in this

chapter, and the spontaneous growth ofthermal fluctuations dominating phasechanges of unstable but nearly ideal (i.e.,defect-free) systems will be emphasized.Although most of the discussion refers tobinary mixtures, an extension to ternarymixtures is often possible.

Unmixing of binary or ternary mixturesis not the only phase change where sponta-neous growth of statistical fluctuations oc-curs. Consider, for example, an alloy A–B,which in thermal equilibrium undergoes anorder–disorder phase transition form a dis-ordered state, where the two species of at-oms A and B are distributed at random overthe available lattice sites, to an ordered ar-rangement. There A and B atoms preferen-tially occupy sites on sublattices (e.g., b-CuZn, Cu3Au, CuAu, Fe3Al, and FeAl). Ifwe quench such an alloy from the disor-dered regime to a state which in equilib-rium should be ordered, the unstable disor-dered initial state may also decay by spon-taneous growth of fluctuations. The dis-tinction from Fig. 6-1a is that the wave-length of the growing concentration waveis not large but coincides with the latticespacing of the superstructure. Neverthe-less, the theoretical treatment of this spino-dal ordering (De Fontaine, 1979) is similarto the theory of spinodal decomposition,and will also be discussed in this chapter,which emphasizes the theoretical aspects(see Wagner and Kampmann (2000) for acomplementary treatment emphasizing thepoint of view of the experimentalist).

One of the basic questions in the applica-tions of materials is to understand the pre-cipitated microstructure that forms in thelate stages of a phase separation process.One of the main characteristics of this mi-crostructure concerns its morphology: theminority phase may either form compactislands well separated from each other or itmay form an irregular interconnected net-

work-like structure (Fig. 6-2). Obviously, acritical volume fraction exists where thisinterconnected net first appears, a so-calledpercolation threshold (Stauffer, 1985). Of-ten it is assumed that the morphology ofwell separated islands must have formedby nucleation and growth, while the perco-lating morphology is taken as the signatureof the spinodal curve. However, such anidentification is misleading, as will be dis-cussed in detail in this chapter: the spinodalcurve at best has a meaning for the initialhomogeneous state and controls the initialstages of phase separation; in the latestages, however, a generally universalcoarsening behavior occurs (which is sen-sitively affected by defects!), irrespectiveof whether the morphology is intercon-nected or not. Since a closely related uni-versal coarsening behavior also occurs influid mixtures and can be studied fairlywell in these systems, we include a discus-sion of phase separation in fluid mixturesthroughout this chapter. Moreover, “poly-mer alloys” can be conveniently producedby quenching from the fluid phase of poly-mer mixtures, and have become a practi-cally important class of materials (Hashi-moto, 1993).

6.2 General Concepts

6.2.1 Phenomenological Thermodynamicsof Binary Mixtures and the Basic Ideasof Phase Separation Kinetics

We first consider the dynamics of phasetransformations induced by an instantane-ous quench from the initial temperature T0

to a final temperature T (the idealization ofthis infinitely rapid cooling will be relaxedin Sec. 6.2.7). The simplest case wheresuch a sudden change of external parame-ters causes a phase transition is a systemundergoing an order–disorder phase tran-sition at a critical temperature Tc withT < Tc < T0 (Fig. 6-3a). The initially disor-dered system is then unstable, and immedi-ately, small ordered domains form. As thetime t after the quench passes, the order pa-rameter in these domains must reach its equi-librium value (± y), and the domain sizemust ultimately grow to macroscopic size.

Alternatively, we consider a solid or liq-uid mixture A–B with a miscibility gap(Fig. 6-3b). Quenching the system at timet = 0 from an equilibrium state in the singlephase region to a state underneath the co-existence curve leads to phase separation;in thermal equilibrium, macroscopic re-gions of both phases with concentrationsc(1)

coex and c(2)coex coexist.


Figure 6-2. Snapshot pic-tures of a two-dimensionalsystem undergoing phaseseparation with a volumefraction of (a) F = 0.21 and(b) F = 0.5. The pictureswere obtained by numericalsolution of the non-linearCahn–Hilliard equation, Eq.(6-14) (Rogers and Desai,1989).

6.2 General Concepts 417

Of course, phase transformations com-bining an ordering process (such as in Fig.6-3a) with an unmixing process (such as inFig. 6-3b) also occur; as we are interestedmainly in the basic concepts of these pro-cesses, we will not consider these morecomplex phenomena here, but will returnto this problem in Sec. 6.4.1.

The basic concepts of the kinetic mech-anisms of the processes indicated in Fig.6-3a and b invoke the idea of extending theconcept of the thermodynamic Helmholtzfree energy F to states out of equilibrium(Gunton et al., 1983; Binder, 1987a). Inthe single-phase region of the A-rich alloy,F first decreases with increasing concen-tration of B atoms cB, owing to the entropyof mixing. Similarly, in the B-rich single-phase region, F decreases with increasingconcentration of A atoms, cA =1– cB. In thetwo-phase coexistence region in equilib-

rium, we have a linear variation of F withcomposition; this reflects the fact that theamounts of the two coexisting phases (withconcentrations c(1)

coex, c(2)coex) change linearly

with cB according to the lever rule.Mean-field type theories (Gunton et al.,

1983; Binder, 1987a) suggest that a freeenergy F¢ Ù F can also be introduced de-scribing single-phase states within the two-phase region. Since F¢ must coincide withF for c(1)

coex and c(2)coex, inevitably a double-

well structure results, as indicated by thedash-dotted curve in Fig. 6-3c. This hypo-thetical free energy F¢ now allows a furtherdistinction to be made: states where(∂2F¢/∂c2

B)T > 0 are called metastable, andstates where (∂2F¢/∂c2

B)T < 0 are called un-stable. The locus of inflection points in the(T, cB) plane,

(∂2 F¢/∂c2B)T = 0 (6-2)

Figure 6-3. (a) Order parameter y of a second-order phase transition vs. temperature, assuming a two-folddegeneracy of the ordered state (described by the plus and minus signs of the order parameter). The quenchingexperiment is indicated. (b) Phase diagram of a binary mixture with a miscibility gap ending in a critical point(Tc, cB

crit) of unmixing, in the temperature–concentration plane. Again the quenching experiment is indicated,and the quenching distances from the coexistence curve (dT ) and from the critical point (DT ) are indicated.(c) Free energy plotted versus composition at temperature T (schematic). For further explanations, see text(Binder, 1981).

defines the spinodal curve c = csB(T ). This

distinction is now linked to the two trans-formation mechanisms shown in Fig. 6-1;it is the unstable regime inside the twobranches of the spinodal curve, wherelong-wavelength fluctuations spontane-ously grow rather than decay. At a laterstage, the inhomogeneous concentrationdistribution thus generated will coarsen. Inthe metastable regime, the system is stableagainst such weak (small-amplitude) fluc-tuations, and localized large-amplitudefluctuations (“droplets” of the new phase)must form in order to start the transforma-tion.

This idea implies, as will be explainedbelow, a singular transition in the kinetictransformation mechanism at the spinodalcurve. However, this spinodal singularity isreally only an artefact of an over-simplifiedtheoretical picture: apart from the veryspecial limit of infinitely weak, infinitelylong-ranged forces in which mean-fieldtheory becomes correct (Penrose and Lebo-witz, 1971; Lebowitz and Penrose, 1966;Binder, 1984b), the transition from the nu-cleation mechanism to the spinodal decom-position mechanism is completely gradual

(Binder et al., 1978). As will be discussedin more detail in Sec. 6.2.5, the spinodalcurve cannot be unambiguously defined.At this point, we recall that the “weak delo-calized long-wavelength fluctuations” can-not be identified in terms of the atomicconcentration variable ci , which undergoesrapid large-amplitude (ci = 0 to ci =1!) vari-ations from one lattice site to the next, butimplies the introduction of the coarse-grained concentration field c (r), Eq. (6-1).

The free energy F¢ of “homogeneous”states in the two-phase region depends onthe length scale L over which short-wave-length concentration fluctuations havebeen integrated (Fig. 6-4). This “coarse-grained” free-energy density fcg is not pre-cisely identical to the true free-energy den-sity f in the single-phase region, since con-centration variations with wavelengths ex-ceeding L contribute to f but are excludedfrom fcg. However, this difference is minor,and in the single-phase region we may con-sider the limit L Æ ∞ and then fcg tends to-wards f uniformly. This is not possible inthe two-phase region, however, where fcg

describes homogeneous states only ifLÁx, the interfacial width; if L Ô x then


Figure 6-4. Schematic plot ofthe coarse-grained free-energydensity fcg(f) as a function of theorder parameter f = (c – ccrit)/ccrit

in a first-order transition fromf1

coex to f2coex, for a “symmetric”

situation with f (f1coex) = f (f2

coex)as in the Ising model. Spinodalsfs

(1,2)(L) defined from inflectionpoints of fcg(f) depend distinctlyon the coarse-graining length Land the interfacial width x (Bin-der, 1987a).


the states that yield dominating contribu-tions to fcg are phase separated on a localscale, and therefore fcg tends smoothly to-wards the double-tangent construction asL Æ ∞ . Hence there is no unique theoreti-cal method of calculating a spinodal curve;remember that mean-field theories, whichyield spinodal curves easily, are only inac-curate descriptions of real systems. Simi-larly, extrapolation procedures by whichspinodal curves are extracted from experi-mental data also involve related ambigui-ties, as will discussed in Sec. 6.2.5.

Hence the coarse-graining alluded toabove means that a microscopic Hamil-tonian ci of the binary mixture is re-placed by a so-called free-energy functionD c (x). For example, the Hamiltonianmay correspond to a standard Ising-typepairwise interaction model (Binder, 1986;De Fontaine, 1979):

where j is the interaction energy between apair of atoms.

Carrying out the coarse-graining definedin Eq. (6-1) we expect that ci/kBT willbe replaced by

where d is the spatial dimensionality of thesystem and r is the range of the effectiveinteraction in Eq. (6-3) (Dj (xi – xj) ∫ j AA

+ j BB – 2j AB):

(6-5)rd

ji j i j

ji j

2

2

2=

∑

∑

− −

−

( ) ( )

( )

x x x x

x x

D

D

j

j

D ( )

[ ( )]/ [ ( )]

ck T

x f c k T r cd

x

x xB

cg B

(6-4)

= d +∫ ∇12

2 2

( )

( ) [ ( ) ( )]( ) ( ) ( )

c c c

c c c cc c

ii j

i j i j

i j i j j i

i j i j

= (6-3)BB

AB

AA

12

1 11 1

≠∑ −

+ − − + −+ − − −

j

jj

x x

x xx x

The term 1–2

r2[—c]2 in Eq. (6-4) accountsfor the free energy cost of inhomogeneousconcentration distributions. Here thecoarse-grained free-energy density fcg(c)resulting from Eqs. (6-3) and (6-1) by car-rying out the restricted trace over the cifor a fixed concentration field c (x) isvery difficult to obtain in practice; qualita-tively the behavior of fcg(c) should be sim-ilar to the mean-field (MF) result for thefree energy of a binary mixture,

(6-6)

with

(6-7)

Near TcMF we can replace fMF(c) by its Lan-

dau expansion,

(6-8)

where A ~ (T/TcMF– 1) < 0 for T < Tc

MF andB > 0. It is well known that the actualcritical temperature Tc does not coincidewith the mean-field prediction. Therefore,it is usually assumed that the parametersappearing in the actual coarse-grainedfree energy fcg(c) are not the mean-fieldparameters A, B,…, but rather these pa-rameters are “renormalized” due to short-wavelength fluctuations, and also r (Eq.(6-5)) may thus be modified. Therefore,fcg(c) and r are not calculated from micro-scopic models such as Eq. (6-3), but aretreated as phenomenological input parame-ters of the theory, which are fitted to ex-periment.

f c f A c c

B c cMF crit

crit

=( ) ( )

( )0

2

4

+ −+ − + …

k Tj

i jB cMF ≡ −∑ Dj ( )x x

11 1

2 1

k Tf c c c c c

TT

c c

BMF

cMF

=( ) ln ( ) ln( )

( )

+ − −

+ −

6.2.2 The Cahn–Hilliard–CookNonlinear Diffusion Equation

Since in the total volume V the averageconcentration,

c– = (1/V ) Ú dx c (x, t) (6-9)

is conserved, the time-depent concentra-tion field c (x, t) satisfies a continuity equa-tion,

(6-10)

where j (x, t) is the concentration currentdensity. Following standard nonequilib-rium thermodynamics (de Groot and Ma-zur, 1962), j (x, t) is assumed to be propor-tional to the gradient of the local chemicalpotential difference m (x, t):

j (x, t) = – M —m (x, t) (6-11)

where M is a mobility that is discussed be-low.

In the thermal equilibrium the chemicalpotential difference is given as a partial de-rivative of the Helmholtz energy F (c, T ):

m = (∂F/∂c)T ; (6-12)

remember that the condition for two-phasecoexistence, the equality of chemical po-tential differences

is the physical content of the double-tan-gent construction shown in Fig. 6-3c. Eq.(6-12) is generalized to an inhomogeneousnonstationary situation far from equilib-rium, where both c (x, t) and m(x, t) dependon space and time, by defining m(x, t) asa functional derivative of the Helmholtzenergy functional D in Eq. (6-4):

m(x, t) ∫ d (D c (x, t))/dc (x, t) (6-13)

Inserting Eq. (6-14) into Eq. (6-13) yields

m(x, t) = (∂ fcg/∂c)T – r2 kBT —2c (x, t)

m m1 21 2= = =coex coex

( / ) ( / )( ) ( )∂ ∂ ∂ ∂F c F cT c T c| |

∂∂

+ ∇ ⋅c tt

t( , )

( , )x

j x = 0

and using this result in the continuity rela-tion, Eq. (6-10), we obtain the Cahn–Hil-liard nonlinear diffusion equation (Cahn,1961):

One immediately obvious defect of thisequation is its completely deterministiccharacter, which implies that random sta-tistical fluctuations are disregarded (apartfrom fluctuations included in the initialcondition, the state at temperature T0 wherethe quench starts). This defect can be rem-edied, following Cook (1970), by adding arandom force term hT (x, t) to Eq. (6-14):

Here hT (x, t) is assumed to be delta-corre-lated Gaussian noise, and the mean-squareamplitude ·h2

T ÒT is then linked to the mobil-ity M via a fluctuation-dissipation relation.

·hT (x, t)hT (x¢, t¢)ÒT

= ·h2TÒT —2 d(x – x¢) d(t – t¢) (6-16)

·h2TÒT = 2 kB T M (6-17)

Eqs. (6-15)–(6-17) constitute the main re-sults of this section, on which all furthertreatment presented here is based. At thispoint we stress the main assumptions thathave been made either explicitly or tacitly:

(i) Effects due to the lattice anisotropy ofthe solid have been ignored. In a crystallinesolid the interfacial free energy betweenco-existing A-rich and B-rich phases will

∂∂

∇

×∂

∂⎛⎝⎜

⎞⎠⎟

− ∇⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+

c tt

M

f c t

cr k T c t

tT

T

( , )

[ ( , )]( , )

( , )

x

xx

x

= (6-15)

cgB

2

2 2

h

∂∂

∇

×∂

∂⎛⎝⎜

⎞⎠⎟

− ∇⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

c tt

M

f c t

cr k T c t

T

( , )

[ ( , )]( , )

x

xx

= (6-14)

cgB

2

2 2



depend on the orientation of the interface;only in the limit T Æ Tc does this interfa-cial free energy become isotropic (Wortis,1985). The isotropic form of Eq. (6-15)outside the critical region would only holdfor phase separation in isotropic amor-phous solids.

(ii) It is assumed that the local concen-tration field is the only slowly relaxing var-iable whose dynamics must be consideredexplicitly after the quench, while all othervariables equilibrate instantaneously. Thisassumption is not true, of course, if phase-separating fluids are considered where thecoupling of hydrodynamic variables andthe resulting long-range hydrodynamicinteractions need to be taken into account(Kawasaki and Ohta, 1978). However, insolids other slow variables may also occur,particularly near the glass transition (inamorphous solids) or near other phasetransformations. A treatment of spinodaldecomposition in the presence of a cou-pling to slowly relaxing structural vari-ables is only possible in very simplifiedcases (Binder et al., 1986).

(iii) It is assumed that the spatial concen-tration variations of interest are small, i.e.,in Eq. (6-4) terms of the order [—2c (x)]2

(and higher) can be neglected in compari-son with the lowest-order gradient term,[—c (x)]2. This assumption is true near thecritical point of unmixing, where the inter-facial width x is much larger than the inter-atomic spacing, but it is not true in the gen-eral.

(iv) The concentration dependence of themobility M is neglected in the derivationyielding Eqs. (6-14) and (6-15). Again, thisapproximation is valid near Tc, since thenthe relevant scale for concentration varia-tions becomes small ((c(2)

coex – c(1)coex)/ccritO1),

but is not true in general.(v) The use of a continuum description

for a solid (Eqs. (6-1) and (6-4)) requires

that a coarse-graining is performed over alength scale much larger than the latticespacing, but less than x, which again re-stricts the region of validity of the theory tothe critical region. On the other hand, fluc-tuations and nonlinear phenomena areknown to lead to a breakdown of meanfieldtheory near the critical point, and nontrivialcritical phenomena arise (Stanley, 1971;Binder, 2001). As we shall see (Secs. 6.2.5and 6.2.6) this also restricts the applicabil-ity of the theory.

6.2.3 Linearized Theoryof Spinodal Decomposition

It may seem unsatisfactory that the basicequation of the theory (Eq. (6-15)) is sup-posed to hold only under fairly restrictiveconditions, but what is even worse is thefact that this equation completely with-stands an analytical solution, and bruteforce numerical approaches where Eq. (6-15) is attacked by large-scale computersimulation are needed (Meakin et al., 1983;Petschek and Metiu, 1985; Milchev et al.,1988; Gawlinski et al., 1989; Rogers et al.,1988; Toral et al., 1988; Oono and Puri,1988; Puri and Oono, 1988). We shall re-turn to these approaches in Sec. 6.2.6, andalso address the complementary approachof Monte Carlo simulations based directlyon the microscopic Hamiltonian, Eq. (6-3)(Bortz et al., 1974; Marro et al., 1975,1979; Rao et al., 1976; Sur et al., 1977;Binder et al., 1979; Lebowitz et al., 1982;Fratzl et al., 1983; Heermann, 1984a, b,1985; Amar et al., 1988). Here we prefer todiscuss the assumption that in the initialstages of unmixing the fluctuation

dc (x, t) ∫ c (x, t) – c–

is small everywhere in the system. Since Lcannot be made arbitrarily large, this as-sumption is not typically true, as we shall

see below (Secs. 6.2.5 and 6.2.6), but nev-ertheless it is instructive to study it!

Given this assumption, we may linearizeEq. (6-15) (or Eq. (6-14) if we also neglectthe thermal noise) in dc (x, t). Then Eq. (6-14) becomes

and, introducing Fourier transformations

dck(t) ∫ ∫ dd x exp(i k · x) dc (x, t) (6-19)

Eq. (6-18) is solved by a simple exponen-tial relaxation,

dck(t) ∫ dck(0) exp[R (k) t] (6-20)

with the rate factor R (k):(6-21)

R (k) ∫ M k2 [(∂2 fcg/∂c2)T,c– + r2 kBT k2]

The equal-time structure factor S (k, t) attime t after the quench:

S (k, t) ∫ ·dc–k(t) dck(t)ÒT (6-22)

where ·…ÒT denotes a thermal average,then also exhibits a simple exponential re-laxation:

S (k, t) ∫ ST0(k) exp[2R (k) t] (6-23)

Here the prefactor(6-24)

ST0(k) ∫ ·dc– k(0) dck(0)ÒT ∫ ·dc– k dckÒT0

is simply the equal-time structure factor inthermal equilibrium at temperature T0 be-fore the quench. Note that R (k) is positivefor 0 < k < kc, with

(6-25)kc ∫ 2 p /lc = [– (∂2 fcg/∂c2)T,c– /(r2 kBT )]1/2

Thus, whereas the structure factor shouldexhibit exponential growth in this region,for k = kc it should be time-independent,S (kc, t) = S (kc, 0). However, neither in ex-

∂∂

∇

×∂

∂⎛

⎝⎜⎞

⎠⎟− ∇

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

tc t M

f c

cr k T c t

T c

d

d

( , )

( )( , )

,

x

x

= (6-18)

cgB

2

2

22 2

periment (Fig. 6-5a, b) nor in simulations(Figs. 6-5c, d) is such a time-independentintersection point observed. Also in theCahn plot, R (k)/k2 plotted versus k2, in-stead of the predicted linear behavior(Fig. 6-6a), 2R (k)/k2 =2 D0 (1 – k2/kc

2) witha negative diffusion constant

D0 = – M(∂2 fcg/∂c2)T,c– (uphill diffusion)

curvature is typically observed (Fig. 6-6b).There are several possible reasons why

the simple linearized theory of spinodal de-composition as it has been outlined so far isinvalid:

(i) Fluctuations in the final state at thetemperature T must be included (Cook,1970). This problem will be considered atthe end of this subsection.

(ii) Nonlinear effects are important dur-ing the early stages of the quench. Thisproblem will be discussed in Secs. 6.2.5and 6.2.6.

(iii) There is already an appreciable re-laxation of the structure factor occurringduring the quench from T0 to T if the cool-ing rate g = – dT/dt is finite (see Sec. 6.2.7).

(iv) The concentration field is coupled toanother slowly relaxing variable (Binder etal., 1986; Jäckle and Pieroth, 1988). Herewe only very briefly outline the idea of theapproach of Binder et al. (1986). Accord-ing to Eqs. (6-23) and (6-21), the maximumgrowth rate of the structure factor occurs at

Rm = R (km), km = kc/ ÷–2 (6-26)

Suppose now the concentration coupleslinearly to a non-conserved variable a (t)describing, for example, structural relaxa-tion, whose fluctuations in the absence ofany coupling would decay exponentially,proportional to exp(–g t). The decay ofconcentration fluctuations will be affected(i) if the coupling between the variablesa (t) and c (x, t) is sufficiently strong, and



Figure 6-5. (a) Neutron small-angle scattering intensity vs. scattering vector k (b: scattering angle) for anAu–60 at.% Pt alloy quenched to 550 °C (Singhal et al., 1978). (b) Time evolution of the structure factorat 541 K in Al–38 at.% Zn after subtraction of the prequench scattering. Solid lines are the best fit to the datausing the LBM theory, Eq. (6-60) (Mainville et al., 1997). (c) Time evolution of the structure factor S (k, t) ac-cording to a Monte Carlo simulation of a three-dimensional nearest-neighbor Ising model of an alloy at criticalconcentration and temperature T = 0.6 Tc. Due to the periodic boundary condition for the 30 ¥ 30 ¥ 30 lattice, k isonly defined for discrete multiples of (2 p)/30; these discrete values of S (k, t) are connected by straight lines(Marro et al., 1975). (d) Time evolution of the normalized structure factor S (k, t) vs. k for the discrete versionof the Ginzburg–Landau model (Eq. (6-4)), namely

where cl is a continuous variable representing the average concentration in the l th cell of size L ¥ L, and thephenomenological constants A, B, and C have been chosen as A/C = – 2.292, ÷-

B/C = 0.972 (C can be scaled outby redefining the cl s). Data are for an N = 40 ¥ 40 lattice with periodic boundary conditions at times t = 0, 10,20, …, 90 Monte Carlo steps (MCS) per site. The arrow indicates the estimate for the wavenumber of maximumgrowth, km(0) = kc /÷-

2 = ÷----–A/C (Milchev et al., 1988).

/ [ ( ) ( ) ] ( )k T A c c B c c C c cl

l ll m

lB crit crit m= ∑ ∑− + − + −⟨ ⟩

2 4 12

2

(ii) if the rates Rm and g are of the sameorder. In the absence of any couplingbetween these variables their fluctuationswould decay independently of each other,with decay rates being given as G+(k) = g(relaxation of a (t) is then independent of k)and G–(k) = – |D0 | k2(1 – k2/kc

2) (spinodaldecomposition). This mode spectrum ofuncoupled structural and concentrationfluctuations is shown by the dashed curvesin Fig. 6-7). However, both relaxation ratesbecome strongly modified if these vari-ables are coupled. The strength of thiscoupling can be related to 1–D∞ /D0, whereD0 and D∞ are the low- and high-frequencylimits of the diffusion constant. Full curvesin Fig. 6-7 show for three (arbitrary)choices of parameters, the relaxation ratesG+(k) and G–(k) for the case of coupledvariables. It is seen that a “mixing” ofinterdiffusion and relaxation occurs; forsmall k the interdiffusion (or spinodal de-composition) is given by G–(k), for large kby G+(k), and for intermediate k both expo-nentials exp[– G+(k) t)] and exp[– G–(k) t)]contribute to the growth and/or decay ofconcentration fluctuations. Plotting themode G–(k) which describes spinodal de-composition for small k in the form of aCahn plot, we encounter pronounced cur-vature.

Such a coupling where Rm and g are ofcomparable size might occur if we studyspinodal decomposition in glasses (see forexample Yokota, 1978; Acuña and Craie-vich, 1979; Craievich and Olivieri, 1981)

or in fluid polymer mixtures near theirglass tansition (e.g., Meier and Strobl,1988). The slow variables are then ex-pected to relax with a broad spectrum ofrates rather than with a single rate g. Fluc-


Figure 6-6. Schematic “Cahn plot”,i.e., R (k)/k2 vs. k2, (a) as predicted bythe linear theory and (b) as it is typi-cally observed. Ideally, the “Cahnplot” should yield a straight line: theintercept with the abscissa occurs atkc

2, and the intercept with the ordinateat D0 (Binder et al., 1986).

Figure 6-7. Mode spectrum G+(k), G–(k) (fullcurves) of an unmixing system coupled to a slow var-iable plotted vs. k2/kc

2 for three parameter choices. Inthe absence of any coupling, the slow variable wouldrelax with a rate G+(k) =g (broken horizontal straightlines) and the unmixing system would relax with arate G–(k) = 4Rm(k2/kc

2) (1– k2/kc2) (broken curves).

All rates are normalized by Rm (Binder et al., 1986).


tuations in the final state also need to be in-cluded in this case (Jäckle and Pieroth,1988).

We now discuss the effect of fluctuationsin the final state for the simplest case, Eq.(6-15), treated in the framework of the lin-earization approximation. After some sim-ple algebra we obtain from Eqs. (6-15) to(6-22) the following equation (Cook,1970):

It is seen that an effective diffusion con-stant for uphill diffusion, defined as

(6-28)D k tk t

S k teffdd

( , ) ln ( , )≡ 12 2

dd

= (6-2 )

cgB B

tS k t M k

f

cr k T k S k t k T

T c

( , )

( , ),

−

×∂∂

+⎡

⎣⎢⎢

⎤

⎦⎥⎥

−⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

2 72

2

22 2

now yields(6-29)

Since in the linear theory S (k, t Æ ∞) Æ ∞,Deff(k, t Æ ∞) reduces to the simple Cahn(1961) result quoted above, but this limit isnever reached owing to nonlinear effects.On the other hand, at t = 0, Eq. (6-29) leadsto a linear relation between Deff(k, 0) andk2 again, because (6-30)

is linear in k2 also. Here the point whereDeff (k, 0) changes sign is not kc, but shiftsto a larger value. This behavior is illus-trated in the upper part of Fig. 6-8 wherethe (rescaled) structure factor S (q,t) and(rescaled) effective diffusion constant

[ ( , )],

S kf

cr k T k k T

T c

0 12

22 2

0

0

− ∂∂

+⎡

⎣⎢⎢

⎤

⎦⎥⎥

= cgB B

D k t Dk

kk T M S k teff

cB( , ) / ( , )≡ − −

⎛⎝⎜

⎞⎠⎟

+0

2

21

Figure 6-8. Scaled structurefunction (left) and normalizeddiffusion constant Deff (q, t) ∫q2 d [ln S (q, t)]/dt (right) vs. qor q2, respectively, for an instan-taneous quench from infinitetemperature to T /Tc = 4/9. Top,Cahn–Hilliard–Cook (CHC)approximation, Eqs. (6-27) and(6-29), bottom based on theLanger–Baron–Miller (LBM)(1975) theory. Ten times t =1,2, …, 10 are shown after thequench. Note that in the CHCapproximation we use therenormalized value m– (Fig.6-18) of (∂2 fcg /∂c2)T, c– instead(∂2 fcg /∂c2)T, c–, m–(T, t = 0) = 0.65in the present units (Carmesinet al., 1986).

Deff (q,t) following from Eqs. (6-27) and(6-29) are plotted against the (rescaled)square of the scattering vector, q2 (whereq = k/kc, t = 2 M r2 T kc

4 t, S = r2 kc2 S). It is

seen that on the level of the Cahn–Hilli-ard–Cook (CHC) approximation there isinitially some shift in the position at whichS (k, t) has its maximum, although this shiftis more pronounced when nonlinear effectsare taken into account, as is approximatedby the theory of Langer et al. (1975), whichwe shall refer to as the Langer–Baron–Miller (LBM) approximation. It is seen thatthe main distinction between the CHC andLBM approximations during the early stagesis the lack of a common intersection point inthe LBM approximation. Also the growth ofS(q,t) with t is generally slower (note thedifference in the ordinate scales!). Qualita-tively, however, the behavior is similar, andthis is also true for the behavior of Deff (q,t).Note that Deff (q,t) is distinctly curved inboth cases (apart from the limit t Æ 0).

6.2.4 Spinodal Decompositionof Polymer Mixtures

Here we consider the modifications nec-essary for describing binary mixtures oflong flexible macromolecules. In a densepolymer melt, the configurations of theselinear chain molecules (we disregard herestar polymers, branched polymers, copol-ymers, and also interconnected networks,although some aspects of the theory can beextended to these more complex situations,see Binder and Frisch (1984)) are randomGaussian coils interpenetrating each other.In terms of subunits (“Kuhn segments”) oflengths sA and sB for the two types ofpolymers A and B, the degree of polymer-ization XA and XB is expressed in terms ofchain lengths NA and NB as XA = NAgA andXB = NBgB, if each subunit contains gA andgB monomers. The mean square end-to-end

distance and gyration radius of these coilsthen is (Flory, 1953; De Gennes, 1979):

·RA2 Ò = s2

A NA, ·RB2 Ò = s2

B NB, (6-31)

·R2gyr,AÒ = 1–

6s2

A NA, ·R2gyr,BÒ = 1–

6s2

B NB

In the Flory–Huggins (FH) approximation(Flory, 1953; Koningsveld et al., 1987) theexpression corresponding to Eq. (6-4) be-comes (De Gennes, 1980)

(6-32)

where F (x) is the volume fraction of A seg-ments, 1 – F (x) is the volume fraction of Bsegments, the mixture is assumed to beincompressible, and the lattice spacing a ofthe Flory–Huggins lattice model is givenby a2/[F (1– F)] ∫ s2

A/F + s2B/(1– F). Note

that unlike Eq. (6-4) the parameter a is notrelated to the range of interaction, butrather reflects the random coil structure ofthe polymer chains. Owing to the connec-tivity of the polymer chains, the entropy ofmixing term is much smaller than in Eq. (6-6) (Flory, 1593):

where the Flory–Huggins parameter c con-tains all enthalpic contributions that lead tounmixing. If c does not depend on volumefraction, then the spinodal curve resultingfrom Eq. (6-33) is given by

2cs (F) = (F NA)–1 + [(1 – F) NB]–1 (6-34)

and the critical point occurs at

Fc A B

c A B

=

=

( / )

( ) // /

N N

N N

+

+

−

− −

1

2

1

1 2 1 2 2c

1

1 11

k Tf

N

N

BFH

A

B

= (6-33)( )ln

( ) ln( )( )

F F F

F F F F+ − − + −c

D ( )[ ( )]/

( )[ ( )]

F F

F FF

xx

x

k Tx f k T

a

d

BFH B= d∫

+−

∇ ⎫⎬⎭

12

11

2 2



These results assume well-defined chainlengths (“monodisperse polymers”), butcan be generalized to a distributionof chain lengths (“polydispersity”; seeJoanny, 1978).

Another modification concerns the con-stitutive kinetic equation: instead of Eq. (6-11) the connectivity of the chains requiresa non-local relationship for the current den-sity JF(x, t) relating to volume fraction F(De Gennes, 1980; Pincus, 1981),

JF (x, t) = – Ú L (x – x¢) —m (x¢, t) dx¢ (6-35)

where L (x – x¢) is a generalized Onsagercoefficient describing polymer–polymerinterdiffusion. The rate R (k) in Eq. (6-21)is then expressed in terms of the Fouriertransform Lk of L (x – x¢), i.e.,

R (k) = k2 Lk [STcoll (k)]–1 (6-36)

where the expression for the effective col-lective structure factor ST

coll (k) can be ex-pressed in terms of the structure factorsSA(k) and SB(k) of the single chains as(Binder, 1983, 1984c, 1987b)

[STcoll (k)]–1 = [F SA(k)]–1

+ [(1 – F ) SB (k)]–1– 2c eff(F, T, k) (6-37)

where 2c eff(F , T, k) is a wavevector-dependent generalization of the term re-sulting from the second derivative of∂2[c F (1 – F )]/∂f2 in Eq. (6-33). The sin-gle-chain structure functions SA(k) andSB(k) are expressed for Gaussian chains bythe Debye function fD(x) as

SA(k) = NA fD(k2 ·R2gyr,AÒ)

SB(k) = NB fD(k2 ·R2gyr,BÒ) (6-38)

where fD(x) ∫ 2[1 – (1 – e–x)/x]/x.Eqs. (6-36) and (6-37) result (Binder,

1983) from random phase approximations(De Gennes, 1979) and agree with thetreatment as given by Eqs. (6-9)–(6-30)(but using Eq. (6-32) instead of Eqs. (6-4)–

(6-6)) only in the long wavelength limit,where kc ·R2

gyrÒ1/2<1 (for both types of

chains). In this limit Lk can be replacedby a constant L0, and the results are justspecial cases of Eqs. (6-9)–(6-30); e.g., fora symmetric mixture (NA = NB, sA=sB=a)we simply find:

Hence the critical wavelength lc is rescaledby a large prefactor, namely the coil gyra-tion radius, similar to the correlation lengthxcoex of concentration fluctuations at thecoexistence curve,

xcoex = (N a2/36)1/2 (c /cc – 1)–1/2 (6-40)

Eq. (6-39), however, is only applicable for“shallow” quenches, where c does notgreatly exceed cs(F). A different behavioroccurs (Binder, 1983) for deep quenches,where c ocs(F) (thus describing the un-mixing of two incompatible polymers):

(6-41)

Now the initial wavelength of maximumgrowth is smaller than the coil radius,and kmO kc, since the gradient-square ex-pansion, Eq. (6-32), is no longer appli-cable.

As is obvious from Eq. (6-36), the relax-ation rate R (k) is the product of two fac-tors, a static factor [ST

coll (k)]–1 which con-tains the thermodynamic singularities atthe critical point (and along the spinodalcurve; cf Eq. (6-39)), and a kinetic factork2Lk, to which we now turn. Here the fac-tor k2 simply reflects the conservation lawfor the concentration, while Lk reflects the

kN a

kN a

c s

m s

≈ ⎛⎝⎜

⎞⎠⎟

≈ ⎛⎝⎜

⎞⎠⎟

62

62

2

1 21 2

2

1 21 4

//

//

[ / ( )]

[ / ( )]

c c

c c

Φ

Φ

218

1

2

2

1 21 2p / ( ( )/ ) ,

/

//l c cc c s

m c

= =

= (6-39)

kN a

k k

⎛⎝⎜

⎞⎠⎟

− F

complicated dynamics of single polymerchains in an entangled polymer melt. De-spite some effort (De Gennes, 1980; Pin-cus, 1981; Binder, 1983; Akcasu et al.,1986) there is no systematic method thatyields information on the k-dependence ofLk; the approximate theories quoted aboveall lead to a long-wavelength limit calledthe slow-mode theory, which is believed tobe an incorrect description of interdiffusionbecause it disregards bulk flow (Krameret al., 1984). Remember that Eq. (6-36) fork Æ 0 can be interpreted as R (k) = k2 Dcoll,where Dcoll = L0/ST

coll (0), i.e., uphill diffu-sion. In the slow-mode theory, the interdif-fusion coefficient of an ideal non-interact-ing mixture is expressed in terms of tracerdiffusion coefficients Dt

A and DtB of an A

chain (B chain) in a pure B (A) environ-ment as

(Dcoll)–1 = (1 – F ) /DtA + F /Dt

B (6-42)

i.e., the slow species controls the interdif-fusion coefficients. In contrast, the fast-mode theory (Kramer et al., 1984) yieldsthe opposite result, that the fast speciescontrols interdiffusion:

Dcoll = (1 – F ) DtA + FDt

B (6-43)

Experimental evidence (for a review ofpolymer interdiffusion, see Binder and Sil-lescu, 1989) seems to favor Eq. (6-43) overEq. (6-42), but we feel that a valid deriva-tion of Eq. (6-43) is still lacking, andmoreover neither Eq. (6-43) nor (6-42) de-scribe interdiffusion in lattice gas modelscorrectly, as computer simulations show(Kehr et al., 1989). Both Eqs. (6-42) and(6-43) are based on a description of trans-port in which in the constitutive equationrelating currents and chemical potentialdifferences (cf. Eq. (6-11)) off-diagnonalOnsager coefficients are neglected; theMonte Carlo simulation shows that this ap-proximation is inaccurate.

If DtA and Dt

B are of the same order, thenEqs. (6-42) and (6-43) also yield Dcoll ofthe same order, and the order of magnitudeof Dcoll (and hence R (k), Eq. (6-36)) is thenpredicted correctly from these equations. Ifthe chain length NA(NB) is less than thechain length N e

A(N eB) between effective

entanglements, the single-chain dynamicsare simply given by the Rouse (1953)model, which implies

DtA ≈ sA WA/NA, Dt

B ≈ sBWB/NB

where WA(WA) are time constants for thereorientation of subunits. In contrast, forNApN e

A, NBpN eB the reptation model

(Doi and Edwards, 1986; De Gennes,1979) implies for strongly entangled chains

DtA ≈ sA WAN e

A/N2A, Dt

B ≈ sBWBN eB/N2

B

(neglecting prefactors of order unity). Aproblem which is not completely under-stood so far is the concentration depen-dence of the parameters N e

A and N eB in

blends. In view of all these uncertaintiesabout the validity of Eqs. (6-42) and (6-43)and the precise values of the constants Dt

A

and DtB to be used in them, it is better to

consider Lk in Eq. (6-36) as a phenomeno-logical coefficient, about which only theorder of magnitude can roughly be in-ferred.

An effect disregarded by the theories butseen in computer simulations (Sariban andBinder, 1989a) is the change in the chainlinear dimensions (Eq. (6-31)) in quench-ing experiments. This also means that theparameter a in Eqs. (6-32) and (6-39) to(6-41) should be treated as a c-dependentquantity, which further complicates thequantitative analysis of experiments.

6.2.5 Significance of the Spinodal Curve

From Eq. (6-25) it is evident that thelinear theory predicts a singular behavior



when the concentration c– approaches theconcentration csp(T ) of the spinodal curve,defined by

(∂2 fcg/∂c2)T,c– =csp(T ) = 0 (6-44)

Here the critical wavelength lc diverges toinfinity. For example, if we adopt Eq. (6-8)with A ~ (T/Tc

MF –1), near the spinodal weobtain, omitting prefactors of order unity,

On the metastable side of the spinodalcurve a similar divergence occurs for boththe correlation length x (c–) of concentra-tion fluctuations and the radius R* of a crit-ical droplet, namely (Binder, 1984b)

Although R* diverges as c– approachescsp(T ), the associated free-energy barrierDF* against the formation of a criticaldroplet vanished there (Fig. 6-9). The max-imum growth rate of spinodal decomposi-

x ( ) ~ * ~ ( / )

( ), ( )

/

/

c R r T T

c T c

c cc c T

1 1 2

1 2

−

×−

−⎡

⎣⎢

⎤

⎦⎥

−

−cMF

sp

coex(2)

coex(1) sp

(6-46)

Á

lc cMF sp

crit sp

sp (6-45)

~ ( / )( )

( )

( )

/

/

r T Tc c T

c c T

c c T

1 1 2

1 2

−−−

⎡

⎣⎢

⎤

⎦⎥−

−

Ê

tion, which according to Eqs. (6-21), (6-25)and (6-26) can be written as

Rm = (1/4) M r2 kBT (2p /lc)4 (6-47)

vanishes as c– Æ csp(T ), and a similar “criti-cal slowing down” (Hohenberg and Halpe-rin, 1977) would occur in the growth rate ofa supercritical droplet. Thus, within theframework of this theory, the spinodalcurve plays the role of a line of criticalpoints. We now wish to investigate whetherthere is a physical signficance to this singu-lar behavior.

In Sec. 6.2.1 we emphasized that the def-inition of fcg is not unique but really in-volves a length scale L over which a coarse-graining of short-range fluctuations is per-formed. This is best seen from the attemptsto calculate fcg explicitly, which can beapproximated by using renormalizationgroup methods (Kawasaki et al., 1981) orby Monte Carlo simulation (Kaski et al.,1984). These treatments show that the posi-tion of csp(T ) depends strongly on thelength scale L (see Fig. 6-10a), and there-fore for systems with short-range forces, towhich these treatments apply, there is nophysical significance to the spinodal singu-larity (Eqs. (6-44) to (6-47)) whatsoever.

Figure 6-9. (a) Character-istic lengths R*, x, lc and(b) nucleation barrier DF*vs. concentration cB (sche-matic). Full curves are thepredictions of the linearizedCahn–Hilliard mean-fieldtheory of nucleation. Dash-dotted curves show, on adifferent scale, the conjec-tured smooth behavior of asystem with extremelyshort-range interaction, forwhich the spinodal singu-larity is completely washedout (Binder, 1981).


Figure 6-10. (a) Monte Carlo results for the dependence of the relative distance of the “spinodal” from the co-existence curve on the size of the coarse-graining cell L. The results refer to a simple-cubic nearest-neighborIsing magnet in the critical region, and are obtained from sampling the magnetization distribution functionPL(M) in L ¥ L ¥ L sub-blocks of a 243 system. Here, Smax is the value at which ln PL(M) has its maximum, andcorresponds to the coexistence curve if we assume ln PL(M) ~ L3 fcg /kBT, with c = (1– M)/2, and the “spinodal”is estimated as inflection point Ms of ln PL(M). By scaling L with the correlation length x, all temperaturessuperimpose on one “scaling function” (Kaski et al., 1984). (b) Extrapolation of the inverse collective structurefunction versus inverse temperature to locate the spinodal temperatures Tsp(c–) or their inverse (arrows). HereMonte Carlo simulation data for a polymer mixture are used, the polymers (A, B) being modelled as self- andmutually avoiding random walks on the simple cubic lattice with NA = NB = N = 32 steps, at a concentration ofvacancies fv = 0.6. Values on the curves are the reduced volume fraction FA/(1– Fv) of monomers of A chains.If two neighboring lattice sites are taken by monomers of the same kind, an energy e is obtained, and thus an en-thalpic driving force for phase separation is created. From Sariban and Binder (1989b). (c) Phase diagram of themodel for a polymer mixture as described in (b), displaying both the true coexistence curve (binodal) and theextrapolated spinodal. From Sariban and Binder (1989b).


Another way of showing this comes froma closer examination of the proceduresby which experimentalists locate spinodalcurves. Eqs. (6-30) and (6-37) imply that

[STcoll (kÆ 0)]–1~ (∂2 fcg/∂c2)T,c– ~ T – Tsp(c–)

where Tsp(c–) is the inverse function ofcsp(T ) in the c–T plane. Thus a plot of[ST

coll (k Æ 0)] vs. temperature should allowa linear extrapolation to locate Tsp(c–) fromthe vanishing of [ST

coll (k Æ 0)] (cf. Fig. 6-10b). However, the extrapolated spinodaldetermined in this way (e.g., Sariban andBinder, 1989b) (Fig. 6-10c), is not physi-cally meaningful as it crosses the true co-existence curve (binodal) near Tc, which isphysically impossible. This happens be-cause, in reality,

[STcoll (k Æ 0)]–1

c– =ccrit~ (T – Tc)g

with g ≈ 1.24, a critical exponent differentfrom the mean-field result g = 1. Hencea linear extrapolation fails near c– = ccrit. Atstrongly off-critical concentrations this ex-trapolation is also ambiguous, because usu-ally actual data cannot be taken deep in themetastable phase for temperatures close toTsp(c–).

This situations is different, however, forsystems with infinitely weak but infinitelylong-range interactions (Penrose and Lebo-witz, 1971); then the mean-field theory iscorrect because statistical fluctuations aresuppressed. At the same time, the lifetimeof metastable states is infinite because ho-mogeneous nucleation is no longer pos-sible (“heterophase fluctuations” are su-pressed at the same times as “homophasefluctuations”). The spinodal curve is thelimit of metastability here.

While such a system with infinitelylong-range interactions is clearly artificial,it makes sense to consider systems withlong but finite range r of the interactions(Heermann et al., 1982; Binder, 1984b;

Heermann, 1984a,b). Although very closeto the critical point Tc such system behavequalitatively like short-range systems (thisis to be expected from the so-called “uni-versality principle” (Kadanoff, 1976)), far-ther from Tc a well-defined mean-field crit-ical region exists. It is this region whereboth the coarse-graining defined in Eq. (6-1) and the linearization approximation ofSec. 6.2.2 make sense, as we will now dis-cuss.

The argument is simply an extension ofthe Ginzburg (1960) criterion for the valid-ity of the mean-field theory for criticalphenomena to nucleation and spinodal de-composition (Binder, 1984b): nonlinearterms in dc (x, t) can be neglected if theirmean-square amplitude in a coarse-grain-ing cell is small in comparison with theconcentration difference squares in thesystem, over which relevant nonlinear ef-fects are felt:

·[dc (x, t)]2ÒT,L [c– – csp(T )]2 (6-48)

We estimate ·[dc (x, t)]2ÒT,L as

·[dc (x, t)]2ÒT,L ≈ ·[dc (x, 0)]2ÒT,L exp[2Rm t]

where Rm is the maximum growth rate de-fined in Eq. (6-26), and the initial mean-square amplitude is related to the correla-tion function of concentration fluctuationsin the initial state at temperature T0. Thus,using Eq. (6-1):

(6-49)

the summations over i and j being restrict-ed to sites within the cell of size L3 cen-tered at x and in the last step the sums areconverted into integrals (one sum Â

iis can-

celled against a factor L3 making use of the

⟨ ⟩ ⟨ ⟩

⟨ ⟩ −

⟨ ⟩ −

∑

∫

[ ( , )] [ ( )]

[ ]

[ ( ) ( ) ]

, ,

,

d dc c

Lc c c

Lc c c

T L T L

i ji j T

T

x x

x x

01

10

2 2

62

32

0

0

0

=

=

= d

translational invariance of the correlationfunction). For distances |x | x the correla-tion function is simply given by a power-law decay, ·c (0) c (x)ÒT0

– c–2 ≈ r–2 x–1, andtherefore the order of magnitude of the in-tegral in Eq. (6-49) is estimated, yielding

·[dc (x, 0)]2ÒT0,L ≈ r –2 L–1 (6-50)

As expected, the mean-square concentra-tion fluctuation is the smaller the larger isthe range of the interactions and the largeris the coarse-graining length L. Since thelargest permissible value for L is L ≈ lc,Eqs. (6-45), (6-48), and (6-50) readilyyield

exp (2Rm t) O r3 (1 – T /Tc)1/2

¥ [c–/csp(T ) – 1]3/2 (6-51)

A similar self-consistency criterion can beformulated in the metastable region, wherethe largest permissible choice for L in theGinzburg criterion

·[dc (x)]2ÒT,L O (csp(T ) – c–)2

now is L = x (c–), as given in Eq. (6-46), andit is found in full analogy with Eqs. (6-48)to (6-51) that

1 O r3 (1 – T /Tc)1/2 [1 – c–/csp(T )]3/2 (6-52)

It is also interesting that Eq. (6-52) can bederived from a completely different argu-ment, namely requiring that the free energybarrier of nucleation D*pkBT (Binder,1984b) (for c– near csp(T ) the Cahn–Hil-liard (1959) mean-field theory of nuclea-tion predicts D*/kBT ~ r3 (1 – T /Tc)1/2

¥[1 – c–/csp(T )]3/2). From Eqs. (6-51) and(6-52) we draw the following conclusions:

(i) For the linearized theory of spinodaldecomposition to be self-consistent, and forthe description of a metastable state nearthe spinodal to be self-consistent (and that ithas a long lifetime owing to a large barrieragainst homogeneous nucleation), the in-

equalities which must be satisfied requirethat

r3 (1 – T /Tc)1/2p1 (6-53)

This condition can be satisfied only fora large range r of interaction, and definesthe mean-field critical region (Ginzburg,1960). As long as Eq. (6-53) holds, theLandau mean-field theory of critical phe-nomena (Stanley, 1971) is quite adequate.However, for systems with rather short-range interactions Eq. (6-53) and henceEqs. (6-51) and (6-52) can never be ful-filled.

(ii) Even for systems with a long but fi-nite range r of interaction, close enough toTc (namely for r3 (1 – T /Tc)1/2 ≈ 1) a cross-over occurs to a non-mean-field critical be-havior, described by the same critical be-havior as short-range systems. This is ex-pected from the principle of universality ofcritical behavior (Kadanoff, 1976). Thenthe nonlinear character of Eq. (6-15) is im-portant during the initial stages of thequench, and the linearization approxima-tion (Eq. (6-18)) is never warranted.

(iii) For mean-field-like systems forwhich Eq. (6-53) holds, Eqs. (6-51) and(6-52) hold only if we do not come tooclose to the spinodal. The region excludedby these inequalities scales as

| c–/csp(T ) – 1 | ~ r –1 (1 – T /Tc)–1/3

In this excluded region the singularities de-scribed by Eqs. (6-45) to (6-47) are essen-tially smeared out, and a gradual transitionfrom nucleation and growth to nonlinearspinodal decomposition occurs. This grad-ual transition can be understood qualita-tively from a cluster dynamics treatment ofphase separation (Binder and Stauffer,1976a; Binder, 1977, 1981; Mirold andBinder, 1977; Binder et al., 1978). Figs.6-11 and 6-12 summarize the main ideasabout this description which is closely



related in spirit to the Langer–Schwartz(1980) and Kampmann–Wagner (1984)treatments of concomitant nucleation,growth, and coarsening (see Wagner et al.,2001). Each state of the system is charac-terized by a cluster size distribution n–l (t),where l is the number of B atoms containedwithin a cluster (Fig. 6-11a) and the barrepresents an average over other “clustercoordinates” (cluster surface area, shape,etc.) which are not considered explicitly.The time evolution of the cluster sizedistribution in a quenching experimentis described by a system of kinetic equa-tions:

Here Sl + l¢, l¢ is a rate factor for a splittingreaction of a cluster of size l + l¢ into two

dd

=

(6

=

=

=

=

tn t S n t

S n t

c n t n t

c n t n t

ll

l l l l l

l

l

l l l

l

l

l l l l l l

ll l l l

( ) ( )

( )

( ) ( )

( ) ( )

,

,

,

,

′

∞

+ ′ ′ + ′

′

−

′

′

−

− ′ ′ ′ − ′

′

∞

′ ′

∑

∑

∑

∑

−

+

−

1

1

1

1

1

1

12

12

--54)

clusters of size l and l¢, and cl, l¢ is the ratefactor for the inverse “coagulation” reac-tion. These rate factors are assumed to beindependent of time t and hence Eq. (6-54)also describes the concentration fluctua-tions in thermal equilibrium, where de-tailed balance must hold between splittingand coagulation reactions:

Sl + l¢, l¢ nl + l¢ = cl, l¢ nl nl¢ ∫ W (l, l¢) (6-55)

where nl is the cluster concentration inequilibrium and W (l, l¢) is a cluster reac-tion matrix in equilibrium. While nl (t = 0)for a random distribution of atoms in thealloy is just the cluster distribution in thewell-known percolation problem (Stauffer,1985), nl and W (l, l¢) are not explicitlyknown, but can be fixed by plausible as-sumptions (Binder, 1977; Mirold andBinder, 1977). Then Eqs. (6-54) and (6-55)can be solved numerically (see Fig. 6-11b).What must happen is that n–l (t) for t Æ ∞develops towards nl, the cluster size distri-bution in the state at the coexistence curve

c(1)coex, with c(1)

coex = l nl . The excess

concentration c– – c(1)coex = l [nl (0) – nl] is

l = 1

∞∑

l = 1

∞∑

Figure 6-11. (a) “Clusters”of B atoms in a binary ABmixture defined from con-tours around groups of Batoms and their reactions.(b) Cluster size distributionn–l (t) for various times t aftera quench from infinite tem-perature to T /Tc = 0.6 atc– = 0.1 for parameters appro-priate for a two-dimensionalIsing model of a binary al-loy. These results were ob-tained from a numerical so-lution of Eq. (6-54), wheretime units are rescaled by anarbitrary rate factor. FromMirold and Binder (1977).

redistributed into macroscopic B-rich do-mains (of concentration c(2)

coex), i.e., occursat cluster size l Æ ∞ for t Æ ∞. For interme-diate times, we find a nonmonotonic clus-ter size distribution (Fig. 6-11b); a mini-mum occurs at the critical size l* of nucle-ation theory (Binder and Stauffer, 1976a),while the broad maximum at larger sizes isdue to growing supercritical clusters whichhave already been nucleated. For l < l* thecluster size distribution is basically theequilibrium size distribution of a slightlysupersaturated solid solution. As time goeson, the peak in n–l (t), representing thesupercritical growing clusters, shifts tolarger and larger cluster sizes, and at thesame time the supersaturation is dimin-ished, until for t Æ ∞ the peak has shiftedto l Æ ∞. This separation of clusters intotwo classes – small ones describing con-centration fluctuations in the supersatu-rated A-rich background and large onesdescribing the growing B-rich domains –can also be shown analytically from Eqs.(6-54) and (6-55) (Binder, 1977). This pic-ture of the phase separation process alsoemerges very clearly from computer simu-

lations (Marro et al., 1975; Rao et al.,1976; Sur et al., 1977).

The description in terms of Eqs. (6-54)and (6-55) contains both nucleation andgrowth and coagulation as special cases(Binder, 1977), and it can be used as a ba-sis for understanding the gradual transi-tion from nucleation to spinodal decompo-sition (Binder et al., 1978). In the meta-stable regime the density of unstable fluc-tuations (“critical” and “supercritical” clus-ters with l < l*) is very small (Fig. 6-12a),because the energy barrier DF*pkBT.Near the spinodal curve, on the other hand,DF*≈ kBT, and hence there is a high den-sity of unstable fluctuations: DF* here nolonger limits the growth, but rather theconservation of concentration. Near grow-ing clusters, c (x, t) locally decreases, andthen no other cluster can grow there. Ow-ing to this “excluded volume” interactionof clusters, a quasiperiodic variation ofconcentration results (Fig. 6-12a, bottom),roughly equivalent to a wavepacket ofCahn’s concentration waves. But the latterare not growing independently, rather theyare strongly interacting; hence it is more


Figure 6-12. (a) Schematic “snapshotpictures” of fluctuations of an unmixingsystem in the metastable regime (top)and in the unstable regime (bottom).Unstable fluctuations, which steadilygrow with increasing time (arrows) areshaded. (b) Structure factor S (q, t) vs.scaled wavevector q at various times tafter a quench from infinite temperatureto T = 0.6 Tc for c– = 0.1. Parameters arechosen for a three-dimensional Isingmodel of a binary alloy. These resultswere obtained from a numerical solu-tion of Eqs. (6-54) to (6-56). FromBinder et al. (1978).


reasonable to consider large clusters asobjects nearly independent of each other(apart from the reactions in Eq. (6-54))rather than as these waves.

The fact that the cluster picture and theconcentration picture are simply two dualdescriptions of the same physical situationbecomes very apparent when the structurefactor S (k, t) is calculated from the clustersize distribution n–l (t). Introducing the con-ditional probability gl (x) that the site x + x¢is taken by a B atom if x¢ is taken by a Batom of an l-cluster, we find (Binder et al.,1978):

(6-56)

Simple assumptions for gl (x) yield S (k, t)as shown in Fig. (6-12b). The resultingS (k, t) is therefore in qualitative agreementwith experimentation (Fig. 6-5a, b), com-puter simulation of microscopic models(Fig. 6-5c,d) and the approximate non-linear theory of spinodal decompositionof Langer et al. (1975) (see Fig. 6-8, bot-tom).

This gradual transition between nuclea-tion and spinodal decomposition alwaysappears close to the critical point in thephase diagram (Fig. 6-13); for systemswith a large interaction range r a mean-field critical regime exists (Eq. (6-53))where this gradual transition is confined toa narrow regime adjacent to the spinodal.Outside this transition regime the linear-ized theory of spinodal decomposition isexpected to hold initially in the unstable re-gime, while a nonclassical nucleation the-ory (where ramified droplets, which mustfirst be compacted before they can grow,are nucleated (Klein and Unger, 1983;Unger and Klein, 1984; Heermann andKlein, 1983a, b)) holds in the metastableregime. This crossover line between clas-

S k t l n t cl

l l( , ) ( ) ( )

exp ( )

= d

i= 1

∞

∑ ∫ −

× ⋅

x x

k x

g

sical and nonclassical “spinodal nuclea-tion” is essentially given by the condi-tion that the free-energy barrier D f * toform a critical nucleus is of the order ofr3 (1 – T /Tc)1/2p1 (Fig. 6-14a). Thereforethis regime of “spinodal nucleation” mustdisappear when the non-mean-field criticalregion is approached (Fig. 6-14b).

It should be emphasized that the phasediagram in Fig. 6-13 showing these variousregimes in the temperature–concentrationplane is relevant only in the very earlystages of phase separation. This should beobvious from Eq. (6-51). The time rangewhere it is valid is, at best, of the order ofRm t ≈ l n r (r being measured in units of thelattice spacing). Therefore the experimentalverification of the linear theory of spinodaldecomposition is a difficult problem (seeSec. 6.4). Clear confirmation of the ideasdescribed above has come from computersimulations of medium-range Ising modelsdue to Heermann (1984a) (see Figs. 6-15and 6-16). Since for the Ising model the re-sult for R (k) is readily worked out, a com-parison between theory and simulation ispossible without any adjustable parameterwhatsoever! Fig. 6-15 shows that there isindeed a regime where initially fluctuationsincrease exponentially with time (Eq.(6-23)) and the observed rates R (k) doagree with the predictions, if we stay far offthe spinodal.

It is not easy to identify physical systemsthat have a large but finite interactionrange. What is really needed is a large pre-factor in the mean-field result for the corre-lation length (Eq. (6-46)), and such a largeprefactor has in fact been identified forpolymers with high molecular weight (Eq.(6-40)). This happens because for polymersthe coefficient of the gradient energy in Eq.(6-32) comes from the random-coil struc-ture, and does not have the meaning of asquared interaction range as in Eq. (6-4).


Figure 6-13. Various regimes in the phase diagram of a binary mixture AB, showing part of the plane formedby temperature T and volume fraction F of the B component (only volume fractions F <Fcrit are shown, sincethis schematic phase diagram is symmetric around the axis F = Fcrit). The horizontal broken line separates thenon-mean-field critical regime (top) from the mean-field critical regime (bottom). The two solid curves are thecoexistence curve (left) and the spinodal curve (right). The two dash-dotted curves on both sides of the spino-dal limit represent the regime where a gradual transition from nucleation to spinodal decomposition occurs. Thelinearized theory of spinodal decomposition (Fig. 6-1a) should hold to the right of these dash-dotted curves. Theregime between the coexistence curve and the left of the two broken curves is described by classical nucleationtheory (compact droplets, Fig. 6-1b). The regime between the right broken curve and the left dash-dotted curveis described by “spinodal nucleation” (ramified droplets). From Binder (1984b).

Figure 6-14. Schematic plots of the nucleation free energy barrier for (a) the mean-field critical regionof a d-dimensional alloy system, i.e., r d (1 – T/Tc)(4–d )/2 o 1 and (b) the non-mean-field critical region,i.e., r d (1 – T/Tc)(4–d )/2 < 1. The gradual transition from nucleation to spinodal decomposition occurs forDF*/kBTc ≈1. From Binder (1984b).


However, we note that for NA = NB = N wemay map Eqs. (6-33) and (6-32) into Eqs.(6-4) and (6-6) with the following identifi-cations (Binder, 1984b):

(6-57)

The condition that the nucleation barrierD*/kBT p1 instead of Eq. (6-52) thenyields

(6-58)

and a similar criterion holds on the un-stable side of the spinodal (Binder, 1983),with 1 being replaced by exp(2 Rm t) onthe left-hand side of Eq. (6-58), as in Eq.(6-51). It follows that for large chain lengthN the spinodal is smeared out over a regionof width N –1/3.

In view of these results, it is gratifying tonote that for polymers, convincing experi-mental demonstrations of the validity ofCahn’s linearized theory of spinodal de-composition have indeed been presented(see Sec. 6.3.4).

6.2.6 Towards a Nonlinear Theoryof Spinodal Decompositionin Solids and Fluids

In the last subsection it was shown thatthe regime of times for which the lineartheory of spinodal decomposition holds isextremely restricted, if it holds at all.Therefore, a treatment of nonlinear effectsis necessary. A systematic approach is onlypossible by an expansion in powers of 1/rfor large r (Grant et al., 1985). This theoryis very complicated and has so far beenworked out only for c– = ccrit. It explicitlyshows the coupling between concentrationwaves with different wavevectors.

1 1 11 21 2

3 2O N c c T//

// ( )−⎛⎝⎜

⎞⎠⎟

−cccrit

sp| |

D D polymer small molecules

c

≡

≡ ≡

1

21 2

32 2

NTT N

r N ac ,

Figure 6-15. Time dependence of the logarithm ofspherically averaged structure factors of a 603 sim-ple-cubic Ising lattice, where each spin interacts withq= 124 neighbors with equal interaction strength J,quenched from infinite temperature to

T = (4/9) TcMF = (4/9) q J/kB and c– = 0.4

The five smallest wavenumbers kn = 2p n/60 are dis-played. Straight lines for short times indicate expo-nential growth and thus yield R (k). From Heermann(1984a).

Figure 6-16. Cahn plot R (k)/k2 vs. k2(R (k) is de-noted as w (k) in this figure), as extracted from datasuch as shown in Fig. 6-15. Crosses are the MonteCarlo results, and straight lines are the predictions ofthe linearized theory. Note that csp(T ) ≈ 0.127 in thiscase. From Heermann (1984b).

So far the most popular (if approximate)approach is the decoupling approximationsuggested by Langer et al. (1975). Exactequations of motion are derived for theprobability distributions r1[c (x)], r2[c (x1),c (x2)], etc. Here r1 is the probability den-sity that, at point x, the concentration c (x)occurs, and r2 is the corresponding two-point function. As expected, the equationof motion for r1 involves r2, and the equa-tion of motion for r2 involves the three-point function r3, etc., so that an infinitehierarchy of equations of motion is gener-ated.

This hierarchy is decoupled by the fol-lowing approximation for the two-pointfunction [d c (x) = c (x) – c– ]:

(6-59)

The motivation for Eq. (6-59) is the fol-lowing: if there were no correlation be-tween concentrations at points x1 and x2,the probability r2 would just be the productof the one-point probabilities. Therefore,the correction of this factorization approxi-mation is made proportional to the two-point correlation function ·dc (x1) dc (x2)ÒT .In this way, Eq. (6-59) yields a closed equa-tion of motion for the probability r1. Thisequation is then solved approximately, as-suming that the coarse-grained free energyhas the Landau form (Eqs. (6-5) and (6-8)).The coefficients A, B and r are adjustedself-consistently such that the resultingstate equilibrium is correctly described inthe non-mean-field critical region.

This approach has also been worked outfor the dynamics of non-conserved orderparameters (Billotet and Binder, 1979) andthe validity of this approximation has beenstudied carefully (see also Binder et al.,1978). The final result can be cast in a form

r r r2 1 2 1 1 1 2

1 22

1 221

[ ( ), ( )] [ ( )] [ ( )]

( ) ( )( )

( ) ( )( )

c c c c

c c

c

c c

cT

T T

x x x x

x x x x

=

× + ⟨ ⟩⟨ ⟩ ⟨ ⟩

⎧⎨⎩

⎫⎬⎭

d dd

d dd

that is similar to Eq. (6-27), namely,

where all nonlinear effects are now con-tained in a correction term a (t), which it-self depends on S (k, t) in a nonlinear way.

As noted above, the coefficients A, B,and r in Eqs. (6-5) and (6-8) are adjustedsuch that the critical behaviors of the co-existence curve, critical scattering inten-sity, and correlation length (at the coexis-tence curve) are reproduced:

(c(1)coex – ccrit)/ccrit = B (1 – T /Tc)b (6-61a)

ccoex = C (1 – T /Tc)–g (6-61b)

xcoex = x (1 – T /Tc)–n (6-61c)

where B, C, and x are the appropriate criti-cal amplitudes and b, g, and n the asso-ciated critical exponents (Stanley, 1971;Binder, 2001). It now turns out that thestrength of nonlinear effects is controlledby the inverse of a parameter f0 which nearTc is expressed as (Billotet and Binder,1979)

f0 ~ x d B2 C –1(1 – T /Tc)g +2b –dn (6-62)

for a d-dimensional system. In the non-mean-field critical region, the hyperscalingrelation dn =g +2b (Kadanoff, 1976) elimi-nates the temperature dependence from Eq.(6-62). In addition, two-scale factor uni-versality (Stauffer et al., 1972) implies thatthe critical amplitude ratio xd B2/C andhence f0 is a universal constant of orderunity (Billotet and Binder, 1979), i.e.,f0 ≈ 9.45. On the other hand, if we considera quench into the mean-field critical re-gion, again adjusting the coefficients A andB in Eq. (6-8) but now using mean-field

dd

= (6-60)

B

tS k t M k

f

ca t r T k S k t k T

T c

( , )

( ) ( , ),

−

× ∂∂

⎛⎝⎜

⎞⎠⎟

+ +⎡

⎣⎢⎢

⎤

⎦⎥⎥

−⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

2 2

2

22 2



results in Eqs. (6-61a–c), namely, b = 1–2,

g =1, and n = 1–2, x ~ r, we obtain

f0 ~ r d (1 – T /Tc)(4–d)/2

for r d (1 – T /Tc)(4–d)/2 p1 (6-63)

It is seen that f0 is now simply proportionalto the parameter appearing in the Ginzburgcriterion, and hence very large. Forrd (1 – T /Tc)(4–d)/2 ≈1 a crossover to the uni-versal constant f0 ≈ 9.45 in the non-mean-field critical region occurs.

Fig. 6-17 shows the time evolution forthe structure factor for two choices of f0.Here again a rescaling of the structure fac-tors S = r2 kc

2 S and of time t = 2 M r2 T kc4 t

was used, while q = 40 k /(2p kc) here. It isseen that for f0 ≈ 9.45 the linear theory isindeed invalid from the start, as expected.

However, for large values of f0 the lineartheory does hold initially, consistent withthe simulations (Fig. 6-15). The same con-clusion emerges from the time dependenceof the rescaled effective second derivativem–(T, t) in Eq. (6-60):

m–(T, t) ∫ 1 + a (t)/[∂2 f /∂c2]T,c (6-64)

(see Fig. 6-18). Nonlinear effects are negli-gible as long as m– = 1, while the decreasein m– is a signal of nonlinear effects, sincea (t) is always negative. The physicalsignificance of m–(T, t) is that it describesthe ratio between the actual kc

2(t), wherethe growth rate in Eq. (6-60) changes sign,and the corresponding prediction of theCahn–Hilliard theory. Fig. 6-18 impliesthat even if the nonlinear effects are verystrong the Cahn–Hilliard prediction for kc

2

differs from the actual kc2(t) by about a fac-

tor of two at most; therefore, the Cahn–Hilliard theory is certainly useful for esti-

Figure 6-17. Plot of ln [S (q, t) – ST (q)]/ST0(q) vs.

scaled time t for five different values of q, for aquench from infinite temperature at ccrit , usingf0 = 9.45 (top) and f0 = 9450 (bottom). From Carmesinet al. (1986).

Figure 6-18. Rescaled effective second derivativem–(Tf , t) (Eq. (6-64)) vs. scaled time t for fourchoices of f0. Only when m–(T, t) ≈ 1 is the Cahn–Hilliard–Cook approximation accurate. For f0 = 9.45the initial value m–(T, 0) = 0.65. From Carmesin et al.(1986).

mating the order of magnitude of the rangeof wavelengths that is unstable initially.

For systems with short-range forces,where f0 ≈ 9.45 applies, the early-timestructure factor for critical quenches (Fig.6-8, bottom) is in reasonable agreementwith both experiment (Fig. 6-5a, b) andsimulations (Figs. 6-5c, d). However, theLanger–Baron–Miller (LBM) (1975) ap-proximation gets worse for strongly off-critical quenches (Binder et al., 1978). Thetheory again exhibits a spurious singularityat a spinodal curve, which does not coin-cide with the mean-field spinodal of theCHC approximation, but occurs at a renor-malized concentration closer to the coexis-tence curve. This shift of the spinodal is notsurprising, since the effective second de-rivative of the potential is also renormal-ized (Fig. 6-18). This spinodal of the LBMapproximation is completely spurious – itsprecise location depends on details of thecoarse-graining procedure. For concentra-tions between the coexistence curve andthis spinodal, i.e., the metastable regime,the structure factor S (k, t) for T Æ ∞ satu-rates at a value S (k, ∞) = kBT /[∂2 f /∂x2)T,c

+ a (∞) + r2 T k2], i.e., at the Ornstein–Zer-nike result for scattering from fluctuationsin metastable equilibrium (Binder et al.,1978).

We now consider fluid binary mixtures.Here the theoretical formulation is com-plicated by another long-range interaction,namely the hydrodynamic backflow inter-action. If the Liouville equation is writtenfor the probability distribution r (c (x), t)that a concentration field c (x) occurs,

(6-65a)

the Liouville operator contains a termA also present in solid alloys and anotherterm HD containing the Oseen tensorT = Tab describing the hydrodynamic

∂∂t

c t c tr r( ( ), ) ( ( ), )x x=

interaction:

and

where h is the shear viscosity. Kawasakiand Ohta (1978) adapted the LBM decou-pling to binary fluids. Their results differfrom the original LBM theory mainlywhere km(t)/km(0) 1; however, at suchlate times neither of these theories is valid,since they can only account for non-linearities during the early stages wherekm(t)/km(0) ≈1. No theory exists which re-liably describes the crossover from theseinitial time regimes to the late stages,where km

–1(t) ~ t is presumed to hold.An interesting extension is to consider

spinodal decomposition of fluid mixturesin the presence of flow, e.g., laminar shearflow (Onuki and Kawasaki, 1978, 1979) orturbulent flow (Onuki, 1984, 1989c). Inweak shear, the scattering pattern is nolonger the ring pattern familiar from stan-dard spinodal decomposition, since con-centration fluctuations become anisotropic.In polymer mixtures additional effectsarise because the polymer coils becomestretched and oriented by the flow (Pistoorand Binder, 1988a, b). No such phenomenaare considered here.

T

x x x x

ab ab

a a b b

h( )

( ) ( )

x xx x

x x

− ′− ′

⎡

⎣⎢

+− ′

− ′ − ′⎤

⎦⎥

= (6-65d)1

81

13

p | |d

| |

HD = d d

(6-65c)

2 ∫ ∫ ′ ∇ ⋅ − ′

× ′∇ ′′

+′

⎡⎣⎢

⎤⎦⎥

x xx

x x x

xx x

ddd

dd

d

cc

cc

Fc

( )( ) ( )

( )( ) ( )

T

= (6-65b)

= d

A HD

A

+

− ∇ +⎡⎣⎢

⎤⎦⎥∫M

c cF

cx

x x xd

dd

dd

d( ) ( ) ( )2



6.2.7 Effects of Finite Quench Rate

So far both the theory and the discussedsimulations have always assumed instanta-neous quenches from an initial temperatureTi to a final temperature Tf . With regard toactual experiments, this is extremely ideal-ized. Often the early stages of spinodal de-composition have already been passed dur-ing such a continuous quench where thetemperature is gradually lowered. While itis believed that the late stages are not af-fected by the “quenching history”, the lat-ter can have a drastic effect on both earlyand intermediate stages of phase separa-tion. Unfortunately, in general, the problemis complicated – the behavior of both thethermodynamic functions and of the mobil-ity M (T ) in the full regime from Ti to Tf

may affect the phase separation behavior.Thus, relatively little theoretical effort hasbeen devoted to this problem (Houston et

al., 1966; Carmesin et al., 1986). Here weonly quote a few results from the modelcalculation of Carmesin et al. (1986), sincethe work by Houston et al. (1966) onlyconsiders fluctuations in the initial stateand not in the intermediate states visited inthe quench, which is an approximation thatusually cannot be justified.

As an example, Fig. 6-19 shows a situa-tion similar to the quench treated in Fig. 6-8, but here the quench is not carried out in-stantaneously from infinite temperature toT /Tc = 4/9, but takes several steps: at t = 0the system is cooled instantaneously toT1/Tc = 0.75185, at t =1 to T2/Tc = 0.67667,at t = 2 to T3/Tc = 0.60148, at t = 3 toT4/Tc = 0.5263, and at t = 4 to T /Tc = 4/9,where the system is later maintained. Thisstepwise quenching is more readily acces-sible to calculation than a fully continuousquench. A further simplification of the cal-culation is to neglect any temperature de-

Figure 6-19. Scaled structurefunction (left) and normalizeddiffusion constant Deff (q, t) ∫q2 d [ln S (q, t)]/dt (right) vs. qor q2, respectively, for the step-wise quenching procedure de-scribed in the text. Top, CHCapproximation; bottom, LBMtheory. Ten scaled times t areshown. From Carmesin et al.(1986).

pendence of the mobility. Comparing Fig.6-19 with Fig. 6-8, characteristic differ-ences are noted: even in the CHC approxi-mation there is no longer a unique intersec-tion point for S (q, t) – S (q, 0) at differenttimes t; apart from t = 0, Deff (q,t) in the“Cahn plot” exhibits pronounced curva-ture, although it depends relatively weaklyon time. Note that in the LBM approxima-tion we can no longer see any distinct shiftof the maximum position of S (q, t) for thetimes shown; this happens because first(at the intermediate temperature steps)smaller-q components become more ampli-fied and later (at the final temperature) am-plification occurs at larger values of q. Thisbehavior just happens to offset the coarsen-ing tendency. This example shows that caremust be taken in drawing any conclusionsabout the validity of the CHC approxima-tion from experimental data – the effects offluctuations in the final state, the gradualonset of nonlinearities, and finite quenchrate effects are interwoven in a compli-cated manner, and very accurate measure-ments and detailed knowledge of thesystem parameters are indispensable fordisentangling all these effects.

These effects are much more dramaticif the mobility has a thermally activatedbehavior, M (T ) ~ exp (–Eact /kBT ), withEact Ô kBTf . As a second example, we con-sider a shallow continuous quench fromTi = 1.01 Tc to Tf = 0.99 Tc during t = 0.001,the temperature being lowered linearlywith time (Fig. 6-20). Although this is asgood an approximation for an instantane-ous quench as is experimentally feasible,nevertheless, for the choice Eact /kBTc

= 900, more relaxation takes place duringthe quench than in the remaining early timeinterval (t &1). The qualitative behavior ofthe CHC and LBM approximations is iden-tical – only the absolute scales of the struc-ture factors differ significantly.

6.2.8 Interconnected PrecipitatedStructure Versus Isolated Droplets,and the Percolation Transition

Whereas so far the description of thestructures formed by phase separation hasfocussed on the behavior of the equal-timestructure factor S (k, t) at time t after thequench, we now concentrate on the de-scription of these structures in real space.In Fig. 6-2 it was illustrated that during lateor intermediate stages two different pat-terns of behavior are present, dependingon the volume fraction F of the minorityphase. For small F, the minority phase isconfined to independent clusters well sep-


Figure 6-20. Structure function S (q, t) vs. scaledwavevector q for scaled times t as indicated in thefigure, for a continuous quench (linear cooling fromTi = 1.01 Tc to Tf = 0.99 Tc during t = 0.001), usingEact /kBT = 900. Top: CHC approximation; bottom:LBM approximation. From Carmesin et al. (1986).


arated from each other; for larger F, wehave a percolating interconnected network(in viewing Fig. 6-2 we expect an isotropicsystem, such as a fluid or glassy mixture; incrystalline solids the shape of the precipi-tates reflects the anisotropy of the interfa-cial free energy between the coexistingphases, and elastic interactions may evenlead to a regular rather than a random ar-rangement of these precipitates).

This difference in morphology of thegrowing structures often – and erroneously– is associated with the distinction betweennucleation and growth versus spinodal de-composition; it is then claimed that the per-colation of the growing structure is thehallmark of spinodal decomposition, whileit is assumed that well separated domainsmust have been formed by nucleation andgrowth. We maintain, however, that thedistinction between nucleation and spino-dal decomposition, meaningful only for theearliest stages of phase separation, mustnot be confused with this distinction inmorphology of the precipitated structures,which is relevant for intermediate and latestages: even if well separated domains oc-cur, they may result from a spinodal de-composition mechanism, and even if a statedecays by nucleation, it may correspond toan interconnected percolating microscopicstructure of B atoms in the A-rich matrix.Hence, the transition regime between nu-cleation and spinodal decomposition andthe line of percolation transitions separat-ing the regime of finite B clusters (Fig.6-11a) from the regime where a percolat-ing cluster occurs which reaches from oneboundary of the system to the opposite onemust in fact cross each other (Fig. 6-21).Fig. 6-21 shows both the molecular-fieldresult for the spinodal and the actual re-gime of gradual transition between nuclea-tion and nonlinear spinodal decompositionof a short-range system, defined approxi-

mately by the region from DF*MF = 10 kBTto DF*MF = kBT, where DF*MF is the mean-field result for the nucleation barrier (thisshift of the transition region from the mo-lecular-field result towards the binodal wasdisregarded in Fig. 6-13). Two differentpercolation transition lines are drawn,which depend on the resolution with whichthe system is observed. Suppose the resolu-tion is very fine, such that individual atomscan be distinguished (this is the situationenvisaged in Fig. 6-11a). Then in the sin-gle-phase region we encounter a percola-

Figure 6-21. Schematic phase diagram (temperaturevs. concentration c–) of a three-dimensional, short-range Ising lattice model of a binary mixture. Sincethe diagram is symmetric along the line c– – ccrit = 0.5,if the meaning of A and B is exchanged, full informa-tion is given only for the regime c–&0.5. Both thecoexistence curve (“binodal”) separating the single-phase region from the two-phase region and themean-field spinodal curve separating metastablefrom unstable states in mean-field theory are shown.The mean-field spinodal is described here by theequation (ccrit – c)/ccrit = ± (1– T/Tc)1/2, Tc being theactual critical temperature. The transition regimefrom nucleation to spinodal decomposition for ashort-range system, as discussed in the text, is alsoindicated. The dash-dotted curves indicate percola-tion transitions, as discussed in the text. From Hay-ward et al. (1987).

tion transition of a correlated percolationproblem at cp

(corr)(T ). This line starts out forT Æ ∞ at the percolation concentration forrandom percolation (cp

(random) ≈ 0.312 forthe simple cubic lattice (Stauffer, 1985))and bends over to the left, until it hits thecoexistence curve at the point T /Tc ≈ 0.96,c– ≈ 0.22 (Müller-Krumbhaar, 1974). Thisline continues in the two-phase region asa transient time-dependent phenomenon,cp

(corr)(T, t) (see Fig. 6-22). Hayward et al.(1987) and Lironis et al. (1989) haveshown that for certain concentrations theconfiguration is percolating for a timeinterval t1 < t < t2, whereas it does not per-colate for 0&t1 and for t7t2. If the con-centration decreases, a critical concentra-

tion is reached where t1 = t2 (for still lowerconcentrations there is no percolation atall), whereas for increasing concentrationanother critical concentration is reachedwhere t2 Æ ∞. For still higher concentra-tions the system percolates as the timeexceeds t1 and then remains percolatingthroughout.

A different behavior occurs if a systemwith a much coarser resolution is studied;at late stages, where the system is phase-separated on a length scale km

–1(t) into thecoexisting phases with concentrationsgiven by the two branches of the coexis-tence curve, c(1)

coex and c(2)coex, respectively. It

now makes sense to consider percolationphenomena on much larger length scalesthan the lattice spacing a. Suppose we di-vide our system again into cells of lineardimension L (Eq. (6-1)), with a L km

–1(t).Most of these cells will then have concen-trations close to either c(1)

coex or c(2)coex. We

may now define clusters consisting ofneighboring cells with concentrations in agiven interval [c(2)

coex – dc/2, c(2)coex + dc/2]

and we may ask whether these clusters arewell separated from each other or if theyform an infinite percolating network. Sincel (t Æ ∞) Æ ∞ we may also take L Æ ∞ inthis limit and therefore dc Æ 0. Hence thereis no longer any ambiguity in this coarse-grained percolation problem. We expectthat this “macroscopic percolation” willoccur at a critical volume fraction fp of theminority phase, which does not depend ontemperature. Therefore, the line of macro-scopic percolation concentrations is simply

cp(macro) = c(1)

coex + (c(2)coex – c(1)

coex) fp

and hence must end in the critical point.This line is also shown schematically inFig. 6-21. Note that this line will be ob-served experimentally by techniques whichare sensitive to the “contrast” (i.e., differ-ence in refractive index) between the two


Figure 6-22. Part of the phase diagram of the simplecubic, nearest-neighbor lattice-gas model, showingthe percolation transition line cp

(corr) (T, t), where thetime t refers to an average over the time interval fromt = 80 to t = 240 MCS per site during phase separa-tion. From Hayward et al. (1987).


coexisting phases, such as observationswith light or electron microscopes.

This phenomenon of “gelation of clus-ters” at cp

(corr)(T, t) into an infinite percolat-ing net has a pronounced effect on the clus-ter size distribution n–l (t), of course, whichwas used in the “cluster dynamics” model-ing of Eq. (6-54). This is shown in Fig.6-23 where n–l (t) is plotted against l forc– = 0.156 in a quench to T = 0, where thesystem develops towards a frozen-in clus-ter size distribution. The curvature of n–l (0)on the log– log plot reflects the exponentialvariation, ln n–l (0) ~ l, while the straight-line behavior occurring at later times ischaracteristic of the power law of percola-tion, n–l (t) ~ l –t . This percolation problemis usually disregarded in treatments basedon Eq. (6-54) or related models (Langerand Schwartz, 1980). Certainly this behav-ior also makes theories of coarsening,where the growing droplets are modeled asessentially spherical, doubtful at volumefractions F close to Fp; note that fromFig. 6-22 and the fact that the critical vol-

ume fraction for continuum percolation isabout 0.16 (Scher and Zallen, 1970), weexpect all these theories only to be reliablefor F 0.16. This percolative behaviorn–l (t) ~ l –t is in conflict with the scaling be-havior that is described in the next section,namely (Binder, 1977, 1989), n–l (t) = l –1

n (l t –d/3), where n () is a scaling function.The extent to which this percolation transi-tion affects S (k, t) is unclear.

6.2.9 Coarsening and Late Stage Scaling

From Figs. 6-5 and 6-12 it is evident thatthe peak position km(t) of the structure fac-tor S (k, t) decreases with increasing timeafter the quench. This decrease already re-flects the onset of a coarsening behavior ofthe domains of A-rich and B-rich phasesthat have formed (Fig. 6-2). For largeenough times, the domains have grown to asize much larger than all “microscopic”lengths (such as the interfacial width).Then a simple power law should hold,

km(t) ~ t –x , t Æ ∞ (6-66)

and the structure factor should satisfy ascaling hypothesis (Binder and Stauffer,1974, 1976b; Binder et al., 1978; Furu-kawa, 1978; Marro et al., 1979)

S (k, t ) – kBT /[(∂2 f /∂c2)T,ccoex(1,2) + r2T k2]

~ [km(t)]–d Sk/km(t) (6-67)

The term subtracted on the left-hand sideof Eq. (6-67) represents the scattering fromconcentration fluctuations within the do-mains, and S (z ) is a scaling function whichwill be discussed later.

Understanding the growth law, Eq. (6-66), and predicting the associated scalingfunction S (z ) has been a longstandingproblem that is still not completely solved(see Bray, 1994, for a recent review). It isnow thought that for solid mixtures, both ind = 2 and in d = 3 dimensions, but in the ab-

Figure 6-23. Cluster size distribution for a systemof L3 lattice site with L = 40. Full symbols give thecluster size distribution n–l (0) corresponding to therandom percolation problem at c– = 0.156 at the sim-ple cubic lattice, and open symbols denote the clustersize distribution averaged over the time intervalsshown. From Hayward et al. (1987).

sence of any elastic interactions (see Sec.6.2.10), a result originally derived by Lif-shitz and Slyozov (1961) and Wagner(1961) holds

x = 1/3 (6-68)

This LSW theory is essentially a mean-field theory valid in the limit of zerovolume fraction F of the new phase[F ∫ (c– – c(1)

coex)/(c(2)coex – c(1)

coex)] and consid-ers the cluster size distribution n–l (t) thatwas discussed in Sec. 6.2.5, showing thatfor F = 0 there exists a solution nl (t) = l –1

n (l t –d/3), which implies that the meancluster “size” (i.e., volume) l

–(t) scales as

l–(t) ~ t d/3 for t Æ ∞ (for a more detailed

outline of this theory, see Wagner et al.,2001). Thus the cluster linear dimensionscales as [l

–(t)]1/d ~ t1/3, and if we can ex-

tend this result to nonzero F we would ex-pect km(t) ~ [l

–(t)]1/d and hence Eq. (6-68)

results. However, despite numerous at-tempts (e.g., Tokuyama and Enomoto,1993; Akaiwa and Voorhees, 1994) eventhe extension of LSW theory to the case ofsmall F is only approximately possible,and the accuracy of these extensions isopen to doubt (Mazenko and Wickham,1995). Different power laws for not so latestages have also been proposed (e.g.,Binder, 1977, invoking a cluster diffusionand coagulation mechanism, and To-kuyama and Enomoto, 1993), and observedin computer simulations where atomic dif-fusion is mediated by a single vacancymoving through the lattice (Fratzl and Pen-rose, 1994, 1997). Only for deep quencheswhere we must take into account that themobility M in Eq. (6-11) is itself concen-tration-dependent and (almost) vanishingin the pure phases, diffusion along inter-faces results in a slower growth law, x =1/4(Puri et al., 1997).

“Cluster dynamics” approaches such asEq. (6-54) (see Binder, 1977; Mirold and

Binder, 1977; and Binder et al., 1978) andvarious extensions (e.g., Langer andSchwartz, 1980; Kampmann and Wagner,1984; Wagner et al., 2001) incorporate theLSW growth law, nucleation and coagula-tion in a phenomenological way. But theydo not take into account the statistical fluc-tuations and the correlations in the diffu-sion field around growing clusters.

Eq. (6-68) has now been confirmed byapproaches based on scaling ideas (Furu-kawa, 1978, 1984, 1985a, 1988; Bray,1998), renormalization group concepts(e.g., Lai et al., 1988; Bray, 1990, 1994),and theories considering fluctuating ran-dom interfaces (Mazenko, 1994; Mazenkoand Wickham, 1995). However, the mostconvincing evidence that Eq. (6-68) is trueboth for critical and for off-criticalquenches in d = 2 and in d = 3 comes fromcomputer simulations (Amar et al., 1988;Gunton et al., 1988; Gawlinski et al., 1989;Huse, 1986; Rogers et al., 1988; Rogersand Desai, 1989; Chakrabarti et al., 1993).

We emphasize that Eq. (6-68) holds bothin d = 2 and in d = 3 dimensions. The situa-tion is different for fluid binary mixtures:the droplet diffusion–coagulation mecha-nism (also called “Brownian coalescence”)predicts (Binder and Stauffer, 1974)

km(t) ~ t –1/d (6-69)

while mechanisms invoking hydrodynamicflow of interconnected structures imply(Siggia, 1979)

km(t) ~ t –1, d = 3 (6-70)

and km(t) ~ t –1/2, d = 2 (San Miguel et al.,1985), in the “viscous hydrodynamic re-gime” (Bray, 1994), while km(t) ~ t –2/3 inthe “inertial hydrodynamic regime” (Furu-kawa, 1985c).

A theoretical problem that is still out-standing is the calculation of the scalingfunction S (z ) in Eq. (6-67) (see Bray



(1994), Puri et al. (1997) and Mazenko(1994, 1998) for recent discussions). Wemention here only a few approaches verybriefly. Rikvold and Gunton (1982) usedEq. (6-56), following Binder et al. (1978)but treating the depletion zones aroundgrowing clusters more realistically. Clearly,their model is only qualitative. More ambi-tious early attempts are due to Furukawa(1984, 1985a, b), Ohta (1984), Tomita(1984) and Tokuyama et al. (1987). Anequation due to Furukawa (1984) has beenextensively compared with experimentaldata on Al–Zn alloys (Komura et al., 1985)

S (z ) ~ z2/(g 2/2 + z2+g ) (6-71)

where g = d +1 for strongly off-criticalquenches (cluster regime) and g = 2d forcritical quenches (percolative regime).S (z ) thus exhibits “Porod’s law” (see e.g.,Glatter and Kratky (1982))

S (k, t) ~ k –(d+1) (6-72)

in the cluster regime only. Eq. (6-71) alsofails to reproduce the exactly established(Yeung, 1988) behavior for small z,S (z ) ~z 4. For systems without hydrody-namics, good empirical forms for S (z )describing the limiting behavior both forsmall z and for large z correctly and ac-counting both for simulation and experi-ment have been constructed by Fratzl et al.(1991), but the derivation of such functionsfrom more fundamental theories is still notfully solved (Mazenko, 1994, 1998; Ma-zenko and Wickham, 1995; Bray, 1994).Fig. 6-27 shows a comparison of sometheoretical predictions for the halfwidth ofthe scaling function S with correspondingexperiments (Kostorz, 1991). It is intri-guing to note that the behavior of S (z ) doesnot change much when the volume fractionf changes from the “cluster regime” atsmaller f through a percolation transitionto the interconnected regime at large f. Re-

call that for fluid binary mixtures even theexponent x is different in these two regimes(Eqs. (6-69), (6-70)). For the percolativeregime of fluid mixtures, Furukawa (1984)has proposed an approximate relation forqm vs. t

where A* and B* are adjustable constants.Again a more precise treatment has to relyon numerical calculations (Koga and Ka-wasaki, 1991, 1993; Puri and Dünweg,1992; Valls and Farrell, 1993; Shinozakiand Oono, 1993; Alexander et al., 1993;Bastea and Lebowitz, 1995).

6.2.10 Effects of Coherent Elastic Misfit

When phase separation occurs on a crys-talline lattice it is often the case that thetwo phases differ slightly in their crystalstructure or their lattice constants, thus in-troducing elastic strains in the crystal. Theresulting elastic interaction is long-rangeand typically anisotropic and may consid-erably change the phase separation pro-cess, e.g., in metal alloys (for a recent re-view, see Fratzl et al., 1999; an extensivetreatment of this problem is also given inKhachaturyan, 1983).

The coherent misfit strain tensor e0ij is

the strain required to transform the (undis-torted) lattice of one phase into the (undis-torted) lattice of the other. In metals, partsof this strain can be relaxed by misfit dislo-cations disrupting the continuity of the lat-tices between the two phases. This processwill not be discussed here. In cases of non-zero misfit strain, e0

ij ≠ 0, an influence onspinodal decomposition is expected if oneof the following conditions is violated:

(i) both phases have the same elasticstiffness tensor;

( ) [ */ * arctan ( */ * )

arctan */ *] *

q A B B A q

B A Bm m

= (6-73)

− −−

1 1

t

(ii) the misfit strain is purely dilatational;(iii) the elastic stiffness tensor is isotropic;(iv) the crystal can be considered infi-

nitely large;(v) the stress depends linearly on the

strains.

If all five conditions are satisfied, thenphase separation can proceed indepen-dently of the misfit strains (Bitter–Crumtheorem, see, for example, Cahn andLarché (1984)). Otherwise, we can expectchanges in the shape of the single-phasedomains, e.g., from spherical to cuboidal orplate-like shapes but also in their spatial ar-rangement and coarsening kinetics. Typi-cally the tendency towards shape changesincreases when single-phase droplets be-come larger because the elastic energy (be-ing proportional to the droplet volume) in-creases faster with the radius than the sur-face energy.

The introduction of elastic misfit effectsinto the theory of spinodal decompositionmeans substituting fcg by fcg + w in Eq. (6-15). The function w is the elastic energydensity stored in the lattice

(6-74)

where lijmn is the elasticity tensor or stiff-ness tensor (which may depend on the alloycomposition).

Deij (x) = eij (x) – e0ij (x) (6-75)

is the difference between the strain at posi-tion x in the elastic equilibrium, eij (x), andthe misfit strain, e0

ij (x). Inserting this intothe nonlinear Cahn–Hilliard equation (Eq.(6-14)), we obtain (Larché and Cahn, 1982;Onuki, 1989a–c)

(6-76)∂∂

∇∂∂

+ ∂∂

− ∇⎧⎨⎩

⎫⎬⎭

ct

Mf

cwc

r k T c= cgB

2 2 2

w e eijmn

ijmn ij mn=12

∑ l D D( ) ( )x x

with

(6-77)

The first term arises from a dependence ofthe elastic constants on composition andthe second one from the composition de-pendence of the misfit strain.

The main difficulty in the use of Eq. (6-77) is the determination of Deij . Since thestrains in the alloy can be assumed to relaxmuch faster to their equilibrium valuesthan the concentration profiles, Deij will al-ways be given by the elastic equilibriumcondition

(6-78)

where the stress tensor is defined byHooke’s law

tij = lijmn Demn (6-79)

As a consequence, the equilibrium strain ateach point in the material is a (non-local)functional of the entire concentration pro-file c (x, t). In the special case when theelastic constants lijmn are independent ofcomposition, the elastic problem can besolved by Fourier transformation (Khacha-turyan, 1966, 1983) to give

(6-80)

or(6-81)

which enters the equations (6-76). Vel(u) isan elastic potential that depends on thestiffness constants lijmn and the misfitstrain e0

ij (Khachaturyan, 1983). The aver-

∂∂

− −∫wc

t V c t c y( , ) ( ) ( ( , ) )x x y y= del3

w V c t c

c t c x y

=

d d

el12

3 3

∫ ∫ − −

× −

( ) ( ( , ) )

( ( , ) )

x y x

y

j

ij

j

t

x∑

∂∂

= 0

∂∂

−

∑

∑

wc c

e e

eec

ijmn

ijmnij mn

ijmnijmn ij

mn

=d

ddd

12

0

l

l

D D

D



age concentration within the specimen iscalled c–.

In the case of isotropic elasticity andmisfit strain, Eq. (6-81) reduces to (Onuki,1989a– c)

(6-82)

where ha is the change of lattice constantwith concentration c, that is

e0ij (c) = ha (c – c–) dij (6-83)

K and nP are Young’s modulus and thePoisson coefficient of the elastic matrix,and dij = 0 if i ≠ j, dii =1 (Kronecker delta).Eq. (6-82) has been introduced by Cahn(1961). The consequence of this equation isa shift of the spinodal line towards lowertemperatures because Eq. (6-44) has to bereplaced by

Hence, the coherent elastic misfit betweenthe phases may stabilize the solid solution,even though the demixing into two inco-herently separated phases would decreasethe total energy (Cahn, 1961). The reasonis that the coherency condition forces thealloy to store a considerable amount ofelastic energy in the lattice, which could bereleased by creating an incoherent boun-dary between two regions both having the(undistorted) equilibrium lattice structure.

For anisotropic stiffness constants lijmn

or anisotropic misfit strains e0ij , the chemi-

cal potential due to elastic interactions,∂w/∂c, will depend on the direction in thecrystal. This has already been recognizedby Cahn (1962) (see also Khachaturyan(1983) and Onuki (1989 a–c)). Numericalsolutions of Eq. (6-76) combined with Eq.(6-81) have shown the development ofstrongly anisotropic domains (Nishimori

∂∂

+ ∂∂

∂∂

+−

2

2

2

2

2

222

10

f

c

w

c

f

c

Ka

cg cg

P= =

nh

∂∂ −

−wc

Kc t ca=

P

21

2

nh [ ( , ) ]x

and Onuki, 1990). A discretized version ofsimilar equations was used by Khachatur-yan and coworkers to study the behavior ofsingle precipitates (Wang et al., 1991) orthe evolution of precipitate morphologies(Wang et al., 1992, 1993). Moreover,Monte Carlo simulations of phase separa-tion have also been performed, includingelastic misfit interactions between un-equally sized atoms on a lattice (Fratzl andPenrose, 1995, 1996; Laberge et al., 1995,1997; Lee, 1997, 1998; Gupta et al., 2001).A further approach is the simulation of thecoarsening of particles with sharp inter-faces to the matrix, describing the elasticmisfit interaction in the framework of mac-roscopic elasticity theory. Recent examplesare the studies by Su and Voorhees (1996),Abinandanan and Johnson (1993, 1996) orJou et al. (1997) (see also the review byFratzl et al. (1999)). The common observa-tions in all these approaches are that thedomains become very anisotropic and typi-cally orient parallel to crystallographic di-rections of the alloy crystal (Fig. 6-24d, e).Moreover, the spatial arrangement of thedomains becomes progressively more peri-odic. Despite these enormous changes withrespect to the case without elastic misfitinteractions, the growth law of a typical do-main size is often still described by the re-sult of the LSW theory (Eq. (6-68)). If ex-ternal stress is applied, an additional reor-ientation of the domains either parallel orperpendicular to the applied stress is ob-served (Laberge et al., 1995, 1997; Weinka-mer et al., 2000). Many of these effects areobserved in real alloys, most notably for thetechnically important nickel-based superal-loys (Maheswari and Ardell, 1993; Conleyet al., 1989; Sequeira et al., 1995; Paris etal., 1995, 1997; Fährmann et al., 1995).

If the elastic stiffness constants lijmn de-pend on alloy composition (see the firstterm in Eq. (6-77)), this can result in anom-

alously slow precipitate growth. The mostspectacular observation, however, is that(at sufficiently late times in the coarseningprocess) the softer phase always “wraps”precipitates of the harder phase (that is, thephase with the higher stiffness) and, there-fore, stays percolated even if it is the mi-nority phase (Onuki and Nishimori, 1991;Sagui et al., 1994; Leo et al., 1998; Orli-kowski et al., 1999), see Fig. 6-24a–c.

Most recently, the possibility of atomicordering within the precipitates has alsobeen included in the theory outlined in thissection (Sagui et al., 1994; Wang andKhachaturyan, 1995), as well as in MonteCarlo simulations on an elastic lattice (Nie-laba et al., 1999). This amounts to studyingdecomposition close to a tricritical point,including the effects of elastic misfit inter-action. With these features, the theoreticalmodels are now capable of describing a sur-prising number of details in the evolution ofalloy microstructures (e.g., Li and Chen,1998). An example is shown in Fig. 6-25.

6.3 Survey of ExperimentalResults

The number of experimental studies de-voted to spinodal decomposition is enor-mous: first, it is a widespread phenomenon


Figure 6-24. Typical snapshot pictures using theCahn–Hilliard equation with elastic interactions (Eq.(6-77)). (a–c) is the case of isotropic elasticity butwhere the elastic stiffness depends on composition.The stiffer phase is shown white with volume frac-tions of 0.3, 0.5 and 0.7, respectively (from Onukiand Nishimori, 1991). Note that the softer phase al-ways “wraps” stiffer particles. (d, e) is the casewhere the elastic stiffness has cubic anisotropy. Thevolume fraction of the white phase is 0.5 and 0.7, re-spectively (from Nishimori and Onuki, 1990).

Figure 6-25. Transmission electron micrographtaken in the (001)-plane of a Ni–Al–Mo wherethe lattice spacing between matrix and precipitates(gamma-prime phase) is different by ha = –0.5%(aged for 5 h at 1253 K). Cube-like precipitates canbe seen, aligned along the elastically soft directions,[010] and [100]. (b) Results from computer simula-tions of an Ising model with elastic interactions on asimple square lattice and with repulsive interactionof like atoms on nearest-neighbor sites (with interac-tion energy J) and attractive interaction of like atomson next-nearest-neighbor sites (energy J/2). The dis-ordered phase (containing mostly A atoms) is shownin black and the ordered phase (consisting of about50% A and 50% B atoms) is shown in white. Theoverall concentration of B atoms was 0.35 and therun was performed at a temperature of T = 0.567 J/kB

on a 128 ¥ 128 lattice and stopped after 106 MonteCarlo steps. (c) The same alloy and heat treatment asin (a), but now with an external compressive load of130 MPa applied along the vertical [010]-direction.(d) The same model, temperature and annealing timeas in (b), but with an additional external load alongthe vertical direction. Experimental data (a and c) arefrom Paris et al. (1997) and simulation data (b and d)from Nielaba et al. (1999).

6.3 Survey of Experimental Results 451

that occurs in diverse systems; second,there has been much interest from a theo-retical point of view in this phenomenon,and therefore many experimental studieshave been carried out in an attempt to testsome of the theoretical concepts.

This section cannot present a completeoverview of all these experiments. Moredetailed reviews of various aspects of theexperimental work can be found in Haasenet al. (1984), De Fontaine (1979), Geroldand Kostorz (1978), Gunton et al. (1983),Goldburg (1981, 1983), Beysens et al.(1988), Kostorz (1988, 1994), Hashimoto(1987, 1988, 1993), and Nose (1987),among others. Here we attempt only togive a few representative examples to illus-trate some of the points discussed in thetheoretical section, and at the same timeshow similarities – as well as differences –between different systems.

6.3.1 Metallic Alloys

Some systems in which spinodal decom-position has been very extensively studiedare Al–Zn alloys (e.g., Hennion et al.,1982; Guyot and Simon, 1982, 1988; Si-mon et al., 1984; Osamura, 1988; Komuraet al., 1985, 1988; Mainville et al., 1997),ternary Al–Zn–Mg alloys (Komura et al.,1988; Fratzl and Blaschko, 1988), Ni-based alloys such as Ni–Al, Ni–Ti, Ni–Cr,and Ni–Mo (Kostorz, 1988; see alsoKampmann and Wagner, 1984), Al–Li al-loys (Furusaka et al., 1985, 1986, 1988; Li-vet and Bloch, 1985; Tomokiyo et al.,1988; Che et al., 1997; Schmitz et al.,1994; Hono et al., 1992), Mn–Cu alloys(Gaulin et al., 1987), Ni–Si alloys (Chen etal., 1988; Cho and Ardell, 1997), Fe–Cr al-loys (Katano and Iizumi, 1984), and vari-ous other ternary alloys, e.g., Cu–Ni–Fe(Lyon and Simon, 1987; Lopez et al., 1993)or Ni–Al–Mo (Fährmann et al., 1995;

Paris et al., 1995, 1997; Sequeira et al.,1995). In all these systems, the structurefactor looks very similar to the early dataon Au–Pt alloys (Singhal et al., 1978) andthose of Al–Zn (Mainville et al., 1997) re-produced in Fig. 6-5. These results are inqualitative agreement with the nonlinearLBM theory and the computer simulations.The hallmarks of the linearized Cahn the-ory (exponential increase in intensity withtime, time-independent intersection pointof S (k, t), maximum position km of S (k, t)independent of time) are never found. In afew cases it has been found that at earlytimes km(t) is nearly independent of time,e.g., in Mn67Cu33 (Morii et al., 1988), andthis fact has been taken as evidence for thevalidity of the Cahn–Hilliard–Cook (CHC)approximation. In view of the fact thatkm(t) almost independent of t can also re-sult as an effect of continuous rather thaninstantaneous quenching (Sec. 6.2.7) froma nonlinear theory, we feel that the questionof whether the CHC theory applies to anyof the metallic systems quantitatively isstill unanswered. Certainly we expect thatthe CHC theory gives a useful order ofmagnitude estimate of km(0) and the initialgrowth rate Rm, as it does for the theoreti-cal models (Fig. 6-5d), provided a spinodalcurve is not close. Experiments by Guyotand Simon (1982) and Hennion et al.(1982) were deliberately carried out choos-ing temperatures both above and below thespinodal curve of Al–Zn, for several con-centrations. The structure factor in bothcases is qualitatively the same, as was pre-dicted (see Sec. 6.2.5). This classical set ofexperiments in our view definitely showsthat the spinodal singularity does not playany role in the unmixing kinetics of Al–Znalloys. Note that it is clear that the effectivepotential in this alloy is predominantlyshort range; however, this alloy does havea significant misfit of atomic radii between

Al and Zn, and thus the “coherent phase di-agram” (where elastic strain fields are notreleased) is depressed by about 30 K belowthe thermodynamic (incoherent) miscibil-ity gap, and the spinodal curves are thensimilarly shifted, providing an exampleof the effects of elastic interactions (Sec.6.2.10). Despite these significant long-range elastic forces, the data for Al–Znspinodal decomposition are surprisinglysimilar to the Monte Carlo results on sim-ple nearest-neighbor Ising models of al-loys, where all such elastic effects are notincluded. In other cases, such as many bi-nary and ternary Ni alloys (see, for exam-ple, Fig. 6-25) elastic misfit interactionslead to highly anisotropic precipitate mor-phologies and to ordered arrangements ofprecipitates (see Sec. 6.2.10).

There is ample experimental evidencefor the scaling of the structure factor at latestages (Eq. (6-67)), and the validity ofthe Lifshitz–Slyozov exponent x =1/3 (Eq.(6-68)). Sometimes smaller exponents arefound, e.g., km

–1(t) ~ t x with x ≈ 0.13 tox ≈ 0.2 (see, for example, Furusaka et al.,1986; Osamura, 1988), but typically dataextending over only 1.5 decades in time areavailable. A general problem is that somemisfit between matrix and precipitates ispresent in many real alloys. Moreover, va-cancy concentrations may be larger thanthe equilibrium value after the quench andthen decrease gradually. Both effects couldaffect the exponent in real alloys. Katanoand Iizumi (1984), Furusaka et al. (1988)and Forouhi and De Fontaine (1987) founda regime of 1.5 decades in time wherex =1/6 whereas after a rather sharp cross-over x =1/3 at later times. Other experi-ments (e.g., Morii et al., 1988) where thekm

–1(t) vs. t relationship is also measuredover about three decades in time find amuch smoother crossover from the initialstages, where the log– log plot of km

–1(t) vs.

t is curved and a well-defined exponent xcannot be identified, to the final Lifshitz–Slyozov behavior, x =1/3. Katano and Ii-zumi (1984) interpret the result x =1/6 interms of the mechanism (Binder and Stauf-fer, 1974) that B-rich clusters in a solidsolution show a random diffusive motionsince B atoms evaporate randomly from thecluster and re-impinge at another boundaryposition, thus leading to a small shift inthe cluster center of gravity. The resultingcluster diffusivity decreases strongly withincreasing cluster size. Assuming then thattwo clusters coalesce when they meet intheir random motions, we arrive at km(t) ~t –1/(d+3) in d dimensions. Although such amechanism certainly exists, it is not clearwhether it ever dominates during a well-defined time interval, since the Lifshitz–Slyozov–Wagner (1961) mechanism com-petes with it and should dominate, at leastfor long times. Monte Carlo simulationsconsidering diffusion via vacanies indicatethat cluster–diffusion–coagulation couldbe important at low temperatures (Fratzland Penrose, 1997). This cluster–diffu-sion–coagulation mechanism was origi-nally proposed to explain the correspond-ing small values of the exponent x seen inMonte Carlo simulations (Bortz et al.,1974; Marro et al., 1975). However, moreaccurate simulations (Huse, 1986; Amar etal., 1988) are rather consistent with a grad-ual approach to the asymptotic Lifshitz–Slyozov–Wagner (1961) law without anintermediate regime characterized by awell-defined different exponent x. In fact,the data can usually be fitted to an equationderived by Huse (1986):

km–1(t) = A + B t1/3 (6-84a)

while the original treatment of Lifshitz andSlyozov (1961) yielded

km–3(t) = A¢ + B¢ t (6-84b)



where A, B, A¢, and B¢ are constants. Eq. (6-84a) can also be interpreted by an effectiveexponent in the growth law,

Xeff (t) ∫ dlog km–1(t)/d (log t)

= 1–3

[1 – A km(t)]

Evidence for this law is found in varioussimulations (Amar et al., 1988; Gunton etal., 1988) and some experiments (Morii etal., 1988; Chen et al., 1988; Alkemper etal., 1999).

There have also been numerous attemptsto characterize experimentally the scalingfunction S () in Eq. (6-67). Typical data forAl–Zn alloys are shown in Fig. 6-26. Al-though the agreement with Eq. (6-71) looksimpressive, the theory (Furukawa, 1984)neglects anisotropy, and so does the dataanalysis on polycrystalline samples of Ko-mura et al. (1985). Single-crystal work (Si-mon et al., 1984), however, shows pro-nounced anisotropy. Anisotropy was alsoseen, for example, in scattering from Ni-based alloys as a result of the coherentelastic misfit (Kostorz, 1988; Fährmann etal., 1995; Sequeira et al., 1995). In earlystages of phase separation, the anisotropyincreases roughly linearly with the meanradius of precipitates, R µ l1/3 (Paris et al.,1995). The reason is that the elastic misfitenergy (which favors anisotropy) is pro-portional to the volume of the precipitate, l,while the surface energy (which favorsround shapes) depends on l 2/3.

The scaling function has been deter-mined experimentally for a number of al-loys and there is a general trend that thewidth of the scaling function decreaseswith increasing volume fraction of precipi-tate phase. A large collection of data isshown in Fig. 6-27, mostly taken from thereview by Kostorz (1991). This decreasecan be understood qualitative as follows: ifL is a typical period in the spatial arrange-

Figure 6-26. (a) Scattering cross-section measuredfor neutron small-angle scattering from Al–10 at.%Zn polycrystals at 18 °C (Komura et al., 1985). (b)Scaling plot of the data shown in (a). Full curve is afit to Furukawa’s (1984) function, Eq. (6-71). FromKomura et al. (1985).

ment of the precipitates, the maximum po-sition of the structure function varies asL–1. For spherical precipitates of radius R,the width of the structure function variesroughly as R–1. Hence, the width of thescaling function behaves approximately asR/L µF –1/3 for small volume fractions F.A similar argument may also be developedfor larger volume fractions, where the mi-crostructure corresponds to percolated do-mains instead of isolated droplets (Fratzl,1991), leading to the prediction shown inFig. 6-27 (broken line).

6.3.2 Glasses, Ceramics,and Other Solid Materials

Early experimental data on glassysystems such as the Na2O–SiO2 systemhave been reviewed by Jantzen and Her-man (1978). Some data have been inter-preted in the framework of the linearizedtheory of spinodal decomposition (e.g.,Yokota, 1978; Acuna and Craievich, 1979;Craievich and Olivieri, 1981), but not all

features of the linear theory can be demon-strated quantitatively in these materials,and thus no real evidence for the signifi-cance of a spinodal curve is present. Whilethe isotropy of these systems is clearly asimplifying feature, coupling to structuralrelaxation sometimes needs to be consid-ered (Yokota, 1978), which implies a sig-nificant complication (Binder et al., 1986).

An investigation of late stages wascarried out by Craievich and Sanchez(1981) for the B2O3–PbO(A2O3) glass atT = 0.65 Tc where the critical point of un-mixing is Tc ≈ 657 °C. The structure factorS (k, t) was obtained in a time range ofabout 12 to 400 min, and Eq. (6-66) wasfound to be obeyed with km

–1(t) ~ t x,x ≈ 0.23. The early stages were studied byStephenson et al. (1991) and were found toqualitatively agree with the Cahn–Hilli-ard–Cook theory (Eq. 6-27). Qualitatively,the behavior is very similar to the resultsfor metallic alloys (Sec. 6.3.1) and to com-puter simulations.

At this point, we also mention the appli-cation of spinodal decomposition of boro-silicate glass-forming melts to produceporous Vycor glass; on cooling the melt be-low its demixing temperature it decom-poses into an SiO2-rich phase and a B2O3-alkali-oxide-rich phase. The latter is acidsoluble and can be leached out with suit-able solvents leaving a fully penetrable mi-croporous glass. The small-angle scatteringfrom such materials has been examined andinterpreted in the framework of the theoryof spinodal decomposition (Wiltzius et al.,1987).

Qualitative observations of phase separ-ation on a local scale attributed to spinodaldecomposition have been reported for a va-riety of glassy materials and ceramics. Ex-amples include rapidly quenched Al2O3–SiO2–ZrO8 (McPherson, 1987), La–Ni–Alamorphous alloys (Okamura et al., 1993),


Figure 6-27. Full width at half maximum (FWHM)of the scaling function normalized such that the posi-tion of the maximum is located at 1. Black stars arefor Al–Ag alloys (Langmayr et al., 1992), all otherdata points are taken from the review by Kostorz(1991) and correspond to Al–Zn, Pt–Au, Cu–Mn,Fe–Cr and Al–Li. Full and broken lines indicate pre-dictions from various models (RG: Rikvold and Gun-ton (1982), TEK: Tokuyama et al., (1987), FL: Fratzland Lebowitz (1989)).


and SiC–AlN ceramics (Kuo et al., 1987).The crystallization behavior of bulk metal-lic glasses, such as Zr–Ti–Cu–Ni–Be, hasbeen found to be strongly influenced by de-composition (e.g., Wang et al., 1998b).

Of course, spinodal decomposition isalso expected to occur in various nonmetal-lic mixed crystals, e.g., oxides such as theTiO2–SnO2 system (Flevaris, 1987; Taka-hashi et al., 1988) and semiconductingGaInAsP epitaxial layers (Cherns et al.,1988; Mcdevitt et al., 1992). Often thesesituations are difficult because of lowercrystal symmetry (the SnO2–TiO2 systemis tetragonal) and strong elastic lattice mis-fit effects (Flevaris, 1987).

6.3.3 Fluid Mixtures

Binary fluids containing small mole-cules such as lutidine–water (Goldburg etal., 1978; Goldburg, 1981, 1983; Chou andGoldburg, 1979, 1981) and isobutyricacid–water (Wong and Knobler, 1978,1979, 1981; Chou and Goldburg, 1979,1981) are classic systems where nonlinearspinodal decomposition was observed vialight scattering techniques. Since the inter-

diffusion in fluid mixtures proceeds muchfaster than in solids, we must considerquenches in a narrow region just below Tc,in order to take advantage of critical slow-ing down. This implies that we must al-ways work in the non-mean-field criticalregion (in the phase diagram in Fig. 6-13),i.e., nonlinear effects are very strong, andso it is not expected that the linearized the-ory of spinodal decomposition will accountfor these systems. In addition, very earlystages are not observable because the un-mixing proceeds too fast. This is best seenwhen working with rescaled variables(Chou and Goldburg, 1979, 1981), defining

qm(t) ∫ km x (t = 0)

and

t = t D (t = 0)/x2(t = 0)

where x (t = 0) and D (t = 0) are the correla-tion length and the interdiffusion constantin the initial state, respectively. Early timesthen mean a t of the order of unity. Fig. 6-28 shows that only scaled times tÊ6 areaccessible, whereas earlier times are ac-cessible for fluid polymer mixtures (seeSec. 6.3.4). The growth rate exponent x

Figure 6-28. Log–logplot of qm vs. t found inthe polymer mixture poly-styrene (PS)–poly(vinylmethyl ether) (PVME)(data labelled system K),compared with isobutyricacid–water (l) and 2,6-luti-dine–water mixtures (m).Systems (K, m) are datafrom Chou and Goldburg(1979). From Snyder andMeakin (1983).

(qm–1(t) ~ t x) crosses over from x near

x ≈1/3 for t =10 to x ≈1 at very late times,as expected (Siggia, 1979) (see Eq. (6-70)).The behavior qm

–1(t) ~ t–1/3, which for off-critical quenches where the structure corre-sponds to well separated clusters is be-lieved to be the true asymptotic behaviorfor t Æ ∞ (Siggia, 1979), is attributed to acluster diffusion and coagulation mecha-nism (Binder and Stauffer, 1974; Siggia,1979). Note that for fluid droplets thediffusion constants decrease in proportionto their radii with increasing droplet size,due to Stokes’ law, and hence the clusterdiffusion–coagulation mechanism neverbecomes negligible in comparison withthe Lifshitz–Slyozov (1961) evaporation–condensation mechanism.

Fig. 6-28 shows, however, that the cross-over to the asymptotic power law is grad-ual. This smooth behavior is approximatelydescribed by the nonlinear theory of spino-dal decomposition of Kawasaki and Ohta(1978). However, this theory can give onlya poor account of the full structure func-tion. This is not surprising, since it is onlyexpected to be accurate for tÁ10 (see Sec.6.2.6). Again, the problem of calculatingthe scaled structure function S () (Eq. (6-67)) arises and is as difficult as in the caseof solids (Furukawa, 1985a).

A problem for binary fluids is the effectof gravity. Since the two coexisting phasesusually differ in density, one phase must goto the top and the other to the bottom of thecontainer. This effect (for a more detaileddiscussion see Beysens et al., (1988)) im-plies a smearing of the critical region forthe unmixing critical point of a sample offinite height, and also affects the late stagesof phase separation (where even hydrody-namic instabilities may set in; see Chanand Goldburg (1987)). The effect of gravity,however, can be strongly reduced if iso-density systems are used (using mixtures of

methanol and partially deuterated cyclo-hexane, a perfect density matching at Tc ispossible; see Houessou et al., (1985)) orspace experiments are performed, wherethe gravitational effect can be reduced by afactor of 104. Beysens et al. (1988) showedthat such experiments agree with the re-sults obtained from isodensity systems. Bysuch means not only can the accuracy ofdata of the sort shown in Fig. 6-28 be sub-stantially improved, but several decadescan be added to a plot of qm vs. t (Fig. 6-29). The full curve in this plot is the func-tion (Eq. (6-73)) proposed by Furukawa(1984), which reduces to Eq. (6-70) forlarge t, i.e., the result of Siggia (1979).More recently, a growth of the mean drop-let size with the power-law exponent 1/3could be followed over more than sevendecades in a microgravity experiment (Per-rot et al., 1994).

Interesting extensions involve spinodaldecomposition in fluid mixtures underweak steady-state shear (Chan et al., 1988;Krall et al., 1992; Lauger et al., 1995;Hashimoto et al., 1995; Hobbie et al.,1996) or periodically applied shear (Bey-sens and Perrot, 1984; Joshua et al., 1985),as well as in strongly stirred mixtureswhere a turbulent suppression of spinodaldecomposition occurs (Pine et al., 1984;Chan et al., 1987; Easwar, 1992). Many ofthese observations can be explained on thebasis of the theoretical considerations ofOnuki (1984, 1986a, 1989c).

Finally, we draw attention to phase sep-aration phenomena in more exotic fluidssuch as surfactant micellar solutions (Wil-coxon et al., 1988, 1995), polymer solu-tions near their Q-point, e.g., polystyrenein methyl acetate (Chu, 1988), gel–geltransitions (such a transition occurs, for ex-ample, in N-isopropylacrylamide gel withwater as a solvent, where this system phaseseparates into two states with different sol-



vent concentrations and hence a differentswelling ratio of the gel; see Hirotsu andKaneki (1988)), and lipid membranes withdissolved protein.

6.3.4 Polymer Mixtures

As pointed out in Secs. 6.2.4 and 6.2.5,mixtures of long flexible macromoleculesare particularly well suited model systemsfor the study of phase separation kinetics:(i) owing to the mutual entanglement of therandom polymer coils, their interdiffusionis very slow for high molecular weights, sothe early stages can be studied; (ii) polymermixtures exhibit a well-defined mean-fieldcritical regime, where the linear theory ofspinodal decomposition should hold, and a

spinodal curve can be defined; (iii) chang-ing the molecular weights while otherparameters (in particular, intermolecularforces!) remain constant allows a morestringent test of theories than for othersystems. For example, it has been possiblerecently to obtain by laser scanning confo-cal microscopy (Jinnai et al., 1997) three-dimensional images of phase-separatedpolymer blends that look very similar topictures obtained from theory (Fig. 6.30).

Therefore, it is gratifying that quantita-tive agreement with the linear theory ofspinodal decomposition was obtained incareful experiments with various poly-mer blends, e.g., polystyrene–poly(vinylmethyl ether) (Okada and Han, 1986; Hanet al., 1988; Sato and Han, 1988), polybu-

Figure 6-29. Behavior of the scaled wavevector qm(t) versus scaled time t. Full curve is from Furukawa(1985), data points refer to mixtures of methanol and cyclohexane in space-flight experiments (F) or to isoden-sity mixtures of methanol and partially deuterated cyclohexane for various quench depths, as indicated. Thelate-time behavior, where qm(t) ~ t –1 is observed, is magnified in the inset. From Beysens et al. (1988).

tadiene and styrene–butadiene random co-polymer mixtures (Hashimoto and Izumi-tani, 1985), and mixtures of deuterated andprotonated polybuta-1,4-diene (Bates andWiltzius, 1989), among others. As an ex-ample, Fig. 6-31 now provides an experi-mental counterpart to Figs. 6-15, 6-16,which were obtained from simulation. Notethat here Eq. (6-23) is used, i.e., fluctua-tions in the final state are neglected – there-fore the Cahn plot bends over for largewavevectors, and R (q)/q2 never becomesnegative, owing to the second term on theright-hand side of Eq. (6-29). Moreover, bycarrying out quenches to different tempera-tures it is shown that both q2

m(0) and the ef-fective diffusion constant D0 vary linearlywith (1– T /Tc), as expected from Sec. 6.2.5.The onset of nonlinear effects can be iden-tified, but it is not accurately described byLBM-type theories (Sato and Han, 1988).


Figure 6-30. Three-dimensional representation of abicontinuous phase-separated polymer blend. Inves-tigation of the blend by laser scanning confocal mi-croscopy gives results very similar to numerical solu-tions of the nonlinear theory of spinodal decomposi-tion (Jinnai et al., 1997).

Figure 6-31. (a) Early stage growth of concentrationfluctuations in a mixture of protonated and deuter-ated polybuta-1,4-diene, with polymerization indexNH = 3180, ND = 3550, at critical volume fractionfD = 0.486 and T = 322 K (Tc = 334.5 K). Four repre-sentative scattering vectors are shown. From thestraight-line fits to these semi-logarithmic plots theamplification factor R (q) is extracted, which isshown as a Cahn plot in (b). From Bates and Wiltzius(1989).


At intermediate time a behaviorkm

–1(t) ~ t x is observed, where x varies withquench depth. This behavior is not under-stood theoretically. In this regime of times,nonlinear effects gradually become impor-tant. A transition stage then occurs, beforethe full scaling behavior of the structurefactor is observed in the final stage. Thescaling plot for qm(t) vs. t is presented inFig. 6-32, and the scaling function for thestructure factor is shown in Fig. 6-33. It isseen that Furukawa’s (1984) function givesa reasonable overall account of the data.The fact that the curves for cyclohexanemethanol and for poly(vinyl methylether)–polystyrene (PVME–PS) mixturesare offset is not considered to be very seri-ous, because there are some uncertaintiesinvolved in the rescaling of k and t in orderto obtain the scaled variables q and t ; onlyin terms of the scaled variables does itmake sense to compare different materialswith each other and to discuss whether atruly universal behavior exists.

A possible interpretation of the devia-tions from universality in the “intermedi-ate” and “transition” stage in Fig. 6-32 isthe fact that Bates and Wiltzius (1989) con-sider deep quenches, for which 1– T /Tc isno longer very small. Only for 1– T /Tc 1can we expect a truly universal behavior. Inany case, comparison of Fig. 6-29 and Fig.6-32 with Fig. 6-28 clearly demonstratesthe experimental progress made during lessthan a decade; the accuracy of the data inFig. 6-32 is substantially better and therange of t is nearly one decade larger.

Figure 6-33 shows the behavior of thescaling function at small and at large for apolymer blend (a) and for a metal alloy (b).This behavior is of the form x with x = –4at large . At small , x is close to 4 in bothpolymers and metal alloys. The decay atlarge follows directly from Porod’s law(Porod, 1951) which is a consequence of

sharp interfaces between the single-phasedomains. The exponent x = 4 at small hasbeen proposed by Yeung (1988) from ananalysis of the Cahn–Hilliard equation. Upto now, there is no theoretical expressionfor the scaling function that is capable ofreproducing all the features in Fig. 6-33.The model of Fratzl and Lebowitz (1989)uses the correct exponents at low and atlarge and predicts a volume fraction de-

Figure 6-32. Log–log plot of qm(t–) versus scaledtime t– for the polymer mixtures in Fig. 6-28 (dif-ferent symbols represent different samples) atT = 298 K. Solid curve is a fit of Furukawa’s (1984)function, Eq. (6-73), to the data in the “final stageregion”. The poly(vinyl methyl ether)–polystyrene(PVME–PS) curve represents results of Hashimotoet al. (1986), and the curve for the cyclohex-ane–methanol small-molecule mixture is taken fromdata of Guenoun et al. (1987). The crossover timefrom the intermediate stage to the transition stage isdenoted as t–s , from the transition stage to the finalstage t–w. Note that the scaled time t– for the small-molecule mixtures is defined differently, sinceD–

eff = D0 is not well defined. From Bates and Wilt-zius (1989).

pendence in reasonable agreement with ex-periments (Fig. 6-27), but it fails to repro-duce the shoulder in the scaling functionappearing on the right side of the maximumfor polymers (Fig. 6-33a) and even somemetal alloys (Fig. 6-33b). Earlier theoriesalso fail to produce this shoulder, exceptfor the approach by Ohta and Nozaki(1989), which predicts a function qualita-tively similar to the data in Fig. 6-33a, butis still quantitatively off.

6.4 Extensions

Basic aspects of the theory of spinodaldecomposition were treated in Sec. 6.2.Only the generic phase diagram (Figs. 6-3b, 6-13, and 6-21) of a binary mixturewith a miscibility gap has been considered.In this section, we briefly mention the re-lated phenomenon of ordering kinetics(Fig. 6-3a) and also discuss cases whereformation of order and unmixing compete.The effects of certain complications (im-purities, effects of finite size and free sur-faces) are briefly assessed.

6.4.1 Systems Near a Tricritical Point

If a binary system exhibits a line of crit-ical points Tc(c–) for an order–disordertransition, a tricritical point Tt (c

–t) may oc-

cur where this critical line stops, and belowthis tricritical point both ordering andphase separations occur simultaneously(Griffiths, 1970; Lawrie and Sarbach,1984). Two order parameters, y (x) andc (x), are simultaneously needed to de-scribe such phenomena. Examples of thissituation occur in some magnetic alloyswhere the order–disorder transition de-scribes a magnetic ordering, 3He– 4He mix-tures (here y describes the superfluid orderparameter and hence must be taken as a


Figure 6-33. (a) Log–log plot for the structure-fac-tor scaling function in the late stages of a quench to313 K for the polymer mixture in Fig. 6-31. The solidcurve is the theoretical prediction of Ohta and No-zaki (1989). The straight line I (q) ~ q–4 demonstratesthe validity of Porod’s law. The inset shows low-qbehavior, as measured in the intermediate stage,where a power law I (q) ~ q 4.5 is found. From Batesand Wiltzius (1989). (b) Similar plot for the scalingfunction in the metallic alloy Al–15 at.% Ag (Lang-mayr et al., 1992). The full line corresponds to themodel function of Fratzl and Lebowitz (1989) for avolume fraction of F = 0.27. This function S

–() in-

creases with 4 at small and decreases with –4 atlarge (Porod’s law).

6.4 Extensions 461

complex variable), and also crystallo-graphic order–disorder phase transitions inalloys and in adsorbed layers at surfacesare known to exhibit such tricritical points.In alloys, the first discussions of phase sep-aration in tricritical systems come from Al-len and Cahn (1976, 1979a, b) in the con-text of Fe–Al alloys. An elegant extensionof the linearized theory of spinodal decom-position to tricritical 3He– 4He mixtureswas given by Hohenberg and Nelson(1979). Nonlinear effects have mostly beenstudied for the model with simple relaxa-tional dynamics of the order parameter(Dee et al., 1981; San Miguel et al., 1981).Here we briefly mention the case of3He– 4He mixtures, since the most detailedexperiments exist for these systems (Hofferet al., 1980; Sinha and Hoffer, 1981; Bendaet al. 1981, 1982; Alpern et al., 1982).3He– 4He mixtures have no practical appli-cations in materials science, but we men-tion them here as a model system for thestudy of phase separation near a tricriti-cal point. The concepts developed (andchecked experimentally) for this modelsystem can be carried over to more com-plex materials such as ternary mixtures,magnetic alloys, etc. Instead of Eqs. (6-4)and (6-8), the expression for the free en-ergy functional is now

where r0, u, v, cn–1, l0, and g are pheno-

menological coefficients that should de-pend only on temperature and not on the lo-cal 3He concentration c, which is measured

D | | | |

| | | |

| |

| |

k Tr u

r

c c c

l c

B

n

= d

(6-85)

∫ ⎧⎨⎩+

+ + ∇

+ + − − +

+ ∇ ⎫⎬⎭

−

x

x

x

12

14

16

12

1212

02 4

6 2 2

1 2 23 4 0

02 2

y y

y y

c g y m m

( )

( )

( )

v

D

from some reference value that fixes theconstant D0, and m3 and m4 are the chemi-cal potentials of 3He and 4He, respectively.For fixed c, T, and D = m3 – m4 + D0 thestatic equilibrium follows from seeking theminimum of

(6-86)

where r = r0 + 2g Dcn, u = u – 2cn g2, andc = cn [D – g |y |2] was eliminated from theequation. For u > 0 there is a second-ordertransition from normal fluid to superfluidat r = 0, whereas for u < 0 the transition isof first order. The critical point occurs foru = 0.

Whereas for the binary mixture a singleLangevin equation for the concentrationfield c (x, t) and y (x, t) had to be derived(Eq. (6-15)), here we need two equations.In fact, the entropy density S (x, t) mustalso be added (Hohenberg and Nelson,1979), and thus the calculation becomescomplicated. However, since y is not a con-served quantity, it relaxes much more rap-idly, hence the assumption is made that it al-ways adjusts to the local equilibrium corre-sponding to the local concentration c (x, t).Instead of the simple exponential functionin Eq. (6-23), additional oscillating termsdue to the “second-sound” mode are found:

S (k, t) = a1 e2R (k) t + a2 e[R (k) – D2 k2] t (6-87)

¥ cos (u2 k t) + a3 e–2D2 k2 t cos (2u2 k t) + a0

where a0, a1, a2, and a3 are constants, u2 isthe second-sound velocity and D2 its damp-ing coefficient. In the spinodal region, R (k)is positive, and not too close to the tricriti-cal point for small k, R (k) – D2 k2 shouldalso be positive. Therefore in addition tothe first term on the right-hand side of Eq.(6-87), which is the analog of Eq. (6-23),there is another exponentially growingterm oscillating in time. This would imply

F r u= n12

14

16

12

2 4 6 2˜ ˜| | | | | |y y y c+ + +v D

a “flickering” component in the scattering.However, the linearized theory of spinodaldecomposition is not expected to hold for3He– 4He mixtures, since neither of theinteraction ranges r and l0 in Eq. (6-85) areexpected to be large. Note, however, thatthe Ginzburg (1960) criterion for tricriticalsystems shows that mean-field theory is es-sentially correct for tricritical points; hencenonlinear effects do not become worsewhen T Æ Tt , and unlike in Eqs. (6-51) and(6-52), we expect no factor depending onthe temperature distance from Tt under theconditions of validity of the linear theory.

For tricritical systems, both experimentsand simulations (Sahni and Gunton, 1980;Sahni et al., 1982; Ohta et al., 1988; Saguiet al., 1994; Wang et al., 1998 a; Gorents-veig et al., 1997; Nielaba et al., 1999) haveshown a nonlinear behavior for all timesaccessible to study, and a scaling behaviorof the structure factor as in Eq. (6-67) againapplies to the late stages.

6.4.2 Spontaneous Growth of OrderedDomains out of Initially DisorderedConfigurations

There are many examples where the dy-namics of a phase change involve the spon-taneous formation of ordered structures.Suppose an alloy such as b-CuZn, which isa prototype realization of Ising-model typeordering (Als-Nielsen, 1976), is quenchedfrom T0 > Tc to T < Tc (Fig. 6-3a). Thenagain the initial homogeneously disorderedstate is thermodynamically unstable, andwe expect that ordered domains will form.Since no sign of the order parameter is pre-ferred, and there is no symmetry-breakingfield, the average order parameter y– in thesystem, y– = (1/V ) ∫ dx y (x, t), will remainy– = 0: domains with both signs of the orderparameter form equally often. Again, the(unfavorable) domain wall energy acts as a

driving force leading to coarsening of thedomain structure, similar to the coarseningdiscussed for spinodal decomposition (Sec.6.2.9).

The description of the initial stages ofthis domain growth process closely par-allels the treatment presented in Eqs. (6-4)to (6-26). Let us assume a free energy func-tional similar to Eq. (6-85), with pheno-menological constants r0, u, v, and r:

and we emphasize that the order parametery is not a conserved quantity. Thus insteadof Eqs. (6-10) and (6-11) we write:

(6-89)

describing a simple relaxational approachtowards equilibrium, with G being an ap-propriate rate constant. Again, fluctuationsmust be added to this equation as in Eqs.(6-14) to (6-17), so that the final result is

where the random force hT (x, t) now satis-fies the relationship

·hT (x, t)hT (x¢, t¢)Ò

= ·h2TÒT d(x – x¢) d(t – t¢)

·h2TÒT = 2 kB T G (6-91)

Again the first step of the analysis is a lin-earization approximation, similar to Eqs.(6-18) to (6-26). For the structure factor,which we now define as follows,

∂∂

∂∂

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

− ∇ +

y yy

y h

( , ) [ ( , )]

( , ) ( , )

x x

x x

tt

f t

r k T t t

T

T

= (6-9 )cg

B

G 0

2 2

∂∂

−y m

m y y

( , ) ˜ ( , )

˜ ( , ) ( ( , ))/ ( , )

xx

x x x

tt

t

t t t

=

=

G

dD d

D

k Tr u

r

B= d (6-88)∫ ⎧⎨⎩

+

+ + ∇ ⎫⎬⎭

x

x

12

14

16

12

02 4

6 2 2

y y

y y[ ( )]v


6.4 Extensions 463

S (k, t) = ·dy–k(t) dyk(t)Ò

dyk(t) ∫ ∫ dx [exp(i k · x)] y (x, t) (6-92)

the result is analogous to Eq. (6-23) (DeFontaine and Cook, 1971):

S (k, t) = ST0(k) exp[2R (k) t] (6-93)

with (6-94)

Note that k in Eqs. (6-92) to (6-94) de-scribes the distance in reciprocal spacefrom the superstructure Bragg spot QB cor-responding to the considered ordering.Suppose now fcg has the form assumedin Eq. (6-88), with u > 0, v > 0, andr0 = r¢ (T – T0) changes sign at T0 (this is thestandard Landau description of a second-order transition; see Stanley (1971) orBinder (2000)). Then

(∂2 fcg /∂y2)y– = r0 + 12uy– 2 + 30vy– 4

is equal to r0 < 0 for y– = 0, i.e., the initialconfiguration is always unstable, and allmodes yk (t) grow in the interval from0 < k < kc, with kc still being given by Eq.(6-25). However, Eq. (6-26) does not holdhere and the maximum growth rate occursfor k = 0, as is obvious from Eq. (6-94).

In a case where u < 0, however, the phasetransition does not occur for T = T0 wherer0 changes sign, but rather at a highertemperature Tc = T0 + 3u2/32r¢v (Binder,1987a). For T0 < T < Tc the state with y– = 0is metastable, and the ordering reactionagain needs a nucleation process (Chan,1977; Fredrickson and Binder, 1989). Thetemperature T0 hence again plays the roleof a spinodal point (“ordering spinodal”;De Fontaine, 1979). Such spinodals forthermally driven first-order transitionshave been calculated for diverse systems,from alloys such as Cu–Au systems (De

Rf

r k T kT

( ),

k ≡ −∂∂

⎛

⎝⎜⎞

⎠⎟+

⎡

⎣⎢⎢

⎤

⎦⎥⎥

G2

20

2 2cg

=

Byy

Fontaine and Kikuchi, 1978) to the meso-phase separation transition of block copol-ymers (Fredrickson and Binder, 1989). It isimportant to realize, however, that the “or-dering spinodal” is also a mean-field con-cept, suffering from the same “ill-defined-ness” as the unmixing spinodal. Takinginto account both statistical fluctuationsand nonlinear effects, as they are both con-tained in Eqs. (6-90) and (6-91), we againexpect to find a smooth transition betweenthe nucleation of order and the “spinodalordering” mechanism (Cook et al., 1969),as described by Eqs. (6-93) and (6-94). Inaddition, nonlinear effects will limit thepredicted exponential growth, as they do inthe case of spinodal decomposition.

This similarity of the behavior also car-ries over to the late stages of growth, wherea scaling similar to Eq. (6-67) applies: butnow km(t) denotes the half-width of thepeak (unlike spinodal decomposition, themaximum growth always occurs at theBragg position QB describing the order,i.e., k ∫ 0). Since the order parameter y inEq. (6-90) is not conserved, a faster growthlaw than Eq. (6-68) is predicted (Lifshitz,1962; Allen and Cahn, 1979b; Ohta et al.,1982; Bray, 1994):

km–1(t) ~ t1/2 (6-95)

In a binary alloy undergoing an order–dis-order transition, or in a layer adsorbed at asurface at constant coverage, the conserva-tion of concentration (or density, respec-tively) may change the growth law (Eq. (6-95)): Sadiq and Binder (1984) suggestedthat the excess concentration (or excessdensity, respectively) contained in certaintypes of domain walls could again lead to amechanism of the Lifshitz–Slyozov type,i.e., Eq. (6-68). The validity of this sugges-tion is not yet confirmed (Binder and Heer-mann, 1985; Binder et al., 1987). Figs.6-34 and 6-35 give some examples of the

behavior found in computer simulations ofa square lattice model of a two-dimen-sional alloy, where there is a repulsiveinteraction between AB pairs on both near-est- and next-nearest-neighbor sites. Forc– = 0.5 the ordering is the two-component(2 ¥1) structure: there are four kinds of do-mains, where in the ground state full rowsof A atoms alternate with full rows of Batoms, and these rows may be oriented ineither the x-direction or y-direction for asquare lattice. Fig. 6-34 shows the growthof these ordered domains in the simulationof a quenching experiment for this model,and Fig. 6-35 the resulting growth of thesuperstructure Bragg intensity. The scalingbehavior of Eq. (6-67) is well verified.Similar results have also been found forsimulations of various other models (seeBinder et al., (1987), Furukawa (1985a),Bray (1994), for review.

Experiments on the kinetics of the for-mation of ordered superstructures havebeen carried out both for two-dimensionalmonolayers adsorbed at surfaces (Wangand Lu, 1983; Wu et al., 1983; 1989; Trin-gides et al., 1986, 1987; Henzler andBusch, 1990), three-dimensional metallicalloys (e.g., Hashimoto et al., 1978; Nishi-hara et al., 1982; Noda et al., 1984; Takedaet al., 1987; Katano and Iizumi, 1988; Ko-nishi and Noda, 1988), dielectric materialssuch as K2Ba(NO2)4 (Noda, 1988) orK2ZnCl4 (Mashiyama, 1988), martensiticmaterials such as KD3(SeO3)2 (Yagi and Lu,


Figure 6-34. “Snapshot pictures” of the computer simulation of the ordering process of a 120 ¥ 120 square lat-tice model of an alloy with repulsive interactions between A–B pairs on nearest- and next-nearest-neighborsites, enn = ennn. A quench from a disordered configuration at infinite temperature to kBT/enn = 1.33 is performedfor c– = 0.5 (a second-order transition to the (2 ¥1) structure occurs in equilibrium at kBTc/enn ≈ 2.1). Time evo-lution occurs via random nearest-neighbor A–B exchanges (i.e., the model of Kawasaki (1972)). Times shownrefer to 10 MCS after the quench (upper part), 200 MCS (middle) and 1700 MCS (bottom). Only B atoms areshown (using four different symbols to indicate whether a B atom belongs to a domain of type 1, 2, 3, or 4, seeFig. 6-35a), A atoms are not shown. From Sadiq and Binder (1984).

6.4 Extensions 465

1988), graphite intercalation compoundssuch as transition metal chlorides in graph-ite (Matsuura, 1988), molecular crystalssuch as CN1–xClx-adamantane (Descampsand Caucheteux, 1988), etc. Of course,there are also cases where phase separationand ordering occur simultaneously, a clas-sic example being Fe–Al alloys whereFe3Al domains precipitate (Allen, 1977;Oki et al., 1973; Eguchi et al., 1984).Again, no attempt at completeness is madehere, we simply quote a number of exam-ples to emphasize the universality of thephenomenon.

As expected from the above discussion,we must distinguish the cases where the“parent phase” from which the orderedstructure forms is metastable or unstable;in the latter case, the structure forms by“spinodal ordering” (De Fontaine, 1979)whereas in the former case it forms by nu-cleation and growth, although again thisdistinction is not very sharp. An exampleof the latter case is Mg3In, where the orderis of Cu3Au type but the transition is verystrongly of first order (Konishi and Noda,1988). The volume fraction of the orderedphase is then found to obey the well-knownAvrami (1939), Johnson–Mehl (1939)equation

F (t) = 1 – exp [– (t /t)n] (6-96)

where n is a constant characteristic of thetype of nucleation process and t a timeconstant that depends on the degree of

Figure 6-35. (a) Four types of domains (1, 2, 3, and 4) in the (2 ¥1) structure of a binary alloy AB on the squarelattice (B atoms are indicated by black circles and A atoms by white circles). (b) Structure factor S (q, t) ofsuperlattice Bragg scattering, at times t after the quench. Since the lattice spacing is set to unity, the Bragg po-sitions of the (2 ¥1) structure are (±p, 0) and (0, ± p). Only the variation with qx near qx = p is shown. Times aremeasured in units of Monte Carlo steps per site. Data are for a computer simulation of a quench from T = • tokBT/enn = 1.33 as in Fig. 6-34. (c) Scaling plot of the data shown in (b), with the structure factor S (q, t) beingrescaled with the maximum intensity S (p, t) and the relative distance q/p –1 from the Bragg position being re-scaled with the half-widths s (t) of the peaks in (b). From Sadiq and Binder (1984).

undercooling of the transition. Here we aremore interested in the “spinodal ordering”as occurs in Cu3Pd (Takeda et al., 1987),Ni3Mn (Katano and Iizumi, 1988), Cu3Au(Hashimoto et al., 1978; Nishihara et al.,1982; Noda et al., 1984), and K2Ba(NO2)4

(Noda, 1988).Fig. 6-36 shows examples of growing

Bragg peaks, i.e. the experimental counter-parts to simulation data such as shownin Fig. 6-35. Whereas for K2Ba(NO2)4 we

find a law km–1(t) ~ t1/4 (Noda, 1988), for

Ni3Mn a crossover from km–1(t) ~ t1/4 at

short times to km–1(t) ~ t1/2 at later times is

found (Katano and Iizumi, 1988), and forAuCu3 a law km

–1(t) ~ t1/2 (i.e., Eq. (6-95))is established (Noda et al., 1984). Whereasthe t1/2 law is expected from various theo-ries (Lifshitz, 1962; Allen and Cahn,1979b; Ohta et al., 1982), a t1/4 law occur-ring over a transient period of time can per-haps be attributed to the “softness” of thedomain walls between the growing orderedregions (Mouritsen, 1986; Mouritsen andPraestgaard, 1988). An interesting obser-vation of Lifshitz–Slyozov type coarsen-ing of the structure factor is also reportedfor colloid crystallization (Schätzel andAckerson, 1993).

In the two dimensional case, Wu et al.(1989) measured a growth exponent x =0.28 ± 0.05 for O on W (110). Since thep (2 ¥1) phase is believed to have an eight-fold ground-state degeneracy in this sys-tem, whereas the theories mentioned aboverefer to a two-fold degenerate orderingonly, the interpretation of this result is notobvious. We note, however, that reasonablescaling of the structure factor is observed(Fig. 6-37). For the system silver on Ge(111), a growth exponent x =1/2 is found(Henzler and Busch, 1990) and also goodscaling of the structure factor is seen. Inthis system, on the other hand, a crossoverto slower growth occurs at later times; thereason for this behavior is not clear – per-haps it is due to pinning of domain walls atdefects (see Sec. 6.4.4). At this point, wenote that a similar slowing down of thegrowth where km

–1(t) basically stops grow-ing further, has also been seen in Ni3Mn(Katano and Iizumi, 1988) and in the phaseseparation of mixtures of flexible andsemi-rigid polymers (Hasegawa et al.,1988). In neither of these systems is thevery slow growth at late stages understood.


Figure 6-36. (a) Time evolution of the 211 super-lattice peak of Ni3Mn annealed at 470 °C for timesup to 34 h. From Katano and Iizumi (1988). (b) Timeevolution of the (–1

2, x , –5

2) superlattice peak of

K2Ba(NO2)4 annealed at 190 K. From Noda (1988).

6.4 Extensions 467

In domain growth studies of O on W (112)Zuo et al. (1989) also found scaling behav-ior at early times but slowing of domaingrowth attributed to random field effects atlater times.

6.4.3 Phase Separation in ReducedGeometry and Near Surfaces

Various cases of reduced geometry maybe of interest; for example, spinodal de-composition of fluid mixtures confined topores in porous media like Vycor glass, orspinodal decomposition or ordering of thinfilms or adsorbed monolayers at surfaces,and phase separation in small particles orgrains in inhomogeneous materials. Allthese problems have seen much recent ac-tivity, and lack of space allows us to giveonly a very brief introduction.

In a porous medium (or gel), a binarymixture at critical composition experiencesa random chemical potential difference (DeGennes, 1984). Using concepts from therandom field Ising model (Villain, 1985;Huse, 1987) we expect both a change in thephase diagram and slow relaxation (Gold-burg, 1988; Goldburg et al., 1995; Falicovand Berker, 1995). However, there is alsothe view that we must rather focus on thewetting behavior on the walls inside astraight cylindrical pore (Liu et al., 1990;Monette et al., 1992; Tanaka, 1993; Zhang

Figure 6-37. (a) Dynamic scaling in the growth ofBragg peaks for W (110) p (2 ¥1) – O at coverageF = 0.5 and T = 297 K. From Wu et al. (1989). (b)Scaled LEED intensities of the ÷-

3 peak of Ag ad-sorbed on Ge (111) at T = 111 °C. From Henzler andBusch (1990).

and Chakrabarti, 1994, 1995). Of course,ultimately we should combine this single-pore behavior with effects due to the ran-dom structure of the pore network.

From Sec. 6.2 it should be clear that thebehavior in bulk two and bulk three dimen-sions is qualitatively similar, although non-linear phenomena are usually more impor-tant for d = 2 than for d = 3. In particular,for polymer mixtures different coils caninterpenetrate each other for d = 3 but notfor d = 2; therefore a mean-field critical re-gime does not exist for two-dimensionalpolymer mixtures.

For thin films that are many atomic di-ameters thick, there is an interesting inter-play between finite size effects (Binder etal., 1987) and surface effects. Usually al-ready in equilibrium at the surface of amixture, the concentration of one compo-nent of a mixture will be enhanced in com-parison with the bulk concentration (“sur-face enrichment” occurs because the inter-actions between a wall and the two compo-nents A and B differ from each other). Atthe coexistence curve, this preference ofthe surface for, say, the B component maylead to the formation of a wetting layer atthe surface (Dietrich, 1988). Inside thetwo-phase coexistence region, surfacesmay hence have a profound effect on thekinetics of phase separation: surface-di-rected concentration waves (with wavevec-tor normal to the surface plane) dominatethe initial growth in the region near the sur-face (first observed by Jones et al. (1991)for a polymer mixture). In later stages ofthe coarsening, we have an interestingcompetition between the domain growth inthe bulk and the possible propagation of awetting layer from the surface into the bulk(see Krausch, 1995; Puri and Frisch, 1997;and Binder, 1998, for recent reviews).

6.4.4 Effects of Quenched Impurities;Vacancies; and Electrical Resistivityof Metallic Alloys Undergoing PhaseChanges

In this subsection, we draw attention to avariety of topics that cannot be discussedhere in any depth, owing to lack of space.

(i) Quenched ImpuritiesThroughout this chapter, only ideal

systems have been considered where phaseseparation is triggered by spontaneousfluctuations. However, real materials al-ways contain impurities, whereas in fluidsthese impurities are mobile (“annealeddefects”) and hence act like a dilute addi-tional component of a multi-componentmixture, in solids such impurities are oftenimmobile (“quenched defects”) at the tem-peratures of interest. This frozen-in disor-der may have two effects: in a metastableregion, free energy barriers for nucleationare reduced and hence heterogeneous nu-cleation is facilitated (Zettlemoyer, 1969).In the late stages of spinodal decomposi-tion or domain growth, there is an impor-tant interaction between such defects anddomain walls. Whereas so far it has beenassumed that domain walls may diffusefreely owing to their capillary wave excita-tions, the randomly spaced defects act likea random potential in which the interfacemoves. Since thermal activation is nowrequired to overcome the barriers of thispotential, at low temperatures the domainwall motion is dramatically slowed owingto such impurities. Simulations of thisproblem have been carried out by Grest andSrolovitz (1985) and Srolovitz and Grest(1985). More work has been devoted to therelated problem of the dynamics of the ran-dom field Ising model (see Villain, 1985;and Nattermann, 1998, for reviews). In allthese systems, we expect that the growthlaw of the ideal system (Eqs. (6-68) or


6.4 Extensions 469

(6-95)) will hold until some characteristiclength lc is reached, which depends on theconcentration ci of the quenched impurities(lc ~ ci

–1/d in d dimensions). At times t sig-nificantly exceeding the time given by thecondition km

–1(t) lc ≈1, a logarithmic growthlaw (Villain, 1985) is expected:

km–1(t) ~ ln t (6-97)

A similar crossover in the relaxation from afast to a slow growth law may also becaused by interface pinning at extended de-fects (e.g., dislocations, grain boundaries)rather than point defects.

(ii) VacanciesIn solid mixtures, vacancies V are cru-

cial for a microscopic description of inter-diffusion, which occurs via a vacancymechanism rather than by direct A–B ex-change (Flynn, 1972; Manning, 1986). Mi-croscopically we therefore need concentra-tions ci

A(t), ciB(t), and ci

V(t) for A atoms, Batoms, and vacancies as dynamic variables,respectively, rather than the single concen-tration variable ci or concentration fieldc (x, t) (Eq. (6-1)). Since ci

V(t) 1, it maybe possible to reduce the problem to Eq.(6-15), where the mobility M then needs tobe related to the jump rates GA and GB ofA and B atoms to vacant sites (and the va-cancy concentration). However, owing tocorrelation effects in the vacancy motion,this is a difficult problem even in the non-interacting case (Kehr et al., 1989). ThereEqs. (6-6) and (6-14) would yield an inter-diffusion coefficient Dint = M/[c (1– c)],i.e., M can be found if Dint can be simplyrelated to GA and GB and the average va-cancy concentration c–V. Although M obvi-ously is proportional to c–V for small c–V,computer simulations (Kehr et al., 1989)show that the simple relationships pro-posed in the literature to relate M to GA andGB are inaccurate.

A simulation of the early stages of spino-dal decomposition in such an ABV model(Yaldram and Binder, 1991) shows that thegeneral features of the structure factorS (k, t) are almost the same as those of thedirect-exchange AB model (Fig. 6-5), andthe two models can be approximatelymapped onto each other by adjusting thetime scales. In real systems, however, thebehavior may be more complicated; the va-cancy concentration does not need to re-main constant, and it may be that many va-cancies are created during the quenchwhich later are annealed out by migrationto surfaces or recombination with intersti-tials. If this happens, the effective mobilityM would itself depend on the time t afterthe quench. Also the vacancy concentra-tions in equilibrium may be different in A-rich and B-rich domains, or may becomepreferentially enriched in domain boundar-ies. This did not occur in the simulation ofYaldram and Binder (1991), since there thestatic properties were assumed with perfectsymmetry between A and B and there wasno energy parameter associated with thevacancies, but this model is still a grossover-simplification of reality. Fratzl andPenrose (1994, 1997) found that vacanciesmay speed up the coarsening by changingthe mechanism. Note that for very low va-cancy content but very late stages a fastergrowth law (x =1/2) due to cluster–clusteraggregation has been suggested (Mukher-jee and Cooper, 1998). These problemsmay confuse the proper interpretation ofreal experiments.

(iii) Electrical Resistivity of MetallicAlloys Undergoing Phase Changes

In metallic alloys the quasi-free elec-trons responsible for electrical conductionare scattered from the atomic disorder. Inthe framework of the Born approximation,treating the scattering as elastic, the excess

resistivity Dr due to concentration fluctua-tions can be represented as a convolutionof the structure factor ·dy–k(t) dyk(t)Ò andthe Fourier transform of the effective po-tential that the quasi-free electrons feel,namely (Binder and Stauffer, 1976b),

(6-98)

where e is the charge of an electron, meff itseffective mass, N is the number of atomsper cm3, ZA and ZB are the atomic numbersof the two constituents, ä is Planck’s con-stant, and k describes the screening of theCoulomb interaction. In Eq. (6-98) it is as-sumed that the Fermi sphere (whose radiusis kF) lies completely within the first Bril-louin zone, and that there is no sublatticeordering.

Using the structure factor from the com-puter simulations of Marro et al. (1975),Binder and Stauffer (1976b) showed thatduring spinodal decomposition a resistivitymaximum occurs. This can be understoodsince in Eq. (6-98) the main contributioncomes from the large k (k of order k or oforder kF). As the coarsening proceeds,however, the structure factor is large onlyat much smaller values of k. Scaling con-siderations (Eq. (6-67)) suggest that

Dr (t) – Dr (∞) ~ km(t) ~ t –1/3

This treatment can again be generalized forthe kinetics of ordering alloys. We refer toBinder and Stauffer (1976a) for a discus-sion of this problem, and of previous theo-ries and related early experiments.

Just as an alternative description for con-centration inhomogeneities in terms of“concentration waves” and their mean-square amplitude S (k, t) in terms of the

Dp

d d d

ra

k

=

d

B A eff

eff F

F

Ne Z Z m

n kk

k

c t c t ck k

2 2 2

2 3 4 3

0 22 2 2

2

4

1

( )( )

( / )

[ ( ) ( ) ( )]

−

×+

× ⟨ ⟩ −< <

−

∫

k

kk k

time-dependent cluster distribution func-tion (see Sec. 6.2.5), the problem of theelectrical resistivity can be rephrased interms of electron scattering from clusters(Hillel et al., 1975; Edwards and Hillel,1977). Experimental resistivity data forAl–Zn alloys can be interpreted qualita-tively along such lines (Luiggi et al., 1980).

At this point, we also mention that elec-trical currents have been found to affect thephase separation behavior of Al–Si alloys(Onodera and Hirano, 1986, 1988) andother alloys. At present, the explanation ofthese phenomena is unclear.

6.4.5 Further Related Phenomena

As a first point of this subsection, wedraw attention to peculiar morphologies ofstructures that may form via phase separa-tion in fluid mixtures with very strongdynamic asymmetry, e.g., solution of poly-mers which are frozen into a glass-likestructure at high density. The resulting vis-coelastic phase separation leads to the for-mation of sponge-like patterns (Tanaka,1994, 1996; Taniguchi and Onuki, 1996) orrigid foams (Hikmet et al., 1988).

Another recent topic of interest is phaseseparation induced by temperature gra-dients (Kumaki et al., 1996) and velocitygradients, i.e. shear (Onuki et al., 1997).The effect of shear flow on the unmixing offluids is in fact three-fold, as recently re-viewed by Onuki (1997): the phase dia-gram is modified (shear-induced mixingdue to a depression of the critical tempera-ture may occur), the rheology of the un-mixed two-phase states is profoundly al-tered, and the morphology of the patternsthat form is changed.

Finally we mention the interplay ofphase separation and chemical reactions(Huberman, 1976; Glotzer et al., 1995; Le-fever et al., 1995). Chemical reactions may


6.5 Discussion 471

terminate the coarsening process and thusfreeze in an inhomogeneous pattern.

6.5 Discussion

The understanding of the dynamics ofphase changes of materials has advancedconsiderably in recent years; the theoreti-cal results now present a fairly clear pictureof at least some basic questions, and alsoquantitatively reliable experiments areavailable for many systems. This chapterhas emphasized the theoretical aspects; itsmain conclusions can be summarized asfollows:

(i) The linearized theory of spinodal de-composition holds only for mean-field typesystems (with long ranges of interactions)or in systems that are equivalent to them,such as polymer mixtures.

(ii) Whereas a spinodal curve is well de-fined in the mean-field limit or for polymermixtures in the limit of chain length NÆ∞,in all other cases the spinodal is an ill-de-fined concept. This implies that the transi-tion from nucleation to spinodal decompo-sition is gradual. For polymer mixtures, thewidth of this transition region may be nar-row, as estimations from Ginzburg criteriashow. Experiments are still needed tocheck the latter point, whereas the gradualtransition from nucleation to spinodal de-composition in metallic alloys over a broadtemperature region has been establishedexperimentally.

(iii) Nonlinear spinodal decompositionduring the early stages can be successfullydescribed qualitatively by the Langer–Baron–Miller theory (for solid mixtures)and the Kawasaki–Ohta theory (for fluidmixtures). Recent experiments, however,show that these theories are not quantita-tively accurate. More work is required to

understand the crossover from these earlystages to the late stage of phase separationwhere the structure factor develops to-wards a scaling limit. It is not clear underwhich conditions power-law behaviorkm(t) ~ t –x can be observed at intermediatestages, and what the appropriate interpreta-tion of the exponent x is.

(iv) In the late stages the structure factorobeys the scaling behavior first suggestedby Binder and Stauffer (1974, 1976b),

S (k, t) ~ [km(t)]–d Sk/km(t)

While the scaling function S () proposedby Furukawa (1984) seems to account wellfor the general shape of experimental dataon solid or polymer mixtures, several de-tails are not reproduced correctly. Good ev-idence for Porod’s law, S () ~ –(d+1) for 1, has been provided. The width of S ()increases when the volume fraction f ofprecipitate phase decreases from that of thecritical mixture (Fratzl and Lebowitz,1989). There we find, as a function of tem-perature and f, both morphologies, inter-connected domains and well separateddroplets. Such a transition can also be ob-served as a function of time in computersimulations (Hayward et al., 1987; Lironiset al., 1989) and experiments on polymermixtures (Hasegawa et al., 1988) (see Fig.6-38).

(v) There is now agreement that in solidmixtures the behavior at late times is givenby the Lifshitz–Slyozov law, km

–1(t) ~ t1/3,even in the percolative regime (Amar et al.,1988). This has recently also been verifiedin a solid– liquid system that satisfies allassumptions of the LSW theory (Alkemperet al., 1999). For critical quenches of fluidmixtures at late times, Siggia’s (1979) re-sult km

–1(t) ~ t holds, but further research isneeded to achieve an understanding of theexponents for the growth of ordered struc-tures in cases with a higher degeneracy of

the ordered phase, and to understand howthese growth laws are affected by defects.Other questions concern the suggestion(Langer et al., 1975; Binder, 1977) that byexpressing km and t as suitably scaledquantities qm and t, e.g., qm(t) = km(t) xand t = D0 t/x2, where x is the correlationlength at phase coexistence and D0 theinterdiffusion constant, a material-inde-pendent universal function should be ob-tained (which must depend, however, onthe volume fraction of the phases coexist-ing in the final equilibrium state). A coun-ter-example for this universality seems tobe the “N branching” found for polymermixtures (Onuki, 1986b; Hashimoto,1988), and more work is needed to under-stand this problem. Clearly such a univer-sality, if it exists, would be useful as it al-lows the phase separation behavior of othermaterials to be predicted if properties suchas x and D0 are known. Of course, we mustdistinguish between different “universalityclasses” (Binder, 1977) for solids and fluidmixtures (cf. Figs. 6-28 and 6-39).

(vi) Apart from crude “cluster dynam-ics” models, there is no theoretical ap-


Figure 6-38. Polarizing optical microscope imagesobtained from the same area of a cast film of a binarymixture of poly(ethylene terephthalate) and a copoly-ester composed of 60 mol% p-oxybenzoate and40 mol% ethylene terephthalate at 50% relative con-centration during isothermal heat treatment at270 °C. Times at which the images are taken are indi-cated (in seconds). From Hasegawa et al. (1988).

Figure 6-39. Comparison of qm vs.t behavior found in metallic alloys(a)–(h) and in organic glasses (i), (j).Here the scaling is done differently tothat in the text, namely qm = km(t)/km(0)and t = – 2 t R [km(0)]. Systems used areAu–60 at.% Pt at 550 °C (a) (Singhal etal., 1978), Al–6.8 at.% Zn at T = 108 °C(b), 116 °C (c), 129 °C (d) (Laslaz et al.,1977), Al–5.3 at.% Zn at T = 20 °C (e),Al–6.8 at.% Zn at 20 °C (f), 90 °C (g),110 °C (h) (Hennion et al., 1982),76B2O3–19PbO–5Al2O3 at 450 °C (i)(Zarzycki and Naudin, 1969), and13.2Na2O–SiO2 at 560 °C (j) (Tomo-zawa et al., 1970). From Synder andMeakin (1983).

6.6 Acknowledgements 473

proach to describe the behavior in the tran-sition region from spinodal decompositionto nucleation. This problem, and theoriesconnecting consistently the early-time spi-nodal decomposition to the scaling behav-ior at late stages, still needs more research.So far much progress has been due to large-scale computer simulation, e.g., Amar et al.(1988) (see Figs. 6-40 and 6-41); it is still achallenge to explain quantitatively simula-tion data such as shown in these figuresor experimental data as shown in Fig. 6-32by analytical theories.

(vii) Finally, we draw attention to thefact that the concepts developed in thepresent context can also be extended tovery different physical phenomena. Forexample, the Lifshitz–Slyozov mechanismcan be identified to describe processes suchas the healing of rough surfaces at low tem-perature (Villain, 1986; Selke, 1987).


This work has profited considerablyfrom a longstanding and stimulating inter-action of K. B. with D. Stauffer and D. W.Heermann. This author thanks them andalso other co-workers, namely C. Billotet,H.-O. Carmesin, H. L. Frisch, J. D. Gun-ton, S. Hayward, J. Jäckle, M. H. Kalos, K.Kaski, K. W. Kehr, J. L. Lebowitz, G. Liro-nis, A. Milchev, P. Mirold, S. Reulein, A.Sariban, and K. Yaldram, for fruitful col-laboration.

The other author (P. F.) is particularly in-debted to O. Penrose and J. L. Lebowitz fora longstanding and fruitful collaboration.Furthermore he is indebted to H. Gupta, C.A. Laberge, F. Langmayr, P. Nielaba, O.Paris, G. Vogl, and R. Weinkamer for fruit-ful interactions.

Figure 6-40. “Snapshot pictures” of the two-dimen-sional nearest-neighbor Ising model of a phase-sep-arating mixture evolving after a critical quench toT = 0.5 Tc at times t = 5000 MCS (a), t = 105 MCS (b)and t = 9.8 ¥ 105 MCS (c). B atoms are shownin black, A atoms are not shown. Data are for a512 ¥512 lattice. From Amar et al. (1988).

Figure 6-41. Log–log plot of the characteristic lin-ear dimensions RG(t) and RE(t) at time t after thequench, for the simulation shown in Fig. 6-40. RG(t)is extracted from the correlation function and RE(t)from the energy relaxation. The full curve is a fit toEq. (6-84a). From Amar et al. (1988).

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7 Transformations Involving Interfacial Diffusion

Gary R. Purdy

Department of Materials Science and Engineering, McMaster University, Hamilton,Ontario, Canada

Yves J. M. Bréchet

Laboratoire de Thermodynamique et Physico-Chimie Métallurgiques, E.N.S.E.E.G.,Institut National Polytechnique de Grenoble, France

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 4827.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4847.2 Equilibrium Properties of Solid–Solid Interfaces . . . . . . . . . . . . 4847.2.1 Structures of Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . 4847.2.2 Structures of Interphase Boundaries . . . . . . . . . . . . . . . . . . . . 4867.2.3 Segregation to an Interface . . . . . . . . . . . . . . . . . . . . . . . . . 4897.3 Interfacial Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4907.4 Forces for Interface Migration . . . . . . . . . . . . . . . . . . . . . . 4927.4.1 Capillary Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4927.4.2 Chemical Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4937.4.3 Mechanical Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4937.4.3.1 Elastic Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4937.4.3.2 Plastic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4947.4.4 Frictional Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4967.4.4.1 Solute Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4967.4.4.2 Particle Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4987.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4987.5.1 The Massive Transformation . . . . . . . . . . . . . . . . . . . . . . . . 4987.5.2 Chemically-Induced Grain Boundary Migration . . . . . . . . . . . . . . 5037.5.3 Discontinuous Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . 5047.5.3.1 Initiation of Discontinuous Precipitation . . . . . . . . . . . . . . . . . . 5057.5.3.2 Theories of Steady Cooperative Growth . . . . . . . . . . . . . . . . . . 5077.5.3.3 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . 5107.5.4 Interface Migration in Multilayers . . . . . . . . . . . . . . . . . . . . . 5147.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5167.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516



A interfacial areab magnitude of the Burgers vectorC0 average concentrationCa concentrationDC local difference from the average concentration CD diffusion constantDV /v ratio of the volume diffusion constant and the interfacial velocityD– b average grain boundary diffusion coefficientdE/dx measure of force between solute atoms and boundaryDFch molar Helmholtz energy change accompanying complete transformationDF ¢ch molar Helmholtz energy difference across an interfaceDF 0

ch fraction of the total available Helmholtz energy dissipated in volume diffusionf (C ) Helmholtz energy density functionalf0 measure of curvature of Helmholtz energy–composition relationshipgB shear modulusH 3ka /(3ka + 4gB)JB flux of solute atomsK = dA/dV interface curvatureL half-thickness of a foil specimenLa

0, Lb0 regular solution parameters for the crystal and boundary phases

l correlation lengthM mobilityNV number of atoms per unit volumena

A mole numberpch interfacial driving force/chemical forceph residual chemical force derived from coherency strainpi solute drag forcepW virtual mechanical forcer1, r2 radii of curvatureri radius of facetS spacing between precipitate lamellaes equilibrium ratio of solute concentration in a grain boundary and that in adja-

cent crystalT /Tm reduced temperatureTm melting temperaturev boundary velocityVf volume fraction of a precipitateVm molar volumeVm

b molar volume of the boundary phaseW applied load per unit areax, y, z coordinatesY elastic constantZ applied tensile stress

482 7 Transformations Involving Interfacial Diffusion


a crystalline phasea, b included, external phasega activity coefficientGB surface excess concentration of solute B per unit aread boundary thicknesse misfitea interaction parameter; a measure of the non-ideality of the a solid solutionh coefficient of lattice parameter change with compositionq angle of tiltk gradient energy coefficientka bulk modulus of precipitatel spacing of parallel edge dislocationsmA, mB solvent, solute chemical potentialsW alignment parameters specific interfacial Helmholtz energysA, sB specific grain boundary Helmholtz energy in pure A, Bsab specific interfacial Helmholtz energy of a/b interfacej phase field parametery order parameter

b.c.c. body-centered cubicCIGM chemically induced grain boundary migrationDP discontinuous precipitationDC discontinuous coarseningf.c.c. face-centered cubich.c.p. hexagonal close-packedSTEM scanning transmission electron microscopy

7.1 Introduction

Many microstructure-determining reac-tions in solids are similar in the sense thatthey occur primarily at an interface, whichtraverses a volume of solid material, andleaves that volume altered structurally orchemically, or both. Indeed, most solid–solid transformations fall into this set; inthis chapter, we will restrict discussion inthe following ways:

1) We will consider only those processeswhose main characteristics are determinedby diffusion within the transformationfront, which may be a grain boundary or aninterphase boundary.

2) We will further restrict attention toprocesses that begin mainly at crystalboundaries, and that lead to heterogene-ously transformed regions. This restrictionimplies that the transformation interfacewill be of relatively high energy, and there-fore that the transformation will often beinitiated with some difficulty.

The subset defined above excludes thosehomogeneous transformations which areinitiated everywhere within a parent crystal(Chap. 5; Wagner et al., 2001), and trulydiffusionless transformations (Chap. 9; De-laey, 2001). However, the massive trans-formation survives, as do the processes ofperitectoid precipitation, chemically in-duced grain boundary migration, and dis-continuous precipitation. The latter two(and similar diffusional processes) are the subject of an excellent monograph byPawlowski and Zieba (1991), which is, un-fortunately, not widely available. It has re-cently been observed that the study of theseand similar phenomena may be advancedthrough the use of artificially preparedmultilayered structures (Klinger et al.,1997b, c, 1998), which may be subject, forexample, to discontinuous homogenization

or to interfacial precipitation. Before dis-cussing these processes, a brief review willbe given of the equilibrium properties oftransformation interfaces and of the dy-namics of interfacial diffusion and inter-face migration.

7.2 Equilibrium Propertiesof Solid–Solid Interfaces

7.2.1 Structures of Grain Boundaries

Grain boundaries constitute a well stud-ied class of solid state defects which arestructural in nature, and which depend fortheir existence on a misorientation betweenotherwise identical crystals. If we restrictattention to macroscopic variables only, the properties of a planar boundary are ex-pected to depend on five geometric vari-ables, three of which define the relative or-ientations of the two crystals, and a furthertwo are required to specify the orientationof the boundary plane. Three additional de-grees of freedom, associated with sublat-tice translations, exist on the microscopicscale appropriate to the modeling of grainboundary structure.

Much of what is known or inferred aboutgrain boundary structure is obtained fromsimulations using molecular statics or mo-lecular dynamics, supported in some casesby high resolution microscopic techniques(Smith, 1986).

The three main modeling techniques thathave been used for surfaces, grain bound-aries and (to a lesser extent) interphaseboundaries, are energy relaxation methods,molecular dynamics, and Monte Carlo sim-ulations (Sutton and Balluffi, 1995). The information expected from these simula-tions concerns both the structural aspects(molecular relaxation, molecular dynam-ics), chemical aspects such as segregation


7.2 Equilibrium Properties of Solid – Solid Interfaces 485

(Monte Carlo), and dynamic aspects suchas diffusion (molecular dynamics). Thestructural information stemming fromthese methods is very much dependent onthe interatomic potential chosen, and espe-cially on the anharmonic part, since thestructure of the interface is usually ratherdistorted. The dynamic aspects of inter-faces include the interface diffusivity andthe interface mobility. While the diffusivityis accessible to these simulations (via mo-lecular dynamics), the mobility is more of aproblem, because the system size is typi-cally constrained by available computa-tional resources. This imposes, for exam-ple, very large curvatures as driving forces,and results in high interface velocitieswhich might not be realistic reflections ofexperimental situations (Sutton, 1995).

The chemical aspects accessible toMonte Carlo simulation also possess twofacets: the thermodynamic equilibrium tobe attained and the kinetics required toreach it. Provided that the interatomic po-tential is reliable, the Monte Carlo methodgives an accurate description of chemicalsegregation (Treglia and Legrand, 1998).However, the simulation of the kineticsneeded to reach this state encounters a newproblem of coupling the bulk diffusion tothe diffusion close to the interface. MonteCarlo simulations with time residence al-gorithms (Martin et al., 1998) have allowedthe inclusion of the role of vacancies inbulk diffusion, but the treatment of vacan-cies in solids with free surfaces or internalinterfaces remains a source of problems(Delage, 1998).

The whole field of surface and interfacethermodynamics and kinetics of metallicalloys with tendencies to ordering or un-mixing has profited from Monte Carlo simulations. The next step (to introduce simultaneous structural relaxations andchemical aspects, e.g., via off-lattice

Monte Carlo methods) still requires furtherdevelopment.

A surprisingly large amount of our infor-mation relates to a rather special kind ofinterface, the symmetric tilt grain bound-ary. As suggested by Fig. 7-1, low angleboundaries are made up of arrays of dislo-cations, and the low angle symmetric tiltboundary is composed of parallel edge dis-locations, whose spacing l is given by

(7-1)

where b is the magnitude of the Burgersvector, and q the angle of tilt. Studies byKrakow et al. (1986) suggest that this de-scription in terms of individual disloca-tions can be extrapolated, with some mod-ification, to high angles, e.g., to 26.5°.More generally, the grain boundary regionis expected to possess the following char-acteristics:

a) The high angle grain boundary is thin,perhaps two or three atomic diameters inwidth. To a first approximation, it may beconsidered a high energy “phase”, con-strained to have constant volume.

b) The high angle grain boundary is peri-odic in structure, like its low angle counter-part. The geometric repeating units are not

l q= b/ sin ( / )2 2

Figure 7-1. A symmetric low angle tilt grain bound-ary. The tilt angle q and the edge dislocation spacingl are related by Eq. (7-1).

lattice dislocations, but are capable of de-scription in terms of nearly perfect com-pact polyhedra (Ashby et al., 1979). Depar-tures from symmetric orientations are oftenaccommodated by mixing two or moretypes of structural units within the bound-ary. It is the view of Sutton and Balluffi(1987) that no general criterion for lowgrain boundary energy can be developed onthe basis of simple geometry.

c) Metastable states, and even degener-ate states of different structure but identicalenergy, may be formed by the relative mi-croscopic translation of one of the crystals(Vitek, 1984).

The computed structures are of greatconceptual value, yielding information thatis inaccessible or accessible only with dif-ficulty through experiment; they should betreated with a measure of caution however,as the computed results depend to some ex-tent on the choice of potential. Further,many (but not all) of these simulations areperformed for a temperature of 0 K. Never-theless, in combination with experiment,they form a valuable tool in the study ofthis complex field.

7.2.2 Structures of Interphase Boundaries

These internal surfaces, unlike grainboundaries, are necessary elements of anequilibrium multiphase system; they arenot defects. They separate phases which ingeneral may differ both chemically andstructurally from one another.

The description of an interface usingcontinuum concepts dates from the seminalpapers by Cahn and Hilliard (1958) and Al-len and Cahn (1977). The Cahn–Hilliardmodel of a stationary coherent interphaseboundary is illustrated in Fig. 7-2. TheCahn–Hilliard and Allen–Cahn treatmentsdeal with two broad cases involving con-served and non-conserved order parame-

ters. The first applies to chemically hetero-geneous solids, and the second to structu-rally heterogeneous materials. The first al-lows us to derive a diffusion equation for theevolution of concentrations, and the other anequation for the evolution of order parame-ters. These two classes of problems, coupledwith intermediate situations, cover the rangeof problems involving relaxed interfaces.

The overall method is to write a Helm-holtz energy functional involving not onlythe dependence of the local composition Cor order parameter y, labeled f (C, y), butalso those gradient terms compatible withthe symmetry of the problem. This leads toterms in (—C )2 or (—y)2. The interfacialenergy between the two regions of differentprescribed order parameter or compositionis then computed in two steps: the concen-tration or order parameter which minimizes


Figure 7-2. Top: A schematic concentration (C )profile through a diffuse coherent interface (compo-nents A and B) in which excess chemical Helmholtzenergy, represented by the quantity Dm, is every-where balanced against gradient Helmholtz energy(F: Helmholtz energy, m e

A and m eB: equilibrium chem-

ical potentials). Bottom: Finite thickness d of equi-librium interface, corresponding to concentrationprofile.


the excess Helmholtz energy is computed,and this excess energy is then taken as theinterfacial energy. In a single model wetherefore get both a “continuum structure”for the interface, and the specific interfa-cial Helmholtz energy.

The time-dependent version of theseequations allows us to treat the motion ofthe interface under a prescribed drivingforce, provided that a relation is assumedbetween the fluxes and the driving forces.

For a conserved order parameter C (suchas concentration), the kinetic equationtakes the form:

(7-2)

where k is a gradient energy coefficient,and for a non-conserved order parameter(such as a long-range order parameter, or acrystal orientation), we obtain:

(7-3)

The interfacial Helmholtz energy for a dif-fuse interface between two concentrationsC1 and C2 is given by

(7-4)

Similarly, assuming a quadratic expan-sion of the Helmholtz energy with two min-ima for order parameters y = 1 and y = –1gives the following expression for the inter-facial Helmholtz energy of an antiphaseboundary:

(7-5)

In each case, the width of the interface scalesas k /f 1/2: the stronger the gradient term com-pared with the local chemical Helmholtz energy contribution, the more diffuse will be the interface. The “structural width” of

s k=83

2N fv ′′

s k= d21

22N f C C

C

C

v ∫ ( )

∂∂

∇ − ′y k y yt

M f= [ ( )]2 2

∂∂

∇ − ∇Ct

M f C C= [ ( ) ]2 2 4k

the interfaces is thus related to the couplingterm weighting the square gradients.

These continuum descriptions are inprinciple well suited for application to dif-fuse interfaces. Their facility of implemen-tation in computer simulations (since theydo not involve front tracing methods) hastriggered much interest under the name of“phase field approaches” (Carter et al.,1997). They have recently been applied tovarious problems involving sharp inter-faces using the following systematics: theinterface is artificially made diffuse withrespect to a “phase field parameter” j. (Forinstance, this parameter can be taken equalto 1 in a solid and to 0 in a liquid if we wishto address a solidification problem involv-ing both solute and heat flow (Wheeler etal., 1992). It can be taken to vary continu-ously from +1 to –1 through a grain bound-ary, mimicking the orientation changeacross the boundary for application to aproblem in grain growth (Chen et al.,1998).) A Helmholtz energy equation isproposed which involves both the concen-tration and the phase field parameter andtheir spatial derivatives. The dynamic equa-tions for the concentrations and the phasefields are solved using continuum methodsand the coupling term containing the phasefield gradient is set equal to zero so that thesharp interface can be recovered.

These methods have been applied to solidification, phase transformations, graingrowth, and chemically induced grainboundary migration. They possess the ad-vantage of relative ease of implementationusing classical numerical techniques, and agreat flexibility in terms of the introductionof such features as anisotropy. They sharethe drawback of all Ginzburg–Landau typephenomenological equations, i.e., that theprecise atomistic interpretation of both themobility and the coupling terms is not al-ways clear.

When a structural difference coexistswith the chemical difference, the situationbecomes more complex. However, if thedifference between phases is small, as isthe case for structurally similar phases withsimilar orientations and lattice parameters,the differences may be accommodated byan elastic component or by an array of mis-fit dislocations. In the case of g precipitatesin b brass, the interfacial energy has beentaken as the sum of structural (dislocation)and chemical energies. This approach per-mits the prediction of weak anisotropies,for example those in Fig. 7-3, after Ste-phens and Purdy (1975).

More complex systems, e.g., interfacesjoining f.c.c. and b.c.c. phases, are gener-ally discussed with respect to a particular

orientation relationship or set of orienta-tion relations, considered to be establishedat nucleation. In these cases, we search fora reason for the strong anisotropy of inter-facial Helmholtz energy commonly ob-served in such systems. The most plausibleapproach involves the search for optimummatching of atomic positions at the inter-face, that is a maximum in coherency, cor-responding to a minimum in the structuralcomponent of the energy. A particularlysimple example is found in the f.c.c./h.c.p.system Al–Ag, where the (111) f.c.c. habitplane corresponds to the structurally simi-lar (001) h.c.p.

Figure 7-4 illustrates a model of thef.c.c./b.c.c. interface due to Rigsbee andAaronson (1979), in which a macroscopi-cally irrational habit plane interface ismade up of rational areas with a high de-gree of coherency, separated by “structuralledges” which serve to displace the inter-face plane by a few atomic dimensions,thereby to increase the degree of cohe-rency. Aaronson and his co-workers em-phasize that these structural ledges are con-sidered intrinsic to the structure of theinterface, and do not provide a mechanismfor interface motion. Other studies haveutilized a more formal geometrical ap-proach (Bollman, 1970; Dahmen, 1981;Zhang and Purdy, 1993a, b) to seek inter-faces of optimal matching, with similarconclusions. Based on evidence from highresolution electron microscopy (e.g.,Zhang et al., 1998), it has become apparentthat the typical interface between phases ofdissimilar structure is often faceted on amicroscopic scale, and that invariantplanes or invariant lines of the transforma-tion, when they exist, are clearly importantin the selection of precipitate habits.


Figure 7-3. The morphologies of equilibrated g pre-cipitates (solid curves) formed in b brass, and thecorresponding Wulff plots (dashed), for the tempera-tures indicated. The anisotropies are consistent witha model that treats the interfacial Helmholtz energyas a sum of structural (dislocation) and chemicalterms (Stephens and Purdy, 1975).


7.2.3 Segregation to an Interface

In system of two or more components, itis often found that the equilibrium compo-sition at an internal surface is differentfrom that of the surrounding phase(s). Asbefore, it is necessary to distinguish be-tween grain and interphase boundaries; forthe case of a grain boundary in a two-com-ponent solid solution, we may consider thatthe high energy defect has its own thermo-dynamic solution properties. A constrainedequilibrium is then possible between theboundary “phase”, which must have an ap-proximately fixed volume, and the adjacentcrystalline phase (a). The constant volumecondition requires, for a virtual change inmole number na

A,

dnaA = – dnb

A ; dnbA = – dnb

B ;

dnbB = – dna

B (7-6)

and the constrained equilibrium is definedby a line tangential to the a Helmholtz-en-ergy curve at the bulk composition C0, anda parallel line tangential to the boundaryHelmholtz-energy curve (Hillert, 1975a).This is illustrated in Fig. 7-5. We then ob-tain for the distribution coefficient

(7-7)sC

C= B

b

Ba

(7-8)

where sA and sB are the specific grainboundary Helmholtz energies in pure Aand B, V b

m the molar volume of the boun-dary “phase”, and La and Lb the regular so-lution parameters for the crystal and boun-dary phases, respectively.

A more comprehensive treatment ofphase equilibria of grain boundary struc-tures is given by Cahn (1982). The pos-sibility of faceting transitions is high-lighted in this work and supported experi-mentally by the studies of Ferrence andBalluffi (1988) and Hsieh and Balluffi(1989).

ln( )

sV

RTL L

RT= B A m

b bs sd

− + −a

Figure 7-4. Rigsbee andAaronson’s model for thef.c.c./b.c.c. interface, inwhich structural ledges areintroduced which maximizethe degree of coherencewithin the interface. Re-printed with permissionfrom Rigsbee and Aaronson(1979), copyright PergamonPress.

Figure 7-5. Graphical interpretation of the con-strained equilibrium between a crystalline solid solu-tion, a, and a grain boundary phase, b. (For detailssee text.)

For interphase boundaries, segregationis also expected, as first shown by Gibbs(1906). His expression for the relationshipbetween the rate of change of interfacialHelmholtz energy with bulk compositionand the interface excess solute concentra-tion, GB, the Gibbs absorption isotherm,may be written as

ds = – GB dmB (7-9)

where GB is the surface excess concentra-tion of solute B per unit area of interface.(The solvent surface excess is zero by defi-nition.)

A growing body of experimental evidencedemonstrates that both grain boundariesand interphase boundaries are generallysites for complex interactions among chem-ical species. Guttman (1995) has reviewedthe elementary mechanisms of intergranu-lar segregation in multi-component sys-tems, and concludes that site competition,as well as attractive and repulsive chemicalinteractions, are all important in determin-ing the degree of interfacial segregation.

7.3 Interfacial Diffusion

Solid–solid interfaces often provide ex-ceptionally rapid paths for diffusion. Pro-vided that there is a significant structuraldifference, so that there is a disordered re-gion linking nearly perfect crystalline re-gions, this is easy to understand in broadterms. The mechanisms and the details ofgrain and interphase boundary diffusionare not fully understood at present, al-though there exists a wealth of experimen-tal information on the former. The reviewsby Peterson (1983) and Kaur and Gust(1988) for grain boundary diffusion, andthat of LeClaire (1986) for the relatedproblem of diffusion along dislocations arerecommended.

Grain boundary diffusion is commonlyinvestigated experimentally in one of twoways:

a) “Random” polycrystalline specimensare exposed to a source of diffusant (e.g., a planar plated or implanted layer), an-nealed, then serially sectioned. If the dif-fusant is a tracer, the technique can be used to give a sensitive evaluation of theproduct D

– bd, where D– b is the average

grain boundary diffusion coefficient, and d the boundary thickness, assumed to represent a region of uniformly enhanceddiffusivity. If a solute is diffused into thepolycrystal, the usual experimentally ac-cessible quantity is the product s D

– b d(Levine and MacCallum, 1960). It is onlyin exceptional cases, where the volume dif-fusion contribution to some part of the pen-etration profile is negligible (type “C” be-havior) that we can obtain in principle amore direct evaluation of the boundary dif-fusivity.

b) Bicrystals of controlled misorienta-tion are sectioned such that the grain boun-dary plane is perpendicular to a source ofdiffusant, which may be of constant com-position, or finite. In most cases, as above,both volume diffusion and boundary diffu-sion will contribute to the penetration pro-file, as schematized in Fig. 7-6. In thismore controlled case, the mathematical solutions of Whipple (1954) or Suzuoka(1964) (depending on whether the source isfixed in composition or finite) are used tounfold the grain boundary transport coeffi-cient from serial sectioning data. The dif-ference here lies in the opportunity to cor-relate misorientation (and, by inference,structure) with diffusion rates. Again, alarge amount of precise data has been ac-quired, mainly for the special case of thesymmetric tilt boundary. This was re-viewed by Peterson (1983).


7.3 Interfacial Diffusion 491

From the consideration of grain bound-ary diffusion data for metals, we can drawa few general conclusions:

a) The activation enthalpy for grainboundary self-diffusion is of the order ofhalf that for volume diffusion. Indeed, agood empirical correlation exists betweengrain boundary diffusivity and reducedtemperature T /Tm for different metal struc-tures. For f.c.c. metals, Gust et al. (1985)obtained (7-10)d D

– b = 9.7 ¥ 10–15 exp (– 9.07 Tm /T ) m3/s

b) For the low angle symmetric tiltboundary, rates of diffusion are aniso-tropic, with faster diffusion occurring par-allel to the dislocation lines. Undissociateddislocations have higher diffusivities thandissociated ones; the diffusivity increaseslinearly with dislocation density (and there-fore with tilt angle q) for angles up toabout 10°.

c) Anisotropic behavior is also obtainedfor higher angle symmetric tilt boundaries.

d) Low angle twist boundaries, com-posed of screw dislocations, are less effi-cient in transporting material than their tiltcounterparts.

e) The mechanism of grain boundary dif-fusion in close-packed metallic systems is thought to involve vacancy exchange.This is a tentative conclusion based on the experimental evidence of Martin et al.(1967), who studied the pressure depen-dence of grain boundary diffusion and de-duced activation volumes from their data,and of Robinson and Peterson (1973), whostudied isotope effects in silver polycrys-tals and bicrystals, and demonstrated thattheir results were consistent with a vacancymechanism. In their computer simulationstudies, Balluffi et al. (1981) and Faridiand Crocker (1980) have provided supportfor this idea by showing that the formationand migration energies for vacancies andinterstitials in the grain boundary favored avacancy exchange process.

Having summarized briefly the presentlevel of understanding of grain boundarydiffusion, it is perhaps worth emphasizingthat the correlation of structure with diffu-sivity remains incomplete, and that much re-mains to be learned about the fundamentalprocesses of interfacial diffusion in solids.

Interphase boundary diffusion is lesswell studied, to the extent that very few reliable data are available in the litera-ture. Kaur and Gust (1988) cite only onesystem for which extensive measurementshave been reported, the Sn–Ge/In system(Straumal et al., 1981). One reason for thelack of direct information about this impor-tant aspect of interfacial diffusion may liein the fact that, unlike grain boundaries, in-terphase boundaries can support local equi-librium concentration gradients only in thepresence of gradients of curvature, temper-ature or stress. The isothermal planar inter-phase boundary is normally isoconcentrate.

Mullins (1957) has considered a numberof important cases of interfacial diffusionin response to variations in curvature. For

Figure 7-6. Schematic composition (C) distributionin the neighborhood of a grain boundary, normal to asurface of fixed composition, C0. d : boundary thick-ness.

the case of substitutional diffusion, andunder the simplifying assumptions of con-stancy of composition of the two phases aand b and site conservation at the interface,we may write for the interfacial flux:

(7-11)

Here, L represents the mobility of atomsalong the interface, Vm is a molar volume,the m’s are chemical potentials within theinterface, sab is the specific interfacialHelmholtz energy and Kab is the interfacialcurvature.

7.4 Forces for Interface Migration

It is useful to define and classify a set ofgeneralized “forces” for interface migra-tion. In most cases, the driving and retard-ing forces are expressed in terms of a nor-mal force per unit area of interface, andtherefore have the units of pressure.

It is conventional to separate the variousforces according to their origin (e.g., surfaceHelmholtz energy, elastic energy, chemicalHelmholtz energy) and to some extent thedivision is arbitrary. Its utility lies in theability to visualize the interplay of differentforces on a moving transformation interface.

It is also true that the definition of the “force” for boundary migration is only apart of the solution to the migration prob-lem. The response function (or mobility) ofthe interface must also be known, and thisfunction is expected to differ greatly fromone interface type to the next.

7.4.1 Capillary Forces

These forces originate from the system’stendency to reduce its total interfacial

J J L

L V

c cK

ab ab ab ab ab

ab ab

a bab

A B A B

m

= =

=

− − ∇ −

−−

∇

( )m ms

Helmholtz energy s A, where s is the spe-cific interfacial Helmholtz energy and Athe interfacial area. For constant s , theforce can be expressed as

ps = K s (7-12)

which is the product of the surface Helm-holtz energy and the interface curvatureK = dA/dV.

For smoothly curved interfaces the curva-ture is the sum of the reciprocals of the twoprincipal radii of curvature [(1/r1) + (1/r2)].For strongly anisotropic or faceted inter-faces, an effective capillary force may stillbe defined, through the derivative dA/dV, al-though care must be taken in its formulation.

For the special case of an equilibriumparticle, the capillary force gives rise to a variation of the specific Helmholtzenergy of the included phase: the resultingrelation between the zero-curvature equi-libria (C0) and the equilibrium composi-tions in the presence of curvature is sum-marized in the Gibbs–Thompson relation-ship. For two components we obtain thelinearized expressions (Purdy, 1971)

(7-13a)

(7-13b)

where a is the included phase and b the external phase, Vm is the molar volume ofthe a phase, V ′m the molar volume of the a phase extrapolated to the composition of

the b phase and , where

gb is an activity coefficient. This last termis a measure of the departure of the parentphase from ideal solution behavior.

An equivalent graphical constructiondue to Hillert (1975a) is shown in Fig. 7-7.

eg

bb

b= 1 +

∂∂

⎛

⎝⎜⎞

⎠⎟ln

lnC

D

D

aa a

a bb

bb b

a bb

CC C V

C C

KRT

CC C V

C C

KRT

=

=

m

m

0 0

0 0

0 0

0 0

1

1

( )

( )

( )

( )

−−

⎛

⎝⎜⎞

⎠⎟

− ′−

⎛

⎝⎜⎞

⎠⎟

se

se


7.4 Forces for Interface Migration 493

The construction is of great conceptualvalue, and permits the straightforward ex-tension to the case of three componentswith different partial molar volumes. Mor-ral and Purdy (1995) have also developedcompact expressions for the general multi-component case. For example, the solubil-ity change of the included a phase (a column vector |DC j

b|) with curvature canbe written in terms of second derivatives

of the Gibbs energy, , of the par-

ent b phase, the molar volume of the aphase, Vm, and the equilibrium concentra-tion differences between the a and bphases, the row vector |DCj

ab|:

(7-14)

For faceted equilibrium shapes, the cur-vature may be replaced by a constant termsi /ri , where the subscript indicates facet i,and ri is the minimum radial distance to theith facet.

D Dab bCG

CC K Vj

j

kj

∂∂

2

2 = ms

∂∂

2

2

G

Cj

k

7.4.2 Chemical Forces

The chemical force for interface migra-tion should be distinguished from thechemical driving force for the transforma-tion DFch, defined by Eq. (5-9) in the chap-ter by Wagner et al. (2001). In this sectionthe case where the chemical force acts di-rectly across the transformation interface isconsidered. Its magnitude is given by

(7-15)

where DF ¢ch is the molar Helmholtz energydifference acting directly across the inter-face, and is generally less than DFch, theHelmholtz energy difference accompany-ing the complete transformation. The forceis derived through a virtual work argument,in which the energy change in displacingthe interface through unit normal distanceis equated to a work term.

The distinction between total and localchemical driving forces is best seen in thecomposition-invariant transformation il-lustrated in Fig. 7-8. Here, the Helmholtzenergy difference for interface migration isgiven by DF ¢, and some fraction DF 0 ofthe total available Helmholtz energy is dis-sipated in the volume diffusion field. In theoften assumed limit of local equilibrium,DF ¢ is implicitly set equal to zero, and allof the Helmholtz energy difference for thetransformation is used in the volume diffu-sion field.

7.4.3 Mechanical Forces

These forces fall into two categories:those that may simply be treated as part ofthe chemical Helmholtz energy of thestrained phase(s) and those that cannot.

7.4.3.1 Elastic Strain Energy

As noted by Wagner et al. (2001, Chap. 5of this volume), elastic strain energy is an

pF

Vchch

m=

D ′Figure 7-7. Effect of interfacial curvature, K, on thetwo-phase equilibrium in a binary system; the com-mon tangent line, which generates equilibrium com-positions at its points of tangency for the planarphase interface, is replaced by two nonparallel tan-gent lines, whose intercepts differ by the amountsshown.

additive term in the Helmholtz energy of aphase. Thus, for homophase variations, thelocal elastic energy density is given byh2DC 2 Y as indicated in Eq. (5-12).

This term generally reduces the totaldriving force for a phase transformation. InEq. (5-12), DC is the local difference fromthe average concentration C0. The equationtherefore has application to any case wherea gradient of misfitting solute exists in anotherwise perfect crystal, and in particularto the case where a thin layer of alteredcomposition is in coherent contact with aninfinite bulk phase of composition C0.

The strain energy density in a coherentsolute profile will be given as an intensivevariable by Eq. (5-12), which therefore hasapplication to the diffusional growth of aprecipitate in a binary solid solution, andalso to the problem of the stability of thinepitaxial deposited layers.

For the more complex case of the strainenergy associated with the formation of asecond phase inclusion, Eq. (5-14) holds,independently of inclusion shape, when thetransformation strain is a pure dilation andthe elastic constants of the two phases aresimilar. For spherical coherent transformedregions, the combined effects of surfaceenergy and (dilatational) transformationstrain may be incorporated in the Gibbs–Thomson equation by adding 6 H e2gB tothe numerator of the term in braces in Eqs.(7-13). Here H = 3ka/(3ka + 4 gB) whereka is the bulk modulus of the precipitate,gB is the shear modulus of the matrix, and eis the misfit (Rottman et al., 1988).

7.4.3.2 Plastic Response

It is important to distinguish betweenmechanically derived forces which origi-nate from elastic strain energy, and thosewhich result in the plastic relaxation of theloaded specimen. In the latter case, theelastic energy may remain approximatelyconstant during the course of the transfor-mation, but the interfaces can experiencevirtual forces, in the sense that the Helm-holtz energy of the loading system is re-duced by interface motion.

As an example, consider the symmetrictilt boundary of Fig. 7-9, which is capableof motion in response to the applied load.Studies of this type of boundary haveshown that the dislocation boundary under-goes normal migration, that is, synchro-nous glide, under an appropriate load. Wecan consider that there is a virtual mechan-


Figure 7-8. Helmholtz energy and interfacial com-position relationships for a composition-invariantphase transformation. The available Helmholtz en-ergy for the transformation is used in part to drive theinterface, DF ¢, and the balance, DF 0, is dissipated inthe volume-diffusion field.


ical force on the interface equal to pw =Wtanq where W is the applied load per unitarea and q is the angle of tilt. In this exam-ple, the applied stress can be thought of as acting directly on the interface. The ex-perimental response of the symmetric tiltboundary is linear, and a simple statementof the mobility of the boundary summa-rizes the kinetics of boundary motion.

If a coherent twin boundary were to re-place the symmetric tilt boundary in Fig. 7-9, the transformation strain would againbe a shear. It is clear, however, that the syn-chronous glide process would not occur.The propagation of the twin will requirethe lateral motion of twinning dislocations,and the interface response function will befundamentally different. This simple ex-ample illustrates the general principle thata knowledge of the force for boundary mi-gration is only part of the empirical de-scription of boundary motion; evidently theresponse characteristics of the boundarymust also be known.

Another class of strain energy stored inbulk material is that associated with struc-tural defects such as dislocations. Thesedefects may be generated and stored duringplastic deformation, and their effect on theHelmholtz energy of the solid provides thebasis for the driving force for recrystalliza-tion. Dislocations may also occur due toplastic relaxation of misfit stresses (as seenin some cases of discontinuous precipita-tion or in some transformations in steels).

The excess Helmholtz energy is tradi-tionally taken to be proportional to the totaldislocation density and is added as a bulkdriving force in the overall thermodynamicbalance. This is probably acceptable formisfit-generated dislocations but, for thecase of dislocations stored during plasticdeformation, this approximation neglectsthe formation of dislocation substructuressuch as subgrains. This relaxation phenom-enon decreases the stored Helmholtz en-ergy and renders its increase with disloca-tion density less than linear due to self-screening effects (Verdier et al., 1997).Again the question of the contribution ofthis stored energy to the overall kinetics ofmicrostructure development is two-fold:we may consider the nucleation step froman initially homogeneous situation, or thegrowth regime for which the stored energyis different on either side of the movinginterface. A simple approximation for thegrowth process would be the addition of anextra driving force when the stored ener-gies differ on either side of the interface(e.g. due to different stages of deformationin two neighboring grains). However, thisapproach is unlikely to be sufficient to ex-plain the nucleation of recrystallization,which is known to occur at grain boundar-ies, and at localized shear or transitionbands (Humphreys and Hatherly, 1995). Asproposed initially by Bailey and Hirsch(1962), nucleation seems to be a random

Figure 7-9. A tilt wall (tilt angle q ) is subject to avirtual force, p, derived from the work done on thespecimen by the loading system.

event associated with local fluctuations insubgrain configuration; it is unlikely to besuccessfully modeled either by a standardnucleation theory or by an averaging ap-proach for the stored energies. Informationon local misorientations (leading to differ-ences in local mobilities and local drivingforces) may be crucial, as has been sug-gested using vertex-based numerical simu-lations (Weygand, 1998).

7.4.4 Frictional Forces

7.4.4.1 Solute Drag

Each type of mobile interface will pos-sess its own characteristic response func-tion. For the simple tilt wall subject to amechanical force, we have noted that theforce–velocity relationship is linear over arange of forces. If a considerable recon-struction of the crystal structure is requiredfor the motion of an interface, as in the mo-tion of a high-angle boundary in a single-component system, we expect that cross-boundary diffusive motion will play a sig-nificant role. Turnbull’s (1951) expressionfor the intrinsic mobility, M, of a grainboundary

(7-16)

is based on a model in which atom jumpsacross the boundary are independent of oneanother. This relation has not been con-firmed by experiment, and this is taken tosuggest that more complex, cooperativemovements of atoms are generally in-volved in interface migration. Again refer-ring to experiment, there exists a consider-able database drawn from measurements of grain boundary motion in high puritymetals. Thus the intrinsic response of high-angle grain boundaries is rather welldocumented, even if the mechanisms of

MD V

b RT=

bmd

2

boundary migration are not fully under-stood.

When two or more components are in-volved, this picture must be extended.Superimposed on the intrinsic response ofthe interface, and frequently masking it, wefind the effects of solutes which are prefe-rentially attracted to or repelled from theinterface. The experiments of Aust andRutter (1959) first demonstrated conclu-sively the powerful retarding effects oftrace amounts of solute on grain boundarymigration.

The idea of a “solute drag” perhaps orig-inated with Cottrell (1953), who noted thatsolute atoms would be attracted to disloca-tions, and would therefore be required todiffuse along with the dislocations, or to beleft behind by dislocations that had brokenfree of their solute atmospheres. Similarideas were put forward by Lücke and hisco-workers (1957) for application to thecase where a solute is preferentially segre-gated to grain boundaries. The most com-plete and informative theoretical treat-ments of this effect are those of Cahn(1962) and Hillert and Sundman (1976).The latter is based on a treatment put for-ward earlier by Hillert (1969). The basicreason for the solute drag force is simple: ifsolute is attracted to the boundary, it willtend to diffuse along with the movingboundary. Depending on the relative ratesof boundary motion and solute diffusion,the solute distribution may become asym-metric, with more solute trailing theboundary than leading it. Because of themutual attraction between the boundaryand the solute, the asymmetric solute dis-tribution leads to a retarding force of theform

pi = NV Ú (C – C0) (dE/dx) dx (7-17)

where NV is the number of atoms per unitvolume, and dE/dx is a measure of the



force between the solute atoms and theboundary. The effects of small amounts ofsolute can be profound. The profiles in Fig.7-10 are due to Cahn (1962). They arebased on a triangular interaction potentialbetween solute and boundary, and corre-spond (a) to a slowly moving boundary, (b)to a boundary experiencing maximum so-lute drag, and (c) to a fast-moving bound-ary. That is, the drag force goes through amaximum as the velocity is increased. If aninitial intrinsic force–velocity relationshipis assumed linear, and the solute drag termis added, the total response function isquickly rendered nonlinear. Under moresevere conditions, the system can even be-come unstable, displaying a region of ve-

locities where is negative.

Hillert’s formulation uses a differentinteraction potential, initially square (rather than triangular), and later a square-topped potential with ramps at the edges.The drag is evaluated as a rate of dissipa-tion of Helmholtz energy due to diffusionin and in front of the moving interface. Themodel lends itself to numerical analysis; it

∂∂pv

is not restricted to dilute solutions, nor is itrestricted to grain boundaries. Figure 7-11uses the simple square well to illustrate theeffect of cross-boundary diffusion in set-ting the degree of asymmetry of the soluteprofile. The drag term may be written inseveral ways. Among the most convenient,Hillert and Sundman (1976) find:

(7-18)

In their approach, the ramps (zones 1 and 3in Fig. 7-12), as well as the centre of theboundary (zone 2) and the parent phase re-gion (zone 4), contribute to the drag. In-deed, it is possible to separate out the con-

pRTV

c cDc c

xi = dm

v∫

−−

( )( )

02

1

Figure 7-10. Computed solute profile in the vicinityof a moving grain boundary: a) at equilibrium, b) at avelocity corresponding to maximum solute drag, andc) at higher velocity. After Cahn (1962).

Figure 7-11. A square-well model for the movinggrain boundary. After Hillert (1969).

tribution of any slice taken parallel to theinterface, and Sundman and Hillert havepresented their results in this way, as com-ponents of the total solute drag. They find,as we might expect, that the drag is reducedfor given velocity by increasing the cross-boundary diffusion coefficient, or by re-ducing the depth of the potential well seenby the solute. The model is capable of gen-erating a variety of informative results. It is not restricted to grain boundary motion,but also finds application to steady statephase transformations, as will be seen later.

It is interesting to note that solute drageffects have been (in some sense) betterstudied and quantified than intrinsic bound-ary friction forces. The solute effects arethought to be additive, and to dominate andeven to mask the intrinsic properties ofgrain boundaries for many cases of practi-cal interest.

7.4.4.2 Particle Pinning

Whereas the contribution of solute dragas a retarding force is by definition of a vis-cous nature, another term may prevent the

interface from moving: a pinning force.The idea of pinning a grain boundary withforeign particles was put forward initiallyby Zener (Smith, 1948). The grain bound-ary will be able to move only if the gradientof bulk stored energy across the interface is sufficient to move the interface to an untrapped configuration. Zener assumed apurely geometrical distribution of particlesalong the grain boundary; this led to a fric-tion force to be overcome that scales asVf /r, where Vf is the volume fraction ofparticles and r their radius. A more refinedapproach due to Hazzledine and Oldershaw(1990) allows us to treat the collective pin-ning problem in a consistent manner (i.e.,to treat both the flexibility of the interfaceand the number of particles along the inter-face). Hazzledine’s result, recently con-firmed by computer simulations (Weygand,1998), indicates that a scaling law in Vf

1/2/rmight be more appropriate.

The consequence of these pinning forceson the overall kinetics of interface migra-tion can be two-fold: on the one hand, theparticles will impose a threshold drivingforce for boundary migration; and on theother hand, they will impose a constant re-tarding force to be subtracted from theavailable Helmholtz energy. The standardmethod is to equate the retarding force withthe threshold force, but this identificationis by no means obvious. Numerical experi-ments on grain growth kinetics do howeverindicate that this simple approach gives agood description of the overall kinetic be-havior (Weygand, 1998).

7.5 Examples

7.5.1 The Massive Transformation

Massive transformations are consideredto include all inhomogeneous, noncoher-ent, thermally activated, composition-in-


Figure 7-12. The Hillert–Sundman (1976) model ofa moving boundary, showing the interaction profile,and some computed solute profiles for vd/2D=0.001(1), 1.61 (2), and 10 (3).

7.5 Examples 499

variant, solid–solid transformations (Mas-salski, 1958). Massive solidification reac-tions are excluded by this definition, buttheir literature is relevant, in part becauseof advances in understanding of the rapidsolidification of silicon-based alloys (Azizand Kaplan, 1988).

The massive transformation generally in-volves a major structural change; it normal-ly begins with the heterogeneous nuclea-tion of a thermodynamically stable or meta-stable daughter phase, at imperfections orgrain boundaries in a supercooled parent, andproceeds by the thermally activated migra-tion of a mobile transformation interface.

Plichta et al. (1984) reviewed the avail-able information on massive transforma-tion nucleation, and concluded that the nucleation event is structurally identicalwith that expected for diffusional transfor-mations among dissimilar phases; grainboundary and triple junction sites are ener-getically preferred, and an orientation rela-tionship between the nucleus and at leastone parent grain is generally set on nuclea-tion. Since interfacial torques are likely tobe present during the nucleation event, theconstruction of Hoffman and Cahn (1972)may be expected to provide guidance in themodeling of the critical nucleus. Experi-ence has shown, however, that the searchfor plausible nucleus shapes in such heter-ogeneous systems seldom leads to unam-biguous results.

The chemical Helmholtz energy change,DFch, for nucleation in a single-componentsystem that undergoes an allotropic trans-formation is obtained directly from theundercooling below the equilibrium tem-perature, as suggested by Fig. 7-13. Forsmall undercoolings, DT, this may be ex-pressed as

DFch = DH DT /T0 (7-19)

with DH as the change of enthalpy.

For two components, the composition-invariant condition need not apply to mas-sive nucleation (although it must holdoverall for massive growth). In the exam-ple in Fig. 7-14, drawn for a temperaturebelow T0, the temperature for which the aand b phases of the bulk composition C0

have the same molar Helmholtz energy, itis clear that the formation of a nucleus ofcomposition C0 would be accompanied bya volume Helmholtz energy changeDF1/Vm. However, the volume Helmholtzenergy change is maximized (DF2/Vm) ifthe nucleus takes composition C ¢. Even inthe single-phase supersaturated region, asimilar argument shows that the volumeHelmholtz energy change will be max-imized for a composition other than C0.

Turning to the process of massive prod-uct growth, if we again consider the allo-tropic transformation of pure element, it isclear that the rate of transformation is de-termined by the undercooling (which setsthe interfacial driving force pch), and by themigration characteristics of the transforma-

Figure 7-13. Helmholtz energy relationships for theallotropic transformation of a pure element.

tion interface. As in the case of the high an-gle grain boundary, the details of the cross-boundary atomic transfer process are notwell understood. As in the grain boundarycase, it has proved difficult to separate theeffects of trace impurities from the intrinsicmigration properties of the noncoherentinterphase boundary. Hillert (1975b) hasestimated the mobility of grain boundariesin pure iron from a series of limiting argu-ments as

M = 0.035 exp (–17 700/T ) m4/(J s) (7-20)

He then used this expression in his anal-ysis of the rates of the a/g transformationof iron with very low carbon contents, asinvestigated experimentally by Bibby andParr (1964) and Ackert and Parr (1971).Using a model in which a fraction of thechemical force is dissipated by solute dif-fusion ahead of the interface, Hillert wasable to predict a number of characteristicsof the massive transformation in pure andnearly pure iron, as summarized in Fig. 7-15.

For richer alloys, massive growth could,in principle, be entirely composition-invar-iant. However, some results (Singh et al.,1985) suggest that solute diffusion aheadof the interface can play a major role insteady interface migration. Referring toFig. 7-16, which is based on the simplemodel of an infinitely thin interface, it isclear that a solute build-up in front of theinterface will result in a reduction of thechemical force on the interface. Havingconceptually released the concentration inthe interface region from its value far fromthe interface (C0), the description of two-component massive growth becomes a freeboundary problem, in which the interfacialdriving force and the velocity must be si-multaneously determined. It is likely thatthis type of approach to local equilibriumoccurs at higher relative temperatures,where the possibility of volume-diffusionloss is strongest, and where, for example,Widmanstätten growth competes with themassive reaction.

Much of our current knowledge of mas-sive growth derives from the pioneeringwork of Massalski and co-workers (Mas-salski, 1958; Barrett and Massalski, 1966).


Figure 7-14. Helmholtz energy relationship for theisothermal nucleation of a massive product in a bi-nary system.

Figure 7-15. Diffusional and diffusionless growthof a from g iron, calculated assuming that carbon dif-fusion is negligible at high growth rates. Reprintedwith permission from Hillert (1975b), p. 12.

7.5 Examples 501

In early papers, it was implicitly assumedthat a pseudo-unary cross-boundary diffu-sion process controlled the rate of massivegrowth; later (Massalski, 1984) it wasnoted that local changes of composition inthe region of the interface, consistent withoverall composition invariance, are pos-sible, even probable, under certain condi-tions.

Because of the speed of transformation,massive reactions are often studied underconditions of continuous cooling. The mostinformative kinetic studies, however, arethose that yield rates of isothermal trans-formation, as exemplified by the work ofKarlyn et al. (1969); they used pulse-heat-ing and rapid quenching to evaluate therates of formation of massive a in a Cu–

38 at.% Zn alloy, which had previouslybeen quenched to retain the b phase. Theyfound:

i) that the transformation occurred onlywithin the single phase a region of thephase diagram and

ii) that steady massive growth (at con-stant isothermal velocity of order 1 cm/s)developed after a delay time of several mil-liseconds.

Both the limited temperature range forgrowth and the delay time were attributedto the pre-existence of solute fields aroundsmall a particles. Only in the single-phasea region would such solute fields be con-sumed during initial growth, thereby per-mitting the development of composition-invariant (massive) growth. They notedthat massive growth in the two-phase a + bregion (below T0) is possible, provided thatsome massive a has first been formed bypulsing into the single-phase region. Withfurther heating these massive regions con-tinued to growth into the a + b region, thusreinforcing the idea that the inhibition ofsuch growth lies in the initiation stage.

In the analysis of growth rates, it was as-sumed that massive growth occurred with-out local composition change. However, indiscussion, impurity drag effects and par-ent-phase solute fields were admitted aspossibilities.

A comprehensive theoretical treatmentof the two-component massive transforma-tion, which applies to a range of transfor-mation conditions, will deal with solutediffusion within the interface and also inthe thin region ahead of it. The model ofHillert and Sundman (1976), an extensionof their treatment of solute drag at grainboundaries, finds application here. Theirformulation of the solute drag involves thecomputation of the dissipation of Helm-holtz energy due to diffusion within the

Figure 7-16. Solute build-up in front of the massiveinterface will result in a reduction of the chemicalforce for interface migration.

interface region (Eq. (7-18)). The solutefield is schematized in Fig. 7-17(a) for asolute that is attracted to the interphaseboundary. The cross-boundary diffusion ofsolute introduces a component of drag (re-gions 1, 2 and 3), and diffusion ahead ofthe boundary provides a further contribu-tion (region 4), which dominates the totaldrag as the velocity approaches zero. The

total drag is optimized for and it

may be that the optimum of dissipation of

vd2

1D

≈ ,

Helmholtz energy defines the most prob-able velocity. A second treatment of the so-lute drag in massive growth (Bréchet andPurdy, 1992) extended Cahn’s (1962) anal-ysis, and demonstrated that a finite dragterm will always be present, however smallthe velocity. This analysis, like that of Hil-lert and Sundman, therefore suggests that athreshold driving force is a natural charac-teristic of such transformations. However,if the solute field in front of the interface isaccompanied by a misfit, in the sense ofEq. (5-12), we expect an additional term inthe force balance, a “pulling” force due tothe elastic energy contained in the coherentcomposition gradient.

The above discussion is based on thepremise that the migration characteristicsof the massive front are those of a non-co-herent interface. This is borne out by nu-merous observations (Massalski, 1984)which indicate, for example, that the trans-formation interface is able to cross grainboundaries in the parent phase without achange in velocity or morphology. Dy-namic observations indicate, however, thatthe motion of the front is irregular, and of-ten accomplished by a lateral process, inwhich diffuse steps move parallel to theinterface plane (Perepezko, 1984). In theview of Menon et al. (1988), lateral pro-cesses are the rule in massive growth.Hence models based solely on diffusionnormal to the boundary may need to bemodified. However, it is interesting thatPerepezko (1984) demonstrated a scalingrelationship for a wide range of alloysystems, which yields a composite en-thalpy of activation for massive propaga-tion of 94 Tm J/mol. This value is similar tothat for grain boundary diffusion, whichsuggests that diffusion within the interfaceis a common rate-determining feature forall such transformations.


Figure 7-17. a) Schematic concentration profilethrough a massive transformation interface, i) atequilibrium, and ii) in motion. b) Computed solutedrag for the massive transformation in part a, for aconstant value of D. X ·

A and X ·B represent the initial

composition of the parent phase expressed as molefractions of the two components A and B. After Hil-lert and Sundman (1976).

7.5 Examples 503

7.5.2 Chemically-Induced Grain Boundary Migration

Chemically-induced grain boundary mi-gration (CIGM), or diffusion-induced grainboundary migration, is a rather recentlyrecognized phenomenon (den Broeder,1972; Hillert and Purdy, 1978). The pro-cess is one in which diffusion along a grainboundary, to or from a sink or source of so-lute, causes the boundary to move. Thesource may be in the solid, liquid or gasstate. A composition change results (Fig. 7-18).

The process is widespread in binary me-tallic systems. King (1987) lists 30 systemsin which it has been detected. The processis capable of generating substantial inter-mixing where little or none might be ex-pected in the absence of grain boundarymotion.

A variation on the conditions in Fig. 7-18 is obtained for a supersaturated solidsolution: an initially planar boundary bowsout between grain boundary precipitates,and simultaneously sweeps solute to theprecipitates (Fig. 7-19). This type of mi-crostructural development was reported byHillert and Lagneborg (1971), and was re-cently supported by microanalytical evi-dence for solute depletion in the volumeswept by the moving boundary by Solor-zano et al. (1984).

CIGM is often seen as symmetry break-ing. The initiation process is not well docu-mented or understood. It is clear that diffu-sion along an initially stationary boundarywill lead to symmetric diffusion profiles,as illustrated in Fig. 7-6, and this has led tothe suggestion that the region next to theboundary will become one of increasedelastic strain energy, provided that a solutemisfit exists, and provided that the solute-enriched or solute-depleted regions remaincoherently connected with their respectivegrains. The strain energy density will thenbe given by Eq. (5-12). Because the elasticconstant Y is a function of orientation, thestrain energy density will in general be dif-ferent on the two sides of the boundary, andthis may lead to boundary displacement.Tashiro and Purdy (1987), in a survey ofbinary metallic systems, found only onesystem in which CIGM could not be in-duced; this was the system with the lowestmisfit, Ag–Mn.

Further evidence for the importance ofsolute misfit in the initiation of CIGM isfound in the work of Rhee and Yoon

Figure 7-18. Schematic representation of chemi-cally-induced grain boundary migration in a thinsample exposed to a vapor source of solute. The frontis often observed to bow against its curvature.

Figure 7-19. Grain boundary bowing between pre-cipitates, which act as solute sinks, and serve to pinthe grain boundary.

(1989), who systematically varied the mis-fit parameter in a ternary system, andshowed that the phenomenon is suppressedwhen the misfit is brought to zero. An ap-proximate parabolic dependence of veloc-ity on misfit was also found, consistentwith Eq. (5-12). This suggests that elasticstrain energy also plays a role in propaga-tion of the grain boundary.

In practice it is difficult to distinguishthe possible driving forces for the process.At the highest temperatures, it is likely thatsolute field stresses will be dominant. Thisappears to be the case for the study of Rheeand Yoon (1989). In the early work of Hil-lert and Purdy (1978), in which thin poly-crystalline iron films were exposed to zincvapor, it was evident that the volume diffu-sion penetration of the parent grains, in-dexed by DV/v, was of atomic dimensions.Hence, it was assumed that the drivingforce was entirely chemical. A comprehen-sive model should take into account allpossible sources of the driving force.

The moving grain boundary is subject toa set of forces, which may in general in-clude chemical, elastic, frictional and cap-illary forces. As in the massive transforma-tion, there exists in principle a degree offreedom corresponding to the concentra-tion in the parent phase immediately adja-cent to the boundary. In the conceptuallimit where no volume-diffusion penetra-tion exists in advance of the boundary, theconcentration profile will be a step, and thefull chemical force pch will act across theinterface. In a second limiting case, a con-centration gradient exists in front of theboundary and an elastically-derived resid-ual force is determined by the coherencyfield. Intermediate cases would then corre-spond to a higher or lower degree of relax-ation of the concentration at the leadingedge of the boundary towards a constrainedequilibrium between strained parent crystal

and unstrained product crystal. These con-siderations are contained in the phenomen-ological treatment due to Bréchet andPurdy (1992), in which the driving force isevaluated over a correlation length (l) oneither side of the boundary. The steadyconcentration profile is C (z). The forcethen becomes:

(7-21)

where the second derivative is a measure ofthe Helmholtz energy composition relation.This approach has the virtue of includingboth possible contributions to the drivingforce; it results in the conclusion that, as inthe massive transformation, a threshold ex-ists, below which no motion is possible.

An ambitious treatment of the problemhas been put forward by Cahn et al. (1998).Their analysis is based on a phase-fieldtreatment of the grain boundary. They lookfor travelling-wave solutions of the equa-tions of motion, and treat their existence as a requirement for forces capable ofcoupling with the boundary motion. Theyfind that coupling with the elastic field isindeed possible. However, their treatmentgives no indication that a purely chemicalforce is effective in moving the boundary.

7.5.3 Discontinuous Precipitation

Discontinuous precipitation, like chemi-cally-induced grain boundary migration,involves the lateral diffusion of solutewithin a sweeping grain boundary. The dif-ference lies in the nature and spacing of thesolute sources/sinks. In the case of discon-tinuous precipitation, these are members ofa regular array of precipitates, whose spac-ing is a free variable, capable of internaladjustment. The reaction is found in a widevariety of precipitation systems, often at

1 120

2

2 02 2 2

lF

CC z C Y C z z

l

∫∂∂

− +⎡

⎣⎢

⎤

⎦⎥( ( ) ) ( )h d


7.5 Examples 505

low homologous temperatures where grainboundary diffusion is expected to be adominant mechanism of material transport.Like CIGM, discontinuous precipitationcan in principle occur in the absence ofvolume diffusion. The reaction is capableof destabilizing microstructure at relativelylow temperatures, and it is therefore ofpractical interest.

Unlike eutectoid reactions, which yieldmorphologically similar products, discon-tinuous precipitation reactions are not as-sociated with a particular feature of thephase diagram, although supersaturation isan obvious prerequisite. The question ofwhich systems will give rise to discontinu-ous precipitation, and which will not, willperhaps be answered by reference to anddetailed understanding of the initiation andgrowth processes.

7.5.3.1 Initiation of Discontinuous Precipitation

A nucleation event, in the classicalsense, is not required. As in CIGM, a pre-existing grain boundary is caused to move,and eventually to become the steady reac-tion front, as suggested by Fig. 7-20. Themechanisms proposed for the initial stagesof grain boundary displacement fall intotwo broad classes: free-boundary mecha-nisms and precipitate-assisted mecha-nisms. Baumann et al. (1981) have indi-cated that free-boundary initiation is domi-nant at higher homologous temperatures,leading eventually to “single-seam” mor-phologies. At lower temperatures, it is ex-pected that boundary motion is initiated byprecipitate–boundary interactions, and thatthese lead to a preponderance of double-seam morphologies. The correlation be-tween temperature and morphology ap-pears to be quite general, with the break oc-curring at half the absolute melting temper-

ature. Gust (1984) found a correlation withthe solvus temperature rather than the meltingtemperature. Duly and Bréchet (1994) de-termined that the proportion of double-seammorphologies decreased from more than1/2 to 0 as the temperature was increased.

In free-boundary initiation, the initialgrain boundary displacement is thought tobe caused by ordinary recrystallization orgrain growth forces (Fournelle and Clark,1972), or by forces derived from solutesegregation (Meyrick, 1976). The subse-quent stages must then include the nuclea-tion of precipitates at the moving boundaryand the evolution of a steady state in whichthe precipitates acquire a uniform spacing,as depicted in Fig. 7-21. Cu–Co alloysseem to require this type of initiation, andreadily undergo discontinuous precipita-tion only when cold-worked or whentreated to give a small parent grain size(Perovic and Purdy, 1981).

Precipitate–boundary interactions werefirst studied in detail by Tu and Turnbull(1969), who showed that boundary torques,

Figure 7-20. Scanning electron micrograph of a dis-continuous precipitation colony formed in Al–22%Zn at 478 K. The original grain boundary positions isindicated by arrows. After Solorzano et al. (1984).

developed when a regular array of precipi-tates, formed on the boundaries in Pb–Snbicrystals, were responsible for the initialdisplacement of the boundary (Fig. 7-22).Eventually, boundary breakaway was en-visaged, with the subsequent embedding ofthe precipitates in the advancing grain. Aa-ronson and Aaron (1972) extended theseideas to include a range of possible geome-tries, each of which resulted in an initialboundary displacement due to capillaryforces associated with the formation ofequilibrium precipitate nuclei at the grainboundaries.

A further class of precipitate–boundaryinteractions, based on CIGM, was pro-posed by Purdy and Lange (1984). In thiscase, initial displacement can occur againstcapillary forces, in response to chemicalforces. If the initial precipitates are closelyspaced, the initial displacement may bepreceded by a period of precipitate coars-ening. The process is suggested by Fig. 7-19. Several reports of grain boundariesbowing between fixed precipitates can befound in the literature (Hillert and Lagne-borg, 1971; Solorzano et al., 1984). Mi-chael and Williams (1986) found that so-


Figure 7-21. Free-boundary initiationand early development of discontinuousprecipitation. Reprinted with permissionfrom Fournelle and Clark (1972), p. 2762.

Figure 7-22. Formation of grain boundary precipi-tates at an initially static boundary (a), and boundarydisplacements in response to capillary forces (b). Re-printed with permission from Tu (1972), p. 2773.

7.5 Examples 507

lute-depleted volumes were left in thewake of bulging grain boundaries in super-saturated Al–4.7 wt.% Cu. Fonda et al.(1998) and Mangan and Shiflet (1997)studied the initiation of discontinuous pre-cipitation in Cu–3% Ti alloys, and demon-strated that, at low supercooling, the initialboundary motion is due to the presence ofWidmanstätten precipitates, which growinto one grain, and cause boundary dis-placement into the other. Duly and Bréchet(1994) examined the initiation of discon-tinuous precipitation over a wide range oftemperatures, initial grain sizes and com-positions in Mg–Al alloys. They concludethat the Tu–Turnbull and Fournelle–Clarkmechanisms dominate at low and high tem-peratures respectively, and that the Purdy–Lange mechanism may be important atintermediate temperatures.

It is now clear that a range of mecha-nisms exist, each mechanism capable of in-itiating boundary motion. At lower temper-atures, precipitation on the static boundaryis a common precursor of discontinuousprecipitation; the initial precipitates are ca-pable of acting either directly, to pull theboundary from its initial location, or indi-rectly through CIGM to accomplish a simi-lar result. We conclude that, for most situa-tions where the supersaturation is signifi-cant, there will be no difficulty in initiatinggrain boundary motion.

7.5.3.2 Theories of Steady CooperativeGrowth

The development of a steady or near-steady growth front, characterized by aregular spacing between precipitate lamel-lae (or rods), S, and a constant velocity, v,is not immediate upon initiation, but is pre-ceded by extended transient growth re-gions. However, steady or near-steady con-ditions appear eventually to prevail, and

the steady state has attracted the attentionof successive generations of theorists. Inthis section, we review only the more ad-vanced theories, while acknowledgingtheir geneology.

Before proceeding, the question of thesteady state should be pursued in more de-tail. It appears that the steady state can beachieved fairly generally, and that it can beapproached from higher or lower supersat-uration, such that the system has little or nomemory, and the steady state is characteris-tic of the isothermal reaction conditionsonly. Nevertheless, the constancy of reac-tion front velocity has recently been ques-tioned by a number of authors, includingKaur and Gust (1988), Mangan and Shiflet(1997), and Fonda et al. (1998). This pointis discussed in Sec. 7.5.3.3.

It is convenient to divide the theoreticaltreatments into two classes depending onwhether the details of the transformationfront are predicted as part of the develop-ment. In the “global” approach of Cahn(1959), which builds on the ideas implicitin Turnbull’s (1951) and Zener’s (1946)treatments, the reaction front is treated as aplanar high-diffusivity path whose mobil-ity is rate determining. Thus the importantinput quantities to the theory include theoverall Helmholtz energy change accom-panying the passage of the front and theinterface mobility. Cahn (1959) first em-phasized the importance of storage ofHelmholtz energy in the product phase, andshowed how to estimate it from a knowl-edge of the interphase boundary energysa,b , the grain boundary diffusivity anddistribution coefficient, s, and the interla-mellar spacing S and velocity v. The Helm-holtz energy balance is written

DF≤ = DFch – 2sab Vm /S (7-22)

Here DFch is the total chemical Helm-holtz energy difference between the parent

and product phases, evaluated some dis-tance from the interface. In determiningthis quantity, the stored Helmholtz energyin the product a phase must be calculatedfrom a knowledge of the composition pro-file in the a lamellae. This is determined in theory from a knowledge of the grainboundary transport properties and the ve-locity and spacing through

(7-23)

where C0 is the initial concentration in theparent phase, C (z) is the local concentra-tion in the a phase behind the transforma-tion front, C3 is the concentration in the aphase in contact with the b lamellae and

(7-24)

Eq. (7-23) is a solution of the moving-boundary diffusion equation, for constantv, S, and boundary concentration C3; thesolution reduces in approximation to a sinefunction when the centerline compositionC (z = 0) is closer to C3 than to C0.

Cahn (1959) next chose C3 = C eab , the

equilibrium composition of a in contactwith b at a planar interface. He was thenable to evaluate the stored chemical Helm-holtz energy in the product phases from Eq. (7-23). To complete the description, he as-sumed a relationship between the totalHelmholtz energy change DF≤ and the ve-locity of the form

v = M≤ DF≤ (7-25)

Here, M≤ is a “global” mobility that dif-fers from the intrinsic grain-boundary mo-bility, e.g., of Eq. (7-16).

The theory described above has beenmodified by others, for example by Aaron-son and Liu (1968), to take partial accountof capillary forces acting on the b lamellae.

aS

s D=

b

v 2

d

C C z

C C

z a S

a

0

0 3 2

−−

( ) cosh ( / )

cosh ( / )= a

Certain aspects of the theory are amenableto experimental verification, for example,the concentration profiles left in the aproduct lamellae have been measured byPorter and Edington (1977) and by Solor-zano and Purdy (1984) using high resolu-tion elemental analysis, to give the trans-port properties of the reaction front,s Dbd, through Eq. (7-23). These samemeasurements allow the evaluation of theamount of the total available Helmholtzenergy for the transformation retained assegregation in the product; this latter quan-tity is also available, albeit in averagedform, through X-ray measurements of thelattice parameter of the product phase.

A heuristic description of discontinuousprecipitation, after Petermann and Horn-bogen (1968), utilizes a rate equation of thetype of Eq. (7-25) coupled with an approx-imate evaluation of the relaxation time forgrain boundary diffusion to the b lamellae,to give a velocity expression which is di-mensionally similar to that of Cahn.

A second class of theoretical descriptionhas been developed by Hillert (1969, 1972,1982). This approach has been termed “de-tailed”, in the sense that a much deeperknowledge of the transformation front isimplicit, even required, for its application.The basis for this treatment is the applica-tion of a local force balance at every pointalong the interface. Thus the possibility israised of determining the interface shapefor steady growth, as a consequence of theinterplay of capillary, chemical, elastic,and frictional forces, as defined in Sec. 7.4.

In simplified form, for application to thecase where the composition profile (step),DC, measured normal to the interface is notrelaxed by volume diffusion, and whereelastic and solute drag effects are negli-gible, the interfacial force balance becomes

v/M = pch – ps (7-26)


7.5 Examples 509

which holds at every point. The solute pro-file in the boundary will be given by Eq.(7-23), and the concentration step (DC) atthe interface will take a minimum value atthe centres of the lamellae, varying contin-uously to a maximum value at the a/a0

interface immediately adjacent to the b la-mellae. Hillert has chosen the limitingvalue of the solute concentration at thispoint, C3 in Eq. (7-23), as the capillary-modified solubility in a in equilibriumwith b under curvature 2/Sb .

For a steady transformation, the right-hand side of Eq. (7-26) is constant alongthe interface, and the curvature must varyto match the chemical force, indexed by thecomposition step DC.

A further local force balance is requiredat the a0/a/b junction. This is depicted by a vectorial balance of surface energies in Fig. 7-23. Thus the angles of intersec-tion and local curvatures are set at the junc-tion, and the variation of the grain bound-ary curvature is required to match the vari-ation of chemical force along the boundary.The theory therefore allows the self-con-sistent calculation of the interface shapefor given values of the spacing S, the veloc-ity v, and the interfacial transport propertys Dbd.

Sundquist (1973) extended Hillert’streatment to include the possibility of

solute drag forces and severely nonplanarshapes, thus allowing for consideration ofmorphological stability of the transforma-tion front.

Neither of the approaches describedabove is sufficient to predict the transfor-mation state that will occur at given super-saturation in a particular system. Each iscapable of generating a unique relationshipbetween v and S, however.

In considering the spacing problem forlamellar eutectoids, Zener (1964) set athermodynamic limit, a minimum spacingfor which all of the available Helmholtzenergy for the transformation is stored asinterfacial Helmholtz energy in the prod-uct. This minimum also exists for discon-tinuous products:

(7-27)

At this (virtual) state, the system is inequilibrium and no growth is possible.

Larger spacings corresponds to finiterates of growth. It has been suggested thatthe steady spacing is one which maximizesthe growth rate (Hillert, 1972) or the inte-gral rate of dissipation of Helmholtzenergy (Cahn, 1959). Solorzano and Purdy(1984) put this latter hypothesis to the test,and found a reasonable level of confirma-

SV

Fmin =m

ch

2sab

D

Figure 7-23. Schematic (plan)view of a steady discontinuousgrowth front, defining thespacings, S, Sa , Sb , relevant tothe problem, and indicating thevectorial balance of interfacialenergies, sa/b, sa0/a, sa0/b, atthe three-grain junction.

tion for two well-characterized systems,Al–22 at.% Zn and Mg–9 at.% Al.

Building upon the local approach of Hil-lert and co-workers, Klinger et al. (1996)have proposed a solution which leads to aselection of the growth velocity without theneed for an optimization principle. Thistreatment relies on the cooperative natureof the growth of the a and b lamellae, andon the different characters of the interfacesbetween the a and b phases and the parenta phase: one is a grain boundary and theother is an interphase boundary. Assuminga definite composition for the b phase leadsto a Mullins (1957) equation for transportalong the a0/b interface determined by gra-dients of curvature. Transport along thea0/a is considered to be driven by concen-tration gradients, according to Cahn’s(1959) formulation. The condition at thetriple junction is taken as a local equilib-rium, corrected for Gibbs–Thompson ef-fects. The requirement that the a and bgrowth velocities match in order to have acooperative growth process is sufficient toselect the spacing and velocity of the front.This approach has the advantage of avoid-ing the introduction of optimization princi-ples, and it has been shown to accuratelydescribe some experimental results for theAl–Zn system. It also predicts that steady-state solutions with invariant front shapesare possible within a limited range of driv-ing forces, suggesting the possibility ofmorphological instabilities outside thisrange.

All of these theories of steady growth as-sume that the motion of the interface iscontinuous. This assumption is probablyacceptable when the growth front is well-rounded, but there is substantial evidencefor faceted interfaces at low supersatura-tions, as discussed in detail in the next sec-tion. In the high-supersaturation regime,the continuum models seem to apply with-

out modification; however, the theoriesneed to take into account the possibility oflateral migration of the growth interface forlower supersaturation. A recent treatmentby Klinger et al. (1997a) takes this into ac-count by describing the motion of the inter-face as intermittent.

7.5.3.3 Experimental Observations

The majority of experimental studieshave focused on microstructures developedthrough steady isothermal growth. Theyare therefore capable of description interms of the theoretical treatments of theprevious section. The types of informationaccessible to experiment are:

a) quantities that may be derived fromconventional metallographic methods: av-erage velocity v and spacing S;

b) chemical information obtained eitherfrom averaging processes such as X-raydiffraction measurements of lattice param-eters of product lamellae or from high res-olution microanalyses of the product;

c) structural information relating to thegrowth interface, as obtained for exampleby transmission electron microscopy.

In addition, for the test of theory we re-quire solution thermodynamic data, inter-facial energies and interfacial diffusion co-efficients. Many alloy systems have beeninvestigated. In this section we will focuson the systems Al–Zn and Mg–Al, forwhich much of the necessary data are avail-able (e.g., Rundman and Hilliard, 1967;Cheetham and Sale, 1974; Hassner, 1974).

Yang et al. (1988) collected velocity andspacing data for Al–Zn alloys, and for arange of reaction temperatures and spac-ings, as summarized in Figs. 7-24 and 7-25.Figure 7-26 gives the average compositionof the a lamellae superimposed on the Al-rich portion of the phase diagram. These


7.5 Examples 511

composite figures include results from thework of Ju and Fournelle (1985), Sureshand Gupta (1986), Cheetham and Sale(1974), Razik and Maksoud (1979), Wata-nabe and Koda (1965), Gust et al. (1984)and Yang et al. (1988). Perhaps the moststriking feature of these data is the extentof residual supersaturation left in the aphase behind the transformation front; asignificant amount of the total Helmholtzenergy for the transformation is stored inthe product phase.

This stored Helmholtz energy, first em-phasized by Cahn (1959), must be esti-mated in order to evaluate the driving forcefor the transformation. This estimation is

readily performed using data of the typeshown in Fig. 7-26. This stored Helmholtzenergy provides a portion of the drivingforce for subsequent discontinuous coars-ening processes, which generally increasethe interlamellar spacing and deplete theproduct a to (near) the solubility limit(Yang et al., 1988).

High resolution microanalysis is capableof providing further insights into the storedHelmholtz energy term (Zieba and Gust,1998). The elemental trace of Fig. 7-27was obtained from a STEM (scanningtransmission electron microscope) X-raymicroanalysis of a thin foil of Al–22 at.%Zn transformed at 428 K. Measurements ofthis type give both local and integral valuesof the stored Helmholtz energy. In additionto the information concerning the soluteconcentration distribution in the a phase,these profiles can be analyzed to yieldrather directly the product s Dbd via Eq.(7-23), (Porter and Edington, 1977; Solor-

Figure 7-24. Grain-boundary velocities for a rangeof reaction temperatures and for several Al–Zn al-loys (index DP denotes discontinuous precipitation).The data are from studies by Yang et al. (1988), Juand Fournelle (1985) [1], Suresh and Gupta (1986)[2], Cheetham and Sale (1974) [9], Razik and Mak-soud (1979) [10], Gust et al. (1984) [18], and Wata-nabe and Koda (1965) [19]. Reprinted with permis-sion from Yang et al. (1988).

Figure 7-25. Interlamellar spacing measurementsfor discontinuous precipitation (DP) products, fromthe same sources as for Fig. 7-24. Reprinted with per-mission from Yang et al. (1988).

zano et al., 1984; Duly et al., 1994a, b).However, in their studies of regular growthin Mg–Al alloys, Duly et al. determinedthat:

a) In certain cases, the concentrationprofiles between parallel lamellae (similarto Fig. 7-27) were well described by

Cahn’s equation with constant values ofC*ab and ÷–

a.b) However, within a given nodule, the

value of (a/Sa2) was generally not constant,

varying by more than an order of magni-tude. This quantity is proportional to theinstantaneous interfacial velocity, and itsvariation is taken to mean that, on anatomic length scale, the interface velocityis irregular. On a larger length scale, typi-cally 0.1 µm, an average velocity can bedefined that is consistent with the steadydiffusion analysis of Cahn.

A major implication of these findings isthat the transformation interface moves bya lateral growth mechanism, resulting in anaverage velocity much less than the instan-taneous velocity accompanying the pas-sage of a growth step. It is here that the recent studies of Shiflet and his co-workers(Fonda and Shiflet, 1990; Fonda et al.,1998) on Cu–Ti alloys are relevant. Theirresults indicate a strong and consistent ten-dency for the transformation front to facet,particularly at low undercoolings, and thisimplies a lateral displacement process,consistent with the findings of Duly et al.


Figure 7-26. Average zinc concentrations in the aphase for discontinuous precipitation (DP) and fordiscontinuous coarsening (DC) reactions in variousAl–Zn alloys. Reprinted with permission from Yanget al. (1988).

Figure 7-27. STEM microanalysisof an a lamella in Al–22 at.% Zn,formed at 428 K. After Solorzanoand Purdy (1984).

7.5 Examples 513

(1994a, b). At lower reaction temperatures,the growth interfaces in Cu–Ti becomesmoothly curved. This is attributed to therapid accumulation of growth ledges withincreasing undercooling.

The smoothly curved interfaces andsteady migration rates seen in high resolu-tion in situ studies of discontinuous precip-itation in Al–Zn (see Fig. 7-28, Tashiroand Purdy, 1989) suggest that, at lowertemperatures, the interface can be consid-ered to move continuously. Indeed, theinterface shapes observed could be ration-alized in terms of a local balance between

chemical (evaluated from microanalyses),capillary and friction forces, realized atevery point on the transformation front. Ittherefore appears that at some stage an ef-fective transition occurs where the accu-mulation of growth ledges is sufficient tomimic the normal migration of the front,and the classical theories of Cahn and Hil-lert then hold to a good approximation.

There is a further set of experimentaldata on discontinuous precipitation that re-lates to the effect of an applied stress on thetransforming system. Sulonen (1964a, b)showed that in some alloy systems, frontswith normals parallel to the tensile axismoved more slowly, and those with nor-mals perpendicular to the stress axis morequickly than those in unstressed alloys. Inother alloy systems, the opposite behaviorwas found. The effect for Cu–Cd alloys isshown in Fig. 7-29. Sulonen proposed an

Figure 7-28. Observed and calculated interfaceshapes for an a lamella formed at 400 K in Al–22 at.% Zn. The numbers on the curves represent different amounts of Helmholtz energy lost owing tocontinuous precipitation in the parent phase.

Figure 7-29. Effect of an applied tensile stress onthe rates of growth of the discontinuous product inCu–Cd alloys (1 kp/mm2 ≈ 9.81 ¥ 106 N/m2). Re-printed with permission from Sulonen (1964b),copyright Pergamon Press.

explanation based on the elastic interactionbetween the applied and coherency (solute)stress fields. Hillert (1972) indicated how aquantitative treatment of such an effectcould be formulated. Sulonen’s result andHillert’s analysis have been widely quotedin support of the existence of a solute (co-herency) field in front of the discontinuoustransformation front, even at temperatureswhere the calculated diffusion distance isnegligible.

Dryden and Purdy (1990) reconsideredthe problem, and included the possible ef-fect of a volume misfit in the transformedregion. It was found, in agreement withHillert, that the sign of the effect could bepredicted on the basis of an elastic solutefield interaction; however, the predicted ef-fects on interface velocities are too smallby four orders of magnitude. The morelikely cause of the coupling is found in theplastic response of the dead-loaded speci-men. The reduction in Helmholtz energy ofthe loading device results in a virtual forceon the transformation interface, as dis-cussed in Sec. 7.4.3.2. For the case of inter-faces with normals parallel to the x, y, andz (tensile) axes, these virtual forces are re-lated by

(7-28)

where px , py , and pz are forces per unitarea, Z is the applied tensile stress, e is thestress-free strain in the transformed vol-ume, and W is an alignment parameterwhich accounts for the possibility that thelamellae in transformed regions boundedby x and y interfaces are aligned with re-spect to the z axis.

It is found that the virtual forces so de-rived, in conjunction with independentlyderived grain-boundary mobilities, are ca-pable of explaining the sign and magnitudeof Sulonen’s data for Cu–Cd alloys (Fig.

p p pZ

x y z= = =− +⎛⎝

⎞⎠ +1 3

2 21 3

W We( )

7-29), and give the correct sign of the ef-fect for the other five systems investigated,for which no quantitative data were ob-tained. We conclude that it is the plastic (orcreep) response of the specimen which ismore plausibly coupled to the migrationrates of differently oriented transformationinterfaces.

7.5.4 Interface Migration in Multilayers

Multilayers have been utilized for criti-cal experiments since the seminal work ofHilliard and his co-workers (1954). Theyprovide an ideal tool for the investiga-tion of the thermodynamics and kinetics ofheterogeneous systems far from equilib-rium. Much of the work to date has beenconcerned with the approach to equilib-rium via bulk diffusion. The recently de-veloped techniques for precisely controlledgrowth (e.g., molecular beam epitaxy)prompt the exploration of the approach toequilibrium via interfacial diffusion andmigration. The theoretical investigationssummarized in this section deal with pos-sible effects, many of which are still to beobserved.

Several classes of problems can be de-veloped, beginning with an A–B multi-layer grown on a bicrystalline substratesuch that a grain boundary penetrates thewhole structure and provides a possiblefast diffusion path. The simplest case oc-curs when A and B are fully miscible. Homogenization can then occur by grainboundary motion, either via a cooperativemechanism, or by a “fingering solution” atthe former A/B interfaces as illustrated inFigure 7-30 (Klinger et al., 1997a). In eachcase, the shape of the moving boundary, aswell as its velocity and the concentrationprofile left in its wake, can be computed interms of driving forces, interface energiesand diffusion coefficients.


7.5 Examples 515

The next situation to be considered iswhen A and B are reactive, and form a stoi-chiometric compound w (or, conversely,when such a compound decomposes toform A and B). Again, for this case, the ve-locity of the cooperative moving front forreaction (or dissolution) can be computedas a function of the energies and mobilitiesinvolved (Klinger et al., 1997c). A solutioninvolving a reaction product layer at theinterface rather than a cooperative growthfront can also be considered (Fig. 7-31),and this may help to explain certain pecu-liarities found in kinetics in reactive multi-layers, in terms of diffusion barrier effects(Klinger et al., 1998; Emeric, 1998).

These theoretical investigations, manystill awaiting experimental confirmation,suggest a wide variety of possible kineticpaths toward thermodynamic equilibrium(involving pattern and velocity selection as well as morphological instabilities) inwell-controlled systems. In these systems,a number of classical hypotheses such asthat of local equilibrium could be checkedquantitatively. It is suggested that the

search for such interface-mediated structu-ral evolution in controlled multilayeredstructures can play a key role in the deeperunderstanding of the more frequently en-countered phenomena described in the restof this chapter.

Figure 7-30. Illustrating three different possibilities for the discontinuous homogenization of a monophasemultilayer containing a mobile grain boundary, assumed initially to bisect the multilayer. Situations correspond-ing to (a) the steady state motion of the boundary; (b) an initially sinusoidal instability of the grain boundary,and (c) a “fingering” instability of the moving boundary, after Klinger et al. (1997b).

Figure 7-31. Schematic representation of the pos-sible reactions of A and B to form a stoichiometricproduct phase w. (a) A cooperative reaction at a sin-gle front; (b) the growth of a product layer at theinterface (after Klinger et al., 1997c).

7.6 Conclusions

In this chapter we have been concernedwith those phase transformations that de-pend on diffusion within a transformationfront. The massive transformation, chemi-cally-induced grain boundary migration,discontinuous precipitation and transfor-mations in multilayers represent differentand distinct facets of the general class ofinterface-diffusion controlled transforma-tions.

In each case, the description of the trans-formation in terms of a balance of forces onthe moving interface has proved a unifyingconcept. For the massive transformation, itis believed that the major forces at play arethe chemical driving force and the oppos-ing frictional forces. In the description ofchemically-induced grain boundary migra-tion, especially at higher temperatures, anelastic (coherency) term due to a gradientof misfitting solute atoms in the parentgrain must be added to the other two typesof force. For discontinuous precipitation,capillary forces must be introduced to op-pose the chemical driving force. We havealso considered transformations in multi-layer heterostructures, where capillaryterms must be accounted, and where inter-facial diffusion is often driven by interfa-cial curvature.

The description offered has been pre-dominantly thermodynamic, rather thandeeply mechanistic. Much remains to belearned about the mechanisms of accom-modation of diffusing atoms within thetransformation interface, about the cou-pling of the driving forces with interfaceresponse, and about the interrelationshipsbetween interfacial structure and kinetics.It has become increasingly clear that manytransformation interfaces are faceted, andremain faceted during migration. In suchcases, the classical models for diffusional

transformations will need to be reformu-lated, and this is only beginning to takeplace.

The thermodynamic structure remains avaluable synthetic framework within whichto place advances in the understanding ofthis fascinating field.

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70.


8 Atomic Ordering

Gerhard Inden

Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Federal Republic of Germany

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 5218.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5238.2 Definition of Atomic Configurations . . . . . . . . . . . . . . . . . . . 5238.2.1 Configurational Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 5238.2.2 Point Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5258.2.3 Point Correlation Functions and Point Probabilities . . . . . . . . . . . . 5268.2.4 Pair Variables, Correlation Functions, and Probabilities . . . . . . . . . . 5278.2.5 Generalized Cluster Variables, Correlation Functions, and Probabilities . . 5298.2.6 Short-Range-Order (sro) Configurations –

Long-Range-Order (lro) Configurations . . . . . . . . . . . . . . . . . . 5298.3 The Existence Domain and Configuration Polyhedron . . . . . . . . . 5318.3.1 F.C.C. Structure, First Neighbor Interactions . . . . . . . . . . . . . . . . 5328.3.2 F.C.C. Structure, First and Second Neighbor Interactions . . . . . . . . . 5358.3.3 B.C.C. Structure, First and Second Neighbor Interactions . . . . . . . . . 5378.4 Ground States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5388.4.1 Pair Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5388.4.1.1 Ground State Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 5408.4.1.2 F.C.C. Structure, First Neighbor Interactions . . . . . . . . . . . . . . . . 5408.4.1.3 F.C.C. Structure, First and Second Neighbor Interactions . . . . . . . . . 5438.4.1.4 B.C.C. Structure, First and Second Neighbor Interactions . . . . . . . . . 5448.4.1.5 Energy Minimum at Constant Composition . . . . . . . . . . . . . . . . . 5448.4.1.6 Canonical Energy of lro States . . . . . . . . . . . . . . . . . . . . . . . 5468.4.1.7 Relevant Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5478.4.2 Effective Cluster Interactions (ECIs) . . . . . . . . . . . . . . . . . . . . 5488.5 Phase Equilibria at Finite Temperatures . . . . . . . . . . . . . . . . . 5518.5.1 Cluster Variation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.5.2 Calculation of Phase Diagrams with the CVM . . . . . . . . . . . . . . . 5548.5.3 Phase Diagram Calculation with the Monte Carlo Method . . . . . . . . . 5548.5.4 Examples of Prototype Diagrams . . . . . . . . . . . . . . . . . . . . . . 5558.5.4.1 F.C.C. Structure, First Neighbor Interactions . . . . . . . . . . . . . . . . 5558.5.4.2 F.C.C. Structure, First and Second Neighbor Interactions . . . . . . . . . 5578.5.4.3 B.C.C. Structure, First and Second Neighbor Interactions . . . . . . . . . 5588.5.4.4 Hexagonal Lattice, Anisotropic Nearest-Neighbor Interactions . . . . . . 5588.5.5 The Cluster Site Approximation (CSA) . . . . . . . . . . . . . . . . . . . 5608.6 Application to Real Systems . . . . . . . . . . . . . . . . . . . . . . . . 5618.6.1 The Au–Ni System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563


8.6.2 The Thermodynamic Factor of Ordered Phases . . . . . . . . . . . . . . . 5658.6.2.1 The B.C.C. Fe–Al System . . . . . . . . . . . . . . . . . . . . . . . . . 5658.6.2.2 The F.C.C. Ni–Al System . . . . . . . . . . . . . . . . . . . . . . . . . . 5678.6.3 Ternary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5698.6.3.1 B.C.C. Fe–Ti–Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5698.6.3.2 B.C.C. Ferromagnetic Fe–Co–Al . . . . . . . . . . . . . . . . . . . . . 5708.6.4 H.C.P. Cd–Mg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5738.6.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5748.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5758.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5788.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

520 8 Atomic Ordering



a, a1, … basic vectors of the structureaa CVM exponentsb, b1, … basic vectors of the structurec, c1, … basic vectors of the structured distance of hyperplane to the originE part of internal energy that depends only on the configurational variablesF Helmholtz energyi number identifying the type of atomk number identifying a neighbor distanceK number of constituentskB Boltzmann constantma number of a-clusters per pointN number of lattice sitesn, m number identifying a lattice siten normal vectorNi number of atoms iN (sr) number of r-clusters with occupation sr

Nr number of equivalent r-site clusterspn

(i) site-occupation operator for atom i on site npij

nm occupation operator for atom pair i, j on sites n, mr number of lattice points within a clusterS entropyT temperatureU, U internal energy, grand canonical internal energyV volumeVij

nm pair energy of atoms i, j on sites n, mW number of possible arrangements of clusters formed for given values of the

correlation functionsWij

(k) pair exchange energy of k-th neighbor atoms I and JW (k) binary case: indices i j omittedWij… l

12…r r-site cluster exchange energy correction termVn cluster expansion coefficientwn

iÆ j transition probabilitiesx hyperplane position vectorxv vertex vectorxi mole fraction of atom of type iz(1…r) coordination number of r-site clusters

a number identifying a clustera ij

n–m Cowley–Warren sro parametersmi chemical potential of atoms in dimension of configurational space

ntot , np total number of configurational probabilities, number of configurational prob-abilities related by partial summations

z1, z2, z3 lro parametersr (sr) N (sr)/Nr

rijnm probability of having atoms i, j on sites n, m

s vector with site operators as componentssn site operator or spin variabletk values of the site operatorj arbitrary functionFi , F energy variables, defining dual space of the configurational variables, their

vector hyperplaneW grand potential

CVM cluster variation methodCSA cluster site approximationEC cluster expansionECI effective cluster interactionlro long-range ordern – m neighbor distance between sites n, m (equivalent to number k)MC Monte Carlo simulationsro short-range order

1 kB-unit = 1 kB K = 13.8 ¥ 10–24 J = 8.6 ¥ 10–2 meV


8.2 Definition of Atomic Configurations 523

8.1 Introduction

Ordered structures occur frequently insolid solutions and exhibit interestingphysical and mechanical properties. Thisexplains the continuous interest in orderingreactions by theoreticians and experimen-talists over recent decades. There are someexcellent overview articles and books (e.g.,Ducastelle, 1991; Turchi and Gonis, 1994;de Fontaine, 1979, 1994; Khachaturyan,1978) on the theoretical aspects of this sub-ject, while the experimental and materialsscience aspects can be found in articles byKear et al. (1970), Warlimont (1974), Kochet al. (1985), Stoloff et al. (1987), Liu et al.(1989, 1992), and Whang et al. (1990).

The ordered phases can be classifiedunder the family of intermetallic phases.As such they have attracted interest in thedevelopment of materials for special appli-cations (e.g., materials for use at high tem-peratures). Thus there is an increasing de-mand for quantitative descriptions of thevariations in their properties. Furthermore,materials with practical relevance rarelyconsist of binary systems. Treatments ofmulticomponent systems are thus espe-cially needed. The most fundamental prob-lems that need to be solved at the start ofany optimization of materials involve thetype of ordered structure, the variation ofthe atomic distribution with temperatureand composition, and the phase equilibria.This chapter is an updated version of theprevious contribution by Inden and Pitsch(1991). The tutorial aspects have beenmaintained as much as possible. It is forthis reason that the basic ideas are still presented in terms of a pair interactionscheme, which can only provide prototyperesults. An up-to-date treatment of realsystems requires more sophisticated ap-proaches that take into account previouslyneglected but very important physical ef-

fects. First-principles calculations of totalenergies, of lattice relaxations and of localrelaxations were still in their infancy tenyears ago, but such theoretical calculationsare now available. They will be discussedin relation to the treatment of real alloys.

The variety of materials exhibiting or-dering is too great to be dealt with com-pletely. The present chapter will thereforebe limited to metallic substitutional alloys,despite the fact that the techniques dis-cussed have also been used extensively inthe field of interstitial alloys, carbides, ni-trides, oxides, and semiconductor systems.

The chapter is organized as follows. InSec. 8.2, a general formalism for describ-ing and characterizing atomic configura-tions in multicomponent substitutional al-loys is presented. The concept of correla-tion functions as independent variables forthe definition of the configurations is intro-duced and used in Sec. 8.3 to derive exis-tence domains; they limit the range of nu-merical values that can be scanned by thecorrelation functions for topologically ex-isting configurations. These existence do-mains are used in Sec. 8.4 to determine theground states. In Sec. 8.5, equilibrium at fi-nite temperatures is discussed in terms ofthe cluster variation method (CVM) andthe Monte Carlo simulation (MC). Finally,in Sec. 8.6, the application to real systemsis discussed.

8.2 Definition of Atomic Configurations

8.2.1 Configurational Variables

Let us consider a crystalline system withN lattice sites and K constituents i = A,B, … . Defining the number of atoms oftype i by Ni gives Â

iNi = N and the mole

fractions xi = Ni /N. The distribution of the

atoms on the lattice sites defines the con-figuration.

In order to specify a configuration, weneed an operator which identifies unequiv-ocally the atomic species on an arbitrarysite n. A convenient means of identifica-tion is to associate an integral number witheach constituent and to define a site opera-tor sn , which takes these integral valuescorresponding to the constituent on site n.Different choices are possible, e.g.,

Any configuration is then specified by thevector s = (s1, s2, …, sN). In total, thereare KN different configurations. For binary

alloys, the choice was first sug-

gested by Flinn (1956). The operator sn issometimes called the spin variable becauseof its correspondence with the Ising modelfor binary alloys if we take sn = ±1. Anyfunction of sn including sn itself is called aconfigurational variable.

Because N is a very large number, it isnot possible to handle such a large amountof information. Therefore, we are forced towork with a reduced amount of informationby considering the configurations of muchsmaller units called clusters. A cluster is defined by a set of lattice points 1, 2, …, r,and a configuration on this cluster is given

s nxx

= B

A−⎧⎨⎩

or generally

for = Afor = B

for =

tt

t

1

2

K

ii

i K

⎧

⎨⎪

⎩⎪

s n

K K

K K

K K

= or or

012

1

123

21

2

101

21

2

−

⎧

⎨⎪⎪

⎩⎪⎪

⎧

⎨⎪⎪

⎩⎪⎪

−⎛⎝

⎞⎠

−

− − +⎛⎝

⎞⎠

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

( )

by sr = (s1, s2, …, sr). The smallest clus-ter is a point, the next one a pair, then atriplet, and so on. On an r-site cluster thereare Kr configurations, a much smaller num-ber than KN. The configurations of the N-point system can then be classified intogroups with the same number of clustersN (sr) with the configuration sr . Instead ofN (sr), it is preferable to work with the re-duced number r (sr) = N (sr)/Nr , where Nr

is the number of equivalent r-site clusterscontained in the system. These fractionsare called cluster probabilities. They spec-ify the configuration in the r-point clusterapproximation and constitute the most im-portant configurational variables. The ap-proximation depends on the size of thelargest cluster taken into account. For athree-dimensional (3-dim.) lattice, it isnecessary to include at least one 3-dim.cluster, otherwise the topological connec-tion of the clusters for space filling cannotbe taken into account correctly. For in-stance, if only pairs are considered, it is notpossible to distinguish a 3-dim. configura-tion from a 2-dim. Bethe lattice with thesame coordination number.

So far r defines a particular set of points1, 2, …, r, and Nr is the number of clustershaving the same orientation in space, thusdiffering from this particular set by a trans-lation in the lattice, or Nr = N. In this in-stance, we speak of oriented clusters andoriented-cluster probabilities. The N-pointsystem usually exhibits more symmetryelements than translation, and clusters ofdifferent orientation then become equiva-lent. Bringing these clusters together givesNr > N.

Suppose that an r-point cluster has beenselected for the description of the config-urations. Then the thermodynamic func-tions derived, for example, with CVM, de-pend on the probabilities of this cluster andalso on the probabilities of all subclusters.



Therefore, the total number of configura-tional probabilities ntot adds up to

because for a point, there are K choices of

an element and choices of a point in

the cluster, for a pair, there are K2 choices

of two elements and choices of a pair

of points in this cluster, and so on. How-ever, these ntot probabilities are not all in-dependent: if the probabilities of the largestcluster are given, then the probabilities ofall subclusters can be derived by partialsummation, e.g., for a 3-point cluster

(8-1)

The number of these partial summations in

Moreover, the sum of the probabilities ofthe largest cluster is equal to 1 by defini-tion. Therefore the number of independentprobabilities is

n = ntot – np – 1 = Kr – 1

that is, n is the dimension of the configura-tional space. When r and K increase, thedifference ntot – np becomes very large.

Of course any choice of n probabilitiesout of the whole set ntot may serve as a setof independent configurational variables.A convenient choice is any selection of nprobabilities out of the Kr probabilities ofthe largest cluster. This is usually done inKikuchi’s natural iteration method (Kiku-chi and Sato, 1974). It is necessary, how-

n p =r

Kr

Kr

rK r

1 2 12 1⎛

⎝⎞⎠ + ⎛

⎝⎞⎠ + … + −

⎛⎝

⎞⎠

−

r s s s

r s s s s ss s

( , , )

( , , , , , )

1 2 3

1 2 3 44

=…∑ …

r

r

r2

⎛⎝

⎞⎠

r1

⎛⎝

⎞⎠

n tot =

=

rK

rK

rr

K

K

r

r

1 2

1 1

2⎛⎝

⎞⎠ + ⎛

⎝⎞⎠ + … + ⎛

⎝⎞⎠

+ −( )

ever, to introduce Lagrange parameters inthe minimization of Gibbs energy in orderto take account of the np consistency rela-tions of Eq. (8-1), and it is not straightfor-ward to make use of efficient minimizationalgorithms. Therefore, effort has been con-centrated in defining a set of independentvariables (or a basis) in the n-dim. spacesuch that all the configurational variablescan be expressed in terms of this basis. Forbinary alloys, Sanchez and de Fontaine(1978) used multisite correlation func-tions as an extension of the pair correlationfunctions introduced by Clapp and Moss(1966). An extension to multicomponentsystems was first proposed by Taggart(1973) using the spin concept, after whichSanchez et al. (1984) suggested Chebychevpolynomials as a basis of the configura-tional space. In the following another basisis developed by a method that is equivalentto the one by Taggart. This basis is simplerand better suited to numerical applicationsthan that by Sanchez et al. (1984).

The procedure is as follows. First, we de-fine a basis in the space of point variables.This yields (K – 1) functions of sn . Then,we consider the (K2 – 1)-dim. space of thetwo-point variables and define a basis bytaking products of the (K – 1) basic pointvariables, and so on. This explains why themost important step is an appropriatechoice of the basis for point variables.

8.2.2 Point Variables

In the case of point variables, r corre-sponds to one point, and the vector s re-duces to one element sn . In order to definea basis of configurational point variables,in particular for the point probabilitiesr (sn), it is helpful to introduce a secondoperator pn

(i) which allows us to count thenumber of sites n with the same type ofatom i for taking averages. This operator

pn(i) is called the site-occupation operator

(Clapp and Moss, 1966) and is defined asfollows:

Because i goes from 1 to K we have K op-erators. Using this defintion we can imme-diately write the following equations:

(8-2)

where the upper indices signify powers of sn or ti . This notation will be usedthroughout this chapter. To distinguish anupper index from a power, the index will beput in parentheses, except if it is a doubleor multiple index, which cannot be con-fused with a power. Let us call M the ma-trix of this system of equations. Its deter-minant is the van der Monde determinant,which is given by

Because all the ti are different numbers,this determinant is different from zero, andthe equations are linearly independent.This confirms that the K – 1 functions sn ,sn

2, …, snK–1 are linearly independent.

Det =Mj i

K

i

r

j i= + =∏ ∏ −

1 1( )t t

=

1 1 1 1

1 2 3

12

22

32 2

11

21

31 1

1

2

3

……………

t t t tt t t t

t t t t

K

K

K K KKK

n

n

n

nK

ppp

p⋅ ⋅ ⋅ ⋅

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟ ⋅

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟− − − −

( )

( )

( )

( )

1 11

1

2

1

2

1

1

1

2

1

=

=

=

=

or

i

K

ni

ni

K

i ni

ni

K

i ni

nK

i

K

iK

ni

n

n

nK

p

p

p

p

=

=

=

−

=

− −

∑

∑

∑

∑

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎛

⎝

⎜⎜⎜⎜⎜⎜

( )

( )

( )

( )

s t

s t

s t

s

s

s

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

pi n

ni( ) =

if an atom of type occupies site otherwise

10

⎧⎨⎩

Det M ≠ 0 and so we can invert the ma-trix M and define R = (Rik) = M –1. FromEq. (8-2) we arrive at

(8-3)

This is the relation that connects the site-occupation operators with the site opera-tors. Because all the elements of the firstrow of M are unity, the inverse matrix Rexhibits the following properties: the ele-ments of the first column of R add up tounity and the elements of all other columns

add up to zero, i.e.,

8.2.3 Point Correlation Functionsand Point Probabilities

So far we have considered the point var-iables for one arbitrary point n out of the Nlattice points of the crystal. As mentionedbefore, we want to reduce the number ofparameters describing an atomic configura-tion by considering averages over equiva-lent clusters, or in this case, points. Thespace group of the structure will define theequivalence of points. We may enumeratethe classes of equivalent points by 1, 2, …and define the number of lattice points ineach class by N(1), N(2), …, N(L). The aver-age of an arbitrary function j (sn , sn

2, …)

i

K

ik kR=∑

11= d .

p R

p R

p R

ppp

p

R

nk

K

k nk

nk

K

k nk

nK

k

K

K k nk

n

n

n

nK

n

n

nK

( )

( )

( )

( )

( )

( )

( )

1

11

1

2

12

1

1

1

1

2

3 2

1

1

=

=

=

or =

=

−

=

−

=

−

−

∑

∑

∑

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

s

s

s

ss

s

⎟⎟⎟⎟⎟



of the point variables sn , sn2, … is defined

by

The index n now stands for one representa-tive point out of the corresponding class.The averages of the site-occupation opera-tors pn

(i) yield the probabilities of finding anatom A, B, …, K on a point n:

In the limit of N Æ ∞, this average cor-responds to the thermodynamic site-occu-pation probability. The (K – 1) functions·sn

K–1Ò are called point correlation func-tions. These functions constitute the basisof the configurational space of point vari-ables, and Eq. (8-4) yields the componentsof the point probabilities with respect tothis basis.

8.2.4 Pair Variables, Correlation Func-tions, and Probabilities

Consider a two-point cluster r = n, m. Wecan immediately introduce a pair-occupa-tion operator that takes the value 1 if anatom of type i occupies site n, and if type joccupies site m. This operator is simply theproduct of the two previously introducedsite-occupation operators pij

nm = pn(i) pm

( j).Using the expressions of the site operators,Eq. (8-3), we obtain

The pair probabilities rijnm are obtained by

taking the average of the operator pijnm over

all equivalent pairs in the crystal. It isworth mentioning that the pairs may benon-equivalent if they differ by their orien-

p R Rnmij

k

K

h

K

ih j k nh

mk=

= =

− −∑ ∑1 1

1 1s s

r s t s( )

, ,

( )n i n

i

k

K

ik nkp R

i K

= = =

= (8-4)

⟨ ⟩ ⟨ ⟩

…=

−∑1

1

1

⟨ … ⟩ …=∑j s s j s s( , , ) ( , , )( )

( )

n n ns

N

s sN

n

2

1

21=

tation, i.e., if n and m are non-equivalentpoints. In this instance, we speak oforiented pair probabilities. In the oppositecase, we speak of isotropic pair probabil-ities:

(8-5)

The pair probabilities can thus be ex-pressed in terms of the already introducedpoint correlation functions, written here inthe form ·sn

h–1smk–1Ò with h =1 and k = 2, 3,

…, K (or k =1 and h = 2, 3, …, K ) togetherwith a second set of functions, ·sn

h–1smk–1Ò,

with h, k = 2, …, K called pair correlationfunctions. The number of pair correlationfunctions is (K – 1)2 if n and m are non-equivalent sites. The total number of pointand pair correlation functions adds up to2 (K – 1) + (K – 1)2 = K2 – 1. This is exactlythe dimension of the configurational spaceif we assume the pair to be the basic clus-ter. Point and pair correlation functions to-gether constitute a basis in this instance.

By virtue of the relation Âj

pn(i) pm

( j) = pn(i)

the pair probabilities are consistent withthe point probabilities.

For the purposes of illustration, let usconsider two examples:

Binary alloy

For a binary alloy, K = 2 and t1 = 1 forthe first element A, and t2 = –1 for the sec-ond element B. The point probabilities fol-low from

which yields

1 1 11 1

1

2s n

n

n

pp

⎛⎝

⎞⎠ −

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

=( )

( )

M R= and =1 11 1

12

1 11 1−

⎛⎝

⎞⎠ − − −

−⎛⎝

⎞⎠

r

s s

nmij

ni

mj

k

K

h

K

ih j k

nh

mk

p p R R= =⟨ ⟩

× ⟨ ⟩= =

− −

∑ ∑( ) ( )

1 1

1 1

and

and consequently(8-6)

For the pair probabilities, we obtain, ac-cording to Eq. (8-5),

or

rrrr

rrrr

ss

s s

nm

nm

nm

nm

nm

nm

nm

nm

n

m

n m

AA

AB

BA

BB

11

12

21

22

= (8-7)

=

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

− −− −− −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⟨ ⟩⟨ ⟩

⟨ ⟩

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

14

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1

r s snmij

k hih jk n

hmkR R=

= =

− −∑ ∑ ⟨ ⟩1

2

1

21 1

r sr s

s r rn n

n nn n n

( )

( )( ) ( )( )

( )

1 12

2 12

1 21

1

=

=and =

+ ⟨ ⟩− ⟨ ⟩

⎧⎨⎩

⟨ ⟩ −

pp

pp

n

n

n

n n

A

B = =⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ −

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

( )

1

212

1 11 1

1s

Ternary alloy

For a ternary alloy, K = 3 and t1 = 1 forelement A, t2 = 0 for B, and t3 = –1 for C.In this instance, the point variables followfrom

and

According to Eqs. (8-2) and (8-5), we ob-tain

(8-8)

For the pair probabilities, we obtain, ac-cording to Eq. (8-5),

⟨ ⟩ −⟨ ⟩ +

⟨ ⟩ + ⟨ ⟩

− ⟨ ⟩

− ⟨ ⟩ + ⟨ ⟩

s r r

s r r

r r s s

r r s

r r s s

n n n

n n n

n n n n

n n n

n n n n

=

=

= =

= =

= =

A

B

C

( ) ( )

( ) ( )

( )

( )

( )

( )

( )

( )

1 3

2 1 3

1 12

2

2 12

2

3 12

2

2 2

R =12

0 1 12 0 20 1 1

−−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

M =1 1 11 0 11 0 1

−⎛

⎝⎜⎜

⎞

⎠⎟⎟


rrrrrrrrr

rrrrrrrrr

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

nm

AA

AB

AC

BA

BB

BC

CA

CB

CC

11

12

13

21

22

23

31

32

33

= =

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

14

0 0 0 0 1 11 0 1 1

0 2 2 0 0 0 0 2 2

0 0 0 0 1 1 0 1 1

0 0 0 2 0 2 2 0 2

4 0 4 0 0 0 4 0 4

0 0 0 2 0 2 2 0 2

0 0 0 0 1 1 0 1 1

0 2 2 0 0 0 0 2 2

0 0 0 0 1 1 0 1 1

− −− −

− −− −

− −− −

− −− −

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⟨ ⟩⟨ ⟩⟨ ⟩

⟨ ⟩⟨ ⟩

⟨ ⟩⟨ ⟩⟨ ⟩

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

1

2

2

2

2

2 2

sss

s ss s

ss ss s

n

n

m

n m

n m

m

n m

n m

(8-9)


8.2.5 Generalized Cluster Variables,Correlation Functions, and Probabilities

The point and pair variables are the sim-plest examples of the general cluster vari-ables. Consider an oriented cluster definedby r points. As before, we introduce a clus-ter-occupation operator by the product ofsite-occupation operators p1

(i1) p2(i2) … pr

(ir).The cluster correlation functions are theaverages over all equivalent clusters in thecrystal. Thus the result for the cluster prob-abilities is

(8-10)

The parameters include the r-

point correlation functions (point, pair,etc.) which provide the characterization ofthe configuration by means of r-site clus-ters, for example, the point correlationfunctions are coded in the form ·smÒ =·s1

0 …s1m …sr

0Ò, ·s 2mÒ = ·s1

0 …s2m …sr

0Ò,etc. and, correspondingly, for the pair cor-relation functions, they are coded ·sm snÒ =·s1

0 …s1m s1

n …sr0Ò, etc. The r-site cluster

correlation functions are obtained whenkn ≠ 1 for all values of kn . If all the r sitesare not equivalent, the total number of r-point correlation functions is (K – 1)r.The total number of correlation functionsNc required for an r-point cluster treatment,including all the subcluster correlationfunctions, is

This is the dimension of the configura-tional space, and the whole set of cluster

Nr

Kr

K

rr

K K Kr r r

c =

= =

11

21

1 1 1 1 1

2⎛⎝

⎞⎠ − + ⎛

⎝⎞⎠ − + …

+ ⎛⎝

⎞⎠ − − + − −

( ) ( )

( ) ( )

n

r

nkn

=

−∏1

1s

r

s s s

12 1 2

1 1 1

11

21 1

1 2 1 2

1 2

1 1 2 2

1 2

……

= = =− − −

⟨ … ⟩

… …

× ⟨ … ⟩

∑ ∑ ∑

ri i i i i

ri

k

K

k

K

k

K

i k i k i k

k krk

r r

r

r r

r

p p p

R R R

=

=

( ) ( ) ( )

correlation functions constitutes a basis ofthis configurational space. If all sites areequivalent, the number of correlation func-tions is much smaller.

8.2.6 Short-Range Order (sro) Configurations – Long-Range Order(lro) Configurations

A configuration is called short-range or-dered if the whole lattice constitutes oneclass of lattice points (i.e., N(1) = N). As aconsequence, all the cluster correlationfunctions are isotropic. A configuration iscalled long-range ordered if at least twoclasses of points are to be distinguished bytheir different average occupation. Theseclasses of lattice points are usually calledsublattices. In Fig. 8-1, the face-centeredcubic unit cell is shown with a subdivisionof the lattice sites into four simple cubicsublattices labeled 1 to 4 with N/4 pointseach. With these sublattices, the super-structures L12, L10 and P4/mmm can bedescribed. Each point on a sublattice is sur-rounded in its first-neighbor shell by fourpoints corresponding to each of the other

Figure 8-1. Subdivision of the face-centered cubic(f.c.c.) unit cell into four simple cubic sublattices fora characterization of the superstructures L12, L10 andP4/mmm. The occupations of the sublattices forthese structures are given in Table 8-1.

sublattices. Therefore these structures arecalled ordered in the first shell. The sublat-tice occupation for these structures is givenin Table 8-1.

In the body-centered cubic structure,four face-centered cubic sublattices with aparameter a = 2a0 can be defined as shownin Fig. 8-2. With these sublattices, thesuperstructures B2, D03, F4

–3m, and B32

can be described. These structures are or-dered in the first two neighbor shells. Theoccupation of the sublattices is again givenin Table 8-1. The complete crystallo-graphic specification of these superstruc-tures is given in Sec. 8.7, Table 8-13.

We must therefore consider the follow-ing variables: the points ·s1Ò, ·s2Ò, ·s3Ò,·s4Ò; the pairs ·s1 s3Ò, ·s1 s4Ò, ·s2 s3Ò,·s2 s4Ò, ·s1 s2Ò, ·s3 s4Ò; the triplets·s1 s2 s3Ò, etc.; and, finally, the tetrahedron·s1 s2 s3 s4Ò – in total, (24 – 1) = 15 vari-ables. Of course, the variables have differ-ent meanings in the two structures. In thef.c.c. lattice, all the pairs are nearest neigh-bors, while in the b.c.c. structure, the pairsof type 1–2 and 3–4 are second-nearest-neighbor pairs. A corresponding distinc-tion among the triplets also has to be made.

Among these variables, we need four vari-ables to define the isotropic f.c.c. sro state:·s1Ò, ·s1 s2Ò, ·s1 s2 s3Ò, and ·s1 s2 s3 s4Ò;one more variable, ·s1 s3Ò, is needed tocharacterize the isotropic b.c.c. sro state.The remaining variables characterize thelro state. For many applications, it is suffi-cient to identify the existence of lro by the


Table 8-1. F.c.c. and b.c.c. superstructures and values of their lro parameters.

Designation/Spacegroup Point probabilities lro parameter Values for maximumlro (binary)

A1 (Cu) A2 (Fe) r1(i) = r2

(i) = r3(i) = r4

(i) x1 = x2 = x3 = 0 r1A = 1 – xB

L10 (CuAu) B2 (CsCl) r1(i) = r2

(i) ≠ r3(i) = r4

(i) x1 ≠ 0, x2 = x3 = 0 r1A = 1, r3

A = 1 – 2 xB

L12 (Cu3Au) r1(i) ≠ r2

(i) = r3(i) = r4

(i) x1 ≠ x2 ≠ 0, x3 = 0 r1A = 1 – 4 xB, r2

A = 1

D03 (Fe3Al) r1(i) ≠ r2

(i) ≠ r3(i) = r4

(i) x1 ≠ x2 ≠ 0, x3 = 0 r1A = 1 – 4 xB, r2

A = 1,xB Ù 0.25 r3

A = 1

P4/mmm D03 (Fe3Al) r1(i) ≠ r2

(i) ≠ r3(i) = r4

(i) x1 ≠ x2 ≠ 0, x3 = 0 r1A = 0, r2

A = 2 – 4 xB,0.25 Ù xB Ù 0.5 0.25 Ù xB Ù 0.5 r3

A = 1

B32 (NaTl) r1(i) = r3

(i) ≠ r2(i) = r4

(i) x1 = 0, x2 ≠ x3 ≠ 0 r1A = 1, r2

A = 1 – 2 xB

F4–

3m r1(i) ≠ r2

(i) ≠ r3(i) ≠ r4

(i) x1 ≠ x2 ≠ x3 ≠ 00.25 Ù xB Ù 0.5

Figure 8-2. Subdivision of the body-centered cubic(b.c.c.) unit cell into four f.c.c. sublattices with twicethe lattice constant, 2a0, for a characterization of thesuperstructures B2, D03 and B32. The occupation ofthe sublattices for these structures are given in Table8-1. The (irregular) tetrahedron cluster 1234 used inthe CVM calculations is also shown.

8.3 The Existence Domain and Configuration Polyhedron 531

difference between the point correlationfunctions or linear combinations betweenthem (e.g., the same combinations as thoseentering the structure factor for X-ray dif-fraction). These linear combinations of thepoint correlation functions are called lroparameters. In the case of the b.c.c. struc-ture, we usually define the lro parameters

x1 = ·s1Ò + ·s2Ò – ·s3Ò – ·s4Òx2 = ·s1Ò – ·s2Òx3 = ·s3Ò – ·s4Ò

Table 8-1 lists the b.c.c. superstructures,together with the values of the parameters.

If there is no long-range order, then allthe positions are equivalent and the pointprobability takes one value rn

(i) = xi . Taking

In practice, the values of the pair correla-tion functions for the sro states are ob-tained from diffraction experiments. Forthis purpose, we usually introduce theCowley–Warren sro parameters a ij

n–m

(Cowley, 1950), which are directly relatedto the measured intensities. They are de-fined as the deviation from the randomstate:

The index n – m stands for the neighbor dis-tance between the positions n and m. Tak-ing again the ti values as before (i.e., ±1for the binary case and 1, 0, –1 for the ter-nary case), we obtain the isotropic Cow-ley–Warren sro parameters:

r r rr r r r a

n mij

nmij

nmji

ni

mj

nj

mi

n mij

−

−+ −= +

= [ ] ( )( ) ( ) ( ) ( ) 1

binary alloys: =

ternary alloys: =

=

AB A B

A B

AC A C

A C

a s s s ss s

a s s s s s s

a s s s s

a

n mn m n m

n m

n mn m n m n m

n mn m n m

n

x xx x

x xx x

−

−

−

−

⟨ ⟩ − ⟨ ⟩ ⟨ ⟩− ⟨ ⟩ ⟨ ⟩

⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩ − −

− ⟨ ⟩ + ⟨ ⟩ +

1

2 4 14

44

2 2 2 2

2 2

( )

mmn m n m n m x x

x xBC C B

B C=

− ⟨ ⟩ − ⟨ ⟩ + ⟨ ⟩ − −s s s s s s2 2 2 22 4 14

( )

the same ti values as in the examplestreated previously in Sec. 8.2.3, we can use Eq. (8-6) and obtain the following values for the point correlation functions:·snÒ = ·s1Ò = xA – xB in the case of a binaryalloy and ·s1Ò = xA – xC, ·s1

2Ò = xA + xC inthe case of a ternary alloy.

If there is not even any short-range or-der, we consider the completely disorderedstate. All the n-point correlation functionsthen take a limiting value; for example, in a binary alloy, ·s1…srÒ takes the value(xA – xB)r.

8.3 The Existence Domainand Configuration Polyhedron

In the preceding section, we developed aset of correlation functions (point, pair,triplet, etc.) for characterizing atomic con-figurations in multicomponent systems.These correlation functions are indepen-dent variables that define the configura-tional space, but they can only vary within arestricted range of numerical values due tocertain consistency conditions, which willnow be discussed. Outside this restricted

range, the numerical values do not defineactually existing atomic configurations.Therefore, this restricted range of valueshas been called the existence domain, orconfiguration polyhedron (Kudo and Kat-sura, 1976). The dimension of the configu-rational space depends either on the num-ber of correlations which are taken into ac-count or, equivalently, on the size of thelargest cluster. In the following sections theexistence domains will be investigated forsome simple examples. The method ap-plied closely follows the procedure devel-oped by Finel (1987). For the sake of sim-plicity, the analysis will be restricted to bi-nary alloys. For multicomponent systems,the procedure is analogous but has to becarried out on a computer.

Existence domains are most useful for ananalysis of ground states. This will be shownin Sec. 8.4. The ground-state analysis will belimited to a finite range of interactions (e.g.pair interactions between first and secondnearest neighbors). Therefore, we are partic-ularly interested in the part of the existencedomain that is the subspace of correlationfunctions required for a treatment with fi-nite-range interactions. This existence do-main in the configurational subspace willbe investigated in detail in this section.

8.3.1 F.C.C. Structure, First-Neighbor Interactions

The first step in this analysis is to make achoice for the basic cluster. This choicethen determines the dimension of the con-figurational space to be dealt with. It is natural to start with the simplest possiblebasic cluster, the nearest-neighbor pair. Theconfigurational variables are then the pointcorrelation function x1 = ·s1Ò = xA – xB,which defines the composition, and thenearest-neighbor pair correlation functionx2 = ·s1 s2Ò = 1 – 4 r12

AB (see Eq. (8-7)).

Adopting again the values ti = ±1, whichwere already used in the examples of Sec.8.2, the variables can both vary in the inter-val [–1, 1] (see Fig. 8-3): x1 takes the value1 for pure A and –1 for pure B, x2 takes 1for each pure component and –1 for an al-loy composed only of A–B pairs. Theseranges, however, are not fully accessible,for example because of the obvious con-straints

r12AA 7 0 , r12

BB 7 0 , r12AB 7 0 (8-11)

The constraints in Eq. (8-11) yield the fol-lowing inequalities:

r12AA 7 0 fi 1 + 2x1 + x2 7 0

r12AB 7 0 fi 1 – x2 7 0 (8-12)

r12BB 7 0 fi 1 – 2x1 + x2 7 0


Figure 8-3. Projection of the configuration polyhe-dron of a regular tetrahedron into the plane of corre-lation functions x1 = ·s1Ò = xA – xB and x2 = ·s1 s2Ò fordifferent choices of clusters: (1) domain abg: nearest-neighbor pair; (2) domain abfh: nearest-neighbor tri-angle; (3) shaded area abcde: nearest-neighbor tetra-hedron. The vertices a and b correspond to the purecomponents A and B, c and e correspond to L12 withcompositions B3A and A3B, and d to L10, compositionAB. The dashed line indicates the random configura-tions. This line separates the existence domain into re-gions of (ordered) configurations with preferential for-mation of unlike pairs (lower part) and configurationswith preferential formation of like pairs (upper part).


If the relations in Eq. (8-12) are taken asequalities, they define a triangle in thespace (x1, x2), as shown in Fig. 8-3. Anypoint inside the triangle abg fulfills theserelations. However, these relations are not sufficient to define the existence do-main. In fact, the vertex g, for instance,corresponds to r12

AA = r12BB = 0 (introduce

x1 = ·s1Ò = 0 and x2 = ·s1 s2Ò = –1 into Eq.(8-7)), which defines a configuration builtup with A–B pairs only. Such a struc-ture cannot be formed in the f.c.c. lattice.In this structure, the nearest-neighborbonds form a triangular network, and it isnot possible to form a triangle with onlyA–B pairs. Point g in Fig. 8-3 thus corre-sponds to a physically impossible state.Obviously, there must be further restrictingrelations.

The conditions r i j12&1 do not introduce

new constraints because they are automati-cally fulfilled together with the inequalitiesin Eq. (8-11). For the present choice of ti

values, this is immediately seen from thedefinition in Eq. (8-7). The general casefollows from Eq. (8-2), which yields the re-lation:

r12AA + r12

BB + r12AB + r12

BA = 1 (8-13)

If the constraints in Eq. (8-11) are fulfilled,then Eq. (8-13) is a sum of positive termswhich can only be equal to one if each termis less than or equal to one.

Hence, we have to deduce further in-equalities from higher-order clusters. Letus consider the nearest-neighbor triangle123 as the next cluster, see Fig. 8-1. The

new variable x3 = ·s1 s2 s3Ò must be intro-duced, and the configurational space isnow three-dimensional. The constraints forthe triplet probabilities are

(8-14)r123

AAA 7 0 fir123

AAB 7 0 fir123

ABB 7 0 fir123

BBB 7 0 fi

Because we are only interested in the do-main spanned in the subspace (x1, x2), wehave to eliminate the variable x3. In orderto converse the positive value of all the ex-pressions, this elimination can only be per-formed by additive combinations of the in-equalities. This yields the three expressionsin Eq. (8-12) already obtained from thepairs, plus one more inequality:

1 + 3x2 7 0

Taken as an equality, this equation definesthe line f–h in Fig. 8-3, and we have to ver-ify again if the new vertices f and h corre-spond to physically accessible states. Pointf corresponds to a state with no triangle oftype AAA, AAB, BBB (i.e., a state built uponly with triangles of type ABB). It is easyto see that this is impossible in the f.c.c.structure. Thus this procedure has to becontinued in order to find further restrict-ing conditions.

The next higher cluster that may be con-sidered is the nearest-neighbor tetrahedronr = 1234, see Fig. 8-1. In this case, the newvariable x4 = ·s1 s2 s3 s4Ò needs to be intro-duced. The tetrahedron probabilities resultin the following consistency relations:

1 3 3 1

1 1 1 1

1 1 1 1

1 3 3 1

1

01

2

3

− −− −− −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

x

x

x

7

1 4 6 4 1

1 2 0 2 1

1 0 2 0 1

1 2 0 2 1

1 4 6 4 1

1

01

2

3

4

− −−

− −− −

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

x

x

x

x

7 (8-15)

r1234AAAA 7 0 fi

r1234AAAB 7 0 fi

r1234AABB 7 0 fi

r1234ABBB 7 0 fi

r1234BBBB 7 0 fi

Taken as equalities, these equations definea closed polyhedron in the space (x1,…, x4).Because we are still interested in the subspace (x1, x2) we have to eliminate x3

and x4. This is done by taking suitable ad-ditive combinations of the inequalities inEq. (8-15). The result is the polygon abcdein Fig. 8-3, which differs from the pre-vious one in that the vertices f and h are removed, and the vertices c, d, e areadded. It is easy to see that all these ver-tices correspond to possible atomic con-figurations in the f.c.c. structure; c and ecorrespond to the L12 structure with stoi-chiometric compositions B3A and A3B, re-spectively, and d corresponds to the L10

structure with composition AB. The exis-tence domain is given by the shaded area inFig. 8-3.

The regular tetrahedron is thus the small-est basic cluster that allows us to define theexistence domain in the configurationspace of an f.c.c. lattice with nearest-neigh-bor interactions. Consequently, this tetra-hedron is also the smallest basic clusterupon which a statistical treatment shouldbe based. It is possible, of course, to take ahigher cluster containing the tetrahedron asa subcluster. In such a case, the dimensionof the configurational space is increasedbut the projection of the domain into thesubspace of point correlations and nearest-neighbor pair correlations remains thesame. If we had considered a basic clusterthat does not contain the tetrahedron, wewould have obtained a polygon with somevertices, which again would not corre-spond to accessible states. The procedurewould have to be continued until a clustercontaining the tetrahedron were finally se-lected.

The procedure of obtaining the verticesof the configuration polyhedron outlinedabove becomes more and more inconven-ient with increasing cluster size, i.e., with

an increasing number of correlation func-tions and, consequently, with more in-equalities that have to be taken into ac-count in order to define the existence do-main. Therefore we look for a simplifica-tion. At first we notice that each one of theconsistency relations, when taken as anequality, defines a face of a polyhedron. In the cases studied before (see, for exam-ple, Eq. (8-14) or (8-15)), the number ofconsistency relations was one more thanthe dimension of the configurational space.The polyhedron was thus built by d + 1faces. Therefore, the polyhedron is a sim-plex, which is necessarily closed and con-vex. Generally, if d is the dimension of the space, a simplex is a polyhedron withd + 1 faces. The intersection of d faces defines a vertex, a simplex thus has d – 1vertices. The coordinates of the vertices are obtained by solving the systems of dequations which can be chosen out of thewhole set of d + 1. The simplificationcomes from a very convenient way of ob-taining these coordinates: Finel (1987)showed that the values obtained from these solutions are the same as those ob-tained simply by dividing the elements ineach column of the matrix by the cor-responding values of the first row. The coordinates of the vertices are the elementsin each row following the first column,which always contains unity and does notdefine a coordinate. Performing this proce-dure with the matrix of Eq. (8-14), we directly obtain the coordinates given in Table 8-2. Taking the projection into thesubspace (x1, x2) means we conserve onlythe first two coordinates. This exactly de-fines the polygon abfh, which was foundbefore by means of the elimination proce-dure.

Applying the same procedure to Eq. (8-15) immediately yields the coordinatesof the configuration polyhedron in the



four-dimensional space (x1, …, x4), see Table 8-3. The vertices abcde in Fig. 8-3 ofthe two-dimensional polygon in the sub-space (x1, x2) can immediately be takenfrom Table 8-3.

8.3.2 F.C.C. Structure, First and Second Neighbor Interactions

First and second neighbor interactionscorrespond more closely to the situationencountered in real alloys (Inden, 1977a)than the previous case does. The internalenergy contains one more term, the second-nearest neighbor interaction. Consequentlywe will finally be interested in the exis-tence domain within the subspace of pointand first and second neighbor pair correla-tion functions. The interesting part of theexistence domain will be a polyhedron inthis three-dimensional space.

The analysis begins with the octahedronr = 234567 in Fig. 8-4, which is defined asa basic cluster because it includes bothtypes of interaction. For this cluster, ninecorrelation functions have to be intro-duced:

x1 = ·s2Ò x6 = ·s2 s3 s4 s7Òx2 = ·s2 s3Ò x7 = ·s3 s4 s5 s6Òx3 = ·s2 s7Ò x8 = ·s2 s3 s4 s5 s7Òx4 = ·s2 s3 s4Ò x9 = ·s2 s3 s4 s5 s6 s7Òx5 = ·s2 s3 s7Ò

x1 again corresponds to xA – xB. The exis-tence domain will be a polyhedron in 9-dim. space (x1, …, x9), which is defined bythe consistency relations for the variousconfigurations on the octahedron cluster:

Table 8-3. Coordinates of the vertices of the config-uration polyhedron shown in Fig. 8-3.

Configuration Correlation functions

Vertex x1 x2 x3 x4

Pure A a 1 1 1 1

L12 (A3B) e 1/2 0 –1/2 –1

L10 (AB) d 0 –1/3 0 1

L12 (AB3) c –1/2 0 1/2 –1

Pure B b –1 1 –1 1

i j k

i j k

…

− − − −− − −

− − −− −

− − −……

AAAAAABAAAAABBAAAABAAAABBBBAAABBAAABBBBAABABBBBABBBBABBBBBBB

::::::::::

=r23 7 612

1 6 12 3 8 12 12 3 6 11 4 4 1 0 0 4 1 4 11 2 0 1 0 4 0 1 2 11 2 4 3 8 4 4 3 2 11 0 0 3 0 0 0 3 0 11 0 4 1 0 0 4 1 0 11 −− − − − −

− − − −− − − −− − − −

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟

2 4 3 8 4 4 3 2 11 2 0 1 0 4 0 1 2 11 4 4 1 0 0 4 1 4 11 6 12 3 8 12 12 3 6 1

1

1

2

3

4

5

6

7

8

9

xxxxxxxxx

⎟⎟⎟⎟⎟⎟⎟⎟

70

Table 8-2. Coordinates of the vertices shown in Fig.8-3.

Vertex x1 x2 x3

a 1 1 1

h 1/3 –1/3 –1

f –1/3 –1/3 1

b –1 1 –1

Taken as equalities, these 10 equations de-fine a simplex in the 9-dim. configurationalspace. From these equations, the vertices ofthe simplex can be calculated either bysolving nine selected equations out of thewhole set of 10 or obtained directly byFinel’s method outlined above.

Let us consider the projection of thissimplex into the subspace (x1, x2, x3). Thisprojection is obtained from columns 2–4of the matrix. The result is shown in Fig. 8-5. A verification of whether or not all thevertices correspond to possible atomic ar-rangements is necessary. For example, ver-tex (1/3, –1/3, 1) does not correspond to anexisting state because it would be builtonly with octahedra of type BAAAAB.This is topologically impossible in thef.c.c. structure. Therefore, we have to consider further constraints, for instancethose resulting from the nearest-neighbortetrahedron, analyzed before. We can thususe the previous results. In the space(x1, x2, x3), the previously obtained exis-tence domain would be the rectangularprism built on the basis given by the shadedarea in Fig. 8-3 (see also Fig. 8-6). If wenow take into account the constraints im-posed by both the tetrahedron and the octa-


Figure 8-4. Tetrahedron and octahedron clusters inthe face-centered-cubic unit cell.

Figure 8-5. Projection of the configuration polyhe-dron of an octahedron cluster r = 234567 (see Fig. 8-4) into the subspace of point and first and secondneighbor pair correlation functions: x1 = ·s2Ò,x2 = ·s2 s3Ò, x3 = ·s2 s7Ò, respectively. Not all the ver-tices correspond to existing states. The shaded arearepresents the section for x3 = 0, i.e. plane (x1, x2).Comparison with Fig. 8-3 reveals that this section de-fines a larger domain. This shows that the octahedrondoes not impose the same constraints as the tetrahe-dron cluster.

Figure 8-6. Configuration polyhedron of a nearest-neighbor tetrahedron cluster 1234 (Fig. 8-4) in thespace (x1, x2, x3). This cluster does not contain sec-ond-nearest neighbor pairs. The polyhedron is thus arectangular prism built with the existence domain inFig. 8-3 as basis.


hedron, we will have to intersect the twopolyhedra. The result of the intersection ofFig. 8-5 and Fig. 8-6 is shown in Fig. 8-7,and the coordinates of the vertices aregiven in Table 8-4. It is easy to show thatall the vertices of this polyhedron corre-spond to existing atomic configurations.Consequently, in the f.c.c. structure withfirst and second nearest-neighbor interac-tions, we have to introduce at least the reg-ular tetrahedron and the octahedron as ba-sic clusters. Fig. 8-7 also illustrates the

statement made in the previous paragraphthat if we reduce the interaction range tonearest neighbors and keep the two basicclusters, the projected existence domain inthe space (x1, x2) does not change due tothe introduction of the octahedron, a largercluster than the tetrahedron.

8.3.3 B.C.C. Structure, First andSecond Neighbor Interactions

The treatment of the b.c.c. structure ismuch simpler than that of the f.c.c. struc-ture because an irregular tetrahedron as a basic cluster already contains first and second neighbor distances, as shown inFig. 8-2. With this cluster, we must in-troduce the variable x1 = ·s1Ò, x2 = ·s1s3Ò,x3 = ·s1s2Ò, x4 = ·s1s2 s3Ò, x5 = ·s1s2 s3 s4Ò.This consistency relations are

Figure 8-7. Intersection of the two configurationpolyhedra corresponding to the tetrahedron and theoctahedron, Fig. 8-4 and Fig. 8-5. All vertices of thispolyhedron correspond to existing configurations onthe f.c.c. lattice. The corresponding superstructuresare indicated. Notice the symmetry with respect tothe stoichiometric composition AB. The vertical sec-tions represent the existence domains for a fixed al-loy composition, here AB and A3B.



x1 x2 x3

Pure A (A1) 1 1 1A5B, AB5 ±2/3 1/3 1/3

A3B, AB3 (L12) ±1/2 0 1A3B, AB3 (D022) ±1/2 0 2/3

A2B, AB2 (Pt2Mo) ±1/3 –1/9 1/9A2B, AB2 ±1/3 0 –1/3AB (L10) 0 –1/3 1AB (L11) 0 0 –1

A2B2 0 –1/3 1/3Pure B (A1) –1 1 1

::::::

i j k l

xxxxx

i j k l

AAAABAAABABABBAABBBABBBB

=r1 2 3 4 4

1

2

3

4

5

12

1 4 4 2 4 11 2 0 0 2 11 0 0 2 0 11 0 4 2 0 11 2 0 0 2 11 4 4 2 4 1

1

0

− −−

−− −− −

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

7

Taken as equalities, these equations definea simplex in the 5-dim. configurationalspace. All the vertices correspond to exist-ing atomic configurations. Its projectioninto the subspace (x1, x2, x3) defines theconfiguration polyhedron shown in Fig. 8-8. The coordinates of the vertices aregiven in Table 8-5.

8.4 Ground States

The description of the configurationswill now be used to derive the stability ofthe configurations at T = 0 K. The moststable state, the ground state, is given bythe minimum of the (grand canonical)internal energy U (V, T, mA, mB, …) forgiven values T, V, mi obtained by variationof the internal configurational variables,i.e. point, pair, triplet, etc. correlation func-tions. In practice it is preferred to considerthe (canonical) internal energy U (T, V, NA,NB, …) with the particle numbers Ni asvariables. The grand canonical energy isthen obtained by the Legendre transforma-tion of U:

The general expression for U can bewritten

(8-16)

The summations have to be taken over allequivalent points, pairs, etc. In this formu-lation the higher-order cluster interactions(beyond pairs) are not the total energies ofthese clusters, but correction terms. De-pending on the structure, the clusters of agiven class may share subclusters. Thissharing of subclusters has to be taken intoaccount if total energies are used.

8.4.1 Pair Interactions

The concept of pair interactions has along tradition and has been supported bythe theoretical work of Gautier (1984), Bieber and Gautier (1984a, b, 1986, 1987),and Turchi et al. (1991a, b), who came tothe conclusion that pair interactions play a

˜( ) ,

( ) ( )

( ) , ,

( )

( ) ( ) ( )

U p p V p

p p V p

nm i jni

mj

ijnm

nmq i j kni

mj

qk

ij knmq

n ini

i

= ∑ ∑ ∑ ∑

∑ ∑

+

× + … + m

U U Ni

K

i i==

+ ∑1

m


Figure 8-8. Projection of the configuration polyhe-dron corresponding to the irregular tetrahedronr = 1234 of the b.c.c. lattice (see Fig. 8-2) into thesubspace of point and first and second neighbor paircorrelation functions x1 = ·s1Ò = xA – xB, x2 = ·s1 s3Ò,x3 = ·s1 s2Ò, respectively. All vertices correspond toexisting ordered atomic configurations.



x1 x2 x3 x4 x5

Pure A (A2) 1 1 1 1 1

A3B, AB3 (D03) ±1/2 0 0 1/2 –1

AB (B2) 0 –1 1 0 1

AB (B32) 0 0 –1 0 1

Pure B (A2) –1 1 1 –1 1

8.4 Ground States 539

dominant role in alloys, while higher-ordercluster interactions are of minor impor-tance. This appears to be a good approxi-mation for the class of alloys with negli-gible relaxation effects. Furthermore, paircorrelations can be determined by diffrac-tion experiments (see, e.g., Schönfeld,1999).

Grouping together all the energy contri-butions from positions nm with the samedistance k, we obtain the following resultsfor the internal energy:

where nk defines a point in the k-th neigh-bor shell of an arbitrary position in sublat-tice n taken as the origin, and zn

(k) is the cor-responding coordination number. We ob-tain

and

where position 0 stands for any position inthe crystal taken as the origin. Therefore,the energy is

(8-17)

This expression contains only the isotropicpair probabilities because we assumed isotropy for the pair interactions. Conse-quently, it is not possible to distinguish lro

˜ ( ) ( ) ( )U N z V Nk i j

kij

kk

ij

ii

i=12 0 0∑ ∑ ∑+r m r

n

nnr r

==

10

Li iN N∑ ( ) ( )

nn

nn nr r

==

10

Lk ij k

kijz N z N

k∑ ( ) ( ) ( )

˜ ( ) ( )

,

( ) ( ) ( )

( ) ( )

,

( ) ( ) ( )

(

U z N p p V

N p

V z N

N

L

k

k

i j

i ji j

k

i

ii

k i ji j

kL

k ij

ii

L

k

k

=

=

=

=

=

n

nn n

nn

nn

nn n

n

n

m

r

m

1

1

1

12

12

∑ ∑ ∑

∑

∑ ∑ ∑

∑ ∑

⎧⎨⎩⟨ ⟩

+ ⟨ ⟩ ⎫⎬⎭

+ )) ( )rni

from sro by energetic arguments. In fact,lro results from a topological constraint.Beyond a critical number of unlike bonds,it is no longer possible to arrange thesebonds on a lattice with the constraint tomaintain the equivalence of all latticepoints.

In Eq. (8-17), no reference state is speci-fied. We will refer the energy to the me-chanical mixture of the pure components in the same crystal structure as the alloysand obtain the following expression fromEq. (8-17):

where Dmi = mi – mi0 and W (k) = – 2V (k)

AB +V (k)

AA + V (k)BB. The W (k) are called exchange-

energy parameters. They take positive val-ues for an ordering tendency in the k-thshell and negative values for a separationtendency. In order/disorder problems, weare not usually interested in the size of thesystem, and the quantity DU/N can then betreated. If we fix temperature and pressureand consider only equilibrium states, thenumber of independent chemical potentialsis reduced by one. For convenience we oftentake DmA + DmB = 0 and define an effectivechemical potential m* = –1

2(DmA – DmB).

Finding the ground states is equivalentto finding the minimum of E, which is de-fined as the part of the internal energy thatdepends only on the configurational vari-ables:

˜˜

( ) ( )EU

Nz W x

k

k k

s

k

s s= = (8-19)=

D + ∑ ∑+1

8 1

1

F

D

D D D D

˜ ˜

( )

( ) ( )

( )

( )

( ) ( ) ( ) ( )

U U N z

x V x V N

N N

Nz W

Nz W

k

k

k k

ii

i

k

k k

k

k kk

= (8-18)

=

A AA( )

B BB( )

= A, B

A B A B

−

× + −

+ + − ⟨ ⟩

− + ⟨ ⟩

∑

∑

∑ ∑

12

2 8

8 8

00

0

0

m r

m m m m s

s s

with xs = ·s0Ò, ·s0 s1Ò, …, ·s0 skÒ and

8.4.1.1 Ground State Energies

Eq. (8-19) shows how the energy E de-pends on the configurational variables xs ,that is, on the composition by x1 = ·s0Ò =xA – xB, and on the pair correlation func-tions by x2, x3, … . Among the variety ofpossible states, there are also those with thelowest energy Emin. These states will nowbe determined using a geometrical inter-pretation of Eq. (8-19).

Firstly, Eq. (8-19) is a linear relation. Thisimplies that all those values xs which fulfillEq. (8-19) define a (hyper)plane in the con-figurational space. It will now be shownthat the energy E is proportional to the dis-tance of this (hyper)plane from the origin.

To demonstrate this, we introduce thevector x = (x1, x2, …, xk–1) and a unit vec-tor n = F /|F | = |F |–1 (m*, z(1) W (1)/8, …,z(k) W (k)/8). Eq. (8-19) can then be writenin vector form:

(8-20)

where d is the distance of the hyperplanefrom the origin. This distance takes posi-tive or negative values depending on thedirections of the two vectors n and x. Thefactor |F | depends only on the chemicalpotential, the exchange energies and thecoordination numbers, that is, it defines thealloy under consideration. If we change thechemical potential (which controls thecomposition) and/or the energy parame-ters, then the direction of n changes (ex-cept if we change all the components Fi bya common factor), as does the value of d.

If, on the other hand, we consider agiven alloy, which means we keep vector Ffixed but vary the values of x, then the nor-mal vector n does not change. This means

( )˜

x n|| |

= =E dFF

Fsk kz W z W= m*, , , .( ) ( ) ( ) ( )1

818

1 1 …

that in this instance the planes of constantE are parallel to each other, and it followsimmediately from Eq. (8-20) that the ex-trema of E must correspond to extreme val-ues of x. Because the origin is inside theexistence domain for the present choice ofti = ±1, the value Emin must be negative.Searching the ground state thus becomesequivalent to finding, for a given alloy(i.e., for a given n = F /|F |), the values of xthat yield the most negative value for thedistance d from the origin. If we recall thatthe existence domain is a closed polyhe-dron, it follows that the ground states mustcorrespond to vertices of the polyhedron.This will now be illustrated with some ex-amples that have already been consideredin previous sections.

8.4.1.2 F.C.C. Structure,First-Neighbor Interactions

With first-neighbor interactions in thef.c.c. structure, the energy E depends only onthe compositional parameter x1 = xA – xB,and on one configurational parameter, thecorrelation function of nearest-neighborpairs x2 = ·s0 s1Ò = 1 – 4r01

AB (see Eq. (8-7)), where position 1 is a nearest-neighborsite to a central site 0. Therefore, the exis-tence domain in Fig. 8-3 is relevant to thiscase. It follows from the foregoing discus-sion that the ground states are given by thevertices in Fig. 8-3 (i.e., pure A, pure B,L12 with compositions A3B and AB3, andL10 with composition AB). Fig. 8-9 illus-trates this point with some examples.

If, for example, F1 = m*/|F | = 0, then thevector F = (0, F2) and, consequently, n isparallel or antiparallel to the x2-axis, de-pending on the sign of F2 = ––12

8W (1). The

lines E = const. are then parallel to the x1-axis. This can also be seen from Eq. (8-20),which reduces to

F2 x2 = E



These lines are shown in Fig. 8-9. If weconsider an alloy with ordering tendency(W (1) > 0 ¤ F2 > 0), the value Emin isreached for the most negative x2 value,which is x2 = –1/3. This point correspondsto the L10 structure AB. If we consider analloy with separation tendency (W (1) < 0¤ F2 < 0), the value Emin is reached for themost positive x2 value, which is x2 = 1. Inthis case, the line E = Emin coincides with aside of the configuration polygon. This linejoins the vertices corresponding to the purecomponents A and B. The ground state isdegenerate: it could be pure A and pure Bin any proportion juxtaposed to each other,or any other arrangement of A and Bwhich, in the limit of N Æ ∞, does not form A–B bonds (e.g., constant number of2-dim. boundaries).

Next, we may consider the case F2 = ––12

8W (1) = 0, which corresponds to an

ideal solution. The lines E = const. are par-allel to the x2-axis in this case, because n isparallel to the x1-axis. Eq. (8-20) now re-duces to

F1 x1 = E

For a positive value of F1 = m* =–12

(DmA – DmB), the most negative value of E is obtained for x1 = (xA – xB) = –1, i.e.,corresponding to pure B, and a negativevalue of F1 corresponds to pure A.

The procedure outlined above can be re-peated for any direction of the vector F,see Fig. 8-9. As long as the lines of con-stant E pass through a vertex of the poly-gon, a unique ground state can be defined.If these lines coincide with the sides of thepolygon, the ground state is degenerate. Itcould then be any mixture of the two statescorresponding to the two vertices definingthe side of the polygon. It was mentioned

Figure 8-9. Lines of constant grand canonical energy E (Eq. (8-20)) for given sets of values F1 = m* =–12 (mA – mB) and F2 = –3

4 W (1) in the case of the f.c.c. lattice and nearest-neighbor interactions. The configurationpolygon is that of Fig. 8-3. Extrema of the energy correspond to lines passing through the vertices of the poly-gon. Depending on the sign of F1 and F2 the line through a vertex corresponds to a minimum or a maximum.The vertices thus define the ground states. For particular F1, F2 values the lines coincide with the sides of thepolygon. For these values, the ground state is degenerate.

before that the proportion of the two statesdiffering in composition is not defined because in these cases of degeneracy theoverall composition of the system is notdefined by a fixed value of the chemicalpotential. We can say, however, that thestate is two-phase.

Finally, we can determine a so-calledground-state diagram, which is shown inFig. 8-10. It is worth noting that the single-phase fields are sectors in the (F1, F2)plane limited by lines originating from theorigin. It is thus sufficient to know the di-rections of those lines. Furthermore, the di-agram is symmetric with respect to F1 = 0.

When higher-order interactions are in-volved, as in the next paragraph, the deter-mination of the ground-state diagram is notthat simple, and it is useful to have a moreanalytical method for its determination.Such a method will now be introduced and,for the purpose of illustration, will be ap-plied to the present very simple case.

The ground-state diagram displays theranges of F1 and F2 values for which agiven state, e.g., L10, is stable. We thusinterpret Eq. (8-19) inversely to that of Eq.(8-20): given a vertex in the configura-tional space, e.g., vertex xv = (–1 3, 0) cor-responding to L10, we define a unit vectorn* = xv /|xv |. Eq. (8-19) can then be rewrit-ten as

(F |n*) = E /|xv | = d* (8-21)

In a method similar to that used with Eq.(8-20), we interpret Eq. (8-21) as the equa-tion of a line in the space (F1, F2) with adistance d* = E /|xv | from the origin. If wefix the value of E, then all vectors F ful-filling Eq. (8-21) must end in the line withthe normal n* and with distance d*. In Fig.8-10, the result is shown for E = –1 (energyunits). It shows that the vectors fulfillingEq. (8-21) describe a polygon. The coordi-nates of the vertices of this polygon aregiven in Table 8-6. This polygon is the dualpolygon to the configuration polygon (Finel, 1987), and the space (F1, F2) iscalled the dual space. A face of the dualpolygon corresponds to each vertex of theconfiguration polygon in the direct space(x1, x2). All vectors F pointing to the sameface of the polygon define alloys that ex-hibit the same ground state given by the as-sociated vertex of the configuration poly-gon. If we vary E, then the polygon shrinksor blows up, respectively, and the verticesof the dual polygon vary along the linesthat define the two-phase states in Fig. 8-10. It is easy to verify that these lines are


Figure 8-10. Ground-state diagram and dual poly-gon (to Fig. 8-9) for the case of an f.c.c. lattice withnearest-neighbor interactions. The axes areF1 = m* = –1

2 (mA – mB) and F2 = –34 W (1). The solid lines

are radius vectors which subdivide the plane into re-gions with a defined ground state. The directions ofthese vectors are defined by the vertices of the dualpolygon to Fig. 8-3. It corresponds to a fixed value ofthe energy, here E = –1 (energy units), and is ob-tained from Eq. (8-21). The dual polygon alreadycontains complete information on the ground-statediagram. For F1, F2 values corresponding to theselines, the ground state is degenerate.


vertical to the faces of the configurationpolygon in Fig. 8-3.

The generalization of these results tohigher dimensions is straightforward. A dualpolyhedron defined by Eq. (8-21) is asso-ciated with the (direct) configuration poly-hedron in such a way that each vertex of thedirect polyhedron corresponds to a face of thedual polyhedron, and vice versa. In order toget the vertices of the dual polyhedron, wehave to solve Eq. (8-21) simultaneously forthe number of vertices of the direct polyhe-dron required to define the (hyper)plane.The application of this in three dimensionswill be shown in the next two paragraphs.

8.4.1.3 F.C.C. Structure, First and Second Neighbor Interactions

With first and second neighbor interac-tions in the f.c.c. structure, the energy Edepends on one more configurational pa-rameter, the second neighbor pair corre-lation function x3 = ·s0 s2Ò = 1 – 4r02

AB (seeEq. (8-7)). The relevant existence domainis now that shown in Fig. 8-7. We can againproceed as previously.

Let us first consider F3 = ––128

W (2) = 0.The normal vector n corresponding to theplanes of constant energy E is perpendicu-lar to the axis x3 in this case. If we consideran alloy with an ordering tendency in thefirst shell, F2 = ––12

8W (1) > 0, then n is

oriented in the half space x2 > 0, and theplanes corresponding to Emin are thosepassing through the vertices pure A, L12

with compositions A3B or AB3, L10 withcomposition AB, and pure B, depending onthe value of x1 = m* = –1

2(DmA – DmB). This

is in accordance with the results alreadyobtained in Sec. 8.4.3 (see Fig. 8-10).However, an additional result can be de-duced from Fig. 8-7: the energy planes passsimultaneously through the vertices corre-sponding to the structures D022 and L12

(both either A3B or AB3), as well as L10

and A2B2. The ground states are thus de-generate in these instances.

The complete ground-state diagram isobtained from the determination of the dualpolyhedron associated with the existencedomain of Fig. 8-7. This dual polyhedron isshown in Fig. 8-11. All F vectors pointingto (or piercing) a given face of the dual

Figure 8-11. Dual polyhedron (to Fig. 8-7) for the case of an f.c.c. lattice with first and second neighbor inter-actions. The axes are F1 = m* = –1

2 (mA – mB), F2 = –34 W (1), and F3 = –3

4 W (2). The dual polyhedron defines theground-state diagram and corresponds to a constant energy E = –1 (energy units). All radius vectors pointing tothe same face correspond to the same ground state, which is labeled for each face, except the top, which is L11.Vectors coinciding with the boundaries of the faces correspond to degenerate ground states.

polyhedron define alloys with the sameground state, which is defined by the corre-sponding vertex of the configuration poly-hedron. The ground-state diagram thus di-vides the dual space into (hyper)conesformed by the vectors from the origin to theborderlines of the faces. Each cone definesa ground state. Due to graphical limita-tions, those cones are not shown in Fig. 8-11, but the ground states are labeled at each face of the dual polyhedron. The

coordinates of the dual vertices are given inTable 8-6. They have been derived by inserting the vertices in Table 8-3 into Eq.(8-21) for Emin = –1.

8.4.1.4 B.C.C. Structure, First andSecond Neighbor Interactions

In the first and second neighbor interac-tions of the b.c.c. structure, the energy Edepends on the same variables (x1, x2, x3)as in the preceding example, and the rele-vant existence domain is that shown in Fig.8-8. We can now construct the dual polyhe-dron, solving Eq. (8-21) for E = –1 and us-ing the vertex coodinates in Table 8-4. Thecoordinates of the dual vertices are given inTable 8-6, and the dual polyhedron isshown in Fig. 8-12.

8.4.1.5 Energy Minimum at ConstantComposition

From a metallurgical point of view itseems more natural to consider the com-position of a system an independent vari-able which we can control, rather than the


Table 8-6. Coordinates of the dual polyhedra shownin Figs. 8-10, 8-11 and 8-12.

Tetra- Tetrahedron– Tetrahedronhedron Octahedron b.c.c.f.c.c. f.c.c.

F1 F2 F1 F2 F3 F1 F2 F3

2 1 ±2 1 0 0 0 –12 3 ±2 3 0 ±2 1 0

–2 3 ±18/7 12/7 3/7 ±2 2 1–2 1 ±2 0 1 ±2 0 10 –1 ±2 4 1 0 –2 1

0 –2 10 0 –1

Figure 8-12. Dual polyhedron (to Fig. 8-8) for the case of an b.c.c. lattice with first and second neighbor inter-actions. The axes are F1 = m* = –1

2 (mA – mB), F2 = W (1), and F3 = –34 W (2). The dual polyhedron defines the

ground-state diagram and corresponds to a constant energy E = –1 (energy units). All radius vectors pointing tothe same face of the polyhedron correspond to the same ground state, which is labeled for each face, except thetop one, which is B32.


chemical potential. However, this is notpossible. We can, of course, control theoverall composition of our system by tak-ing the appropriate amounts of the puresubstances, closing the system against ma-terials exchange, and producing a certainconfiguration (e.g., by heat treatment). Ifwe thereby equilibrate the system at agiven temperature, the system will attainits energy minimum by internal reactions.These reactions can be the formation of or-dered atomic arrangements (homogeneoussingle-phase state) or the formation of het-erogeneous microstructures. In the lattercase, we have to introduce new variables inaddition to the configurational variables,for example, the volume of producedphases, describing the advance of these re-actions. At equilibrium, the driving forcefor the reactions must be zero, that is, thechemical potentials of all the reaction prod-ucts must be the same. Because the overallcomposition is fixed, the amount of thephases present is also fixed. This is differ-ent from a grand canonical treatment. Theenergy in this case is given by the canoni-cal energy E, which is given by the expres-sion

(8-22)

This equation is similar to Eq. (8-19), ex-cept that the number of variables is reducedby one. The geometrical considerations canbe applied as before. In Fig. 8-7, two sec-tions corresponding to the compositionsx1 = xA – xB = 0 and 0.5 (AB and A3B) areshown as shaded areas. Since in this in-stance we are confined to a planar section,the loci of constant energy are lines, andthe minimum energy corresponds to thelines passing through the vertices of thepolygon. We will consider the case x1 = 0.5in detail, and we thus have to consider thepolygon SLDMK in Fig. 8-7.

E E x xs

k

s s= ==

˜ −+

∑1 12

1

F F

The states corresponding to L and D(L12 and D022) are ground states becausethey are vertices of the configuration poly-hedron. S is not a vertex in this figure, butlies on the line joining the states pure Aand pure B. This line corresponds to con-figurations that do not contain any A–Bpairs. There is no other configuration pos-sible except for pure A and pure B taken to-gether in the proportion A3B. Of course,the planar interface, which is necessarilyformed, contains A–B pairs. In the thermo-dynamic limit of N Æ ∞, however, the rela-tive number of these pairs approaches zeroand can be neglected.

The point K lies on the line joining thevertices pure A and L11 (AB). Here againthe two limiting phases can be taken to-gether in the right proportion to build up atwo-phase configuration at point K. Thepossibility of another homogeneous ar-rangement needs to be verified, though. ButK is not a vertex of the configuration poly-hedron in Fig. 8-7. Hence, an additionalconfiguration cannot be characterizedwithin the configurational space created by the basic clusters of the octahedron andtetrahedron.

The point M is on the line joining thevertices A2B and A5B. Hence, at first, amixture of the two limiting phases can beattributed to this point. But a homogeneousconfiguration also lies here, the structureclassified Amm2 in the appendix, whichhas been observed in Monte Carlo simu-lations performed at fixed composition (Gahn, 1982). This structure is built upwith alternating cube planes which are oc-cupied either by pure A or by an equalamount of A and B in an arrangement iden-tical to the cube planes in the L11 structure,see Fig. 8-13. It is interesting to notice thestructural relationship between this Amm2structure at point M and the arrangement atK (pure A plus L11 (AB)): at M the stack-

ing sequence is one cube plane of pure Afollowed by one cube plane of L11. If weincrease the stacking distance (i.e., thenumber of cube planes of each phase), thecorresponding configurational point movesfrom M toward K. At K, the thickness ofeach package of cube planes approachesinfinity. The displacement of the configu-rational point is caused by the decreasingnumber of AB pairs that cross the “inter-faces” between the two structural units.

8.4.1.6 Canonical Energy of lro States

Instead of imposing a given composi-tion, we may also impose the state of order(e.g., the most perfect lro) and calculate thecanonical energy for these states as a func-tion of composition. These are most stablestates under the constraint of homogeneity.Consider the f.c.c. structure with nearest-neighbor interactions and the b.c.c. struc-ture with first and second neighbor interac-tions as simple examples. In Table 8-1, thepoint probabilities for most perfect lro (atoff-stoichiometric composition) are given.From these probabilities, the numericalvalues of the correlation functions for thedifferent superstructures in their most per-fect lro state can be derived directly using

Eq. (8-5). These values are given in Table8-7 for the b.c.c. structures. Introducingthese values into the expression for the ca-nonical energy

yields the energies of formation of the var-ious lor phases. In Fig. 8-14a, b, the ener-gies of formation of b.c.c. structures areshown for two particular choices of inter-change energies. It can be seen in Fig. 8-14a that the canonical energy of the moststable lro phase for an interchange energyratio W (2)/W (1) = 0.5, namely off-stoichio-metric D03, is a polygon. This means thatall the homogeneous ordered states that canbe formed at every composition are degen-erate with the two-phase mixtures A + D03

and D03 + B2. This is different from Fig. 8-14b where W (2)/W (1) = –1. The canoni-cal energy of the most stable lro phase, off-

D DU U N xN

W W

NW

NW

=

= (8-23)

˜

[ ]

[

]

[

( ) ( )

( )

( )

−

− +

+ ⟨ ⟩ + ⟨ ⟩+ ⟨ ⟩ + ⟨ ⟩

+ ⟨ ⟩ + ⟨ ⟩

1 1

1 2

11 3 1 4

2 3 2 4

21 2 3 4

88 6

4

38

F

s s s ss s s s

s s s s


Figure 8-13. Atomic distribution on four subse-quent (0 0 1) planes of the superstructures L11 (AB)and Amm2 (A3B). The structure Amm2 is composedof alternating (0 0 1) planes of pure A and L11.

Table 8-7. Numerical values of the correlation func-tions for the b.c.c. superstructures in their most per-fect lro state.

B2 D03 D03 B32(xBÙ0.5) (xBÙ0.25) (0.25ÙxBÙ0.5) (xBÙ0.5)

·s1Ò 1 1 1 1·s2Ò 1 1 1 1–4xB

·s3Ò 1–4xB 1 3–8xB 1·s4Ò 1–4xB 1–8xB –1 1–4xB

·s1s3Ò 1–4xB 1 3–8xB 1·s1s4Ò 1–4xB 1–8xB –1 1–4xB

·s2s3Ò 1–4xB 1 3–8xB 1–4xB

·s2s4Ò 1–4xB 1–8xB –1 (1–4xB)2

·s1s2Ò 1 1 1 1–4xB

·s3s4Ò (1–4xB)2 1–8xB – (3–8xB) 1–4xB


stoichiometric B2, is a concave curve. Inthis instance, the most stable state is a two-phase mixture of A + B2. From the canoni-cal energy, we are thus given a hint ofwhether the ground state will be two-phase,or if there is a degeneracy with single-phase states. The canonical energy for thef.c.c. lro structures L12 and L10 can be de-rived in a similar way. The result is shownin Fig. 8-14c for the ratio W (2)/W (1) = –1.Here again, the canonical energy of themost stable lro phase is concave, and onlythe two-phase mixtures A + L12 andL12 + L10 are ground states.

8.4.1.7 Relevant Literature

The ground state analysis of f.c.c. andb.c.c. alloys with first and second neigh-bors was done by Kanamori (1966), Rich-ards and Cahn (1971), Allen and Cahn(1972, 1973), and de Ridder et al. (1980).Multiplet interactions were included byCahn and Kikuchi (1979), Sanchez and deFontaine (1981). The ground states of thef.c.c. lattice with up to fourth neighborinteractions were determined by Kanamoriand Kakehashi (1977), fifth neighbor inter-actions were treated by Kanamori (1979),Finel and Ducastelle (1984) and Finel(1984, 1987). The ground states of the hex-agonal lattice with anisotropic interactionshave been deduced by Crusius and Inden(1988) and Bichara et al. (1992a) from theknown ground states of the planar hexago-nal lattice with up to third neighbor interac-

Figure 8-14. Canonical energy DU of different b.c.c. and f.c.c. long-range ordered structures as afunction of composition for fixed values of the inter-change energies. (a) b.c.c., ordering tendency be-tween first and second neighbors: W (1) = 2W (2) > 0.(b) b.c.c., ordering tendency between first neighborsand separation tendency between second neighbors:W (1) = –W (2) Û 0. (c) f.c.c. ordering tendency be-tween nearest neighbors: W (1) > 0, W (2) = 0.

tions (Kudo and Katsura, 1976; Kanamori,1984).

8.4.2 Effective Cluster Interactions(ECIs)

Up to this point only pair interactionshave been considered with the exchangeenergies W (k) as dummy parameters. Theinternal energy DU, with reference to thepure components in the same crystal struc-ture, has been written in terms of an expan-sion of these W (k) (Eq. (8-19)). This is aspecial case of the more general cluster ex-pansion (CE) of the internal energy DU interms of the correlation functions:

(8-24)

where n refers to subclusters of the basiccluster a (n = 1 for point, n = 2 for pairclusters, etc.) (Sanchez et al., 1984). Thisequation is exact if a represents the entiresystem. It is expected, however, that Eq.(8-24) converges rapidly such that the se-ries can be truncated. For a general discus-sion of the method see Laks et al. (1992),de Fontaine (1994) and Sanchez (1996).

The parameters Vn are supposed to em-brace all possible energy sources that dependon configuration, including all ground-state features like electronic energy, long-range elastic and Coulombic interactions.But they might also contain configuration-dependent excitation energies such as thosearising from vibrational and electronic ex-citations when finite temperatures are con-sidered, making the model interaction en-ergies temperature dependent. It should beappreciated that the parameters Vn repre-sent energy contributions, not only from di-rect interactions within the cluster, but alsofrom the interactions of much longer rangeoutside the cluster range. They are thuscalled effective cluster interactions (ECIs).

DUN

V xn n=0

a

∑

It may be useful to spell out explicitlythe pure configurational part of the Vn . Inthe case of pair interactions the Vn withn ≥ 2 are equivalent to the pair exchange

energies, i.e., In this case

the specific lattice type is only introducedin terms of the coordination numbers.

In the case of higher-order cluster inter-actions the lattice type is not only specifiedby the coordination numbers of the basiccluster and its subclusters, but most impor-tantly by the type of clusters they sharewith each other. Subclusters which are notshared define no new correlations. They arealready taken into account by those of theirparent cluster. We could thus expect that theVn of such clusters do not show up in theexpansion, for example triplet correlationsin the tetrahedron expansion of the f.c.c.lattice. This, however, is not true becausethe Vk in the expansion, Eq. (8-24), not onlycontain terms from the k-point clusters, butalso terms from higher clusters (n > k).

This becomes transparent if the Vn areexpressed in terms of the cluster exchangeenergies, defined as

(for example, W 1234ABBB = – 4VABBB + VAAAA +

3VBBBB). It should be noted that, unlike thepairs, these cluster-exchange energies donot represent the total energy of the clus-ters, only successive corrections to thecluster energies. If total energies were con-sidered, the sharing of subclusters wouldhave to be taken into account.

Considering the f.c.c. lattice and a trun-cation at the regular tetrahedron cluster(a = 4), Eq. (8-24) gives:

(8-25)

D ˜ ( )( )

( ) ( ) ( )

UN

V V x x

V x V x V x

j

j j j

= A B0 1

2 2 3 3 4 4

+ −

+ + +

W rV Vij lr

i j lk i

l

k k k……

… …− + ∑12 ==

Vz

Wn

nn=

( )( ) .

8



where j indicates specific states, e.g. thefive vertices a–e of the configuration poly-hedron in Table 8-3. The Vn expressed interms of the cluster exchange energies aregiven by

Taking for example all the triplet termsWijk = 0 does not mean V3 = 0, becausethere remain tetrahedron terms.

In the irregular tetrahedron approxima-tion of the b.c.c. lattice, the cluster expan-sion of the internal energy is written as:

(8-27)

where the CE parameters Vn correspond to:

V

zW

zW

0

11

22

12

8 8

= A B

AB AB

( )

( )( )

( )( )

D Dm m+

− −

D ˜ ( )( ) ( )

( ) ( ) ( )

UN

V V x x V x

V x V x V x

j j

j j j

= A B0 1 2 2

3 3 4 4 5 5

+ − +

+ + +

Vz

W

zW W

zW W W

Vz

W W

zW W

V z Wz

W

0

11

1

21 1

12 8

24

6432

12 24

642 2

18 24

=

=

=

A B

triplet

BAA BBA

tetra

BAAA BBAA BBBA

A B

triplet

BAA BBA

tetra

BAAA BBBA

triplet

( )

( )

( ) ( )

( )

(

( )( )

( ) ( )

D D

D D

m m

m m

+ −

− +

− + +⎛⎝

⎞⎠

− − −

− −

+ BAABAA BBA

tetra

BBAA

triplet

BAA BBA

tetra

BAAA BBBA

tetra

BAAA BBAA BBBA

=

(8-26)

=

+

+

−

+ −

− +⎛⎝

⎞⎠

W

zW

Vz

W W

zW W

Vz

W W W

)

( )

( )

( )

643

24

642 2

6432

3

4

These relations show that cluster interac-tions not only influence the correlations oftheir own cluster but also those of the sub-clusters. The system of equations, Eq. (8-24), can be solved for the different Vn . Theidea is that the ECIs derived from the ener-gies of ordered configurations are appli-cable to any configuration, i.e. for lro, sroand for the disordered state. This is equiva-

zW W

W Wz

W W

W W

Vz

W W

1

722

2

2564 4

2 4

12 72

2

=

triplet

ABA AAB

BBA BABtetra

BAAA BABA

BBAA BBBA

A B

triplet

ABA AAB

(

)

(

)

( )

(

D Dm m

− ++ +

− ++ +

− −

⋅ + −− −

− −

+ +

+

+

⋅ − − +

+ −

W W

zW W

Vz

Wz

W W

zW

Vz

Wz

W W W W

zW W

Vz

BBA ABB

tetra

BAAA BBBA

AB

triplet

AAB BBA

tetra

BBAA

AB

triplet

ABA AAB BBA BAB

tetra

BABA BBAA

triplet

=

=

=

2

2568 8

8 722 2

2568

8 722 2

2568 4

2

11

3

22

4

)

( )

( )

( )

( )

( )

( )( )

( )( )

7272

2 2

2568 8

2564 4

2 45

( )

( )

(

)

⋅ + − −

+ −

−− +

W W W W

zW W

Vz

W W

W W

ABA AAB BBA AAB

tetra

BAAA BBBA

tetra

BAAA BABA

BBAA BBBA

=

(8-28)

lent to saying, in the pair approximation,that the W (k) can be used for all configura-tions. Since the only source of error in Eq.(8-24) is the truncation, it is imperative tocarefully select the truncation.

The ingredients for the determination ofthe ECIs are values of DU(j), which maybe obtained from various sources (see e.g. Zunger, 1994). The simplest approachis to take experimental data of the corre-sponding quantities (e.g. Inden (1975a, b,1977a, b; Oates et al., 1996)). This requiresa truncation at a level imposed by the avail-ability of data and in most cases does notprovide satisfactory results.

The second approach is to derive DU(j)from first principles calculations, treatingthe random alloy by the Coherent PotentialApproximation (CPA) and the electronicband structure with the Tight-Bindingmethod (Gautier et al., 1975; Ducastelleand Gautier, 1976; Treglia and Ducastelle,1980; de Fontaine, 1984; Sigli and San-chez, 1988; Sluiter and Turchi, 1989a, b)or with the Korringa, Kohn and Rostokermethod (Gyorffy and Stocks, 1983; John-son et al., 1990). The approaches includethe Generalized Perturbation Method andthe Concentration Wave Method. The or-dering contributions are treated as pertur-bations of the disordered states. The pertur-bation method is limited to alloys with sim-ilar atomic species and provides essentiallypair interactions.

The third approach is the inversionmethod of Connolly and Williams (1983)(Ferreira et al., 1988, 1989; Wei et al.,1990, 1992; Terakura et al., 1987) . In thisapproach special atomic configurationslike in ordered compounds are selected andtheir total energy is obtained from directelectronic structure calculations.

It has been pointed out by Zunger (1994)that it is necessary to account for two typesof relaxation, “volume relaxation” when

molar volumes are not constant, and “localrelaxation” when atoms deviate from theirideal lattice positions. Ferreira et al. (1987,1988) and Wei et al. (1990) have shownthat, if the volumes depend on composi-tion but only weakly upon configuration,the energy of formation can be split intotwo additive terms, the elastic energy re-quired to hydrostatically deform the pureconstituents from their equilibrium vol-umes to the alloy volume, and the pureconfigurational or “chemical” term. TheECIs can be determined either from a setof as many ordered compounds as there areunknowns, or from a larger set, in whichcase the ECIs are obtained by a fitting pro-cedure (Lu et al., 1991, 1995). It must beemphasized that the ECIs are “effective”and their values depend on the level oftruncation.

Cluster expansions for systems withstrong lattice relaxations converge slowlyin real space. This has led to an alternativeroute, treating the CE in the reciprocalspace (e.g. Laks et al., 1992).

Finally, it should be mentioned that pairinteractions may also be obtained from thediffuse scattering of X-rays or neutrons(for a recent review, see Schönfeld, 1999).The procedure is the inverse of the CVM(see Sec. 8.5.1) or the MC technique (seeSec. 8.5.3). Instead of calculating equilib-rium configurations using given energy pa-rameters, experimental data of sro are takenas input in order to obtain the interactionparameters. The methods have been pre-sented by Priem et al. (1989a, b) for CVM,and Gerold and Kern (1987) and Livet(1987) for MC.


8.5 Phase Equilibria at Finite Temperatures 551

8.5 Phase Equilibriaat Finite Temperatures

Two ways of determining the equilibriaat finite temperatures have proven to bemost useful (see also the chapter by Binder,2001), CVM introduced by Kikuchi (1950,1951) and the MC technique (Binder, 1979,1986; Binder and Stauffer, 1984; Mourit-sen, 1984; Binder and Heermann, 1988).

The CVM is based on an analytical cal-culation of the configurational entropy S.The equilibrium states at constant volumeand temperature are obtained by a mini-mization of the Helmholtz energy

F (V, T, Ni) = U – T S

for the equilibrium at fixed composition, orby minimization of the grand potential

W (V, T, mi) = U – T S – Âi

mi Ni

for the equilibrium with exchange of atoms,i.e., for given values of the chemical poten-tials.

The MC method simulates the configu-rations in a computer crystal. At a giventemperature and fixed chemical potentials,atoms are exchanged with a reservoir of at-oms with a probability defined in such away that the equilibrium state is reachedafter a sufficient number of atomic replace-ments. In the case of fixed composition, at-oms are selected pairwise and interchangedwith each other according to an equivalentprobability. This method yields the equilib-rium configuration but not the thermody-namic functions. On the other hand, we ob-tain complete and fine-scale informationabout the atomic configurations and thecorrelations at large distances, the limita-tion being imposed only by the size of thecomputer crystal. Both methods will be ap-plied in this section.

8.5.1 Cluster Variation Method

In the CVM, the entropy is evaluated ac-cording to S = kB ln W, where W is thenumber of arrangements that can be formedfor given values of the correlation func-tions. The CVM develops an approximateexpression for W, taking into account onlythe correlations up to the basic cluster size.Therefore, the basic clusters are consideredto be uncorrelated independent species.The first step in determining the entropy isto write W as the number of possible ar-rangements of this basic cluster:

(8-29)

where Na is the total number of a-clustersin a system with N points, and Na (s) is thenumber of a-clusters with a given configu-ration specified by s ; the term aNa

is ashort notation which will be used hereafterfor abbreviation. The ratio Na (s)/Na willbe indicated by ra

(s).Eq. (8-29) overestimates the number of

arrangements: two overlapping a-clusterscannot be permuted independently becausethey must fit together with their over-lapping units. This becomes obvious if we look at the high temperature limit of Eq.(8-29), which is W∞ = 1Na

, while the exact expression of the limit of W is givenby the number of configurations of Npoints: W∞ = 1N .Therefore, a correction isneeded in order to obtain the correct high-temperature limit. Correcting for the pointswould correspond to the generalized quasi-chemical approximation (Yang, 1945; Yangand Li, 1947, 1949). Based on geometricalconsiderations, Kikuchi (1950, 1951) intro-duced a correction in such a way that thehigh temperature limit is not only obtainedfor the point cluster but also for all subclus-ters of a. A short time later Barker (1953)

WNN N=

!=

def

a

aa

a

![ ( )]

ssss∏

arrived at the same result by a more mathe-matical treatment, and the continued inter-est in the derivation of the entropy formulais manifested in a number of papers, e.g.,Hijmanns and de Boer (1955), Burley (1972),Sanchez and de Fontaine (1978), Gratias etal. (1982) and Finel (1987). For a dis-cussion of the hierarchy of the cluster ap-proximations, see Schlijper (1983, 1984,1985).

Following Barker, we define ma as thenumber of a-clusters per point: Na = ma N.Then we can write (in Stirling’s approxi-mation):

We now want to correct for the overlapwith the first subcluster a – 1, which hasone point less than a. Due to the indepen-dent variation of the a-clusters, the over-lapping cluster a – 1 has also been counted

times, where is thenumber of times the a – 1 cluster is con-tained in a. The correct number of timesthe a – 1 cluster should be counted, pre-suming it were an independent species, is

. Therefore, we must correctEq. (8-29) as follows:

Using the identity aa =1, which is valid forthe basic cluster, and aa –1 determined fromthe equation

W can be written as

W Nm a

Nm a= ( ) ( )a aa a a a− − −1 1 1

m a m m na a a a aa

− − −−−1 1 1

1=

W Nm N

m

Nm n

Nm

Nm m n

=

=

( )( )

( )

( ) ( )

a aa

a a

aa

a aa

a a a aa

−−

−

−

−

−−−

1

1

1

1

1

11

Nm( )a a− −1 1

naa −1

Nm n( )a a a

a−

−1

1

( )

( )( )

(( )

( )

( ) ( )ar r

a

a

a

a

a

a

a a aN

m

m

Nm

m N

m N

N

N=

!!

=!

!

=ss

ss

ss

ss∏ ∏

The same reasoning now needs to be fol-lowed with the next subcluster a – 2. Forthis cluster, the number of arrangements ofthe basic cluster a, as well as the correctionterm for a – 1, have already been counted.We can therefore write

with aa –2 determined from the equation

Continuing this reasoning leads to

(8-30)

and to the following recursion scheme forthe exponent an , using the identities nn

n =1and aa =1:

(basic) cluster a: ma = ma naa aa

We thus obtain for the entropy in Stirling’sapproximation

(8-31)

The values of ma and na –ma –n are to be de-

rived by geometrical considerations of thelattice. This procedure is straightforwardeven for large clusters. For the purpose ofillustration, the full set of these values andthe “CVM exponents” an are given in Table8-8 both for the f.c.c. structure in the tetra-hedron–octahedron approximation and forthe b.c.c. structure in the irregular tetra-

S k N m a= B=

− ∑ ∑n

a

n n n nr r1 ss

ss ss( ) ( )ln

a

a

n

a a aa

a

a aa

a

a a aa

a

a aa

a a aa

a

nm

a n

a n a mn

a m

−+

−+ +

− − −−

−−

− − −−

−

− −−

−−

=

−

− − −∑

1

2

1

1 1 11

11

2 2 22

2

1 12

11

0

:

:

:

m m n a

m n a

m m n a

m n a m n a

m m n a

=

=

= with ÙÙ Ùn a

W Nm a=

=n

an n n

1∏ ( )

m m n a m n am a

a a aa

a a aa

a

a a

−−

− −−

−

− −

++

22

1 12

1

2 2

=

W Nm a

Nm a

Nm a

= ( ) ( )

( )

a aa

a a a a

a a

−× −

− −

− −

1

2

1 1

2 2



hedron approximation. In the latter case,the na –m

a –n have been omitted for the sake ofbrevity.

For the entropy, we thus obtain the fol-lowing:

f.c.c.

=

=

(8-32)

B

B

S k

k N

N N N

N N

ij k

ij k i j k

i j k l

i j k l i j k l

i j k

i j k

ln

ln

ln

23 7 1234 12123 1

2

8

1 2 6

8 1

23 7 23 7

1234 1234

123

…

− [+

−

……

……

…∑

∑

∑

r r

r r

r lnln

ln ln

r

r r r r

123

12 12 1 16

i j k

i j

i j i j

i

i i+ − ]∑ ∑

The entropy equations (8-32) and (8-33)are valid for the sro state. In the lro state,the equivalence of certain clusters isbroken. For example, Eq. (8-33) has to bewritten in the lro state with four sublattices(Fig. 8-2):

b.c.c.

=

=

(8-33)

B

B

S k

k N

N N N

N N

ij k l

i j k l i j k l

i j k

i j k i j k

i j

i j i j

ln

ln

ln

ln

1234 12 13123 1

6

12

3

6 3 4

12 1

1234 1234

123 123

12 12

− [−

+

+

∑

∑

∑

r r

r r

r r

44 13 13 1 1

ln lnik

ik ik

i

i i∑ ∑− ]r r r r

Table 8-8. Numerical values of ma , na–na–m , and the CVM exponents an , for the f.c.c. structure in the tetrahe-

dron–octahedron approximation and for the b.c.c. structure in the irregular tetrahedron approximation.

F.C.C. Tetrahedron–Octahedron Fig. 8-4

Cluster 23 … 7 23456 3456 2347 1234 237 123 27 12 1a = 6 5 4a 4b 4c 3a 3b 2a 2b 1

ma = 1 6 3 12 2 12 8 3 6 1n6

a = 1n5

a = 6 1na

4a = 3 1 1na

4b = 12 4 0 1na

4c = 0 0 0 0 1na

3a = 12 6 4 2 0 1na

3b = 8 4 0 2 4 0 1na

2a = 3 2 2 1 0 1 0 1na

2b = 12 8 4 5 6 2 3 0 1n1

a = 6 5 4 4 4 3 3 2 2 1aa = 1 0 0 0 1 0 –1 0 1 –1

B.C.C. Irregular Tetrahedron Fig. 8-2

Cluster 1234 123 12 13 1a = 4 3 2a 2b 1

ma = 6 12 3 4 1aa = 1 –1 1 1 –1

8.5.2 Calculation of Phase Diagramswith the CVM

Once the grand canonical energy U, Eq.(8-18), and the entropy, Eq. (8-31) havebeen derived, the equilibrium at any tem-perature and the chemical potentials can bedetermined by minimizing the grand poten-tial W = U – T S with respect to the configu-rational variables. Two different tech-niques of minimization can be found in theliterature: the natural iteration method, introduced by Kikuchi (1974), in whichminimization is achieved with respect tothe occupation probabilities of the largestcluster; and minimization by means of the Newton–Raphson and steepest-descenttechniques with respect to the correlationfunctions as independent variables (e.g.Sanchez and de Fontaine, 1978; Mohri etal., 1985; Finel, 1987). The natural itera-tion method converges even with startingvalues that are far from the equilibriumvalues, but it is slow. The second method ismore sensitive to the quality of the startingvalues, but it is much faster.

A phase diagram is determined by calcu-lating the equilibrium state for two phasesand comparing the values of the corre-sponding grand potentials. In this way, themost stable phase is obtained. The transi-tion between different lro states of from lroto sro can be of first order or higher (usu-ally second) order. A first-order transitionis detected by the intersection of the grandpotentials of two phases for a certain valueof the chemical potentials (e.g. Kikuchi,1977b, 1987). This value yields a differentcomposition for each phase which definesthe tie-line of the two-phase equilibria.Three-phase equilibria are recognized af-

terwards by the superposition of all thetwo-phase equilibria. More sophisticatedtechniques have not yet been applied be-cause phase diagrams of multicomponentsystems beyond ternary have not yet beenanalyzed. A second-order transition ismore difficult to define. In this instance,there is no metastable extension of the ex-istence range of the most-ordered phase be-yond the transition point, and the grand po-tentials meet at this point with a continuousslope. In such a case, the second Hessian ofthe grand potential has to be analyzed(Kikuchi, 1987). Often this is circum-vented by showing that there is no meta-stable extension of the most-ordered phasewithin given accuracy limits.

8.5.3 Phase Diagram Calculationwith the Monte Carlo Method

In the MC method the configuration of acrystal with a limited number of points(typically several 104 to 105) is stored in acomputer. In order to minimize the boun-dary effects of the finite size of this crystal,periodic boundary conditions are usuallyapplied. These boundary conditions mustbe consistent with the superstructures wewant to treat so that an integer number ofunit cells fits into the crystal. This deter-mines both the shape we have to give to thecomputer crystal as well as the number ofpoints. In the b.c.c. structure with first andsecond neighbors, the superstructures arecubic, and there is no difficulty. In the f.c.c.structure, the superstructures have differentsymmetries and the unit cells have differ-ent extensions in different crystal direc-tions, as shown in the appendix. The phasediagram calculation with the MC method is


S k N N N N N N N

N N N N N N N N

= B ln

/ /

/ / / /1234 12 34 13 14 23 24

123 124 134 234 1 2 3 4

6 3 2 3 2 1 1 1 1

3 3 3 3 1 4 1 4 1 4 1 4 (8-34)


usually carried out in the grand canonicalscheme, i.e., for a given value of the chem-ical potentials. This ensures a homogene-ous single-phase equilibrium state. For thissingle-phase state, the configuration can beanalyzed, and a composition determined.Even in this simple case, problems may appear due to defects such as antiphaseboundaries (APBs), which are produced inthe equilibration process in the same man-ner as in real alloys (see, e.g., Gahn, 1986;Ackermann et al., 1986; Crusius and Inden,1988).

The Monte Carlo process can be startedfrom any configuration and composition,even from the pure components. An arbi-trary site n is selected at random and theatom i on this position is replaced by anatom j according to a transition probabilitywn

iÆ j which has to fulfill certain criteria inorder to guarantee the convergence towardthe equilibrium state from any initial con-figuration (Chesnut and Salsburg, 1963;Fosdick, 1963; Binder, 1976). A sufficientcondition for this convergence is the fulfill-ment of the detailed balance equation andof normalization of the transition probabil-ity:

(8-35a)

(8-35b)

Different expressions for the transitionprobability which fulfill this criterion havebeen suggested. The choice is made ac-cording to the most efficient expression in terms of computer time. The followingexpression fulfills the conditions of Eq. (8-35):

(8-36)

wk T

jn

in

k Tjn

in

ni j

j

K→

− ⎧⎨⎩⎫⎬⎭

− ⎧⎨⎩⎫⎬⎭

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

− ⎧⎨⎩⎫⎬⎭

− ⎧⎨⎩⎫⎬⎭

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥∑

= B

= B

exp

exp

1

1

1

W W

W W

r rni

ni j

nj

nj i

j

K

ni j

w w

w

→ →

→∑

=

==1

1

where is the grand canonical energy

when an atom of kind i occupies site n. Be-cause the sum of the probabilities over allcomponents adds up to 1, the interval [0, 1]can be subdivided into corresponding inter-vals. The atom i on site n is then replacedby the atom j, which corresponds to theinterval selected by a random number be-tween 0 and 1. If the random number se-lects the interval corresponding to i, no ex-change is made. In a Monte Carlo simula-tion, the above-mentioned steps have to beperformed many times, and it is necessaryto use efficient programming techniquesfor data storage and saving of computertime (for general aspects see Binder (1976,1979, 1985), Binder and Stauffer (1984);for multispin coding see Jacobs and Rebbi(1981), Zorn et al. (1981) and Kalle andWinkelmann (1982)).

8.5.4 Examples of Prototype Diagrams

The following examples represent proto-type diagrams. The interchange energiesW (k) used in the calculations are selected insuch a way that the phase diagrams displaytypical features. Calculations for real al-loys will be presented in the next section.We will start with the systems treated in theprevious sections.

8.5.4.1 F.C.C. Structure,First Neighbor Interactions

In this instance we take the case of an ordering tendency in the first shell, i.e.,W (1) > 0, W (2) = 0. The phase diagram, asobtained by CVM in the tetrahedron–octa-hedron approximation (Finel and Ducas-telle, 1986), together with the results fromthe MC method (Ackermann et al., 1986;Diep et al., 1986; Gahn, 1986), are shownin Fig. 8-15. The two diagrams do not coin-

W in

⎧⎨⎩⎫⎬⎭

cide quantitatively, but their main featuresagree qualitatively. In particular, they agreeas to the existence of a triple point at finitetemperature, i.e. at

t triCVM = kB Ttri

CVM/(W (1)/4) ≈ 1.5

(Finel and Ducastelle, 1986) and att tri

MC ≈ 0.8 (Ackermann et al., 1986) and 0.9(Diep et al., 1986). This point gave rise tosome controversy (discussed by Kikuchi,1986), caused by an earlier MC study (Binder, 1980; Binder et al., 1981) whichindicated that the phase boundaries extra-polate to 0 K, and that a triple point doesnot exist. The existence of this triple pointis now well confirmed and has been furthercorroborated by the studies of Lebowitz etal. (1985) and Finel (1994). The diagram inFig. 8-15 replaces that of the earlier MC byBinder. The MCs were not performed atsufficiently low temperatures to detect theP4/mmm phase obtained in the CVM (called L¢ by Finel, 1984).

The CVM diagram in the tetrahedron–octahedron approximation, Fig. 8-15, dif-fers only slightly from that previously cal-culated in the tetrahedron approximation(van Baal, 1973; Kikuchi, 1974), whichgave a triple point at t tri

CVM ≈ 1.6. Thehigher-order cluster approximation doesnot lead to a strong shift of the triple pointto lower temperatures, contrary to an ear-lier result by Sanchez et al. (1982) whofound t tri

CVM ≈ 1.2. The difference resultsonly from certain approximations in thenumerical treatment made by Sanchez etal., which were avoided by Finel (1987). InFinel’s work, a more sophisticated CVMcalculation was also made using the tetra-hedron–octahedron for the ordered phasesand the quadrupole tetrahedron for the srostates. In that approximation. Finel ob-tained complete agreement with the MCwork.


Figure 8-15. F.c.c. structure: calculated prototype phase diagrams for the case of nearest-neighbor interactions,W (1) > 0, W (2) = 0. (a) CVM calculation in the tetrahedron–octahedron approximation (Finel and Ducastelle,1986), (b) Monte Carlo simulation (Ackermann et al., 1986).


8.5.4.2 F.C.C. Structure, First and Second Neighbor Interactions

In this section, the case of an orderingtendency in both shells will be treated first,namely, W (1) > 0 and W (2)/W (1) = 0.25. Theresults of the tetrahedron–octahedronCVM (Sanchez and de Fontaine, 1980) andof MC simulations (Bond and Ross, 1982)are shown in Fig. 8-16. The two diagrams

agree fairly closely. Phase diagrams forother values of the ratio W (2)/W (1) aregiven by Mohri et al. (1985) and Binder et al. (1983).

Fig. 8-17 shows the results for an order-ing tendency between first neighbors and aseparation tendency between second neigh-bors, W (1) > 0, W (2) = –W (1). This situationcorresponds to the calculation presentedpreviously, in Fig. 8-14c. As expected

Figure 8-16. F.c.c. struc-ture: calculated prototypephase diagrams for the caseof first and second neighborordering interactions,W (1) = 4W (2) > 0. (a) CVMcalculation in the tetrahe-dron–octahedron approxi-mation (Sanchez and deFontaine, 1980). (b) MonteCarlo simulation (Bond andRoss, 1982).

Figure 8-17. F.c.c. structure:calculated prototype phase dia-grams for the case of an order-ing tendency between firstneighbors and a separation ten-dency between second neigh-bors, W (1) = –W (2) > 0. (a) CVMcalculation in the tetrahedron–octahedron approximation (Mohri et al., 1985). (b) MonteCarlo simulation (Binder et al.,1983).

from this diagram the low temperaturestates are two-phase states and the phaseboundaries extrapolate to the pure compo-nents and to the stoichiometric orderedphases. A complete series of prototype diagrams calculated with the tetrahedron–octahedron CVM is presented by Mohri etal. (1985).

The case of a separation tendency be-tween first and second neighbors has beenanalyzed with the MC technique: Kutner et al. (1982) analyzed the case W (1) < 0,W (2) = 0 with the grand canonical simula-tion, while Gahn et al. (1984) analyzed thecases W (2) = 0 and W (2) ≠ 0 with a spe-cial canonical simulation. The results forW (2) = 0 are identical in both treatments.The resulting miscibility gaps are shown inFig. 8-18 (on a reduced scale in order toshow the variation in shape). It is foundthat the effect of second neighbor interac-

tions on the shape of the miscibility gap issmall. The shape obtained differs signifi-cantly from the miscibility gap which isusually calculated with the regular solution(i.e., point approx.) model.

8.5.4.3 B.C.C. Structure, First andSecond Neighbor Interactions

A series of prototype phase diagrams,calculated with the tetrahedron CVM forvarying strengths of ordering tendency inboth neighbor shells, was first presented byGolosov and Tolstick (1974, 1975, 1976).Simultaneously, Kikuchi and van Baal(1974) presented a diagram correspondingto the ratio W (2)/W (1) = 0.5, which is closeto the situation encountered in Fe–Si andFe–Al. Figs. 8-19a, b and c display a seriesof diagrams calculated with the tetrahedronCVM and with the MC method (Acker-mann et al., 1989) for different strengthsand signs of the interchange energies. Theresults of both methods are in good agree-ment. In these diagrams, second-order tran-sitions are indicated by a hachure. Thesesecond-order transitions turn into first-or-der transitions at tricritical points, and thetopology of the phase boundaries close tothese points exhibit the characteristics de-rived by Allen and Cahn (1982).

8.5.4.4 Hexagonal Lattice, AnisotropicNearest-Neighbor Interactions

The ordering reactions in the hexagonalcrystal structure have been studied with the MC method (Crusius and Inden, 1988;Bichara et al., 1992b). In order to simulatethe situation for c/a ≠ 1.633, i.e., for non-close packing, the interchange energiesbetween nearest neighbors within a basalplane, W (11), and between two such planes,W (12), were given different values. The resulting phase diagrams for two sets ofinterchange energies are shown in Figs.


Figure 8-18. Miscibility gaps according to MC cal-culations for various values of the interchange ener-gies between first and second neighbors. The dia-gram is symmetric with respect to the equiatomiccomposition, W (1) < 0, W (2) = 0 (Kutner et al., 1982).W (2) = 0 and W (2) ≠ 0 (Gahn et al., 1984).


8-20a and b. The phase A2B is a two-dimensional structure built with three sub-sequently ordered planes, but with no cor-relation between them. Therefore a two-di-mensional characterization has been givenin Sec. 8.7 in addition to the three-dimen-sional one that is to be considered for caseswith W (12) ≠ 0 (see Appendix, Table 8-13).

In the case W (12) = 0, the phase A2B isthe only stable superstructure. The transi-tion A2B ´ A3 is second order in this in-stance. For m* = 0 (xB = 0.5), no phase tran-sition has been observed down to the re-duced temperature t = kBT /(W (11)/4) = 0.6.The extrapolation of the transition temper-ature goes to 0 K at this composition. Thisis consistent with the exact solution which

Figure 8-19. B.c.c. structure: calculated prototype phase diagrams for ordering or separation tendenciesbetween first or second neighbors. The lines correspond to the CVM calculation in the tetrahedron approxima-tion, the points correspond to the Monte Carlo simulation (Ackermann et al., 1989). The hachure indicates a second-order transition. (a) W (1) = 2W (2) > 0, (b) W (1) = –W (2) > 0, (c) W (1) = –W (2) < 0.

is known for m* = 0 (Houtappel, 1950; Ne-well, 1950; Wannier, 1950). The diagramfor W (12) = W (11) corresponds to the hexag-onal close-packed structure, see Bichara et al. (1992b). This diagram is not shownbecause it is exactly the same as the one inFig. 8-15, except that the phases L12 andL10 have to be replaced by the phases D019

and B19, respectively. The reason for thisis that the f.c.c. and hexagonal close pack-ings cannot be distinguished by nearest-neighbor interactions only.

8.5.5 The Cluster Site Approximation(CSA)

Notwithstanding the successes of theCVM in the calculation of phase diagramsand thermodynamic properties, a majordisadvantage is the large number of con-figurational variables, and thus of non-linear equations, to be solved in order tominimize the Helmholtz energy. The num-ber of independent variables in a K-com-ponent system and an r-site CVM is Kr – 1. This high number increases stronglywith the size of the basic cluster, even forbinary systems. Real systems, however,may involve many more than three compo-nents.

The CSA suggested by Oates and Wenzl(1996) is a revival of the quasi-chemicaltetrahedron approximation (Yang and Li,1947). A system with N points is decom-posed into Na clusters of type a in such away that the clusters share only points. Theimportant result of this assumption is thatonly point correlation functions are used inthe Helmholtz energy minimization. The acluster probabilities are obtained after thecalculation from the quasi-chemical equi-librium between the atoms (points) andmolecules (clusters). If a contains r points,the number of independent variables is(K –1) (r –1). Thus with increasing clustersize or number of components the numberof independent variables in the CSA is sig-nificantly smaller than in the case of CVM.This is the main advantage.

Following the concept of the CVM, theentropy of the CSA in the r-site cluster ap-proximation can be written

(8-37)

S k

r

S r S

N

r

N

r

N

=

=

B=

=

ln

( )

( )

( )

12

1

1

1

1

…

− −

∏

∏

g

n

n

g

g

n

n

g g


Figure 8-20. Hexagonal structure: calculated proto-type phase diagrams for the case of anisotropic near-est-neighbor interactions, W (11) within the basalplane and W (12) between basal planes. Monte Carlocalculations (Crusius and Inden, 1988). The hachureindicates a second-order transition. (a) W (12) > 0,W (12)/W (11) = 0.8, (b) W (11) > 0, W (12) = 0.

8.6 Application to Real Systems 561

where g is the coordination number of tet-rahedra per lattice point. In the case of thef.c.c. lattice and tetrahedron cluster thisnumber is g =1, i.e. half the value used inthe CVM.

The principal reason for the lack of at-tention to the CSA is its inability to obtainthe correct topology of prototype phase di-agrams as obtained from CVM or MC cal-culations. The tetrahedron approximationof the f.c.c. lattice is a prominent example.The CSA with g =1 gives lro regions L12

and L10 around the stoichiometric compo-sitions, but between these regions the dis-ordered phase is predicted to be stabledown to 0 K. Oates et al. (1999) haveshown that a simple correction of the valueof g allows us to compensate for this. Theauthors were able to reproduce the f.c.c.prototype phase diagram of the tetrahedronCVM by taking g =1.42.

Several successful results have been ob-tained with the CSA; those for the Cu–Ausystem will be mentioned here. Using thevalue g =1.42 as a constant entropy correc-tion term for the tetrahedron treatment off.c.c. alloys, Oates et al. (1999) applied theCSA to the Cu–Au system using the tetra-hedron interactions from Kikuchi et al.(1980). They obtained exactly the samephase diagram as Kikuchi with the CVM.The diagram is shown in Fig. 8-21a. Fur-thermore, taking the many-body interac-tions and also the elastic energy arisingfrom atomic size mismatch from Ferreira etal. (1987), Oates et al. obtained again thesame phase diagram as Ferreira with theCVM, see Fig. 8-21b. This diagram alsoshows the effect of the elastic terms: theycontribute positive corrections to the con-figurational energy, thus removing the de-generacy of the ground states. Therefore, at0 K, the phase boundaries meet at the stoi-chiometric compositions and go to the purecomponents, contrary to the prototype dia-

grams in Figs. 8-15, 8-16 and 8-19a wheresuch effects were not included.

These results indicate that the CSA maybe a very useful tool when it comes to tech-nological problems. It remains to be seenwhether the CSA can be applied as suc-cessfully to other structures and to largerclusters.

8.6 Application to Real Systems

In contrast to prototype systems, realsystems exhibit very complex properties

Figure 8-21. Cu–Au phase diagrams calculatedwith the CSA using the constant entropy correctionfactor g = 1.42 (Oates et al., 1999). a) Calculatedwithout any size mismatch using the pair and CuCu-CuAu and CuAuAuAu tetrahedron interactions de-rived by Kikuchi et al. (1980) using the tetrahedronCVM. The phase diagram is the same as that of theCVM. b) Calculated with size mismatch using themany body interactions and the elastic energy termsderived by Ferreira et al. (1987) using the tetrahe-dron CVM. The phase diagram is the same as that ofthe CVM.

even at T = 0 K. With the progress made infirst-principles calculations, it has becomeobvious that higher-order pair and multisitecluster interactions have to be used in theconfigurational part of enthalpy, and that itis mandatory to take lattice and local relax-ations into account. This will be illustratedusing the Au–Ni system as an example.When it comes to T > 0 K the cluster inter-actions have to embrace all configuration-dependent excitation energies such as arisefrom thermal vibrations and electronic ex-citations. The inclusion of these effectsmeans that the cluster interactions becometemperature dependent. There is still somedisagreement as to whether vibrationalcontributions can (Ozolins et al., 1998b;van de Walle et al., 1998) or cannot (Craie-vich and Sanchez, 1997; Craievich et al.,1997a, b) be included in the cluster expan-sion.

However, even the apparently most care-ful first-principles calculations (Ozolins etal., 1998a) remain insufficiently accuratefor technological purposes. For the fore-seeable future it seems clear that simplerapproaches will continue to be important,e.g. for technological phase diagram calcu-lations and also for other applications re-lated to the thermodynamic properties. Thesystems Fe–Al and Ni–Al will be takenhere as examples for calculations of phasediagrams and of the thermodynamic factorfor diffusion.

Surveys of the abundant literature de-scribing applications to mainly binary al-loy systems can be found in the reviews byInden and Pitsch (1991) and by de Fontaine(1994).

At this stage multicomponent systemspresent apparently insurmountable prob-lems for first-principles calculations. Thiseven holds for ternary systems. Rubin andFinel (1993) studied ternary Ti–Al–X(X=W, Nb, Mo) systems with first and sec-

ond neighbor pair interactions and used apower series expansion in composition forthe disordered state. No ternary interac-tions were included. The agreement of theresults for the limitrophe binary systemswith experimental data is satisfying only ina few of the cases studied, and very little ispresented with respect to ternary isother-mal sections. McCormack et al. (1996,1997) studied the systems Cd–Ag–Au(1996) and Cu–Al–Mn (1997) using firstand second neighbor pair interactions, butfailed to present phase equilibria for ter-nary and limitrophe binary systems thatcould be checked by experiments. In the following the two ternary systemsFe–Ti–Al and Fe–Co–Al will be dis-cussed. The Fe–Co–Al system will betreated as a magnetic system with spin 1/2given to Fe, Co and Al. This leads in fact toa six-component system.

In multicomponent systems the numberof points within a chosen basic cluster maynot be large enough to accommodate allcomponents. Then the question arises as towhat extent this is detrimental to the qual-ity of the approximation. In order to givesome idea of this effect, a magnetic spin7/2 system with f.c.c. structure has beenanalyzed by Schön and Inden (2001) usingthe tetrahedron CVM and Monte Carlosimulations. This spin problem is equiva-lent to an eight-component alloy problem.

The calculation of phase equilibria andthermodynamic properties of multicompo-nent systems is of paramount importancefor technological design and materials de-velopment. The use of both a cluster ex-pansion for the energy and the CVM forconfigurational entropy introduces severecomputational problems because of thelarge number of cluster types that must beconsidered in both formalisms. It also hasto be emphasized that in practical applica-tions with coexisting phases of different



crystal structures, the calculation of theatomic configurations has to be performedfor each of the structures, and within eachstructure with the same level of approxima-tion for all subsystems in order to achieveconsistency. That is to say, isolated sophis-ticated treatments of particular systemswill provide deep insight into the physics,but they are only of limited use for solvingtechnological problems.

Recently, the quasi-chemical tetrahedronapproximation has been adapted to over-come the problems of obtaining the cor-rect topology of order–disorder phase dia-grams for f.c.c. systems and to treat latticerelaxations and excitations. This methodseems to offer the possibility of handlingorder–disorder effects with sufficient qual-ity within a format that can be used in ther-modynamic databases of multicomponentsystems. The Cd–Mg system will be takenas an example for illustration, because ithas been extensively analyzed: all possiblecontributions to the ground states havebeen treated by first-principles calculation,and the phase diagram has been calculatedby CVM (Asta et al., 1993).

8.6.1 The Au–Ni System

This system has been analyzed inten-sively over the last decade (Renaud et al.,1987; Eymery et al., 1993; Wolverton andZunger, 1997; Wolverton et al., 1998; Ozolins et al., 1998; Colinet and Pasturel,2000). It has attracted much interest becauseof a phase separation tendency and positiveenthalpies of mixing at low temperatures,ordering type of short-range order at hightemperatures, and a large lattice mismatchof about 15% between the constituents. Itmay be considered a key system for check-ing the quality of the theoretical methods.

EXAFS experiments (Renaud et al.,1987) have shown that the atoms arestrongly displaced relative to the regularundistorted lattice. The distribution ofAu–Au distances was found to be narrowand weakly asymmetric, while those forNi–Ni were found to be widely distributedand highly asymmetric. Eymery et al.(1993) came to the same conclusion on thebasis of their theoretical work. From thecalculated average nearest-neighbor dis-tances, partial tetrahedral volumes werecalculated as a function of composition.The results are shown in Fig. 8-22. The size

Figure 8-22. Variation of theaverage tetrahedral volumesAu4, Au3Ni, AuNi, AuNi3 andNi4 in random Au1–xNix solidsolutions at T = 0 K versus Niconcentration [redrawn fromEymery et al. (1993)]. The vol-umes were calculated from thecalculated average nearest-neighbor distances. The differ-ence between the various vol-umes and their variation withcomposition reflects the latticerelaxations. The calculationswere based on a tight-bindingsecond moment approximation.

distribution of the tetrahedral volumes wasalso calculated. The results for the equi-atomic composition are shown in Fig. 8-23.The local relaxations become evident bythe deviation of the average positions ofthese distributions from the correspondingaverage volumes in Fig. 8-22, which areshown in Fig. 8-23 by the arrows. These re-sults clearly indicate that both lattice andlocal relaxations must not be ignored. Thisfact becomes also visible in the enthalpy offormation of the random alloys differing bya factor of two between unrelaxed and re-laxed states.

Wolverton et al. (1997, 1998) consideredthe Au–Ni system and analyzed the ques-tion of up to what order the pair interactionscheme has to be driven in order to repro-duce experimental findings. The CE wasseparated into three parts: the pair interac-tions with arbitrary distance summed in thereciprocal space representation (Laks et al.,1992), the multi-atom interactions in real-

space representation, and the constituentstrain energy. The authors concluded thatpair interactions up to about the 15th shellare needed. Ozolins et al. (1998a, b) exam-ined the same question for the series of no-ble metal alloys Cu–Au, Ag–Au, Cu–Agand Au–Ni. While in Ag–Au and Ag–Cuthe first three neighbor pair interactions aredominant, the same is not true for the othersystems, particularly not for Au–Ni, wherenot only pair interactions up to the 10thshell, but also triplet and four-point clusterinteractions in increasing distances have tobe taken into account.

Most recently, Colinet et al. (2000) tookup to fourth neighbor pair and triplet inter-actions into account, the tetrahedron inter-actions turning out to be almost negligiblein their treatment. They achieved goodagreement in the entropy of mixing, fairlygood agreement in the enthalpy and Gibbsenergy of mixing and calculated the phasediagram by means of the tetrahedron–octa-hedron CVM. The comparison between thecalculated and experimental miscibilitygap is shown in Fig. 8-24.


Figure 8-23. Histogram of the calculated partial tet-rahedral volumes Au4, Au3Ni, AuNi, AuNi3 and Ni4in a random alloy at T = 0 K with xNi = 0.5 [redrawnfrom Eymery et al. (1993)]. The volume distributionsare broad and partially asymmetric. The average vol-umes, taken from Fig. 8-22, are shown by the arrows.The difference between these average values and thecenters of gravity of the distributions is remarkableshowing the importance of local relaxations.

Figure 8-24. Miscibility gap of the Au–Ni system.Experiments from Bienzle et al. (1995). The calcula-tions were performed using pair and triplet interac-tions in the tetrahedron–octahedron CVM (Colinet etal., 2000).


8.6.2 The Thermodynamic Factor of Ordered Phases

8.6.2.1 The B.C.C. Fe–Al System

Iron aluminides such as Fe3Al and FeAlhave received considerable attention ascandidates for high temperature structuralmaterials due to their low cost, highstrength and good oxidation resistance.Schön and Inden (1998) assessed the sys-tem in order to derive ECIs from experi-mental enthalpies of formation, correctedfor b.c.c reference states and converted intothe paramagnetic state:

j DHj [J/mol of atoms]

Fe3Al (D03) –18 650FeAl (B2) –27 940FeAl(B32) –21 570FeAl3(D03) –18 650

Introducing these enthalpies and the valuesof the correlation functions of the configu-rations (see Table 8-5) into Eq. (8-24), the

following parameters are obtained (in unitsof kB K)

V0 = –2190, V1 = 0, V2 = 1680,V3 = 457, V4 = 0, V5 = 52.5

or equivalently(8-38)

W (1) = 1680, W (2) = 740, W 1234FeAlFeAl = –140

With these parameters the irregular tetrahe-dron CVM yields the b.c.c. phase diagramshown in Fig. 8-25. The calculated criticaltemperatures match the high temperatureexperimental data. However, at lower tem-peratures, the tricritical point and the two-phase region A2 + B2 are not obtained andthe agreement with the experimental phaseboundaries A2 + D03 is poor. This is notsurprising in view of the large relaxationeffects, which are to be expected in thissystem as seen from the variation of latticeparameter as a function of composition andof state of order, Fig. 8-26. On the otherhand, recent experimental data on chemicalpotential measurements at 1000 K (Kley-kamp and Glasbrenner, 1997) are very well

Figure 8-25. The b.c.c. phase dia-gram of Fe–Al calculated with theirregular tetrahedron CVM usingthe parameters given in Eq. (8-38)(Schön and Inden, 1998). Second-order transitions are indicated bybroken lines. Three temperaturesare indicated at which the thermo-dynamic factor of diffusion isshown below in Fig. 8-28. Experi-mental points: neutron diffractionand cp from Inden and Pepperhoff(1990), dilatometry from Kösterand Gödecke (1980), TEM andDTA from Ohnuma et al. (1998a).

reproduced by the calculations, see Fig. 8-27.

A very important quantity for the treat-ment of diffusion is the thermodynamicfactor defined for a binary system A–B as

This quantity is smooth in a random solidsolution, but shows a complicated variation

F =d

dB B

B

xxm

in ordered alloys. Fig. 8-28 shows this vari-ation at three different temperatures. Closeto stoichiometric compositions of orderedphases, F varies dramatically. The amountof variation depends on the degree of lro,reaching up to an order of magnitude atleast in almost fully lro states, as observedat T = 650 K for B2 at the compositionxAl = 0.5. At second-order transition pointsF changes discontinuously.


Figure 8-26. Variation of thelattice parameter of Fe–Al alloysas a function of composition andstate of order. Experiments: Lihland Ebel (1961).

Figure 8-27. Chemical potentialof Fe and Al at 1000 K in b.c.c.Fe–Al alloys calculated with thetetrahedron CVM using the parameters given in Eq. (8-38) (Schön and Inden, 1998). Experi-mental data from Kleykamp andGlasbrenner (1997).


8.6.2.2 The F.C.C. Ni–Al System

The variation of the thermodynamic fac-tor is even more pronounced in the Ni–Alsystem because there the order–disordertransitions occur at much higher tempera-tures than in Fe–Al, as shown in the cal-culated f.c.c. phase diagram in Fig. 8-29 (Schön and Inden, 1998). The ECIs werederived from the ground-state energies ofL12–Ni3Al, L12–NiAl3, and L10–NiAl,obtained by Pasturel et al. (1992) fromfirst-principles calculations:

j DHj [J/mol of atoms]

Ni3Al (L12) –48 000NiAl (L10) –56 000NiAl3(L12) –22 000

Introducing these enthalpies and the valuesof the correlation functions of the configu-rations (see Table 8-3) into Eq. (8-24), thefollowing parameters are obtained (in unitsof kB K)

V0 = –4630, V1 = –1563, V2 = 5051,

V3 = 1563, V4 = –421

or equivalently

W (1) = 3370, W 1234NiAlAlAl = – 4810,

W 1234NiNiNiAl = 1440 (8-39)

The calculated thermodynamic factor ofdiffusion is shown in Fig. 8-30 for the twotemperatures indicated in Fig. 8-29. Due tothe high degree of lro, the variation of F inthe range of stoichiometric composition is

Figure 8-28. Thermodynamic factor of diffusion in b.c.c. Fe–Al alloys, calculated as a function of composi-tion at three temperatures, 650 K, 1000 K and 1400 K. The calculations were performed with the irregular tetrahedron CVM using the parameters given in Eq. (8-38) (Schön and Inden, 1998). At second-order transi-tions, F changes by a step. Close to the stoichiometric compositions xAl = 0.25 and 0.5 the value of F increasesas a function of the degree of lro. For example, the temperature of T = 650K is only slightly below the D03/B2transition temperature, see Fig. 8-25. Therefore, at xAl = 0.25 the increase is small compared to xAl = 0.5 whereF increases by more than an order of magnitude because the distance from the B2/A2 transition temperature islarge and the degree of lro almost maximum. With increasing temperature this effect becomes smaller.

particularly pronounced. Variations of thesame order of magnitude are obtained forthe B2 phase.

It is worth mentioning that these dra-matic variations have repercussions in thecomposition profiles of diffusion couples.An example is shown in Fig. 8-31, where acontrast has been observed in the micro-graph within the B2 phase region as it is

usually observed at phase boundaries. Theelectron microprobe analysis reveals asteep change of composition at this boun-dary, which again could be misinterpretedas a tie-line. This is an important fact to berealized, because phase diagram determi-nations in multicomponent systems aremost conveniently performed by means ofdiffusion couple experiments.


Figure 8-29. Phase diagramof the f.c.c. Ni–Al systemcalculated with the regulartetrahedron CVM (Schönand Inden, 1998) using theparameters given in Eq. (8-39). Two temperatures areindicated for which the ther-modynamic factor has beencalculated, Fig. 8-30.

Figure 8-30. Thermody-namic factor of diffusion inf.c.c. Ni–Al alloys, calcu-lated as a function of com-position at two tempera-tures, 1500 K and 2350 K(Schön and Inden, 1998).The calculations were per-formed with the regulartetrahedron CVM using the parameters given in Eq.(8-39). At the stoichiomet-ric compositions, F in-creases by more than anorder of magnitude.


8.6.3 Ternary Systems

The b.c.c. states of the two systemsFe–Ti–Al and Fe–Co–Al will now betreated by taking the energy descriptionfrom the limitrophe binaries in order to seewhat can be obtained by extrapolating fromthe binaries into the ternary system withoutternary interaction terms.

8.6.3.1 B.C.C. Fe–Ti–Al

Ohnuma et al. (1998b) studied orderingand phase separation in the b.c.c. phase ofthe ternary system, with particular empha-sis on ternary miscibility gaps between dis-ordered and ordered phases. The theoreti-cal analysis was based on the irregular tet-rahedron CVM. The analysis of the Fe–Tiand Ti–Al binary systems led to the energy

parameters given in Table 8-9. The param-eters for Fe–Al were taken as given in Eq.(8-38).

The calculated isothermal sections at1173 and 1073 K are shown in Figs. 8-32and 8-33. From the metallurgical point of

Figure 8-31. Diffusion couple Ni77Al23/Ni40Al60

annealed at 1273 K for 100 h (Kainuma et al., 1997).a) Microstructure showing a change in contrastwithin the B2 phase field. The regions labelled b1

and b2 do not represent two different phases. Thepseudo-boundary between b1 and b2 comes from asteep change in composition. b) Composition profilemeasured by electron probe microanalysis.

Table 8-9. Atomic exchange energy parameters inunits of kB K (Ohnuma et al., 1998b).

A–B WAB(1) WAB

(2) W 1234ABAB

Ti–Fe 1580 –1050 1800Al–Ti 2420 1200 0

Figure 8-32. Calculated isothermal section of theb.c.c. Fe–Ti–Al system at 1173 K using the energyparameters in Table 8-9. Second-order transitions areshown as broken lines. The enlarged part of the sec-tion shows the good agreement obtained between ex-periments and calculation (Ohnuma et al., 1998b).

view it is interesting to see the opening ofternary miscibility gaps between orderedand disordered phases. This allows us toproduce coherent two-phase equilibria withinteresting mechanical properties. Guidedby the calculations, Ohnuma et al. (1998b)also performed experiments in order toconfirm the existence of the two-phasefields experimentally. The results areshown in the enlarged parts of the sectionsof Figs. 8-32 and 8-33. The calculated tie-lines agree very well with the experimentalresults.

8.6.3.2 B.C.C. Ferromagnetic Fe–Co–Al(six-component system with spin 1/2 for Fe, Co and Al)

This system has been studied by Colinetet al. (1993) in the irregular tetrahedron approximation. The binary system Fe–Alwas discussed in Sec. 8.6.2.1, but withouttaking magnetic effects into account. Be-cause the Curie temperature of metastableb.c.c. Fe–Co alloys goes up to about1500 K, the magnetic effects cannot be disregarded in this ternary system. A spin1/2 treatment has been taken in this in-stance. The magnetic exchange energiesare defined in the same way as the atomicequivalents: JAB = – 2J ≠Ø

AB + J ≠≠AB + J ØØ

AB, withA, B ŒFe, Co, Al and the J ≠Ø

AB etc. beingproportional to the corresponding ex-change integrals. The energy parametersare given in Table 8-10.

The phase diagram of the b.c.c. Fe–Cosystem calculated with the energy parame-ters given in Table 8-10 is shown in Fig. 8-34. All transitions are of second orderand have been obtained with the secondHessian method. The calculation withoutmagnetic interactions is also shown. In thisnon-magnetic case the critical temperatureof the B2/A2 transition is lower than in theferromagnetic case. The magnetic interac-tions strengthen the atomic ordering in thisinstance.

In the Co–Al system, the b.c.c. phase isstable only around the equiatomic compo-sition, and there the B2 structure is stable


Figure 8-33. Calculated isothermal section of theb.c.c. Fe–Ti–Al system at 1073 K using the energyparameters given in Table 8-9. Second-order transi-tions are shown as broken lines. The enlarged part ofthe section shows the good agreement obtained be-tween experiments and calculation (Ohnuma et al.,1998b).

Table 8-10. Atomic and magnetic exchange energyparameters in units of kB K (Colinet et al., 1993).

A–B WAB(1) WAB

(2) JAA(1) JBB

(2) JAB(1)

Fe–Co 500 0 –163 –218 –24Fe–Al 1680 740 –163 0 –38Co–Al 3600 1500 –218 0 –38


up to the melting point. The D03 structurehas not been observed within the b.c.c.phase field. This imposes an upper boundas to the value of W (2)

CoAl that controls thevalue of the critical temperature D03/B2.Based on these considerations and on ex-perimental values for the critical tempera-ture B2/A2 in some ternary Fe–Co–Al al-loys, the values in Table 8-10 have beenderived by Ackermann (1988).

The b.c.c. phase diagram calculated inthe tetrahedron approximation (Colinet etal., 1993) is shown in Fig. 8-35, super-posed to the stable diagram. This includesthe equilibria with the liquid phase, the

f.c.c. phase, and intermetallic compounds.The B2 phase exists up to very high tem-peratures in a metastable state, far withinthe liquid phase. This tendency is consis-tent with the trend obtained experimentallyfrom data on ternary alloys (Ackermann,1988).

Figs. 8-36 and 8-37 show the isothermalsections of the ternary system Fe–Co–Alat 1000 K and at 600 K. The ordered re-gions are separated by first-order and sec-ond-order transitions. At 600 K the mag-netic interactions become important, butthey are still small compared with thechemical interactions. Prior to these calcu-lations, Ackermann (1988) performed anMC calculation of the isothermal section at700 K using the same chemical interactionparameters as those in Table 8-10. Her dia-gram and the corresponding CVM diagramagree very well (see Inden and Pitsch,1991).

Figure 8-34. Phase diagram of the Fe–Co system.Second-order transitions are indicated by a hachure.Heavy lines: CVM calculation in the (irregular) tetra-hedron approximation using the interchange energiesgiven in Table 8-10 (Colinet et al., 1993); lowercurve: no magnetic interactions, upper curve: withmagnetic interactions. Experiments: (˘) Masumotoet al. (1954), (Ñ) Eguchi et al. (1968), (¸) Oyedeleand Collins (1977).

Figure 8-35. Phase diagram of the Co–Al system.Second-order transitions are indicated by a hachure.Heavy lines: CVM calculation in the (irregular) tetra-hedron approximation using the interchange energiesin Table 8-10 (Colinet et al., 1993). Light lines:Phase diagram according to Hansen and Anderko(1958).

Ackermann (1988) also analyzed theFe–Co–Al system experimentally. Hereonly one comparison will be shown. Fig. 8-38 shows the calculated and experimen-tal miscibility gap in a vertical sectionwhich happened to be quasibinary (the tie-lines are within the section).

The agreement between experiment andcalculation is surprisingly good, though notreally satisfactory. The calculated miscibil-ity gap ends with a “horn” at the tricriticalpoint. This is not obtained in the experi-ments. However, we should recall that thetwo coexisting phases A2 and B2 are coherent with different lattice parameters.It has been shown by Williams (1980,1984) and Cahn and Larché (1984) that theboundary of so-called “coherent” miscibil-ity gaps (with elastic contributions) islower than the so-called “incoherent” gaps(no elastic contributions). Therefore, elas-tic effects due to the precipitation of coher-ent particles depress the phase boundary.This is only one of the effects to be takeninto account. A second one is due to the lattice and local relaxations within both ofthese phases, as already pointed out in Sec.8.6.2.1 for Fe–Al. The arguments present-ed for Fe–Al are also valid for theFe–Co–Al system with Fe and Co beingvery similar. These relaxation effects tendto shift the miscibility gap to higher tem-peratures, opposite to the first effect. A fullcalculation is needed to evaluate the com-bined result.

Simultaneously with the work by Acker-mann (1988) the miscibility gap was ex-perimentally observed by Miyazaki et al.(1987) using transmission electron micros-copy (TEM). These authors, however,speculated that the miscibility gap was dueto magnetic effects and assumed the tie-lines to be oriented towards the Fe–Co binary system, i.e. perpendicular to thoseshown in Fig. 8-37. The contradiction be-


Figure 8-36. Isothermal section of the phase equi-libria of ternary b.c.c. Fe–Co–Al alloy atT = 1000 K, calculated with the CVM in the (irregu-lar) tetrahedron approximation using the energy pa-rameters given in Table 8-10 (Colinet et al., 1993).Second-order transitions are indicated by a hachure.

Figure 8-37. Isothermal section of the phase equi-libria of ternary b.c.c. Fe–Co–Al alloys atT = 700 K, calculated with the CVM in the (irregular)tetrahedron approximation using the energy parame-ters given in Table 8-10 (Colinet et al., 1993). Sec-ond-order transitions are indicated by a hachure. Atthis temperature ferromagnetic (fm) and paramag-netic (pm) states have to be distinguished.


tween the results of Miyazaki et al. andthose from the CVM and MC pointed outby Inden and Pitsch (1991) was removedby the TEM–EDS analysis of Kozakai andMiyazaki (1994) confirming the calculateddirection of the tie-lines.

8.6.4 H.C.P. Cd–Mg

The Cd–Mg system is a prototype order-ing system for h.c.p. alloys in the same wayas Cu–Au plays this role for f.c.c. alloys.This system has been studied using first-principles methods by Asta et al. (1993).The configurational energy was evaluatedby a cluster expansion up to seven-pointclusters, as required for a treatment withthe octahedron–tetrahedron CVM. Latticerelaxation and vibrational energy weretaken into account, but no local relaxations.The lattice relaxation cluster expansionwas based on experimental results for dis-ordered alloys, and the other ECIs were ob-tained from total-energy calculations on or-dered phases. In total 32 energy terms wereused. The ECIs were then used in the tetra-

hedron–octahedron CVM to calculate thephase diagram, which is shown in Fig. 8-39a. The topology of the diagram is correctbut it does not have the accuracy necessaryfor technological purposes.

Zhang et al. (2000) tried the CSA in thetetrahedron approximation. Because thec/a ratio is not ideal (f.c.c. and h.c.p. lat-tices become equivalent in the ideal case),two irregular tetrahedra should be consid-ered (Onodera et al., 1994). Zhang et al.(2000) tried the irregular tetrahedra, butfound that the regular tetrahedron CSAgave an equally good description. Thevalue of g was varied in such a way that thetwo congruent maxima which appear at thecompositions A3B and AB3 for g = 1.42 (valid for the ideal h.c.p. and f.c.c., see Sec.8.5.5) moved towards the mid-compositionuntil, at g = 1.8, the maxima disappeared,just as observed in the Cd–Mg system.Starting with energies taken from Asta etal. (1993), which proved to be excellent,only slight changes were applied in orderto get an optimum description of all theproperties that can be checked with avail-

Figure 8-38. Vertical section ofthe phase diagram showing theternary miscibility gap as a func-tion of composition according to experiment and MC (Acker-mann, 1988), (a) Experiments(¯) miscibility gap, (É) criticaltemperature of lro. (b) Calcula-tion with the MC method: (¯)miscibility gap and critical tem-perature of lro.

able experimental data: the phase diagram,the enthalpy of mixing of the disorderedstate, the enthalpies of formation of or-dered phases and the chemical potentials ofCd and Mg. The cluster energies used were(the values of Asta et al. are given in brack-ets):

Cd3Mg: –9.3 (–10.3); CdMg: –13.7 (–13.5);

CdMg3: –9.95 (–9.5) in kJ/(mol of atoms).

The calculated phase diagram is shownin Fig. 8-39b. The agreement between cal-culation and experiment is very good, notonly for the phase diagram but also for allthe other properties mentioned above.

8.6.5 Concluding Remarks

The examples treated above representthe large group of real systems that can beanalyzed using these techniques. Similarfindings were obtained for oxide systems,e.g., by Burton (1984, 1985), Burton andKikuchi (1984), Kikuchi and Burton (1988),Burton and Cohen (1995), Tepesch et al.(1995, 1996), Kohan and Ceder (1996).

At present, it can be concluded that theCVM and MC techniques supply a treat-ment of sufficient sophistication to cor-rectly handle the statistical aspects of theequilibria in solid solutions.

Much progress has been made in thefield of first-principles calculations of totalenergies, including lattice and local relaxa-tions, and sometimes including excitations.From this the energy parameters of the sta-tistical models can be calculated. The re-sults for binary systems are numerous, butthe field of multicomponent systems is stillto be explored. With increasing numbers ofcomponents the cluster size also has to beincreased.

For the solution of metallurgical prob-lems, the phase equilibria between allphases have to be considered. The order–disorder equilibria within a given crystalstructure are only one part of this task.Other phases, such as the liquid phase orintermetallic compounds, have to be in-cluded. These aspects have been discussedat recent workshops on the thermodynamicmodeling of solutions and alloys (e.g. Cac-ciamani et al., 1997).


Figure 8-39. Calculated phase diagrams of the hexagonal Cd–Mg system. a) Phase diagram calculated from “first principles” (redrawn from Asta et al., 1993). b) Phase diagram calculated with the CSA using almost thesame parameters as in a) and a value g = 1.8. Experimental data from Frantz and Gantois (1971).

8.7 Appendix 575

8.7 Appendix

Table 8-11. Superstructures of the f.c.c. lattice.1

Designation 2 Spacegroup Basis 3 Equivalent positions Occupation

Positions

A1 (Cu) Fm3m a = a0 1 0 0 (0 0 0; 1/2 1/2 0; 1/2 0 1/2; 0 1/2 1/2) A/B

A5B B2/m a = a0/2 [1 1–

2] (0 0 0; 1/2 0 1/2)+Fig. 8-40 b = a0/2 [2 2

–2] (0 0 0) 2 B

c = a0/2 [3 3 0] (0 1/2 0) 2 A(0 0 ±1/3) 4 A

(0 1/2 ±1/3) 4 A

D1a (Ni4Mo) I4/m a1 = a0/2 [3 1–

0] (0 0 0; 1/2 0 1/2)+Fig. 8-41 a2 = a0/2 [1 3 0] (0 0 0) 2 B

c = a0/2 [0 0 2] (x y 0; x– y– 0; y– x 0; y x– 0) 8 Ax = a0 ÷ææ2/5 y = a0/ ÷æ10

L12 (Cu3Au) Pm3m a = a0 1 0 0 ––(0 0 0) B

(1/2 1/2 0; 1/2 0 1/2; 0 1/2 1/2) 3 A

–– P4/mmm a1 = a0 [1 0 0] ––a2 = a0 [0 1 0] (0 0 0) Bc = a0 [0 0 1] (1/2 1/2 0) A B

(1/2 0 1/2; 0 1/2 1/2) A

D022 (Ti3Al) I4/mmm a1 = a0/2 [0 1 0] (0 0 0; 1/2 1/2 1/2)+Fig. 8-42 a2 = a0/2 [0 0 1] (0 0 0) 2 B

c = a0 [2 0 0] (0 0 1/2) 2 A(0 1/2 1/4; 1/2 0 1/4) 4 A

A3B Amm2 a = a0/2 [0 2 2] (0 0 0; 1/2 0 1/2)+b = a0/2 [0 1

–1] (0 0 0) 2 B

c = a0/2 [0 0 2] (0 0 1/2) 2 A(1/4 1/2 1/4; 1/4 1/2 3/4) 4 A

A2B (Pt2Mo) Immm a = a0/2 [1 1–

0] (0 0 0; 1/2 1/2 1/2)+Fig. 8-43 b = a0 [1 0 0] (0 0 0) 2 B

c = a0/2 [3 3 0] (0 0 ±1/3) 4 A

L10 (CuAu) P4/mmm a1 = a0/2 [1 1 0] ––a2 = a0/2 [1 1

–0] (0 0 0) B

c = a0 [0 0 1] (1/2 1/2 1/2) A

L11 (CuPt) R 3–m a1 = a0/2 [2 1 1] ––

Fig. 8-44 a2 = a0/2 [1 2 1] (0 0 0) Ba3 = a0/2 [1 1 2] (1/2 1/2 1/2) A

A2B2 I41/amd a1 = a0 [0 1 0] (0 0 0; 1/2 1/2 1/2)+Fig. 8-45 a2 = a0 [0 0 1] (0 0 0; 0 1/2 1/4) B

c = a0 [2 0 0] (0 0 1/2; 0 1/2 3/4) A

1 Figures 8-40 to 8-45 represent the original f.c.c. unit cell and the unit cells of the superstructures; 2 “Struk-turbericht” designation; 3 In terms of vectors of the f.c.c. structure.


Figure 8-40. Original f.c.c. unit cell and unit cell ofthe superstructure A5B (B2/m), indicated by boldlines.

Figure 8-43. Original f.c.c. unit cell and unit cell ofthe superstructure A2B (Pt2Mo), indicated by boldlines.

Figure 8-41. Original f.c.c. unit cell and unit cell ofthe superstructure D1a (Ni4Mo), indicated by boldlines.

Figure 8-42. Original f.c.c. unit cell and unit cell ofthe superstructure D022 (Ti3Al), indicated by boldlines.

Figure 8-45. Original f.c.c. unit cell and unit cell ofthe superstructure A2B2 (I4/amd), indicated by boldlines.

Figure 8-44. Original f.c.c. unit cell and unit cell ofthe superstructure L11 (CuPt), indicated by boldlines.

8.7 Appendix 577

Table 8-12. Superstructures of the b.c.c. lattice.

Designation1 Spacegroup Basis 2 Equivalent positions Occupation

Positions

A2 (Fe) Fd3c a = a0 1 0 0 –– A/B

D03 (Fe3Al) Fm3m a1 = a0 [2 0 0] ––a2 = a0 [0 2 0] (0 0 0) 4 Ba3 = a0 [0 0 2] (1/2 1/2 1/2) 4 A

(1/4 1/4 1/4; 3/4 3/4 3/4) 8 A

F 4–

3 m a1 = a0 [2 0 0] ––a2 = a0 [0 2 0] (0 0 0) 4 Ba3 = a0 [0 0 2] (1/2 1/2 1/2) 4 A

(1/4 1/4 1/4) 4 A/B(3/4 3/4 3/4) 4 A

B2 (CsCl) Pm3m a = a0 1 0 0 ––(0 0 0) A

(1/2 1/2 1/2) B

B32 (NaTl) Fd3m a1 = a0 [2 0 0] (0 0 0; 0 1/2 1/2; 1/2 0 1/2; 0 1/2 1/2)+a2 = a0 [0 2 0] (0 0 0; 1/4 1/4 1/4) 8 Aa3 = a0 [0 0 2] (1/2 1/2 1/2; 3/4 3/4 3/4) 8 B

1 “Strukturbericht” designation; 2 In terms of vectors of the b.c.c. structure.

Table 8-13. Superstructures of the hexagonal lattice.

Designation1 Spacegroup Basis 2 Equivalent positions Occupation

Positions

A3 F63/mmc a = a0 [1 0 0] ––b = a0 [0 1 0] (1/3 2/3 3/4) 2 (A/B)c = a0 [0 0 1]

D019 (A3B) P63/mmc a = a0 [2 0 0] ––b = a0 [0 2 0] 6 h (5/6 2/3 1/4) 6 Ac = c0 [1 0 0] 2 c (1/3 2/3 1/4) 2 B

B19 Pmma a = c0 [0 0 1–] ––

b = a0 [0 1 0] 2 f (1/4 1/2 5/6) 2 Ac = a0 [2 1 0] 2 e (1/4 0 1/3) 2 B

A2B P21/m a = a0 [1 1–

0] 2 e (1/2 1/6 1/4) Ab = a0 [1 2 0] 2 e (1/6 1/2 1/4) Ac = c0 [0 0 1] 2 e (5/6 5/6 1/4) B

A2B p6m a = a0 [1 1–] 2 b (1/3 2/3) A

(2-dim.) b = a0 [1 2] 1 a (0 0) B

1 “Strukturbericht” designation; 2 In terms of vectors of the h.c.p. structure.


In writing this chapter the author hasbenefited from the work of past students,Claudio G. Schön, Sabine Crusius andHelen Ackermann, and from intense co-op-eration with colleagues, Ryoichi Kikuchi,Catherine Colinet and Christophe Bichara.This is gratefully acknowledged. The con-cept of multicomponent correlation func-tions was worked out in co-operation withAlphonse Finel when writing the versionfor the first edition of this volume (Indenand Pitsch, 1991). Thanks are due to AlanOates for very helpful and enlighteningdiscussions.

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9 Diffusionless Transformations

Luc Delaey

Departement Metaalkunde en Toegepaste Materiaalkunde,Katholieke Universiteit Leuven, Heverlee-Leuven, Belgium

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 5859.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5879.2 Classification and Definitions . . . . . . . . . . . . . . . . . . . . . . . 5909.3 General Aspects of the Transformation . . . . . . . . . . . . . . . . . 5939.3.1 Structural Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5939.3.2 Pre-transformation State . . . . . . . . . . . . . . . . . . . . . . . . . . 5979.3.3 Transformation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 5999.3.4 Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6009.3.5 Shape Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6009.3.6 Transformation Thermodynamics and Kinetics . . . . . . . . . . . . . . . 6049.4 Shuffle Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 6079.4.1 Ferroic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6099.4.2 Omega Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6109.5 Dilatation-Dominant Transformations . . . . . . . . . . . . . . . . . . 6109.6 Quasi-Martensitic Transformations . . . . . . . . . . . . . . . . . . . 6119.7 Shear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 6139.8 Martensitic Transformations . . . . . . . . . . . . . . . . . . . . . . . 6159.8.1 Crystallography of the Martensitic Transformation . . . . . . . . . . . . . 6159.8.1.1 Shape Deformation and Habit Plane . . . . . . . . . . . . . . . . . . . . 6159.8.1.2 Orientation Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 6169.8.1.3 Morphology, Microstructure and Substructure . . . . . . . . . . . . . . . 6189.8.1.4 Crystallographic Phenomenological Theory . . . . . . . . . . . . . . . . 6209.8.1.5 Structure of the Habit Plane . . . . . . . . . . . . . . . . . . . . . . . . . 6239.8.2 Thermodynamics and Kinetics of the Martensitic Transformation . . . . . 6249.8.2.1 Critical Driving Force and Transformation Temperatures . . . . . . . . . 6249.8.2.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6309.8.2.3 Growth and Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6329.8.2.4 Transformation Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . 6349.9 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6349.9.1 Metallic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6349.9.1.1 Ferrous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6359.9.1.2 Non-Ferrous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6379.9.2 Non-Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6399.10 Special Properties and Applications . . . . . . . . . . . . . . . . . . . 6419.10.1 Hardening of Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641


9.10.2 The Shape-Memory Effect . . . . . . . . . . . . . . . . . . . . . . . . . 6419.10.3 High Damping Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6479.10.4 TRIP Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6479.11 Recent Progress in the Understanding of Martensitic Transformations 6499.12 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

584 9 Diffusionless Transformations



a lengthA factor (containing elastic terms)a, b, c constantsAd retransformation temperature (deformation induced)Aj amplitude of perturbation with polarization jAs starting temperature for austenite formationB pure strains associated with lattice correspondenceC lattice correspondenceC cubic sequenceC¢ elastic shear constantc* size of critical nucleusCij eigenvalues of elasticity tensorc/a axial ratioc/r thickness/radius ratio of nucleuse order parameterE electric fielde1, e2, e3 principal strainsEe strain energyF applied forceDG difference in chemical Gibbs energyDG* Gibbs energy of nucleationDgs surface Gibbs energy per unit volumeG* elastic state functionGa, Gb, Gg Gibbs energy of phases a, b, gGc, Gchem chemical Gibbs energyGelast elastic Gibbs energyGsurf surface Gibbs energyGtot total Gibbs energyGd defect Gibbs energyGi interaction Gibbs energyH magnetic fieldH* elastic state functionDH, DH* enthalpy changel molar lengthMd deformation-induced martensitic temperatureMf martensite finishing temperatureMs martensite starting temperatureP inhomogeneous lattice-invariant deformationq wave vectorr radius of a plater lattice vectorR rigid body rotationR rhombohedral sequence

r* size of critical nucleusS strain matrixDS entropy changeT temperatureT0 equilibrium temperatureTc critical transition temperatureTN Neel temperatureTTW temperature at which twins appearv volume of a plateVm molar volumex atomic fraction of elementsx, y lattice vectors

a name of phaseb name of phaseg name of phaseG interfacial energyd0 shear straine name of phasee0 strain associated with the transformationj surface to volume ratios stresssa applied stress

ASM American Society for Materialsb.c.c. body-centered cubicb.c.t. body-centered tetragonalf.c.c. face-centered cubicf.c.t. face-centered tetragonalG–T Greninger–Troianoh.c.p. hexagonal close packedHIDAMETS high-damping metalsHP habit planeHRTEM high-resolution transmission electron microscopyIPS invariant plane strainK–S Kurdjurnov–SachsLOM light optical microscopyN–W Nishiyama–WassermannPTFE polytetrafluoroethyleneSMA shape-memory alloysSME shape-memory effectTRIP transformation-induced plasticity



9.1 Introduction

Diffusionless solid-state phase transfor-mations, as the name suggests, do not re-quire long-range diffusion during the phasechange; only small atomic movements overusually less than the interatomic distancesare needed. The atoms maintain their rela-tive relationships during the phase change.Diffusionless phase transformations there-fore show characteristics (such as crystal-lographic, thermodynamic, kinetic and mi-crostructural) very different from those ofdiffusive phase transformations.

Martensitic transformations, becausesome of the properties associated with themsometimes lead to specialized applications,are considered to be an extreme class ofdiffusionless phase transformation and wetherefore in this chapter concentrate onmartensite. Because of the similarity ofsome of the transformation characteristics,a number of other diffusionless solid-statephase transformations have sometimesbeen designated erroneously as marten-sitic. To avoid misinterpretations, Cohen et al. (1979) proposed a classificationscheme that identifies broad categoris ofdisplacive transformations showing fea-tures in common with martensitic transfor-mations, but distinct from them. Their clas-sification scheme, reproduced in Fig. 9-1,will largely be followed here. Martensitictransformations are here only a subclass ofthe broader class of displacive/diffusion-less phase transformations.

The classification proposed by Cohen etal. (1979) is discussed first, and subsequentsections deal with general aspects of thecrystallography, thermodynamics and ki-netics of the different displacive transfor-mations. Although it is not the purpose togive full details of all materials that exhibitthis type of transformation, the most im-portant material systems in which such

transformations have been observed arepresented.

A martensitic transformation can be de-tected by a number of techniques; some arein situ methods, whereas others are step-by-step measurements. The results are usu-ally plotted as a change in a physical prop-erty versus temperature (see the schematicrepresentation in Fig. 9-2), from which thetransformation temperatures can be deter-mined. Some of these plots can be trans-lated into the volume fraction of martensiteformed versus the temperature. Suchcurves allow us to determine the transfor-mation temperatures as indicated in thesefigures.

In situ detection becomes limited if thetransformation temperature is above roomtemperature, and dilatometry then seems tobe the most appropriate technique providedthat quenching – which is needed to avoidalterations in the sample due to diffusion –is possible inside the dilatometer. There arefar more ways of following the transforma-tion when the transformation temperatureis below room temperature – preparing thesample and carrying out the measurementscan take some time and at room tempera-ture diffusion is then almost negligible.During slow cooling after water quench-ing, the techniques frequently used includedilatometry, electrical resistivity and mag-netic measurements, calorimetry, in situmicroscopy, acoustic emission, elastic andinternal friction measurements, positronannihilation, and Mössbauer spectroscopy.Some of the less common techniques usedto study martensitic transformations werereviewed by Fujita (1982).

If crystallographic information is re-quired, X-ray and electron or neutron dif-fraction are used. X-ray diffraction meas-urements by Fink and Cambell in 1926 (lat-tice parameter of C-steel martensite), byKurdjurnov and Sachs in 1930, by Ni-

shiyama in 1934 and by Greninger andTroiano in 1949 (orientation relationshipbetween austenite – the parent phase – andmartensite) represent breakthroughs in thestudy of martensitic transformations.

The most frequently used techniqueswill now be briefly discussed and illus-trated.

If a sample is polished into the marten-site (= parent phase), a surface relief ap-pears. An edge-on section of such a sampleis shown in Fig. 9-2a (see Hsu, 1980). Theorigin of the surface relief is indicated bythe white arrows. Owing to the macro-scopic martensite shear (the two thinner ar-rows), a surface relief is obtained. This is


Figure 9-1. Classification scheme for the displacive/diffusionless phase transformations as proposed by Cohenet al. (1979).


Figure 9-2. Some examples of how to see or measure the presence and growth of martensite (see text for de-tails).

explained further in Sec. 9.8.1.1 (Fig.9-19).

The electrical resistance shows at thetransformation temperatures a deviationfrom linearity versus temperature. This isshown in Fig. 9-2b for the martensitictransformation in an Fe–Ni alloy and aAu–Cd alloy. The difference in the resis-tance ratio for the two different materials isobvious and remarkable (see Otsuka andWayman, 1977). Measuring the electricalresistance while cooling or heating a sam-ple is a very convenient and relatively easyand accurate technique of determining thetransformation temperatures Ms , Mf , As

and Af .The heat exchanged with the surround-

ings is becoming a more popular method ofdetermining the transformation tempera-tures. An example is shown in Fig. 9-2c(Nakanishi et al., 1993). A DSC (differen-tial scanning calorimetry) curve allows anyparticular behavior of the martensitic sam-ple to be detected (for example, effects oc-curring during heat treatments and/or def-ormation steps).

A number of martensitic transformationsand materials are characterized by a so-called shape-memory effect (see Sec.9.11.2). Figure 9-2d (courtesy of MemoryEurope) shows the displacement of aspring made of a NiTi shape-memory alloy.The spring controls a small valve in a cof-fee-making machine. At the temperature As

the hot water starts to drop onto the coffeepowder. This valve is completely open assoon as the temperature Af is reached. Thetemperature range between As and Af seemsto be most suitable for making the best cupof coffee. On cooling, the valve closesagain. The “displacement– temperature”curve measured on cooling does not coin-cide with the heating curve.

During a martensitic transformation, notonly is the shape of the sample changed but

also the specific volume, which allows thetransformation temperatures to be deter-mined by dilatometry (Fig. 9-2e, fromYang and Wayman, 1993).

Changes in mechanical properties arealso measured while the sample is trans-forming. The Young’s modulus exhibits adip between the two transformation tem-peratures Ms and Mf , as clearly visible inFig. 9-2 f for four different alloys (see Su-gimoto and Nakaniwa, 2000).

9.2 Classification and Definitions

A structrual change in the solid state istermed “displacive” if it occurs by coordi-nated shifts of individual atoms or groupsof atoms in organized ways relative to theirneighbors. In general, this type of transfor-mation can be described as a combinationof “homogeneous lattice-distortive strain”and “shuffles”.

A lattice-distortive deformation is a ho-mogeneous strain that transforms one lat-tice into another; examples are shown inFig. 9-3. The homogeneous strain can berepresented by a matrix according to

y = S x (9-1)

where the strain S deforms the lattice vec-tor x into a lattice vector y. This strain is homogeneous because it transformsstraight lines into other straight lines. Aspherical body of the parent phase will thusbe transformed into another sphere or intoan ellipsoidal body. The actual shape of theellipsoid depends on the deformation in thethree principal directions. If a sphericalbody is completely embedded inside thematrix phase and is undergoing the strainS, the volume and shape change associatedwith this deformation will cause elasticand, sometimes, plastic strains in the parent


9.2 Classification and Definitions 591

and/or product phases. The lattice-distor-tive displacements therefore give rise toelastic strain energy. In addition, an inter-face separating the phases is created, gen-erating an interfacial energy. As is obviousfrom Fig. 9-1, the relative values of theseenergies play an important role in the clas-sification scheme.

A shuffle is a coordinated movement ofatoms that produces, in itself, no lattice-distortive deformations but alters only thesymmetry or structure of the crystal; asphere before the transformation remainsthe same sphere after the transformation.Shuffle deformations produce, in the idealcase, no strain energy and thus only interfa-cial energy. Two examples of the shuffledisplacement are given in Fig. 9-4. Shuffledeformations can be expressed by “lattice

Figure 9-3. Examples of the lattice-distortive defor-mations of a cubic lattice: (1) a dilatation in the threeprincipal directions transforms the lattice into an-other cubic lattice with larger lattice parameters; (2)a shear along the (001) plane leads to a monocliniclattice, and (3) an extension along the [001] axiscombined with a contraction along the [100] and[010] axis results in an orthorhombic lattice.

Figure 9-4. Examples of shuffle displacements in:(a1) strontium titanate; oxygen, strontium, ti-tanium; (a2) the displacement of some of the oxygenatoms can be represented by an alternating clockwiseand anti-clockwise rotation around the titanium at-oms; and (b) the (111) planes in a b (b.c.c.) latticeand the collapsed (0001) planes in the hexagonal wlattice (Sikka et al., 1982).

wave modulations” as

Dx = Aj (q) exp (i q · r) (9-2)

where r is a lattice vector, q is the wavevector giving the direction and inversewavelength of the modulation, A is the am-plitude of the perturbation, and j denotesthe polarization of the wave. An alternativedescription is represented by relative dis-placements of the various atomic sub-lat-tices that specify the structures of the twophases in terms of corresponding unit cells,which are not necessarily primitive.

Cohen et al. (1979) subdivided the dis-placive transformations into two maingroups, according to the relative contribu-tion of the two above-mentioned atom dis-placements and hence the ratio of the inter-facial/strain energy. In this context, theydistinguish between “shuffle transforma-tions” and “lattice-distortive transforma-tions”. Since the latter give rise to elasticstrain energy and the former only to inter-facial energy, major differences are foundin the kinetic and morphological aspects ofthe transformation, which justifies the dis-tinction. Shuffle transformations are notnecessarily pure; small distortive deforma-tions may additionally occur. They there-fore also include those transformations in-volving dilatational displacements, in addi-tion to the pure shuffle displacements, pro-vided that they are small enough not to al-ter significantly the kinetics and morphol-ogy of the transformation.

The lattice-distortive transformationsthemselves are subdivided according to therelative magnitudes of the two componentsof the homogeneous lattice deformation,i.e., the dilatational and the deviatoric (shear) components (see Fig. 9-3). The in-itial and the isotropically dilated sphereshave no intersection and it is therefore notpossible to find a vector whose length hasnot been changed by the transformation.

On the other hand, the ellipsoid obtainedafter a pure shear intersects the originalsphere; hence a set of vectors exist, whoselengths remain unchanged. Such a defor-mation is said to be characterized by an“undistorted line“. An undistorted line canonly result from a homogeneous latticedeformation if the deviatoric or shear com-ponent sufficiently exceeds the dilatationalcomponent.

Cohen et at. (1979) thus consider a trans-formation as deviatoric-dominant if an in-variant line exists. A further subdivisionwas made between phase transformationswith and without an invariant line, orbetween “dilatation-dominant” and “devi-atoric-dominant” transformations.

If the magnitude of the lattice-distortivedisplacements is large in relation to that ofthe lattice vibrational displacements, highelastic strain energies are involved. How-ever, if they are comparable, the strain energies will be small and hence will notdominate the kinetics and morphology ofthe transformation. In the latter case wedeal with “quasi-martensitic transforma-tions”. The former, i.e., the “martensitictransformations”, are therefore those dis-placive or diffusionless phase transforma-tions where the lattice-distortive displace-ments are large enough to dominate the ki-netics and the morphology of the transfor-mation. “Martensite” is the name nowgiven to the product phase resulting from amartensitic transformation. Because of thevolume change and the strain energy in-volved with the transformation, martensitictransformation requires heterogeneous nu-cleation and passes through a two-phasemixture of parent and product; it is a first-order diffusionless phase transformation.Consequently, the forward and reversetransformations are accompanied by an ex-othermic and endothermic heat effect, re-spectively, and forward and reverse trans-


9.3 General Aspects of the Transformation 593

formation paths are separated by a hystere-sis.

Among the various diffusionless phasetransformations which exist in solid-to-solid phase transformations, martensitictransformations have received much atten-tion in the past. Historically, the term “mar-tensite” was suggested by Osmond in 1895,in honor of the well-known German metal-lurgist Adolph Martens, as the name for thehard product obtained during the quench-ing of carbon steels. It was found that thetransformation to martensite in steel wasassociated with a number of distinctivecharacteristic structural and microstructu-ral features. During the last few decades itwas recognized that martensite also formsin numerous other materials, such as super-conductors, non-ferrous copper-based al-loys, zirconia, which recently became apopular research subject for ceramists,physicists, chemists and polymeric and bi-ological scientists.

Martensitic transformations have beenthe subject of a series of international con-ferences held in various places throughoutthe world. A list is given at the end of thischapter.

This increasing interest not only has aca-demic origins but can to a large extent alsobe attributed to a number of industrial ap-plications such as maraging, TRIP (trans-formation-induced plasticity) and dual-phase steels, applications involving theshape-memory effect, the high damping ca-pacity, and the achievement of transforma-tion toughening in ceramics.

It may also be of interest to draw atten-tion here to the “massive phase transforma-tions”. Although this type of phase trans-formation, which occurs upon fast cooling,is composition-invariant and the transfor-mation interface has a relatively rapidmovement, it does not fall into the presentcategory. Massalski (1984) defines mas-

sive transformation as a non-martensiticcomposition-invariant reaction involvingdiffusion at the interfaces (see also thechapter by Purdy and Bréchet (2001)).

“Bainite transformations” are also nottreated in this chapter, although they doshow some martensitic characteristics, butcombined with diffusional processes. Forfurther information, the reader is referredto Aaronson and Reynolds (1988) for an in-troductory review and to Krauss (1992).

9.3 General Aspectsof the Transformation

The various diffusionless phase transfor-mations have a number of features in com-mon, such as the crystallographic aspectsof the structural changes, the pretransfor-mation state, the transformation mecha-nisms, the microstructure and the shapechanges that result from the transforma-tion, and the thermodynamic and kineticaspects. The more general aspects aretreated in the following section before dis-cussing separately each subclass of trans-formation.

9.3.1 Structural Relations

This section is concerned with somecrystallographic aspects of the structuralchanges. It is always useful to first deter-mine a unique relationship – a lattice corre-spondence (C ) – between any vector in theinitial lattice and the vector that it becomesin the product lattice. A lattice correspon-dence thus defines a structural unit in theparent phase that, under the action of a ho-mogeneous deformation, is transformedinto a unit of the product phase. Such a cor-respondence therefore tells us which vec-tors, planes and unit cells of the productphase are derived from particular vectors,

planes and cells of the parent phase, with-out regard to their mutual orientation. Forevery structural change there exist manyways of producing a lattice correspon-dence; the one that involves minimalatomic displacements and which reflectsthe experimentally observed orientation re-lationships most closely should be se-lected.

The actual relationship between labelledvectors, planes, etc., before and after trans-formation (including their mutual orienta-tion) is given by the lattice deformation.Mathematically this lattice deformation isfactorized into a pure strain and a pure rotation, so the correspondence indicateswhat is the pure strain. Knowing the princi-pal axes of the strain ellipsoid, the direc-tions of undistorted lines, if any exist, canthen easily be found. In 1924, Bain pro-posed such a lattice correspondence for thef.c.c.-to-b.c.c. (or b.c.t.) transformation iniron alloys; it is referred to in the literatureas the prototype Bain correspondence and/or Bain strain. Since then, lattice corre-spondence values (C ) and their associatedpure strains (B) have been published for anumber of other structural changes; someexamples are given in Fig. 9-5.

In the original Bain strain, a tetragonalcell is delineated into two adjacent f.c.c.unit cells. Then, it is contracted along z byabout 20% and is expanded along x¢ and y¢by about 12%. In another example, thetransformation from a NaCl-type structureinto a CsCl-type structure, a contraction of40% along the [111] body diagonal and a19% isotropic expansion in the perpendicu-lar (111) plane is needed; the volumechange is about 17%.

Homogeneous strains alone, however, donot always describe the structural transfor-mation completely. Additional shufflesmay be needed to obtain the exact atom ar-rangements inside the deformed unit cell.

A special situation arises for some mate-rials, when the structural change can beachieved by a pure deformation that leavesone of the principal directions unaltered.Such a situation is found in some polymers.It occurs, for example, in polyethylene,which has an orthorhombic and a mono-clinic polymorph with chains parallel to thez-axis; these strong covalently bondedchains are unlikely to be distorted by thetransformation; consequently, the pure def-ormation along the z-axis (e3= 0) is zero.During transformation the chains are dis-placed transversely in such a way that one


Figure 9-5. Some examples of lattice correspon-dence and homogeneous deformation (expansionsand contractions) for (a) f.c.c. to b.c.c. or b.c.t. (afterBain, 1924) and (b) NaCl- to CsCl-type structures(the Shôji–Buerger lattice deformation) (Kriven,1982).


of the remaining principal strains becomespositive (e2 > 0) and the other negative(e1> 0) (Bevis and Allan, 1974).

The strain ellipsoid for the above exam-ple has a special shape. The cone of undis-torted vectors of the product phase degen-erates into a pair of planes, which rotate inopposite directions in the pure strain.Hence either of them may be invariant ifthe total deformation of the lattice includesa suitable rotation. Because all the vectorsin this plane are undistorted, the transfor-mation is said to be an “invariant planestrain (IPS)” type. The pure Bain strain isthen equivalent to a simple shear on that in-variant plane. Because this invariant planeis also a matching plane between the ma-trix and the product phase and both phaseshave to be present at the same time, a rigidbody rotation (R) over an angle q is re-quired in order to bring the product and theparent phases in contact along that plane,the habit plane.

A similar situation is found in structu-rally less sophisticated systems, namely thef.c.c. to h.c.p. transformation in cobalt.Both phases are close packed and a simpleshear on the basal plane transforms the cu-bic stacking into a hexagonal stacking. Be-cause the atomic distances in the basalplane do not change significantly duringthe transformation, the plane of simpleshear is the plane of contact or the habitplane (HP). This is the case, however, onlyif there is zero volume change in the trans-formation, i.e., in the case quoted above ifthe h.c.p. phase has an ideal axial ratio of1.633.

The situation becomes more complicatedif none of the principal strains is zero, butof mixed sign. To achieve matching alongthe plane of contact in cases where the twophases coexist, a deformation is needed ad-ditional to the pure Bain strain in order tohave an invariant plane. Because the final

lattice hase already been generated by theBain strain, this additional strain should bea “lattice-invariant strain”. Slip and twin-ning in the product phase or in both phasesare typical lattice-invariant strains; bothdeformation modes are shown schemati-cally in Fig. 9-6. The diffusionless phasetransformation can in this case be repre-sented by an analog consisting of a purelattice strain (B), a rigid lattice rotation (R)and an inhomogeneous lattice-invariantdeformation (P). The last factor is also as-sociated with a shape change, which mac-roscopically can be considered as homoge-neous. Such a combination is typical ofmartensite and is discussed in Sec. 9.8.1.4.

In cases where the lattice-invariant shearis twinning (as opposed to faulting or slip),type I twinning, where the twin plane orig-inates from a mirror plane in the parentphase, has been assumed. Otsuka (1986)carefully analyzed for a number of systemsthe possibility of a type II twinning as analternative inhomogeneous shear. In type IItwinning, the shear direction stems from atwo-fold symmetry direction of the parentphase. In a table, Otsuka (1986) compiledall the twinning modes observed in marten-

Figure 9-6. Schematic representation of (a) the ho-mogeneous lattice deformation, (b) the inhomogene-ous lattice-invariant deformation (slip and twinning),and (c) the lattice rotation.

site and found that most of them are type Ior compound but that type II twinning hadonly recently been observed. According toNishida and Li (2000), five different twin-ning modes exist in TiNi and other shapememory alloys such as Cu-Al-Ni, Cu-Snetc., namely the 111 type I, the 011type I, the ⟨011⟩ type II, the (100)-com-pound and the (001)-compound twins.Type II twinning has recently receivedmuch attention as a mechanism for latticeinvariant shear in some alloys. Sincetype II twins have irrational twin boundar-ies, the physical meaning of an irrationalboundary is still a controversial problem. Ithas proposed that an irrational boundaryconsists of rational ledges and steps, theaverage being irrational. Thereafter, Haraet al. (1998) carried out a careful study toobserve ⟨111⟩ type II twin boundaries in aCu-Al-Ni alloy by HRTEM, but they couldnot observe any ledges or steps. The boun-dary is always associated with dark straincontrast, and the lattice is continuousthrough the irrational boundary. Nishidaand Li (2000) also made extensive studieson ⟨011⟩ type II twin boundaries in TiNi byHRTEM, but they did not observe ledges orsteps either. Based on these experimentalresults, it is thus most likely that the type IIthin boundary is irrational even on a micro-scopic scale, and the strains at the boun-dary are elastically relaxed with wide twinwidth. To confirm this interpretation, Hara

et al. (1998) carried out computer simula-tions by using the molecular dynamicsmethod. The result showed that the irra-tional thin boundary did not show anysteps. Thus, the above interpretation for anirrational twin boundary is justified. Ot-suka and Ren (1999) have pointed outagain the importance of type II twinning inthe crystallographic aspects of martensite.They also stress the role that martensite ag-ing has on the rubber-like behaviour ofmartensite, a point that has been a long-standing unsolved problem. They showedthat the point defects play a primordialrole. The deformation mechanisms of thecold deformation of NiTi martensite havebeen thoroughly analyzed by Liu et al.(1999a, b). They also found an interplaybetween type I and type II twinning.

The cubic to tetragonal transformations,which occur in a number of metallic andnon-metallic systems, need some specialattention. The volume change with thesetransformations is sometimes very small oreven absent, and the c/a ratio does notchange abruptly but progressively (Fig.9-7); the transformation is then said to be“continuous”. The c/a ratio can be smalleror larger than unity, depending on compo-sition and temperature. The shape changeassociated with the transformation is smallenough in many systems, especially inthose belonging to the quasi-martensites,for elastic accommodation alone to be suf-


Figure 9-7. Temperaturedependence of c/a as mea-sured during the cubic totetragonal transformation;(a) second and (b) first-or-der phase transformation.


ficient for lattice matching. It is, however,possible for c and a to change abruptly withzero volume change.

Based on the crystallographic aspectsdiscussed above, a list of the most typicalcharacteristics of the diffusionless phasetransformations can be compiled (Table9-1).

9.3.2 Pre-transformation State

Diffusionless structural changes areachieved by atom displacements, such asshuffles and shears. The new atomic con-figuration is already prepared in some ma-terial systems at temperatures above thetransition temperature. Atoms in the parentphase then become displaced more easilytowards their positions in the new phasebecause the restoring force that is felt bythe displaced atoms diminishes on cooling.In certain cases the restoring force even vanishes at the phase transition tempera-ture.

Certain shuffle transformations resultfrom a vibrational instability of the parentphase and are therefore called “softmode”

phase transformations. A soft mode is, in simple terms, a vibrational mode, thesquare of whose frequency tends towardzero as the temperature approaches that ofthe phase transition. The average staticatom displacements resemble the frozen-inpattern of the vibrational displacements ofa certain vibrational mode. According toVallade (1982), “the crystal lattice vibra-tions can within the harmonic approxima-tion be separated into independent planewaves (phonons) characterized by a set of collective atomic displacements corre-sponding to a well defined frequency. Theenergy involved is a function of the squaresof the momentum and of the eigenfrequen-cies of the mode. The eigenfrequencies de-pend only on the mass of the atoms and onthe force constants. It is clear that the van-ishing of one eigenfrequency correspondsto the lack of restoring force for the mode:the amplitude can then grow without anylimit and the lattice is mechanically un-stable. Stability can be recovered only by changing atomic equilibrium positionswhich, in turn, changes the force con-stants“.

Table 9-1. A schematic overview of some characteristics typical of the various types of diffusionless phasetransformations.

Characteristics Structural change Pure lattice deformation

Type of Principal strains Volume changediffusionlesstransformation type* order ** Sign Value

Shuffle C or D F All zero Zero Zero up toS 10–5

Dilatational D F All positive or Large Large:All negative 10–1

Quasi-martensitic C or D F Mixed sign Small Small:S 10–4–10–3

Martensitic D F Mixed sign or Large Small or large:Zero and +, – 10–2–10–1

* C: continuous, D: discontinuous** F: first order, S: second order

As regards SrTiO3, the rotation-vibra-tional modes of the oxygen atoms are fro-zen into the low-temperature positions be-low 110 K. The temperature dependence ofthe softening, expressed by the square ofthe eigenfrequency of the mode, is repre-sented schematically in Fig. 9-8 for a sec-ond- and first-order phase transformation.Usually, the low-temperature phase alsoshows a soft mode as the temperature israised towards Tc.

Lattice softening can also be treated interms of a static approach in which thestability of the lattice is examined whensubmitted to small static or quasi-static ho-mogeneous strains. The free energy is thenexpressed as a function of the elastic con-stants; for a lattice to be stable when sub-mitted to small homogeneous strains, thefree elastic energy must increase for allpossible strains. For a cubic crystal this ismathematically expressed by saying thatall the eigenvalues of the elasticity tensormust be positive, in other words C44>0,(C11–C12) > 0, and (C11+ 2 C12) > 0.

The tendency toward mechanical in-stability can also be studied through exam-ination of the phonon dispersion curves,which gives the relationship between thewavevector q of the vibrational mode andits eigenfrequency. The lattice instabilitycan correspond to uniform (q = 0) or mod-ulated (q = non-zero) atom displacementsand the soft phonon may belong to an opticor an acoustic branch; an example of ameasured dispersion curve is given in Fig.9-9. The longitudinal acoustic branch inzirconium shows a dip at about 2/3 [111],which is the mode needed to transform thehigh-temperature b.c.c. structure of Zr intothe omega structure. The slope of the trans-verse acoustic branch of Nb3Sn is very flatat the origin on approaching the transitiontemperature of 46 K (Shapiro, 1981). Thiscorresponds fairly well with the experi-mental observation of a vanishing value ofthe elastic shear constant C ¢= (C11 – C12)/2.The atom displacements induced by thissoft mode coincide exactly with those as-sociated with the deformation from cubic


Figure 9-8. Temperaturedependence of the squaredfrequency of the softeningmode for (a) a second-orderand (b) a first-order transfor-mation, Tc and Tt being thecritical and the transforma-tion temperatures, respec-tively. (c) Phonon energy ofSrTiO3 measured below andabove Tc (after Rao and Rao,1978).


to tetragonal structure. For Cu–Zn–Al nosoft mode is present at 2/3 [111], althoughthe LA branch shows an anomalous dip;the branch measured perpendicular to itproves that the point in the reciprocal spaceis a saddle point and not a minimum. Thebranch TA2 [110] (polarization [11

–0]),

however, shows a small slope correspond-ing to a low value of C ¢ (Guénin 1982).Transformation models have been pro-posed for Cu–Zn–Al taking into accountboth the anomalous dip and the smallslope.

In a number of materials undergoing a cubic to tetragonal transformation, atweed-like pattern is observed in the parentphase by transmission electron micros-copy. This tweed contrast is characterizedby a ·100Ò direction of the modulation, atype of 110 ·11

–0]Ò shear strain and a

modulation which is incommensurate withthe parent phase. It is still debated whetherall the pre-transformational or precursoreffects are evidence of stable or metastablemodulated phases or whether they are well-defined artefacts determined by the kinet-ics of nucleation and the growth process.

A long-standing issue with b Cu-, Ag-and Au-base alloys that has now been re-solved is the appearance of extra maximain the electron diffraction patterns of theparent phase from quenched alloys. Overthe years these maxima have been givenvarious interpretations, often with an over-emphasis as possible pre-martensitic ef-fects. Systematic investigation, however,established that these effects are in fact ob-tained in martensitic structures located onthe surface of the thin foil and extendinginwards to a depth of 1 µm.

When considering martensitic transfor-mations, the role played by the combina-tion of lattice defects and of lattice instabil-ities is of particular interest; the large defor-mations present around the defects may in-

duce a localized lattice instability (Guéninand Gobin, 1982), which may trigger thenucleation of martensite on further coolingor stressing.

Pre-transformational lattice instabilitiesand soft modes and their relation to diffu-sionless phase transformations have beenreviewed by Delaey et al. (1979), Nakani-shi (1979), Vallade (1982), Nakanishi et al.(1982), and Barsch and Krumhansl (1988).The validity of the soft phonon or soft elas-tic stiffness approach to martensite is a dif-ficult and somewhat controversial subject.

9.3.3 Transformation Mechanisms

It should be emphasized that the pure lat-tice distortions considered above do notnecessarily imply the actual path the atomsfollow during the transformation. For sec-ond-order phase transformations, there is acontinuous change throughout the crystalwith decreasing temperature starting at Tc.Following the soft-mode concept, the ap-pearance of the new phase is considered asthe freezing of a particular wavelength vi-bration. The Bain-type strain for a second-order cubic to tetragonal transformation,for example, is equivalent to two 110·11

–0Ò shear strains whose corresponding C¢

Figure 9-9. Phonon dispersion curves for b.c.c. Zr.A pronounced dip occurs in the longitudinal (l) pho-non dispersion curve in the vicinity of q = 2/3 [111] (Sikka et al., 1982).

the fully formed product. The exact mecha-nisms for the various types of martensitictransformations are still under debate.

9.3.4 Microstructures

The microstructures that result from diffusionsless phase transformations showtypical features, which are related to thecrystallograph of the transformation. Thetransformation is associated with a reduc-tion in symmetry; consequently, differentequivalent orientational states of the prod-uct phase are formed. A single crystal ofthe parent phase thus transforms to a col-lection of the product-phase crystals, calledvariants, that are separated by interfaces.The higher the symmetry of the parentphase and the lower the symmetry of theproduct phase, the greater is the number ofequivalent paths of transformation. Thenumber of equivalent orientations or vari-ants is also determined by the symmetryelements that are maintained or broken dueto the Bain strain. The collection of vari-ants constitutes the microstructure.

The order of the transformation (whetherfirst or second order) also determines themicrostructure: in the former case parent/product or heterophase interfaces in addi-tion to product/product or homophase inter-faces are created, whereas in the latter onlyproduct/product interfaces are formed. Inthe former case the first plates formed cangrow to a larger extent than those formedlater, which can lead, for example, to mi-crostructures with fractal characteristics.

Fig. 9-10 shows a selection of character-istic microstructures obtained through dif-fusionless phase transformations.

9.3.5 Shape Changes

If we could transform a single crystal ofthe parent phase into a single crystal of the


shear constant vanishes at Tc. The trans-formation mechanism is therefore not acombination of expanding and contractingatom movements, but a lock-in of long-wavelength shear-type movements on110 planes in the ·11

–0Ò directions in this

scheme.As for the martensitic transformations,

the situation is not so straightforward. TheBain-type strains are concerned only withthe correspondence between initial and fi-nal lattices; they do not give the actual ob-served crystal orientation relationshipsbetween them. Based on the experimen-tally determined orientation relationships,different transformation mechanisms havebeen proposed, such as shears on the planesand along the directions involved in the orientation relationship. However, theseshear mechanisms have been found to betoo simple to be consistent with the experi-mental facts. More recently, a transforma-tion mechanism has been proposed formartensitic transformations of b.c.c. toclose-packed structures; a condensing stateof some soft phonon modes combined witha homogeneous shear explains the varietyof structures that are found. For the sametransformations, Ahlers (1974) proposed atwo-shear mechanism; the first shearcreates the close-packed planes, whereasthe close-packed structure is obtained bythe second shear.

Martensitic transformations are first-order phase transformations that occur bynucleation and growth. The growth stagegenerally takes place by the motion ofinterfaces converting the parent phase tothe fully formed product phase. Two typesof paths have to be considered for the caseof nucleation, the “classical” and the “non-classical” nucleation paths (Olson and Co-hen, 1982). The latter involves a continu-ous change in structure whereas the formerinvolves a nucleus of the same structure as


Figure 9-10. A selection ofmicrostructures obtained bydiffusionless phase transfor-mations: (i) schematic repre-sentation of the microstruc-ture of (a) martensite andaustenite, (b) the Dauphinétwins between the low tem-perature a- and the hightemperature b-phase ofquartz, and (c) the zig-zagdomain structure in neody-mium pentaphosphate whichundergoes an orthorhombicto monoclinic transforma-tion (after James, 1988); (ii) optical and transmissionelectron micrographs of (a, b)the twinned orthorhombicYBa2Cu3Ox high-tempera-ture superconductor (cour-tesy H. Warlimont, 1989),(c) the domain in SiO2 at thetransition temperature a to b(846 K) (courtesy Van Ten-deloo, 1989), and (d) thefractal nature of the marten-site microstructure in steel(courtesy Hornbogen, 1989).

product phase, the macroscopically visibleshape change would clearly reflect theBain-type strain; it is then the maximumtransformation-induced shape change thatcan be achieved. Depending on the symme-try relationship, this deformation can beobtained in as many orientations as productvariants exist.

In a martensitic transformation, the mac-roscopic shape change associated with theformation of a single martensite plate is notonly the result of the Bain strain but also ofa lattice-invariant deformation. The totalmacroscopic shape change is mainly ashear deformation along the habit plane ofthe martensite variant. The martensite platecontains either a large number of stackingfaults or has twins inside. It is therefore nota single crystal. If the single martensiteplate has twins inside and is subjected afterthe transformation to an externally appliedstress, an additional shape change is ob-tained by detwinning. Only then is the finalproduct a single crystal of the productphase. Fig. 9-11a shows the changes inshape after transforming a b-Cu–Zn–Alsingle crystal into a single martensite vari-ant and Fig. 9-11b shows the shape changeafter partially transforming an iron whisker.

The transformed sample usually containsa very large number of single-product do-mains arranged in a special configuration.In some systems the domains are arrangedsuch that the shape changes are mutuallyaccommodated. Because each product variant is associated with a differentlyoriented shape change, applying a mechan-ical stress during the transformation willpromote the formation of those variantsthat accommodate the applied stress. Thisprovides a resolved shape change in the di-rection of the applied stress. This is thefundamental concept of the shape-memoryeffect, as will be explained further in Sec.9.10.2.


Figure 9-11. Macroscopic shape change associatedwith martensite: (a) a Cu–Zn–Al single crystal beforeand after transforming to martensite and (b) a par-tially transformed Fe whisker (courtesy Wayman,1989).

a

b


If a single crystal of the product phase ismechanically strained it can either be trans-formed to another or be deformed to a dif-ferently oriented single crystal of the prod-uct phase (Fig. 9-12). Similar behaviour isalso typical of a number of ferroelastic materials; the reorientation is there referredto as “switching” (Wadhawan, 1982). Theswitching force in these materials is notonly a mechanical stress but can also be anelectric or magnetic field, the domains be-ing either electrically or magnetically po-larized.

In first-order phase transformations, asshown above, the full transformation shapechange is induced locally and is graduallyspread over the whole sample within asmall temperature interval, whereas in asecond-order phase transformation thesample changes its shape homogeneouslyand continuously as soon as the criticaltransition temperature Tc is reached.

Until now, shape changes have been dis-cussed that are induced by the forwardtransformation. It is evident that if a sam-ple of the low-temperature phase, a singlecrystal or a polyvariant, is heated to tem-peratures above the reverse transformation

temperature, similar shape changes are ob-served, provided that the reverse transfor-mation is also diffusionless. The situationfor second-order phase transformations isstraightforward; the sample whose shape ischanged during the forward transformationand possibly after deformation below Tc re-verts back to its original shape above Tc ina homogeneous and continuous way. Forfirst-order transformations, the reversetransformation is more complex and notyet well understood. Much depends onwhether the forward transformation iscompleted or not, and whether the growthof the martensite plate occurs by bursts orunder thermoelastic conditions (see be-low). Occurrence of the reverse transfor-mation does not necessarily imply that theoriginal shape is restored. Depending onthe crystal symmetry of the product phase,more than one crystallographically equiva-lent path can be followed for the reversetransformation. The shape changes that oc-cur during the reverse transformation are atthe origin of the shape-memory effect andare discussed in Sec. 9.10.2.

Figure 9-12. A series of macrographs representing the shape change while mechanically straining a Cu–Al–Nimartensite single crystal; (a) to (e) increasing with time (Ichinose et al., 1985).

9.3.6 Transformation Thermodynamicsand Kinetics

A diffusionsless phase transformationmay be of second or first order. The formeris generally dealt with in the phenomeno-logical Landau theory, while the latter is treated by classical thermodynamics(Kaufman and Cohen, 1958). The Landautheory has, however, been extended to alsocover first-order phase transformations(the Devonshire–Ginzburg–Landau theory)and has been applied by Falk (1982) tomartensitic transformations. The reader isreferred to the chapter by Binder (2001) foran introduction to those theories.

The chemical driving force occupies akey position in the classical thermodynam-ics of first-order diffusionless phase trans-formations, a subject that is introduced in thefirst chapter of this volume (Pelton, 2001). Inthe following section, therefore, only thoseaspects directly relevant to diffusionlessphase transformations will be dealt with.

Because no chemical compositionchange is associated with a diffusionlessphase transformation, the parent and prod-uct phases have the same homogeneouschemical composition and hence they aretreated as a single-component system. Forthose phase transformations whose structu-ral change is easily described by a dis-placement parameter, a phenomenologicaldescription of the free energy as a functionof the order parameter in terms of the Lan-dau theory leads to some interesting con-clusions. In the following, the free energyis discussed as a function of temperatureand composition. Other possible intensivethermodynamic state variables include ex-ternal pressure, mechanical stress, andmagnetic and electrical field strength.

The changes in chemical Gibbs energy,DG, as a function of temperature and com-position are shown schematically in Fig.

9-13 for first-order diffusionless phasetransformations between a parent phase,denoted P, and a product phase, denoted M.The product phase may be one of the low-temperature equilibrium phases or a meta-stable phase.

Taking again the martensitic transforma-tion as an example, the transformationstarts at Ms , which is lower than T0, andfinishes at Mf . This means that a higherdriving force is needed to complete thetransformation. On heating a fully marten-sitic stress-free single crystal, the reversetransformation sets in at a temperature As ,


Figure 9-13. Schematic representation of the molarGibbs energy (a) as a function of temperature butconstant composition and (b) as a function of compo-sition for an Fe–Ni alloy with T4 > T3 > T2 > T1 and T2 = T0 for XNi = X (after Mukherjee, 1982).


which is higher than T0. The differencebetween the forward and the reverse trans-formation temperatures is the transforma-tion hysteresis. The true first-order equilib-rium temperature, T0, which is calculatedfrom DG = 0, can thus only be bracketedfrom experimental data for the forward andthe reverse transformation temperatures,and is not necessarily halfway between Ms

and As.Strain energy resulting from the transfor-

mational shape change and interfacial en-ergy have been omitted from the free-en-ergy curves in Fig. 9-13. These two non-chemical-energy terms have to be consid-ered, however, in the overall free-energybalance. The strain energy associated withthe formation of a single domain of theproduct phase is proportional to the volumeof that domain. The interfacial energy isnot directly related to the volume of thetransformed domain but merely to its sur-face-to-volume ratio, and, in the case of ananisotropic interfacial energy, also to theorientation of the interface. Both quantitiesare positive and thus consume part of thechemical driving force for a forward trans-formation. Both terms will, however, in-crease the driving force for the reversetransformation, provided that the inter-facial coherence is not lost. The reversetransformation might start below T0 if anegligible net driving force is required fornucleation.

Considering the Gibbs energy per unitmolar volume, the total Gibbs energychange per unit molar volume for the for-mation of a single domain of the productphase embedded in the matrix phase isgiven by

DGtot = DGchem + (DGelast + j DGsurf) (9-3)

where j is the surface-to-volume ratio of the single domain. The two terms in parentheses are then the non-chemical con-

tributions to the Gibbs energy change,DGnon-chem, and Eq. (9-3) then becomes

DGtot = DGchem + DGnon-chem (9-4)

The transformation then proceeds untilDGtot becomes minimum or, if the phaseboundary is mobile, until the total force atthe parent-to-product interface is zero. Ifthe advancing direction of the interface isx, we can write

[∂ (DGtot)/∂x] dx = 0 (9-5)

or

[∂DGchem/∂x]dx + [∂DGnon-chem/∂x]dx = 0

The sum of the non-chemical restoringforces is then identical with the chemicaldriving forces. The difficult task now is to find expressions representing the non-chemical terms. Three thermodynamic ap-proaches have been worked out, dealing es-sentially with the influence of the two non-chemical contributions on the transforma-tion behavior (Roitburd, 1988; Ball andJames, 1988; Shibata and Ono, 1975, 1977).

According to Roitburd (1988), the strainenergy, which arises in crystals owing to adiffusionless phase transformation, can de-crease if the crystals are subdivided intodomains arranged such that a maximumcompensation of the individual strain fieldsis achieved. In order to determine which ar-rangements are energetically most favor-able, Roitburd calculates the strain energyfor arbitrary domain arrangements, andthen minimizes this energy. The formula-tion of this problem is complex and canhardly be solved in general, but he suc-ceeded for some specific cases.

Ball and James (1988) do not assumeany geometric restrictions on the shape orarrangements of the domains; they foundthis necessary to determine microstructuresoccurring in complex stress fields, or to explore new and unusual domain arrange-

ments. The general aim of their work wasto develop mathematical models, using cal-culus of variations, capable of predictingthe microstructure, especially the micro-structural details at the interface betweenthe parent and the product phases. At-tempts have been made to predict the pos-sible interfaces between austenite and mar-tensite from a minimization of a Gibbs en-ergy function, which depends on the defor-mation gradients of all possible domainvariants and on temperature. A deforma-tion or domain is then termed stable if it minimizes the total energy. They show,among others, that a martensite–austeniteinterface can exist as an energy-minimiz-ing sequence of very fine twins. A furtherexample of an intriguing application is the formation of triangular Dauphiné twinsin quartz, which become finer and finer inthe direction of increasing temperature. AGibbs energy function accounting for thisbehaviour could be constructed.

Shibata and One (1975, 1977) use theEshelby theory; the principle of their calcu-lation is in a corrected version (Christian,1976) illustrated schematically in Fig.9-14. An embedded part of the parentphase is cut out (step a) and is allowed totransform stress-free into the product phase(step b). A lattice-invariant deformation isapplied (step c) and the transformed crystalis subjected to forces along its surface suchthat it is deformed to the original shape (step d). The thus deformed part of theproduct phase is introduced in the emptyspace of the parent phase (step e), and theforces are removed, creating internalstresses in both the product and the parentphase. The total energy is then calculatedas a function of all possible lattice orienta-tions, taking into account the actual elasticconstants and the modes of lattice-invari-ant deformation, twinning or slip.

The total Gibbs energy of the system istherefore a function not only of the intrin-sic energies of the stress- and defect-freeparent and product phases, but also of thearrangement of the domains. The non-chemical component of the total Gibbs en-ergy of the transforming system is loweredby an appropriate rearrangement of the mi-crostructure and/or by irreversible plasticdeformation.

If the structural change can be repre-sented by an order parameter e, the Gibbsenergy of the system can then, according tothe theory of Landau–Devonshire, be rep-resented by

G = G0 + a (T – T1) e2 – B e4 + Ce6 (9-6)

where a, B and C are constants and T1 > 0.It can be shown that the high-temperaturephase becomes unstable with respect to any fluctuation of e, as soon as the temper-ature reaches T1 on cooling, and hence isthermodynamically metastable between T0

and T1. Accordingly, the low-temperature


Figure 9-14. The necessary steps in calculating theelastic stresses induced by a transforming ellipsoid(Christian, 1976). (For details see text.)

9.4 Shuffle Transformations 607

phase cannot exist at temperatures higherthan T2, which is the temperature abovewhich the low-temperature phase becomesunstable with respect to any fluctuation in e.

That additional undercooling is neededfor further transformation below Ms is due(in part) to the non-chemical contributions,which increase with increasing volumefraction of transformed product.

An interesting aspect of the diffusionlessphase transformations that are accompa-nied by a volume and shape change is therole played by external stresses, e.g., hy-drostatic or uniaxial. Both thermodynamicsand experiments show that the transforma-tion temperatures are affected by the appli-cation of stresses. According to Wollants etal. (1979), the relationship between a uni-axially applied stress s and the transfor-mation temperature T depends on thetransformation entropy and the transfor-mational strain in the direction of the ap-plied stress. This relationship, the Clau-sius–Clapeyron equation for uniaxiallystressed diffusionless first-order phasetransformations, is

ds /dT = – DS/e = – DH*/[T0 (s) e] (9-7)

where DH* = DH – FDl = DH – s e Vm =T0 (s) DS is itself a function of the appliedload, e = Dl/l, l is the total “molar length”of the sample, and F is the applied load(s = F/A). This equation is similar to thatrelating the equilibrium temperature to thehydrostatic pressure, except for the nega-tive sign on the right-hand side of Eq. (9-7).This relationship between ds and dT is ex-perimentally constant for most of the diffu-sionless transformations, which means thatthe thermodynamic quantity DS is, withinthe experimental scatter, independent oftemperature and stress. Knowing the trans-formation strain, uniaxial tensile tests arevery useful for determining the transforma-tion entropy.

The most relevant thermodynamic datafor the various diffusionless phase transfor-mations are presented in Table 9-2.

9.4 Shuffle Transformations

Shuffle transformations from a distinctclass of diffusionless phase transitions. Atthe unit-cell level the atom displacementsare intercellular with little or no pure strainof the lattice. The role of elastic strain en-ergy in shuffle-phase transformations issufficiently small that the transformationcan either occur continuously from the par-ent to the product phase or that it is com-pletely controlled by interfacial energy. Inthe former case the transformation is sec-ond order whereas in the latter it is a first-order phase transformation.

Cohen et al. (1979) gave three exampleswhich clearly illustrate the shuffle transfor-mations. The displacive transformation instrontium titanate is the prototype exampleof a pure shuffle transformation. The asso-ciated strain energy is so small that thetransformation occurs continuously. The b-to-w transformation in some Ti and Zr alloys shows, in addition to the shuffle displacements, small homogeneous latticedistortions. These distortions are smallenough for the transformation mechanismand the resulting microstructure to be dom-inated only by shuffling. In ferroelectrictransformations, which are accomplishedby shuffling, the interfacial energy is con-stituted largely by electrostatic interactionenergies and is therefore dependent on theorientation of the domain interfaces. Theinterfacial energy in those materials isstrongly anisotropic and controls the poly-domain structure.

Phase transformations that can be en-tirely described by shuffle diplacementsare often found where the change in crystal

structure is such that the point group towhich the crystal structure of the productphase belongs is a subgroup of that of theparent phase. In other words, some symme-try elements of the high-temperature phaseare lost on cooling below the transitiontemperature Tc. Because of this group/sub-group relationship, the product phase pos-sesses two or more equally stable orienta-tional states in the absence of any externalfield. The change in crystal structure caneasily be described by an order parameterwhich itself is related to the shuffle dis-

placement. For strontium titanate, the orderparameter would then simply be the rota-tion angle that describes the displacementof the oxygen atoms around the titaniumatoms (see Fig. 9-4). For convenience, theorder parameter is taken as zero for thehigh-temperature configuration and as non-zero for the low-temperature phase. Themajority of such transitions are found inchemical compounds (e.g., Rao and Rao,1978). As soon as the critical temperatureTc is reached on cooling, the order parame-ter changes continuously. The thermody-


Table 9-2. The elastic shear constant and some thermodynamic data characterizing the diffusionless phasetransformation (Delaey et al., 1982b).

Elastic shear constant C ¢ Trans- Thermodynamic quantitiesnear the Ms temperature formation

strain Heat of Change Chemical Transfor-C ¢ (1/C ¢) (dC ¢/dT ) transforma- in driving mation

(1010 Pa) (10–4 K–1) tion entropy force tempera-(J/mol) (J/(K · mol)) (J/mol) ture hys-

teresis (K)

Ferrous g Æ a¢ 2–3 negative ≈10–1 ≈2000–300 5.8 150–450 200–400g Æ e 3–10 (positive 600–1800

for Ni > 30%)

Co alloys negative ≈10–3 ≈400–500 ≈0.2 4–16 40–80rare-earthalloys

Ti and Zr 0.1 negative ≈2 ¥ 10–2 ≈4000 ≈1.0 ≈25 –alloys

b Cu–Ag–Au 0.5–1 4–20 ≈10–2 ≈160–800 0.2–3.0 ≈8–20 10–50alloys (positive)

In alloys 0.05–01 ≈1000 ≈10–3 ≈0 – ≈1.5 1–10(positive)

Mn alloys positive ≈10–3

(strongly)A 15 com- 0.5 ≈1000–3000 ≈10–4

pounds (positive)

Fe–Pt ≈320 – ≈16 ≈20–200(ordered) (ordered)

Fe–Pd ≈1 ≈100 ≈10–3 ≈1200 ≈1200alloys (positive) –10–2 (disordered) (disordered)

9.4 Shuffle Transformations 609

namics of such transformations are then inthe temperature range close to Tc, which isdealt with by a Landau approach.

9.4.1 Ferroic Transformations

Usually a phase transition that is domi-nated by shuffling is associated with achange in some physical properties, such asspontaneous electrical polarization, strainand magnetization. Because the crystalsymmetry of the parent phase decreasesduring the phase transition, two or moreequivalent configurations of the productphase are formed. In the absence of any ex-ternal field, the average polarization of theproduct phase is zero. However, under asuitably chosen driving force, which maybe an electrical field (E ), a mechanicalstress (s), or a magnetic field (H), the do-main walls of the product phase move,switching the crystal from one domain or-ientation to the other. Owing to the applica-tion of a uniaxial stress, for example, oneorientation state can be trensformed repro-ducibly into the other, and the crystal isthen said to be “ferroelastic”. The materialsexhibiting this property are called ferro-elastic materials. Similarly, we can defineferroelectric and ferromagnetic materials.According to Wadhawan (1982), “phasetransitions accompanied by a change of the

point-group symmetry are called ferroicphase transitions. We refer to a crystal asbeing in a ferroic phase if that phase resultsfrom a symmetry-lowering ferroic phasetransition”.

Not all ferroelastic phase transitions be-long to shuffle transformations as definedin Fig. 9-1. Indeed, in addition to shuffledisplacements, as for example those in-volved in the cubic to tetragonal transitionin barium titanate, the lattice may becomehomogeneously distorted. For the exampleconsidered here, the lattice distortion oc-curs discontinuously at the transition tem-perature; the lattice parameters changeabruptly (Fig. 9-15). Even below this tran-sition temperature, the lattice continues tobe homogeneously distorted. In caseswhere this lattice is tetragonal, the c/a ratiosteadily increases. For barium titanate thechange in c/a continues until the tempera-ture for another first-order phase transitionis reached. Many such phase transforma-tions are encountered in chemical com-pounds. In some cases the amount of spon-taneous strain is not large enough to con-trol the microstructure. In others, the strainenergy associated with the transformationwill be dominant. The transformation isthen, according to Fig. 9-1, quasi-marten-sitic or martensitic. In ferroelectric materi-als, the interfacial energy also has to be

Figure 9-15. Temperature de-pendences of the lattice parameters of the differentphases of BaTiO3.

taken into account and may even becomethe dominant parameter controlling the microstructure. A ferroic ferroelastic phasetransformation can thus be a shuffle, aquasi-martensitic, or a martensitic phasetransformation, a clear discrimination isonly possible by analyzing all the transfor-mation characteristics, and this not afterbut during the transformation.

9.4.2 Omega Transformations

The omega transformation is known tooccur as a metastable hexagonal or trigonalphase in certain Ti, Zr and Hf alloys oncooling from the high-temperature b.c.c b-phase solid solution or as a stable phaseunder the influence of high hydrostaticpressures or shock waves. The w-phasecannot be suppressed by quenching andforms as small cuboidal or ellipsoidal par-ticles with a diameter of 10–20 nm. Its lat-tice is obtained by collapse of one pair of(111) planes of the parent b.c.c. b-phase,

leaving the two adjacent planes unaltered(Fig. 9-4b). The collapse can be repre-sented as a short-wavelength displacementof atoms. The displacement of the atomsoccurs over a distance approximately equalto 2/3 ·111Ò. Each lattice site can thus beassociated with a forward, zero or back-ward displacement that can be representedby a sinusoidal wave dividing the repeatdistance along a [111] direction into sixparts. The collapse is not always complete,and then results in a “rumpled” plane. If thecollapse is incomplete the crystal structureof the w-phase is trigonal; if the collapse iscomplete, it is hexagonal. The figure alsoshows that reversing the direction of the dis-placement will not lead to a collapse of the111 planes. Moreover, it can be shownthat a 2/3 [111] displacement wave is equiv-alent to a 1/3 [1

–12] displacement wave.

If the displacement is taken as the orderparameter in a Landau-type approach, thetransformation is seen to be first order andthe Gibbs energy as a function of this orderparameter has an asymmetric shape. Con-sequently, no negative values of the orderparameter are then allowed (Fig. 9-16).The b-to-w transformation can also be par-aphrased in terms of a soft mode. The lat-tice tends to a mechanical instability for a2/3 ·111Ò longitudinal mode. This tendencycan be shown when measuring the phonondispersion curves by inelastic neutron scat-tering. Such curves are reproduced in Fig.9-9a for zirconium; a clear dip is visible atthe 2/3 [111] position.

The omega transformation has been re-viewed by Sikka et al. (1982).

9.5 Dilatation-DominantTransformations

A transformation is regarded by Cohenet al. (1979) as dilatation dominant if no


Figure 9-16. Gibbs energy change for the b.c.c. to wtransformation as a function of the order parameterfor various reduced temperatures (after de Fontaine,1973).

9.6 Quasi-Martensitic Transformations 611

undistorted line can be found in the lattice-distortive deformation. The f.c.c.-to-f.c.c.¢transformation in cerium is considered asthe prototype for a dilatational dominanttransformation. Below 100 K cerium un-dergoes a pure volume contraction of about16%; the ellipsoid of the f.c.c.¢ phase thusfalls completely inside that of the high-temperature f.c.c. phase. The low-tempera-ture cubic to tetragonal transformation intin also appears to be dominated by dilata-tion, although some deviatoric componentsare present; the volume expansion is about27%. The deviatoric component is notlarge enough to let the original sphereintersect with the dilated ellipsoid.

The name “dilatational diffusionlessphase transformation” has been used byBuerger (1951) but with a different mean-ing. In the systems he considers, for exam-ple the CsCl-to-NaCl transitions in manyalkali metal halides, he defines the termdilatational as follows: “the transformationcan be achieved by a differential dilatationin which the structure expands along thetrigonal axis and contracts at right-anglesto the axis”. Although the volume changein these and other related inorganic sys-tems may be very large (up to 17%), thetransformation is, in the context of Fig. 9-1,clearly not dilatation dominant but devi-atoric dominant. See Kriven (1982) for amore detailed review of these dilatationaldominant transformations.

9.6 Quasi-MartensiticTransformations

The quasi-martensitic and the marten-sitic transformation are both deviatoricdominant and are characterized by an un-distorted line. The morphologies of theproduct phases of the two transformations

are very similar (large plates, occurrence ofvariants and twins). A distinction betweenthe two transformations cannot be made bysimply judging only the product morphol-ogy, but rather a knowledge is required ofthe morphological relationships betweenparent and product phases during the trans-formation itself. It may be adequate to sayfirst what a quasi-martensitic transforma-tion is: a quasi-martensitic transformationis not a martensitic transformation, whichitself is “a first-order phase transformation,that undergoes nucleation, passes through atwo-phase mixture of the parent and prod-uct phases, and which product grows with atransformation front in a plate-like or lath-like shape being indicative of a tendencytoward an invariant-plane interface” (Co-hen et al., 1979). If a deviatoric dominanttransformation does not satisfy the abovecriterion, it should not be designated asmartensitic but as quasi-martensitic.

Three aspects are common to most of thematerials that transform quasi-martensiti-cally: (1) the lattice distortion is small anddeviatoric dominant and the change in lat-tice distortion is continuous or nearly con-tinuous; (2) a banded internally twinnedmicrostructure gradually builds up on cool-ing below Tc; and (3) a mechanical latticesoftening is expressed by elastic shear constants approaching zero as Tc is ap-proached. Because of the small lattice dis-tortion at the transformation, the ratio ofthe strain energy to the driving energy fortransformation is small; this ratio has beenused by Cohen et al. (1979) as an alterna-tive index to differentiate quasi-martensitictransformations from martensitic.

The three aspects are now illustrated bytaking the manganese-based magnetostric-tive antiferromagnetic alloys as an example(see Delaey et al., 1982a). One of the fourpolymorphic states of manganese is thegamma f.c.c. phase which is stable only at

high temperatures. Alloying with elementssuch as Cu, Ni, Fe, Ge, Pd and Au stabi-lizes the f.c.c phase and the latter can be re-tained by quenching. However, owing tothe antiferromagnetic ordering, the latticesbecome homogeneously distorted. This or-dering to the Mn atoms starts at a temper-ature TN, which is the Néel temperature for the paramagnetic to antiferromagnetictransition. The transformed product phasehas a banded microstructure containingfine twins. The temperature at which thisbanded microstructure is formed does notalways coincide with the transformationtemperature TN. Vintaikin et al. (1979) di-vide these antiferromagnetic alloys intothree classes according to the relative posi-tions of the temperature TN and the temper-ature TTW. The latter is the temperature atwhich the banded microstructure sets in.Depending on the type of lattice distortion,the alloys are grouped into three classes,each class being characterized by the rela-tive positions of the two temperatures. A

schematic representation of the phase dia-gram of the Mn-based alloys is given inFig. 9-17a, showing the temperature–com-position areas in which the various crystalstructures and microstructures are ob-served. The accompanying variation in thelattice parameters as a function of tempera-ture for the three classes of Mn-based al-loys is given in Fig. 9-17b.

The changes in lattice parameters showthat the transformation is almost second or-der, except for some alloys of class I and IIIwhere the transformation is weakly first order. A phase transformation is called “weakly first order” whenever the height ofthe discontinuous jump in the correspond-ing thermodynamic property is very small.The formation of the twinned banded mi-crostructure extends over the entire volumeof the sample quasi-instantaneously and isvisible in polarized light because of thenon-cubic structure of the product phase.Similar microstructures are observed inother quasi-martensitic product phasessuch as V54–xRu46Osx (Oota and Müller,1987). The microstructure, if properlyoriented with respect to the prepolishedsurface, exhibits a surface relief effect thatis enhanced as the temperature decreasesbelow TTW. This surface relief proves thatthe transformation is accompanied by ashape change associated with each domain.Because of the continuously changing lat-tice parameters, accommodation stressesare built up as the temperature decreases.An appropriate arrangement of these do-mains reduces the overall stored elastic en-ergy; further changes in microstructure aretherefore expected to occur even below thetransition temperature.

Class II alloys do not exhibit the twinnedbanded microstructure immediately belowTN. In the temperature region between TN

and TTW, broadening of some of the X-raydiffraction peaks is observed, which is


Figure 9-17. Schematic representation of (a) thephase diagram and (b) the variation of the lattice pa-rameters for the three classes of Mn-based alloys(Delaey et al., 1982a).

9.7 Shear Transformations 613

attributable to a chaotic distribution of thea and c axes with small undercooling. AtTTW the banded structure becomes visible(point A in Fig. 9-17) and a tetragonalstructure can now be clearly detected by X-ray diffraction. If the sample is now heated,only the banded microstructure disappears,not at A but at a temperature B that coin-cides with the Néel temperature. Thisproves that on cooling, very small, submi-croscopic tetragonal regions are firstformed as soon as TN is reached. Hocke andWarlimont (1977) have shown that whenthe distortion |c/a – 1| becomes greaterthan 0.005, a critical value is obtained atwhich the elastic strain is relaxed throughcoalescence of the small distorted regionsinto large banded twinned regions. Thus, atTTW there is not a phase transformation buta stress relaxation in the microstructure,which results in a twinned microstructure.The lattice distortive phase transformationitself occurs at TN, followed immediately (class I) or after some undercooling (classII) by a domain rearrangement and macro-scopic twinning.

Similar conclusions can be drawn forother quasi-martensitic transformations, asfor example in the iron–palladium alloys;the Pd-rich f.c.c. phase transforms on cool-ing first to an f.c.t phase and at lower tem-peratures to a b.c.t. phase. The f.c.c.-to-f.c.t. transformation, although sometimesregarded as martensitic, shows all the char-acteristics of a quasi-martensitic transfor-mation.

Because the formation of each single do-main is associated with a shape change andthus with accommodation stresses, the ap-plication of an external stress to the trans-formed product will result in a macro-scopic shape change. As the domain boun-daries, which for the Mn-based alloys coin-cide with the antiferromagnetic boundar-ies, are mobile, the banded structure will

gradually disappear and the product phasebecomes a single domain maximizing theshape change. The shape change thus ob-tained is gradually recovered on heatingthe sample and is completely recovered atTN and not (as in the case of Mn-based alloys of class II) at TTW, but at the point Bin Fig. 9-17. The quasi-martensitic alloysthus also exhibit the shape-memory effect.

Some of the materials characterized byshuffle displacements during the phasetransition may develop elastic strains astransformation proceeds. As in ferroelec-trics, for example, in addition to the elasticstrain energy, the dipole interaction energyalso contributes to the polydomain forma-tion. If the elastic strain energies are onlyslightly dominating, the transformation isquasi-martensitic; if, however, the elasticstrain energy is largely dominating, thetransformation can be martensitic.

Sometimes it becomes difficult to differ-entiate between martensite and quasi-mar-tensite, as for example in In-based alloys.In particular, if quasi-martensitic samplesare cooled in such a way that a temperaturegradient is created across the sample, theproduct phase and the parent phase thencoexist and apparently the transformationgoes through a two-phase region, the tworegions being separated by a blurred orplanar interface. Such observations do not,of course, facilitate the distinction betweenquasi-martensite and true martensite.

9.7 Shear Transformations

In this section we discuss a special groupof phase transformations, the so-calledshear transformations or polytypic transi-tions, which strictly belong to the marten-sitic transformations. According to Vermaand Krishna (1966), “polytypism may bedefined, in general, as the ability of a sub-

stance to crystallize into a number of dif-ferent modifications, in all of which twodimensions of the unit cell are the samewhile the third is a variable integral multi-ple of a common unit. The different poly-typic modifications can be regarded asbuilt-up of atom layers stacked parallel toeach other at constant intervals along thevariable dimension. The two unit-cell di-mensions parallel to these layers are thesame for all the modifications. The thirddimension depends on the stacking se-quence, but is always an integral multipleof the layer spacing. Different manners ofstacking these layers may result in struc-tures having not only different morpholo-gies but even different lattice types andspace groups”. Some random disorder oflayers (faulted sequences) is almost alwayspresent. Polytypic transitions are then tran-sitions among different polytypes; themovement of partial dislocations along thebasal plane constitutes the transition mech-anism, thereafter the name shear trans-formations. Polytypic transformations arefound in a variety of inorganic compoundsand also in metals and alloys.

Polytypic phases are constructed bystacking basic units in a cubic, hexago-nal or rhombohedral sequence. The stack-ing sequences are described by three keylayer positions, X, Y and Z; a cubic se-quence (C) is represented by the sequenceXYZXYZ …, a hexagonal (H) by, for example, XYXZ … or XYXZXYXZ …,and a rhombohedral (R) by, for example,XYZYZXZXYXYZYZXZXY … . Manyother stacking variants are possible andunit cells containing as many as 126 or 144layers have been reported. Each unit itselfcan contain a single layer, as in cobalt andits alloys, or two as in silicon carbide.

Transitions between different modifica-tions can be achieved either by a simpleshear, a shear combined with shuffle dis-

placements or the movement of partial dis-loctions along the basal plane. For exam-ple, the transition between a 2 H and a 3 Cstacking is easily performed by a shear,whereby the basic units are kept together inpairs (Fig. 9-18). This shear results in alarge deviatoric shape change. It should bekept in mind that the interlayer spacingneed not be constant, as is observed in thef.c.c.–h.c.p. changes in metals; the trans-formation may involve small changes andthus be IPS (invariant plane strain) ratherthan simple shear transitions.

The same transitions can be achieved bythe generation and movement of closelyspaced and repeatedly arranged partial dis-locations. The passage of a positive partialdislocation shifts the crystal in the direc-tion X Æ Y Æ Z Æ X and negative partialdislocations shifts the crystal in the direc-tion X Æ Z Æ Y Æ X. The following dis-tribution of partial dislocations on the unitlayers in the direction perpendicular to thelayers is proposed by Liao and Allen(1982) (a layer without a partial dislocationis denoted by a dot):

· – + · for 2 H Æ 4 H· – – – · · for 6 H Æ 3 C

The polytypic shear transformation fromone modification to the other is thus ac-complished by a coordinated propagationof groups of partials along the interfacebetween the two phases. The lateral dis-


Figure 9-18. Mechanism of the f.c.c.-to-h.c.p. trans-formation (Nishiyama, 1978).

9.8 Martensitic Transformations 615

placement of the interface, which is thethickening of the new phase generated bythe movement of those partials, is then dueto the formation of partial dislocations andtheir outward movement. During the tran-sition from a cubic to a hexagonal 2H se-quence, it has been implicitly assumedabove that the glide of the layers would al-ways occur in the same direction. There ex-ist, however, three different directions fortransforming an X stacked unit into a Zstacked unit. If the glide occurs alternatelyin these three directions, no shape changeresults from such a mechanism, as shownin Fig. 9-19 (Bidaux, 1988).

9.8 Martensitic Transformations

The characteristics necessary and suffi-cient for defining a martensitic transforma-tion are (a) displaciveness of the lattice-distortive type involving a shear-dominantshape change, (b) diffusion not required forthe transformation, and (c) sufficientlyhigh shear-strain energy in the process todominate the kinetics and morphology dur-ing the transformation (Cohen, 1982). Thedefinition is thus not based on the identityof the transformation product itself (its

structure, specific morphology or proper-ties), but rather on how it forms.

The crystallographic and thermody-namic aspects are fully discussed in the lit-erature and have already been introducedin a more general context in the above sec-tions; only a brief overview is given here.

9.8.1 Crystallography of the MartensiticTransformation

The relevant experimental observableparameters of the martensitic transforma-tion are the shape deformation, the habitplane, the crystallographic orientation rela-tionships, and the characteristic micro-structures.

9.8.1.1 Shape Deformationand Habit Plane

When a sample of the parent phase iscooled to below Ms , a relief gradually ap-pears on a prepolished surface of the par-ent-phase crystal. The surface relief disap-pears on heating to temperatures above As ,provided that no diffusion-controlled trans-formation interferes.

The martensite phase usually takes theform of plates; the plane of contact be-tween the parent and the martensite phasesis called the “habit plane”. A schematicrepresentation of such a martensite plateembedded in the matrix is shown in Fig. 9-20. During the formation of martensite,straight lines (for example, scratches on theprepolished surface) are transformed intoother straight lines and planes are trans-formed into other planes. No discontinu-ities are observed at the points of deflec-tion. This distortion can thus be repre-sented as a “linear homogeneous transfor-mation” of vectors and can be expressed bya matrix formulation. The macroscopicshape deformation can be decomposed into

Figure 9-19. Two different mechanisms, (a) and (b),to transform an h.c.p. to an f.c.c. structure (Bidaux,1988).

a component normal to the habit plane anda shear component parallel to a shear direc-tion located in this interface. The latter iscalled “macroscopic shear” and quantifiesthe shape deformation, whereas the formerrepresents the volume change associatedwith the transformation. A careful analysisof the surface relief reveals that the habitplane itself is unrotated and that any vectorin this interface is also left unrotated andundistorted by the shape change. The habitplane is thus essentially “undistorted” andthe macroscopic shape change associated

with the formation of martensite is thus an“invariant plane strain” deformation, ab-breviated to IPS. The most general invari-ant plane strain deformation, as observedin most martensitic transformations, can beachieved by combining an extension and asimple shear.

The habit plane and the direction of mac-roscopic shear are, with few exceptions,not simple low-indexed crystallographicplanes or directions of the parent or prod-uct phase. They are usually represented in a stereographic projection as shown sche-matically in Fig. 9-20.

9.8.1.2 Orientation Relationship

The next most important observable pa-rameter is the crystallographically well-de-fined “orientation relationship” that existsbetween the lattices of the parent and themartensite phases. It is described either bythe angles between certain crystallographicdirections in both phases or by specifyingthe parallelism between certain planes anddirections. This parallelism does not needto be rigorous, however, experimental re-sults usually deviate slightly. Nevertheless,the fact that such crystallographic parallel-ism is observed yields important informa-tion concerning the possible mechanismsexplaining the change in crystal structure.Some of those relationships observed insteels received great attention in the earlymartensite literature. Depending on the alloy composition, the f.c.c. austenite insteels transforms either to a b.c.c. or b.c.t.martensite or to an h.c.p. martensite, whichitself may further transform into b.c.c. mar-tensite. As regards the transformation off.c.c. austenite to b.c.c. or b.c.t. martensite,the orientation relationships are as follows:

– the Kurdjumov–Sachs (K–S) relations:

(111)P//(011)a¢ and [01–1]P//(11

–1)a¢


Figure 9-20. Schematic representation of (a) a sin-gle martensite plate embedded in a single crystal ofthe matrix phase, (b) a twinned plate and the positionof habit and twin plane, and (c) their stereographicrepresentation.


– the Nishiyama–Wassermann (N–W) re-lations:

(111)P//(101)a¢ and [12–1]P//(101

–)a¢

– the Greninger–Troiano (G–T) relations(here the planes and directions are nolonger exactly parallel):

(111)P ≈ (011)a¢ and [1–01]P ≈ [1

–1–1]a¢

Table 9-3. The crystallographic observables of the martensitic transformations in some metals and alloys(courtesy G. Guénin et al. 1979*).

Alloy system Structural change Composition wt.% Orientation relationship Habit plane

Fe–C f.c.c. 0–0.4% C (111)PΩΩ(101)M (111)P

Ø [110]PΩΩ[111]M

b.c.tetr. K–S relationship0.55–1.4% C K–S relationship (225)P

1.4–1.8% C Idem

Fe–Ni f.c.c. 27–34% Ni (111)PΩΩ[101]M

Ø [121]PΩΩ[101]M ≈ (259)P

b.c.c. N-relationship

Fe–C–Ni f.c.c. 0.8% C–22% Ni (111)P ≈1° of (101)M (3, 10, 15)P

Ø [121]P ≈2° of [101]M

b.c.tetr. G–T relationship

Fe–Mn f.c.c. 13 to 25% Mn (111)PΩΩ(0001)e (111)P

Ø [110]PΩΩ[1210]eh.c.p. (e-phase)

Fe-Cr-Ni f.c.c. 18% Cr, 8% Ni (111)PΩΩ(0001)eΩΩ(101)a¢ e (111)P

Ø [110]PΩΩ[1210]eΩΩ[111]a¢ a¢ (111)P

h.c.p. (e), b.c.c. (a¢)

Cu–Zn b b.c.c.Æ 9 R 40% Zn (011)PΩΩ?(11––

4)M ≈ (2, 11, 12)P

Cu–Sn idem 25.6% SN [111]PΩΩ[110]M ≈ (133)P

Cu–Al b.c.c. 11.0 to 13.1% Al (101)P at 4° of (0001)M 2° of (133)P

Ø [111]PΩΩ[1010]M

h.c.p. distorted 12.9 to 14.7% Al (101)PΩΩ(1011)M 3° of (122)P

[111]PΩΩ[1010]M

Pure Co f.c.c. (111)PΩΩ(0001)M (111)P

Ø ·110ÒPΩΩ[1120]M

h.c.p.

Pure Zr b.c.c. (101)PΩΩ(0001)M (596)P

Ø [111]PΩΩ[1120]M (8, 12, 9)P

Pure Ti h.c.p. (334)P

(441)P

Pure Li Burgers relations

* Gobin, P. F., Guénin, G., Morin, M., Robin, M. (1979), in Transformations de Phases à l’État Solide-Trans-formations Martensitiques. Lyon: Dep. Gènie Phys. Mat., INSA

Concerning the transformation of f.c.c. toh.c.p. austenite and that of h.c.p. to b.c.c.martensite (the b.c.c. to h.c.p. relation isknown as the Burgers relation), the follow-ing relations apply:

(111)P//(0001)e//(101)a¢

and

[11–0]P//[12

–10]e//[111

–]a¢

Taking the N–W relations as an exam-ple, any one of the four crystallographi-cally equivalent 111 austenite planes,(111), (1

–11), (11

–1) and (111

–), can be the

plane of parallelism. In each such planeany one of the three ·12

–1Ò directions, which

happen to be directions of the Burgers vec-tors, can be chosen. This therefore resultsin 12 different orientations of an a¢-crystalin one austenite crystal. These differentlyoriented martensite crystals are called“variants”. It can easily be shown that theK–S relations lead to 24 variants.

Orientation relationships and the orien-tation of the habit plane change from onealloy system to another, and within a givenalloy system from one composition to an-other. The observable crystallographic pa-rameters are summarized in Table 9-3 for a large number of alloy systems; a morecomplete list of these and other crystallo-graphic characteristics of various marten-sites is given by Nishiyama (1978).

9.8.1.3 Morphology, Microstructureand Substructure

Because the martensitic transformationis a first-order phase transformation, bothphases, the parent and the martensitephase, coexist on cooling in a temperaturerange between Ms and Mf and on heatingbetween As and Af . Martensite thus occursin physically isolated regions, the morphol-ogy of which is typical of the transforma-

tion. This morphology is easily observedby light optical microscopy (LOM) and themutual arrangement of these regions con-stitutes the microstructure at the LOMlevel. Electron microscopic analysis re-veals that also at the submicroscopic levelmartensite is characterized by a typicalsubstructure. The morphological, micro-structural and substructural aspects of mar-tensite are briefly discussed below.

The martensite regions are generallyplate-shaped, i.e. one lateral dimension ismuch smaller than the other two. If the twolarger dimensions are nearly equal they arecalled “plates”, and if they are very un-equal “laths”. A typical lath in low-carbonsteel (with a carbon content less than 0.4%)has dimensions 0.3 ¥ 4 ¥ 200 µm3. How-ever, martensite formed into the parentphase does not always appear as a geomet-rically well-shaped plate. Martensite platesthat form near a free surface or in a singlecrystal as a result of a single-interfacetransformation may show the idealizedplate-like shape. In such a single-interfacetransformation the habit plane extendsfrom one side of the crystal to the other(see Fig. 9-11). Because of the shapechange and the high elastic stresses that arecreated, a thick plate cannot terminate in-side a parent crystal. As is frequently ob-served, lenticular shapes or groupings ofdifferently oriented martensite plates willreduce these elastic stresses.

In the case of lenticular martensite, thehabit plane is no longer a plane but acurved surface and the normal average ofthe lenticular plate is then taken as the or-ientation of the habit plane. Sometimes thisorientation is visible as a “midrib” in somemartensites (Fig. 9-21). It is believed thatthe martensite could grow to a certain ex-tent as a plate, but that lenticular shapes areformed owing to the high elastic stressesthat are building up. The high stresses may



trigger other plates to form in the vicinityof the plate formed earlier, giving rise to an“autocatalytic growth” of martensite.

A multivariant martensite arrangementis the most commonly observed micro-structure. Often variants are arranged insome recognizable patterns and at timesnumerous variants present in a regular ar-ray give the impression of a martensite col-ony. The latter is typical of the “massive”microstructure, consisting of a packet ofparallel martensite “laths” separated bymore or less wavy interfaces. Each lath inthe packet maintains the same variant ororientation relationship with the parentcrystal. A single grain of the parent phasecan transform into one or more such pack-ets. The “plate” martensite arrangementwhich is observed in the same alloys differsfrom the lath configuration, because adja-cent martensite plates are generally notparallel to each other.

Diagrams have been constructed forcases where there is a variety in morphol-ogy, as for example for Fe–Ni–C alloys.Maki and Tamura (1987) showed that themorphology of the a¢-martensite in thesealloys is related to the transformation tem-perature and the carbon content (Fig. 9-22).

Distinct martensite plate arrangementscan also be recognized in alloys pertainingto the b-Hume–Rothery alloys. Schroederand Wayman (1977) classified these ar-rangements into spear, fork, wedge and di-amond forms. Each representation carrieswith it a definite crystallographic relation-ship between the variants constituting the arrangement. Grouping of martensiteplates in such arrangements will lead to aserious reduction in the elastic stresses. Byanalysis of the crystallography of the platesin a single group, it can be shown that therespective macroscopic shape changes an-nihilate each other (Tas et al., 1973). Such

group formation is then called “self-ac-commodation”.

In these b-Hume–Rothery alloys, threetypes of martensite form, the 3R-, 9R- and2H-types. A detailed analysis of the micro-structure reveals that the martensite vari-

Figure 9-21. Transmission electron micrograph of amartensite in steel showing the twinned midrib(courtesy C. M. Wayman, 1989, University of Illi-nois, Urbana (IL)).

ants form in six different groups, eachgroup consisting of four variants. The habitplanes of these four variants are locatedaround the same 110b pole, whereas eachbasal plane is located close to one of fourother 110b poles. The total macroscopicdeformation of such a group is to a first ap-proximation completely compensated. Amore complete reduction in the three-di-mensional strain is obtained if the totaltransformation strain is calculated for thesix groups as an entity. As the martensitetransformation involves a very small vol-ume change in all b-Hume–Rothery alloys,the strain accommodation is thus almostcomplete.

The shape change associated with a mar-tensite plate creates stresses in both theparent and the martensite phases. If thesestresses exceed the flow stress for plastic

deformation, strain accommodation is thenaccomplished not only by elastic but also byplastic deformation in one or both phases.

Partitioning of the parent crystal, withfiner plates forming subsequently in thepartitioned region, frequently occurs andillustrates the fractal nature of the transfor-mation (Fig. 9-10). It should be mentioned,however, that not all martensite micro-structures show fractal characteristics(Hornbogen, 1988).

Until now only the more macroscopicobservable features of the microstructureof martensite have been discussed. Trans-mission electron microscopy reveals thatthe substructure of martensite is also char-acteristic. It consists, depending on the al-loy system and alloy composition, of regu-larly spaced stacking faults (e.g. Cu-baseb¢-type martensite), twins with a constantthickness ratio (e.g. Fe–30% Ni), disloca-tions (e.g. Fe–20% Ni–5% Mn), stackingfaults and twins in the same martensiteplate (e.g. Cu–Ga), or twins in the midribregion surrounded by dislocations.

9.8.1.4 Crystallographic Phenomenological Theory

The formal phenomenological theoriesof martensite formation predict the crystal-lographic characteristics, such as the shapedeformation, the orientation of the habitplane, the orientation relationship betweenparent and product phase, and the ampli-tude of lattice invariant deformation. Thisprediction is obtained from the sole knowl-edge of the structures and lattice parame-ters of the two phases and with the basic as-sumption that the interface between parentphase and martensite is undistorted on amacroscopic scale.

The observation of the K–S and N–Worientation relationships led us to origi-nally believe that a martensite was formed


Figure 9-22. Relationship between a¢-martensitemorphology and Ms temperature as a function of car-bon content in Fe–Ni–C alloys (Maki and Tamura,1987).


by shear on those planes and directionsspecified in the orientation relationships.However, it was found that the shear mech-anisms proposed by the K–S and N–W relations are not consistent with these ex-perimental observations. The observationsmade by Greninger and Troiano (1949) onFe–22% – Ni–0.8% C martensite were thekey to the mathematical development ofthe crystallographic theory of martensite.They found that martensite plates exhibiteda surface relief that can be described by ahomogeneous shear along the habit plane,but this homogeneous shear could nottransform the f.c.c. lattice of the parentphase into the b.c.t. lattice of the marten-site. If the f.c.c. lattice had undergone thesame homogeneous deformation, the struc-ture of the martensite would have beentrigonal. They therefore suggested that twotypes of shear are involved in the marten-sitic transformation: a “first” simple shearwhich is responsible for the macroscopicshape change, and a “second” shear whichneeds to be added to obtain the structuralchange but which should produce no ob-servable macroscopic change in shape.Two years later, Bowles (1951) showedthat the shape deformation may be any in-variant plane strain. This opened the way tothe formulation of the general theory of thecrystallography by Wechsler et al. (1953)and, independently, by Bowles and Mack-enzie (1954). Almost equivalent theorieswere later developed by Bullough andBilby (1956) and Bilby and Frank (1960).The reader may consult the following moreelaborate reviews of these theories: Way-man (1964), Christian (1965), Nishiyama(1978) and Ahlers (1982).

The basic assumption in the crystallo-graphic theories is that the interface be-tween the product and the parent phases isundistorted, which means that any vectorthat lies in this interface on the side of the

martensite would be a vector of the samesize and the same orientation in the parentphase before transformation. As indicatedin Sec. 9.3.1, the macroscopic shapechange of an invariant-plane transforma-tion can be represented by a combinationof a pure lattice deformation (B), the so-called Bain strain, a rigid lattice rotation(R), and an inhomogeneous lattice-invari-ant deformation (P). The pure lattice defor-mation either increases or decreases somevectors in length. According to Wayman(1964), “the essence of the crystallographictheory of martensitic transformations is tofind a simple shear (of a unique amount, ona certain plane, and in a certain direction)such that vectors which are increased inlength due to the lattice deformation arecorrespondingly decreased in length due tothe simple shear, and vice versa. Such vec-tors which remain invariant in length tothese operations define the potential habitplane. Physically speaking, the ellipsoidgenerated from the initial sphere by the lat-tice deformation is distorted by the simpleshear into another ellipsoid which becomestangential to the initial sphere, the points oftangency being related along a diameter.”This is clearly illustrated in Fig. 9-23,where the problem becomes two-dimen-sional, because one of the principal axes ofthe lattice deformation is taken as normalto the plane of shear.

Some complementary remarks concern-ing the crystallographic theory should bemade. The input data for the calculationsare (i) the lattice parameters of the parentand martensite phases, (ii) the lattice corre-spondence, and (iii) the lattice-invariantshear. The output of the calculations is thenthe amount of inhomogeneous shear re-quired to obtain the invariant plane condi-tion, the macroscopic shape change, andthe orientation relationship. Because of thelattice symmetries, differently oriented

Brain relations and inhomogeneous shearsystems lead to a number of crystallo-graphically equivalent solutions. Becauseof the observed orientation relationships,the Bain relationship is fixed for mostcases. However, larger unit cells are some-times chosen, especially for those marten-sites where the crystal structure has a largeunit cell compared with that of the parentphase. The only variable in these calcula-tions is the choice of the inhomogeneousshear system. The orientation of the habitplane is found to be very sensitive to thechoice that is made. For the f.c.c. to b.c.c.or b.c.t. transformation, the twin shear(112)M[111

–]M gives a (3 15 10)P habit

plane, whereas a (011)M[1–1–1]M shear re-

sults in a (111)P habit plane.The phenomenological theory as ex-

plained above is based on one active shearsystem. However, for some alloy systemsthis is not adequate. For example, a single(112)M twinning system is not able to ex-plain the 225P habit plane in some steels,and even two inhomogeneous shear sys-tems do not give agreement with the ex-perimental observations. Similar disagree-

ments have been observed in other alloysystems. To test critically the validity ofthe crystallographic theories, all crystallo-graphic parameters should be measuredand compared with the theoretical predic-tions. Agreement should be obtained forthe complete set of parameters. For themartensites that are twinned, this includesa careful determination of the normal to thetwinning plane K1 relative to the parent lat-tice. In Cu–Al–Ni, for example, inconsis-tencies up to 12.5° have been found.

In cases where the lattice-invariant shearis twinning (as opposed to faulting or slip),type I twinning, where the twin plane orig-inates from a mirror plane in the parentphase, has been assumed. Otsuka (1986)carefully analyzed for a number of systemsthe possibility of a type II twinning as analternative inhomogeneous shear. In type IItwinning, the shear direction stems from atwo-fold symmetry direction of the parentphase. Otsuka (1986) compiled all thetwinning modes observed in martensiteinto a table and found that most of them aretype I or compound but that type II twin-ning had only recently been observed. According to Nishida and Li (2000), fivedifferent twinning modes exist in TiNi and other shape-memory alloys such asCu–Al–Ni, Cu–Sn etc., namely the 111type I, the 011 type I, the ·011Ò type II,the (100)-compound and the (001)-com-pound twins. Type II twinning has recentlyreceived much attention as a mechanismfor lattice-invariant shear in some alloys.Because type II twins have irrational twinboundaries, the physical meaning of an ir-rational boundary is still a controversialproblem. It has been proposed that an irra-tional boundary consists of rational ledgesand steps, the average being irrational.Thereafter, Hara et al. (1998) carried out acareful study using HRTEM, to try and ob-serve ·111Ò type II twin boundaries in a


Figure 9-23. Production of an undistorted plane byshear such that the shape ellipsoid touches the unitsphere along one of its principal axes (Christian,1965).


Cu–Al–Ni alloy, but they were unable toobserve any ledges or steps. The boundaryis always associated with dark strain con-trast, and the lattice is continuous throughthe irrational boundary. Nishida and Li(2000) also carried out extensive studies on·011Ò type II twin boundaries in TiNi usingHRTEM, but they did not observe ledges orsteps either. Based on these experimentalresults, it is thus most likely that the type IIthin boundary is irrational even on a micro-scopic scale, and the strains at the boun-dary are elastically relaxed with wide twinwidth. To confirm this interpretation. Haraet al. (1998) carried out computer simula-tions by using the molecular dynamicsmethod. The result showed that the irra-tional thin boundary did not show anysteps. Thus, the above interpretation for anirrational twin boundary is justified. Ot-suka and Ren (1999) have pointed outagain the importance of type II twinning inthe crystallographic aspects of martensite.They also stress the role that martensite aging has on the rubber-like behavior ofmartensite, a point that has been a long-standing unsolved problem. They showedthat the point defects play a fundamentalrole. The deformation mechanisms of thecold deformation of NiTi martensite havebeen thoroughly analyzed by Liu et al.(1999a, b). They also found an interplaybetween type I and type II twinning.

As already mentioned in Sec. 9.3.6, abetter and more complete agreement can beachieved when the strain energy terms,both bulk and interfacial, are included inthe calculation.

9.8.1.5 Structure of the Habit Plane

In a number of alloys, especially those inwhich the so-called thermoelastic marten-sites are formed, the interface betweenmartensite and the parent phase is mobile,

even at very low temperatures. This obser-vation shows that the interface migrationmust be accomplished without appreciablethermal activation. The interface is thus“glissile”. In searching for models to ex-plain the structure and mobility of theinterface, we are concerned with the idealand the actual interface morphology. Acareful experimental analysis of the inter-face structure is therefore required if wewant to verify the various models that havebeen proposed. As the models treat theinterface on an atomistic scale, the sub-structure of the interface should be studiedby conventional and by high-resolutiontransmission electron microscopy. Thesame holds for martensite-to-martensiteinterfaces, which in some alloys are alsomobile. Recently, atomistic imaging of themartensite/austenite and martensite/mar-tensite interfaces have been obtained. It istherefore not surprising that both aspects,the observation of interface substructuresand the atomistic models, are treatedjointly in the literature. For further readingconcerning the interface structures and thegrowth mechanism of martensite we referto the review papers by Christian (1982),Christian and Knowles (1982), and Olsonand Cohen (1986). A summary of these pa-pers is given below.

Let us first introduce the kinds of mar-tensite interfaces concerned: glissile andnon-glissile martensitic interfaces, with thelatter subdivided into the coherent andsemi-coherent interfaces. The two struc-tures, martensite and the parent phase, aresaid to be “fully coherent” if both latticeshave a matching plane parallel to the inter-face. If a fully coherent interface is dis-played, the crystal undergoes a shape de-formation leaving all vectors in the inter-face invariant. In general, the two phasesdo not have a plane of atomic fit, so thatfully coherent martensite interfaces are ex-

ceptional. A fully coherent martensitic in-terface is, for example, that between f.c.c.and h.c.p. structures with lattice parame-ters such that a (f.c.c.) = ÷–

2 a (h.c.p.); theatomic arrangement in the basal planes,which constitute the interface between thetwo structures, are identical. Such transfor-mations are found in Co and its alloys andin some Fe-based alloys. The situation atsemi-coherent interfaces becomes morecomplex. The models predict the presenceof dislocations to correct the mismatchalong the interface. If this coherent inter-face moves, it is suggested that not all vec-tors are left invariant and that the move-ment of dislocations causes shear in theproduct phase. Fig. 9-24 shows the slip as-sociated with the interface dislocations.

Internally twinned martensite has beenreported to show a zig-zag parent–marten-site interface, as observed by conventionalelectron microscopy in, for example,Ti–Mn and Cu–Al–Ni. Fine parallel stria-tions have been observed in the interfacebetween austenite and both the b¢-type andthe g¢-type Cu–Al–Ni-martensite. Thesestriations have been accounted for in termsof interfacial dislocations resulting fromrandom faulting on the basal plane of theb¢-type martensite and the twinning planesof the g¢-type martensite. High-resolution

electron micrographs show that the inter-face between martensite and the parentphase and also the intervariant interfacesand the interfaces between the internaltwins in one martensite plate contain dis-continuities (“steps”) on an atomic scale,the nature of which has not yet been com-pletely unravelled. These steps can be con-sidered as resulting from a small deviationof the ideal habit plane, and would then becomparable to those observed along theinterfaces of tapered twins.

An exact understanding of the structureof the interfaces involved in the martensitictransformation (the parent–martensite, theintervariant, and the twin/twin interfaces)is therefore essential in determining themechanism of transformation and the mo-bility of the interfaces.

9.8.2 Thermodynamics and Kineticsof the Martensitic Transformation

9.8.2.1 Critical Driving Forceand Transformation Temperatures

A quantitative thermodynamic treatmentof the martensitic transformation requires aprecise knowledge of the thermodynamicequilibrium temperature T0 and of thechange in Gibbs energy at the transforma-


Figure 9-24. Three-dimen-sional representation of asemi-coherent martensiteinterface; the vectors OA aredistorted into O¢A¢ but thelarge vectors OZ = O¢Z¢ areinvariant (Christian, 1982).


tion temperature Ms . Both can be calculated and/or derived from measureddata, as is shown here for two examples:the martensitic transformation in Fe–X(X = Ni, Ru, …) and in Cu–Zn–Al alloys.In the former example, both the parent andthe martensite phases have the same struc-ture as the equilibrium phases and hencethe data for the equilibrium phases can betaken. In the latter example, both structuresdiffer from that of the equilibrium phases,which requires a more elaborate calculation.

For the Fe–X alloys, the Gibbs energyper mole of the parent austenite phase gand of the martensite phase a are Gg andGa, respectively. The change in Gibbs en-ergy per mole, DGgÆa, which for a mar-tensitic transformation g Æ a is availableto the system at any temperature T, is then

DGgÆa|T = Ga – Gg (9-8)

This quantity is negative for temperaturesat which the a-phase is the more stable andpositive for temperatures at which the g-phase is the more stable. There is a charac-teristic temperature T0 corresponding to thethermodynamic equilibrium between bothphases, such that

DGgÆa|T =T0= 0 (9-9)

Because the transformation creates interfa-cial and elastic energies, the martensitictransformation g Æ a or a Æ g does notstart at T0, but at a temperature below orabove T0, respectively. It is therefore nec-essary to undercool or overheat, respec-tively, until Ms or As is reached. At thesetemperatures the Gibbs energy changeDGgÆa is sufficiently large to induce theforward or reverse transformation, respec-tively. DGgÆa (at Ms) is then the criticalchemical driving force.

The martensite phase, represented by M,is to be regarded as the a-phase embedded

in the g-phase. Because of the shape andvolume changes associated with the trans-formation, elastic strain energy also has tobe considered. The Gibbs energy is thuscomposed of chemical Gibbs energy, Gc,and strain energy, Ee, so that the Gibbs en-ergy change accompanying the transforma-tion may be written as

GgÆM = DGcgÆa + DEe

aÆM (9-10)

At temperatures below Ms , where bothphases coexist and thus are in equilibrium,DGgÆM |T = 0. DGc

gÆa|T is then exactlyequal, but opposite in sign, to the sum of allnon-chemical energies DGnc

aÆM |T . If thesurface energies are neglected in compari-son with the high strain energies, the non-chemical energy equals DEe

aÆM |T , andapproaches zero at T = Ms . The strain en-ergy stored in the material is the sum ofthat produced by shearing and by volumechange. The former depends on thestrength of the parent phase and thus alsoon the grain size, hence Ms also depends onthe grain size, as shown by Hsu and Xiao-wang (1989).

The necessary undercooling (T0 – Ms)and superheating (As – T0) vary for differ-ent alloy systems, and for certain materialseven with composition. A precise thermo-dynamic definition of Ms and As cannot begiven, however, if the non-chemical ener-gies, DGnc

aÆM, are not known. We can thenonly say that Ms or As is the temperature atwhich the quantity DGgÆa (at T = Ms orT = As , respectively) is sufficiently nega-tive or positive, respectively, to have a rea-sonable chance of nucleation.

Two approaches are found in the litera-ture for calculating the critical chemicaldriving force. The first is based on the ex-perimentally determined Ms temperatures(Kaufman, 1965) and the other on a theo-retical model for the non-chemical energies(Hsu, 1985).

The first approach has been used for fer-rous alloys, which can be classified intotwo systems: those with g-loops and thosewith stabilized g-phases (see Fig. 9-25).Figure 9-25 also gives the Ms temperaturesfor the a- and e-martensite. If A is the al-loying element for iron, the molar chemicalGibbs energy for the austenite phase (Gg)can be written as

Gg = (1 – x) GgFe + x Gg

A + Ggm (9-11)

where x represents the atomic fraction ofthe element A in solid solution in the g-austenite, (1 – x) the atomic fraction ofiron, Gg

Fe the chemical Gibbs energy ofpure iron as f.c.c. g-phase, Gg

A the chemicalGibbs energy of element A as f.c.c. phase,and Gg

m the Gibbs energy of mixing of the g-phase. Similarly, the Gibbs energy of thea-phase can be given as

Ga = (1 – x) GaFe + x Ga

A + Gam (9-12)

where GaFe and Ga

A are the Gibbs energies ofpure iron and pure element A as a b.c.c. a-phase, respectively, and Ga

m is the Gibbsenergy of mixing of the martensite phase.The change in chemical Gibbs energy ac-companying the martensitic transformationgÆa then becomes

DGgÆa = (1 – x) DGFegÆa + x DGA

gÆa

+ DGmgÆa (9-13)

The quantity DGFegÆa represents the Gibbs

energy change for transformation g Æ a ofpure iron and can be assessed experimen-tally from the measured heat of transforma-tion and the specific heat of both phases.The quantity DGA

gÆa cannot usually be ob-tained from experiments because the ele-ment A does not always exist in the twomodifications g and a; it must therefore beestimated from thermodynamic models forsolid solutions. The quantity DGm

gÆa is the


Figure 9-25. Schematic dia-grams for ferrous alloys thatform a g-loop (Fe–Cu, Cr,Mo, Sn, V, W) and that g-loop forms a stabilized aus-tenite phase (Fe–C, Ir, Mn,N, Ni, Pt, Ru); (a) equilib-rium diagrams; (b) Ms tem-perature diagrams (Kraussand Marder, 1971).


difference in Gibbs energy of mixing andcan in principle be measured experimen-tally through activity measurements; if not,it must also be estimated.

The equilibrium temperature T0 and thecritical driving force and Ms and As can becalculated from Eq. (9-12). Such calcula-tions have been performed by Kaufman(1965) for the iron–ruthenium alloy sys-tem, which is of particular interest becausethe g-phase transforms martensitically intotwo phases, the a-b.c.c. and the e-hexago-nal phases. Both phases also occur as equi-librium phases, as shown in Fig. 9-26 to-gether with observed Ms and As tempera-tures and the calculated T0 temperatures.The undercooling for the a-martensite formation is strongly composition depen-dent, whereas it is independent of composi-tion for the hexagonal martensite. Thecomputed T0 curves are seen to lie betweenthe appropriate transformation temperaturecurves. The calculated driving forces,DGgÆa and DGgÆe, for both transforma-tions are plotted as a function of temperaturefor various compositions. The intersectionsof these curves with the temperature axiscorrespond to the theoretically deduced T0

temperatures for the appropriate composi-tions. When the appropriate experimentallyderived Ms and As are cross-plotted, the crit-ical driving forces for the g/a and g/e mar-tensitic transformations are obtained. Thelatter is seen to be smaller, which is consis-tent with the closer lattice correspondenceof the former transformation.

Hsu (1985) presented a model by whichmore accurate computations of the non-chemical part DGaÆM are possible forFe–C, Fe–X and Fe–C–X alloys. This en-abled him to obtain from Eq. (9-9) thetheoretical Ms temperatures, which are ingood agreement with the observed values.

In the second example, martensite for-mation in Cu–Zn alloys, the change in

Figure 9-26. (a) The iron–ruthenium phase diagramand (b) the T0 and Ms and As temperature diagrams(after Kaufman, 1965).

Gibbs energy can, according to Hsu andZhou (1989), be described as

DGb¢–M = DGb¢–b + DGb–a + DGa–a¢

+ DGa¢–M (9-14)

where b¢ – M represents the transformationfrom the the ordered b.c.c. phase to the or-dered 9R-type of martensite, b¢ – b the or-der–disorder transition, b – a the transfor-mation from the disordered b.c.c. to thedisordered f.c.c. phase having the samecomposition, a – a¢ the disorder–ordertransformation in the f.c.c. phase, anda¢ – M the transition from the ordered f.c.c.phase to the ordered martensite phase. As-suming a simplified relationship betweenthe degree of ordering and temperature,Hsu and Zhou found good agreementbetween the calculated and observed Ms .Their calculations show that ordering ofthe parent phase, which cannot be sup-pressed even by severe quenching, stronglyinfluences T0.

It is known that martensite may also beinduced by an external stress at tempera-tures above Ms . The problem now is to cal-culate the change in T0 due to changes instress. As a first approximation, it is as-sumed that the driving force DGm

PÆM |T =Ms

required for nucleation remains constantwith temperature and thus independent ofstress. Patel and Cohen (1953) calculatedthe work done on the stressed specimen;their treatment provides a good under-standing of how an applied stress that is de-composed into a shear stress along thehabit plane and a normal stress perpendicu-lar to it, affects the transformation temper-ature. At Ms

s, which is the martensite starttemperature when cooling under an appliedstress s, the chemical Gibbs energy changeequals the transformation work of the ex-ternal stress:

DGsPÆM = 1/2sa (9-15)

¥ [d0 sin 2q ± e0 (1 + cos 2q )] Vm

where d0 is the shear strain, sa the appliedstress, q the angle between the stress axisand the normal to the operative shear plane,e0 the corresponding strain associated withthe transformation, and Vm the molar vol-ume. The quantity DHPÆM can be meas-ured by calorimetry and DSPÆM can beevaluated from stress-induced transfor-mation experiments or calculated from Eq.(9-16). The temperature T0 can be calcu-lated thermodynamically or obtained moreor less accurately from the relationships

DGPÆM = DHPÆM – T DSPÆM

DHPÆM = T0 DSPÆM (9-16)

and

T0 = (As – Ms)/2 = (Af – Mf)/2 (9-17)

However, it should be noted that the deter-mination of T0 does not always obey thesesimple relationships and that the calorimet-


Figure 9-27. Gibbs energy G* versus temperatureand force for stressed samples: P and M representfree energy surfaces for parent and martensite, re-spectively (Wollants et al., 1979).


rically measured heats do not always re-flect the exact heats of transformation.

As stress itself is also a state variable in-dependent of temperature, it should be con-sidered in the thermodynamic treatment asexplained by Wollants et al. (1979). To de-scribe the thermodynamic state of a uniax-ially stressed crystal, they introduced the“elastic” state functions H* and G*, whichincorporate the effect of stress as follows:

H* = U + P V – F l = H – F l

= H – s e Vm (9-18)

G* = U + P V – T S – F l = G – F l

= G – s e Vm (9-19)

where F is the applied force and l the “mo-lar length” of the crystal. Fig. 9-27 illus-trates how the equilibrium temperature andforce change when one of the variables ischanged; P and M represent the Gibbs energysurfaces of the parent phase and of marten-site, respectively. At the two-phase equilib-rium G*P = G*M and if, at constant hydro-static pressure, the intensive variables F(or s) and T are changed in such a way that there is thermodynamic equilibriumbetween martensite and the parent phase,then dG*P = dG*M, or

– SP dT – l P dF = – SM dT l M dF

so that dF/dT = – [DS/Dl]PÆM, or, sincethe molar work FDlPÆM = s ePÆM Vm andDSPÆM = DH*(s)/T0 (s), it also followsthat

(9-20)ds /dT = – [DH*(s)]/[T0 (s ) ePÆM Vm]

where Vm is the molar volume (Vm = V P =V M). Eq. (9-20) is the Clausius–Clapeyronequation for a uniaxial stress, which is sim-ilar in form to that for hydrostatic pressure,except for the negative sign.

The change in critical stress necessary toinduce martensite can be obtained fromtensile tests carried out at different temper-

atures (see Fig. 9-28). The elongation re-sulting from the transformation is orienta-tion dependent (Fig. 9-29). From data suchas those shown in Figs. 9-28 and 9-29, wecan calculate DSP–M. It is evident that foreach crystal orientation the slope ds /dT isdifferent.

Figure 9-28. Results of tensile tests for inducingmartensite in a Cu–34.1 Zn–1.8 Sn (at.%) alloy (Pops, 1970).

Figure 9-29. Orientation dependence of stress–strain curves for martensite formation in aCu–Al–Ni alloy (Horikawa et al., 1988).

Similarly, martensite can be induced bymagnetic fields. By taking into account thecomposition, the influence of grain boun-daries, and crystal orientation, the mag-netic invar effect and the austenite magne-tism, Shimizu and Kakeshita (1989) pro-posed an equation that describes the shiftof Ms as a function of the magnetic field.

9.8.2.2 Nucleation

Martensitic transformations are first-or-der phase transformations and hence occurby nucleation and growth. In most in-stances, except in the case of thermoelas-ticity (see below), the growth of a marten-site plate proceeds so rapidly that the trans-formation kinetics are dominated by thenucleation event. Various mechanisms ofmartensite nucleation have in the past beenproposed and can be considered under twosubheadings. In the first group of models,the so-called localized nucleation models,concepts of diffusional nucleation kineticsare applied; the second group of models isbased on lattice instability considerations

concerning both static and dynamic latticeinstability. All nucleation models can fur-ther be divided into classical and non-clas-sical. The former model involves latticeperturbations of fixed amplitude and vary-ing size, whereas the latter considers per-turbations of varying size (Olson and Co-hen, 1982b).

In the classical nucleation theory; mar-tensite nuclei form along a path of constantcomposition and structure and the state ofthe nucleus is given by its size. Because themartensitic transformation involves shearstrains, it can be shown that the strain en-ergy is minimized for a disc-like nucleus,but then the surface energy becomes verylarge. The critical nucleus, assuming anoblate spheroidal shape (Fig. 9-30), willthen have an aspect ratio (c/r) such that forany change in shape, the decrease in strainenergy will be exactly balanced by an in-crease in interfacial energy. The interfacialGibbs energy per plate is

v Dgs = 2p r2 G (9-21)

where v is the volume of the plate, Dgs thesurface Gibbs energy per unit volume, andG the interfacial energy. The strain energyper plate is

v Dge = (4/3) p r2 c (Ac/r) (9-22)

where Dge (= Ac/r) is the strain energy perunit volume and A is a factor to be deducedfrom linear elasticity and thus a function ofthe elastic constants and of the shear anddilatational strains. The chemical Gibbsenergy change per plate is

v Dgc or (4/3) p r2 cDgc (9-23)

If the nucleation occurs at a lattice defect,we have to consider also the Gibbs energyGd due to the defect and the nucleus–de-fect interaction energy Gi . According toOlson and Cohen (1982), the total Gibbsenergy describing the formation of a classi-


Figure 9-30. Shape of a nucleus of a martensiteplate.


cal martensitic nucleus becomes(9-24)

G (r, c) = Gd + Gi + v (Dgc + Dge + Dgs)

and is given schematically in Fig. 9-31.Three cases are considered in calculatingthe critical free energy for nucleation DG*and the critical nucleus size r* and c*.

In the case of homogeneous nucleation,Gd and Gi are zero, and on inserting thenecessary quantities into Eq. (9-24) wefind that the barrier DG* is too high by sev-eral orders of magnitude. Even assuminglocal compositional fluctuations or the ex-istence of pre-existing embryos does notgive full satisfaction. It was therefore soon

recognized that homogeneous nucleationof martensite is impossible. Recently,much progress has been made in the under-standing of nucleus formation at lattice de-fects. DG* and therefore also the criticalsize of the nucleus can be reduced by as-suming the nucleation at a defect. Undercertain special conditions, this heterogene-ous nucleation may even be barrierless.Such a case applies to the f.c.c.-to-h.c.p.transformation, which may take place bydissociation of a number of properlyspaced total dislocations present in the ma-trix phase into partial dislocations separ-ated by stacking faults. The stacking-fault

Figure 9-31. Schematicnucleus Gibbs energy (G)curves for nucleation via aclassical path: (a) homoge-neous, (b) heterogeneous,and (c) barrierless nuclea-tion (Olson and Cohen,1982b).

Figure 9-32. Electron mi-crographs of the nucleationand early growth stagearound inclusion particles ina Ti–Ni–Cu alloy (Saburiand Nenno, 1987).

energy is temperature dependent and be-comes positive below T0, which results in abarrierless nucleation.

In a number of alloy systems, softeningof certain elastic constants is observed andit is then argued that, although the homoge-neous soft-mode concept is definitely notadequate to describe the nucleation of mar-tensite, stresses and strains present arounddefects of the lattice can induce a local me-chanical instability. Such a model is called“the localized soft-mode concept” (Guéninand Clapp, 1986). In this model the latticeGibbs energy is a function of pure strainsand therefore of second- and third-orderelastic constants. The third-order constants(which relate the strain energy to theamount of strain) introduce anharmonicterms into the strain energy and may lead tomechanical instability. The region of me-chanical instability, or the “strain spino-dal”, is so defined that any further increasein strain will make the lattice unstable withrespect to a decomposition into strained re-gions. In these zones a nucleus can developwithout generating any strain energy, andthe only resisting term remains the surfaceenergy. This results in a reduced criticalsize of the nucleus, which is further de-creased as the temperature is lowered ow-ing to the increase in chemical drivingforce.

In situ electron microscope observationshave been made on the nucleation and earlystages of growth of martensite, as shown inFig. 9-32. Martensite nucleates at stressconcentrations, the nucleation takes placerepeatedly at the same place, and the straincontrast disappears as nucleation andgrowth proceed and reappears when mar-tensite disappears.

At present, the nucleation models are be-ing further refined by molecular dynamiccalculations.

9.8.2.3 Growth and Kinetics

A distinction is made between the kinet-ics of a single martensite plate and the glo-bal kinetics, which expresses the volumefraction of the parent phase that is trans-formed. According to the observed kinet-ics, martensitic transformations can be di-vided into two distinct classes: athermaland isothermal martensite. In athermal mar-tensite, the transformation progresses withdecreasing temperature, whereas in iso-thermal martensite, the transformation pro-gresses with time at a constant temperature.

The growth may be “thermoelastic” or ofthe “burst” type. The latter is the morecommon mode. It consists of the formationof comparatively large amounts of marten-site (typically 10–30 vol.%) in “bursts”that are caused by autocatalytic nucleationand rapid growth of numerous plates. Eachindividual martensite plate is completelyformed with a speed higher than 105 cm/sand the transformation progresses by theformation of new plates. The global kinet-ics of the transformation are therefore es-sentially controlled by the nucleation fre-quency. The thermoelastic growth mode ischaracterized by the formation of thin, par-allel-sided plates or wedge-shaped pairs ofplates (Fig. 9-33), which form and growprogressively as the temperature is loweredbelow Ms and which shrink and disappearon reversing the temperature change. Thisbehavior arises because the matrix accom-modates the shape deformation of the mar-tensite plate elastically, so that at a speci-fied temperature the transformation frontof the plate and the matrix are in thermody-namic equilibrium. Any change in temper-ature displaces this equilibrium and, there-fore, the plate grows or shrinks. A com-plete mechanical analog of this thermoelas-tic behavior is the pseudoelastic behavior.The growth or shrinkage of individual mar-



Figure 9-33. Thermoelastic behavior in Ag–Cd alloys showing thegrowth of self-accommodating groups of martensite plates (Delaey et al., 1974).

He

ati

ng

Co

olin

g

Figure 9-34. Schematic representation of some relevant features (volume-transformed product or transforma-tion strain) experimentally observed in hysteresis curves corresponding to thermally induced and stress-inducedthermoelastic transformations: (a, e) single interface transformation in a single crystal; (b, f) multiple interfacetransformation; (c, g) discontinuous jumps (bursts), (d, h) partial cycling behavior.

tensite plates is then a direct function of theincrease or decrease in stress. More elab-orate thermodynamic treatments of ther-moelasticity can be found in the papers byDelaey et al. (1974), Salzbrenner and Co-hen (1979), Ling and Owen (1981) and Or-tin and Planes (1989).

The best quantitative understanding ofthe kinetics of the martensitic transforma-tion is obtained from isothermal transfor-mations, because they permit both the nu-cleation and the transformation rates to bedetermined. In those alloys exhibiting iso-thermal martensite (Thadhani and Meyers,1986), it is shown that at each temperaturethe transformation starts in the austeniteand proceeds as a function of time. Thetransformation exhibits a C-curve behav-ior. Isothermal martensitic transformationkinetics consist of two effects: an initial in-crease in the total volume fraction of mar-tensite, which is attributed to an autocata-lytic nucleation of new martensite plates,followed by a decrease due to the compart-mentalization of the austenite into smallerand smaller areas.

9.8.2.4 Transformation Hysteresis

Hysteresis behavior is one of the pecu-liar characteristics of both the thermal andstress-induced martensitic transformations.In several studies the origin of the fric-tional resistance opposing the interfacialmotion of martensite plates has been inves-tigated and described. From a practicalpoint of view, the hysteresis phenomenonis an important problem in the applicationof shape-memory alloys. In general hyster-esis appears when, on passing through a lo-cal extreme value (maximum or minimum)of any control parameter such as tempera-ture or stress, one or more state variablesdo not follow the original path in statespace. When all the state variables, includ-

ing the control parameter, return to theiroriginal values, a closed loop is formed(Fig. 9-34). The loop is always contoured insuch a sense that it encloses a positive area,representing the energy lost in the cyclicprocess. Therefore, hysteretic behavior isalways related to an energy-dissipative pro-cess. The dissipated energy is much smallerin thermoelastic martensitic transforma-tions than in burst-type transformations.

9.9 Materials

9.9.1 Metallic Materials

A classification of the diffusionless dis-placive transformations in metallic materi-als is given in Table 9-4, where the alloysystems are subdivided into three groups.The origin of the martensitic transforma-tion in the first group lies in the allotropictransformation of the pure solvent. Theparent phase of the alloys of this group thusdoes not show any remarkable mechanicalinstability. The second group consists ofthe b b.c.c. Hume–Rothery alloys, whichare characterized by a moderate lattice in-stability in the temperature range aboveMs . The third group is characterized by adrastic mechanical instability of the parentphase. Because the transformation is onlyweakly first order (by this we mean a dis-continuous jump in the corresponding ther-modynamic property whose height, how-ever, is very small) or even second order, itis in this group of alloy systems that wefind, in addition to the martensitic, thequasi-martensitic transformations.

Traditionally, the ferrous and non-fer-rous martensites have been treated separ-ately in the literature. Before going into de-tail it will be an advantage to first compareand contrast ferrous and non-ferrous mar-tensites and to do it in such a way that we


9.9 Materials 635

can establish criteria to differentiate thetwo groups. Although not every detail oftransformation behavior will be discussed,a set of criteria have been chosen as shownin Table 9-5. A more detailed review ofmartensite in metallic systems is given byNishiyama (1978).

9.9.1.1 Ferrous Alloys

Martensitic transformations in ferrousalloys have been studied extensively, espe-cially the crystallography and morphologywhich have been reviewed by Muddle

(1982). Depending on alloy composition, adistinction is made among the various mar-tensites, based either on the crystallogra-phy, morphology or growth characteristics.

Essentially, three different crystal struc-tures appear: the b.c.c. or b.c.t. a¢-marten-site, the h.c.p. e-martensite, and the long-range ordered f.c.t. martensite. In plain car-bon steels martensite is regarded as asupersaturated, interstitial solid solution ofcarbon in b.c.c. iron (ferrite), with a crystalstructure that is a tetragonally distortedversion of the ferrite structure. The tetrago-nality is linearly dependent on the carbon

Table 9-4. Classification of metallic alloy systems showing diffusionless displacive transformations (Delaey et al., (1982).

1. Martensite based on allotropic transformation of solvent atom

1. Iron and iron-based alloys

2. Shear transformation, close packed to close packed1. Cobalt and alloys f.c.c. Æ h.c.p., 126 R SF*2. Rare earth and alloys f.c.c., h.c.p., d.h.c.p., 9 R(3. MnSi, TiCr2 NaCl Æ NiAs, Laves)

3. Body centered cubic to close packed1. Titanium, zirconium and alloys b.c.c. Æ h.c.p., orth. f.c.c tw, d*2. Alkali and alloys (Li) b.c.c. Æ h.c.p.3. Thallium b.c.c. Æ h.c.p.

4. Others: plutonium, uranium, mercury, Complex structuresetc. and alloys

2. b-b.c.c. Hume–Rothery and Ni-based martensitic shape-memory alloys

1. Copper-, silver-, gold-, b-alloys(disord., ord.) b.c.c. AB, ABABCBCAC, ABAC

2. Ni–Ti–X b-alloys b.c.c. Æ 9 R, AB tw, SF*Nickel b-alloys (Ni–Al) b.c.c. Æ ABC tw, SF*Ni3–xMxSn (M = Cu, Mn) b.c.c. Æ AB tw*(Cobalt b-alloys, Ni–Co–X)

3. Cubic to tetragonal, stress-relaxation twinning or martensite

1. Indium-based alloys f.c.c. Æ f.c.t., orth. tw, tws*2. Manganese-based alloys f.c.c. Æ f.c.t., orth. tws*3. A 15 compounds, LaAgxIn1–x b-W. Æ tetr.4. Others: Ru–Ta, Ru–Nb, YCu, LaCd

* SF: stacking faults; tws: (stress relaxation) twins; d: dislocated

content. This a¢-martensite is also found ina number of substitutional ferrous alloys,the martensite being either b.c.c. or b.c.t. Incertain alloy systems with an austenitephase of low stacking fault energy, a mar-tensitic transformation to a fully coherenth.c.p. product (e-martensite) is observed.

In some long-range ordered alloys, as inFe–Pt and Fe–Pd, f.c.t. in addition to b.c.t.martensite is observed.

As far as the morphology is concerned,plate, lath, butterfly, lenticular, banded,thin-plate and needle-like martensite canbe distinguished.


Table 9-5. A qualitative comparison between ferrous and non-ferrous martensites (Delaey et al., 1982b).

Ferrous martensite Non-ferrous martensite

Interstitial and/or substitutional Nature of alloying Substitutional

Martensitic state in interstitial Hardness Martensitic state is not muchferrous alloys is much harder harder and may even be softer

than the austenite state than the austenite state

Large Transformation hysteresis Small to very small

Relatively large Transformation strain Relatively small

High values near the Ms Elastic constants of Low values near the Ms

the parent phase

Negative near the Ms Temperature coefficient Positive near the Ms

in most cases of elastic shear constant in many cases

Self-accommodation is not obvious Growth character Well developed self-accommodatingvariants

High rate, “burst”, athermal Kinetics Slower rate, no “burst”,and/or isothermal transformation no isothermal transformation,

thermoelastic balance

High Transformation enthalpy Low to very low

Large Transformation entropy Small

Large Chemical driving force Small

No single interface Growth front Single interface possibletransformation observed

Low and non-reversible Interface mobility High and reversible

Low Damping capacity Highof martensite

9.9 Materials 637

A distinct substructure, crystallographicorientation of the habit plane and austeniteto martensite orientation relationship areassociated with each morphology, as sum-marized in Table 9-6.

Because the carbon atom occupies octa-hedral interstices in the austenite f.c.c. lat-tice, special attention is drawn to theFe–X–C martensite. In the martensite lat-tice, those interstitial positions are definedby the Bain correspondence (Fig. 9-35).Only those at the midpoints of cell edgesparallel to [001]B and at the centers of thefaces normal to [001]B are permitted. Thispreferred occupancy affords an explanationof the observed tetragonality c/a, the de-gree of which is a function of the carboncontent:

c/a = 1 + 0.045 (wt.% C) (9-25)

Careful X-ray diffraction of martensite,freshly quenched and maintained at liquidnitrogen temperature, has shown signifi-

cant deviations from the above equation.The tetragonality is abnormally lower forX = Mn or Re and abnormally higher forX = Al or Ni. Heating to room temperatureof the latter martensite results in a loweringof the tetragonality. The formation of do-mains or microtwinning in the former al-loys and ordering of the Al atoms in the lat-ter have been put forward as the origin ofthe abnormal c/a ratio. This behavior hasmoreover been related to the martensiteplate morphology (Kajiwara et al., 1986,1991). Kajiwara and Kikuchi found that inFe–Ni–C alloys the tetragonality is abnor-mally large and depends on the microstruc-ture. It is very large for a plate martensite,while it is normal or not so large for a len-ticular martensite. They conclude that “themartensite tetragonality is dependent onthe mode of the lattice deformation in themartensitic transformation. If the latticedeformation is twinning, the resulting c : ais large, while in the case of slip it is small”(Kajiwara and Kikuchi, 1991).

9.9.1.2 Non-Ferrous Alloys

A classification base of the non-ferrousalloy systems exhibiting martensite is

Table 9-6. Summary of substructure, habit plane(H.P.) and orientation relationship (O.R.) for the fourtypes of a¢-martensite (Maki and Tamura, 1987).

Morphology Substructure H.P. O.R.* Ms

Lath (Tangled) (111)A K–S Highdislocations

Butterfly (Straight) (225)A K–Sdislocations

andtwins

Lenticular (Straight) (259)A Ndislocations

and or ortwins (3 10 15)A

(Mid-rib) G–T

Thin-plate Twins (2 10 15)A G–T Low

* K–S: Kurdjumow–Sachs relationship, N: Nishiyamarelationship, G–T: Greninger–Troiano relationship

Figure 9-35. Schematic representation of the Baincorrespondence for the f.c.c. to b.c.t. transformation.The square symbols represents the possible occupiedpositions of the interstitial carbon atoms (Muddle,1982).

given in Table 9-4, while the alloying ele-ments are given in Table 9-7. A review ofthe non-ferrous martensites was given byDelaey et al. (1982a).

Two typical examples of the first groupare the cobalt and titanium alloys. Thestructure of the cobalt-based martensites is in general hexagonal close-packed, butmore complex close-packed layered struc-tures have been reported, such as the 126R,84R and 48R structures observed inCo–Al alloys. Because the transformationis a result of an f.c.c. to h.c.p. transforma-tion, the basal planes of the martensitephase are parallel with the (111) planes ofthe parent phase and constitute the habitplane. The structure of the martensite in Ti-based alloys is also hexagonal but that ofthe high-temperature phase is b.c.c. Bothplate and lath morphologies are encoun-tered in titanium, and also in the similarzirconium-based alloys. Slip is suggestedas the lattice-invariant deformation modein lath martensite, whereas the twiningmode is observed in the plate martensities.

A typical example of the second groupare the copper-, silver- and gold-based alloys, which have been extensively re-viewed by Warlimont and Delaey (1974).Depending on the composition, three typesof close-packed martensite are formedfrom the disordered or ordered high-tem-perature b.c.c. phase, either by quenchingor by stressing. The factors determining theexact structure are the stacking sequence ofthe close-packed structure, the long-rangeorder of the martensite as derived from theparent b-phase ordering, and the deviationsfrom the regular hexagonal arrangementsof the martensite. The last factor is due todifferences in the sizes of the constituentatoms. The stacking sequence of the mainthree phases are ABC, ABCBCACAB andAB, respectively.

One of the interesting findings is the suc-cessive stress-induced martensitic transfor-mations in some of the b-phase alloys dis-cussed above, as shown clearly if we plotthe critical stresses needed for the transfor-mation (Fig. 9-36). Stressing a single crys-tal of Cu–Al–Ni at, for example, 320 Kwe find the parent b1-to-b¢1, the b¢1 – g¢1,


Figure 9-36. Critical stresses as a function of tem-perature for the various stress-induced martensitetransformations in a Cu–Al–Ni alloy (Otsuka andShimizu, 1986).

Table 9-7. Schematic representation of some non-ferrous martensitic alloy systems; the Co-, Ti- andZr-based terminal solid solutions, the intermetallicNi-, Cu-, Ag- and Alu-based alloys, the antiferro-magnetic Mn-based alloys and the In-based alloys(Delaey et al., 1982).

9.9 Materials 639

the g¢1 – b≤1, and finally the b≤1 – a¢1 marten-site.

The other typical example of the secondgroup is the Ni–Ti-based alloy system;both prototype alloy systems constitute theshape-memory alloys (SMA). The occur-rence of a so-called “pre-martensitic” R-shape has long obscured the observations.The review by Wayman (1987), illustratesthe complexity of the transformation be-havior. During cooling, the high-tempera-ture ordered b.c.c. phase (P) transformsfirst to an incommensurate phase (I) and onfurther cooling to a commensurate phase(C), and finally to martensite. The P-to-Itransformation is second order, whereas theI-to-C transformation is a first-order phasetransformation involving a cubic-to-rhom-

bohedral (the so-called R-phase) structuralchange. At still lower temperatures therhombohedral R-phase transforms into amonoclinically distorted martensite. TheR-phase also forms displacively and can bestress-induced, and shows all the character-istics of a reversible transformation.

Concerning the third group, in only afew cases, as in In–Tl, has definite proofbeen provided to justify the conclusion thatthe transformation is martensitic. Most ofthe transformations in these systems haveto be classified as quasi-martensitic.

9.9.2 Non-Metals

Inorganic compounds exhibit a varietyof crystal structures owing to their diverse

Table 9-8. Non-metals with lattice deformational transformations (Kriven, 1982).

Inorganic compoundsAlkali and ammonium halides MX, NH4X (NaCl-cubic ¤ CsCl-cubic)

Nitrates RbNO3 (NaCl-cubic ¤ rhombohedral ¤ CsCl-cubic)KNO3, TlNO3, AgNO3 (Orthorhombic ¤ rhombohedral)

Sulfides MnS (Zinc-blende-type ¤ NaCl-cubic)(Wurtzite-type ¤ NaCl-cubic)

ZnS (Zinc-blende-type ¤ wurtzite-type)BaS (NaCl-type ¤ CsCl-type)

MineralsPyroxene chain silicates Enstatite (MgSiO3) (Orthorhombic ¤ monoclinic)

Wollastonite (CaSiO3) (Monoclinic ¤ triclinic)Ferrosilite (FeSiO3) (Orthorhombic ¤ monoclinic)

Silica Quartz (Trigonal ¤ hexagonal)Tridymite (Hexagonal, wurtzite-related)

Cristobalite (Cubic ¤ tetragonal, zinc blende-related)

CeramicsBoron nitride BN (Wurtzite type ¤ graphite-type)

Carbon C (Wurtzite type ¤ graphite)Zirconia ZrO2 (Tetragonal ¤ monoclinic)

OrganicsChain polymers Polyethylene (CH2–CH2)n (Orthorhombic ¤ monoclinic)

CementBelite 2 CaO · SiO2 (Trigonal ¤ orthorhombic ¤ monoclinic)

chemistry and bonding. Compared withmetals, the relatively low-symmetry parentstructures have fewer degrees of freedomon transforming to even lower symmetryproduct structures, or vice versa. Many ofthese transformations involve changes inelectronic states with relatively small volumechanges. They tend to proceed by shuffle-dominated mechanisms. However, sheartransformations involving large structuralchanges in terms of coordination numberor volume changes have also been reportedin inorganic and organic compounds, min-erals, ceramics, organic compounds, andsome crystalline compounds of cement.Some of the most prominent examples aregiven in Table 9-8 (Kriven, 1982, 1988).

Because of its technological interest as atoughener for brittle ceramic materials, zir-conia is considered as the prototype ofmartensite in ceramic materials. On cool-ing, the high-temperature cubic phase ofzirconia transforms at 2370 °C to a tetrago-nal phase. On further cooling, bulk zirconiatransforms at 950 °C to a monoclinic phasewith a volume increase of 3%. The lattertransforms on heating at about 1170 °C.The monoclinic to tetragonal phase trans-formation is considered to be martensitic.The Ms temperature can be lowered sub-stantially even below room temperature byalloying or by reducing the powder size.Small particles of zirconia, embedded in asingle-crystal matrix of alumina, remainmetastable (= tetragonal) at room tempera-ture for particle diameters less than a criti-cal diameter (Rühle and Kriven, 1982).These metastable particles can transform tothe monoclinic phase under the action of anapplied stress, and it is this property that isexploited in toughening brittle ceramics(see Becher and Rose (1994)).

Polymorphism is known to occur in several crystalline polymeric materials. Inmost of these systems the transformation

depends strongly on thermal activation.However, in PTFE (polytetrafluoroethy-lene) the conditions for no or weak thermalactivation are fulfilled, and the transforma-tion can then be regarded as a diffusionless


Figure 9-37. (a) Helix structure of the a- and b-modification of PTFE, and (b) dilatometric deter-mination of relative volume changes and tempera-ture range of transformations of PTFE (Hornbogen,1978).

9.10 Special Properties and Applications 641

or martensitic transformation (Hornbogen,1978). This polymer crystallizes as parallelarrangements of molecular chains parallelto the c-axis. The atoms along the chainsare arranged as helices; the period alongthe c-axis in the a-modification is 13C2F4

units while it is 15 units in the b-modifica-tion. The transformation from the a- to theb-helix occurs at about 19 °C (Fig. 9-37).Relaxation of the helix during this transfor-mation does not lead to an extension of thespecific length of the molecules in the c-di-rection. The diameters of the molecule andthus the lattice parameters in the a-direc-tion increase, which leads to an increase of about 1% in the specific volume. Theobserved shape change can be increased ifthe molecules have been aligned by plasticdeformation. An analysis of the shapechanges leads to the conclusion that thePTFE transformation is diffusionless by afree volume shear, a type of transformationnot yet known in metallic and inorganicmaterials.

Biological materials consisting of crys-talline proteins also undergo martensitictransformations in performing their lifefunctions. In a review entitled “Martensiteand Life”, Olson and Hartman (1982) dis-cuss some examples. The tail-sheath con-traction in T4 bacteriophages can be de-scribed as an irreversible strain-inducedmartensitic transformation, while polymor-phic transformations in bacterial flagellaeappear to be stress-assisted and exhibit ashape-memory effect.

9.10 Special Propertiesand Applications

9.10.1 Hardening of Steel

Much of the technological interest con-cerns martensite in steels. In a review onstrengthening of metals and alloys, Wil-

liams and Thompson (1981) consider mar-tensite as one of the most complex cases ofcombined strengthening. The hardness ofmartensite in as-quenched carbon steel de-pends very much on the carbon content. Upto about 0.4 wt.% C the hardening is lin-ear; retained austenite is present in steelcontaining more carbon, which reduces therate of hardening. Solute solution harden-ing by the interstitial carbon atoms is verysubstantial, whereas substitutional solidsolution hardening is low. For example,Fe–30 wt.% Ni martensites, where the car-bon content is very low, are not very hard.The hardening of martensite is not due onlyto interstitial solute solution hardening,however. The martensite contains a largenumber of boundaries and dislocations,and the carbon atoms may rearrange duringthe quench forming clusters that cause ex-tra dislocation pinning. The various contri-butions to the strength of a typical C-con-taining martensite are given in Table 9-9,from which it becomes evident that inter-stitial solid solution hardening is not themost important cause. Because many mar-tensitic steels are used after tempering,lower strengths than those shown in Table9-9 are found.

9.10.2 The Shape-Memory Effect

A number of remarkable properties havetheir origin in a martensitic phase trans-

Table 9-9. Contribution to as-quenched martensitestrength in 0.4 wt.% C steel (Williams and Thomp-son, 1981).

Boundary strengthening 620 MPaDislocation density 270 MPa

Solid solution of carbon 400 MPaRearrangement of C in quench 750 MPa

Other effects 200 MPa

0.2% yield strength 2240 MPa

formation, such as the shape-memory ef-fect, superelasticity, rubber-like behaviorand pseudoelasticity. The most fascinatingproperty is undoubtedly the shape-memoryeffect. Review articles were published byDelaey et al. (1974), Otsuka and Shimizu(1986), Schetky (1979), and Junakubo(1987). Recently, Van Humbeeck (1997)prepared a review on Shape Memory Mate-rials: “State of the Art and Requirementsfor Future Applications”. His review con-tains 104 references to recent articles onthe topic.

A metallic sample made of a commonmaterial (low-carbon steel, 70/30 brass,aluminum, etc.) can be plastically de-formed at room temperature. The macro-scopic shape change resulting from thisdeformation will remain unchanged if thesample is heated to higher temperatures.The only observable change in propertymay be its hardness, provided that the tem-perature to which the sample has beenheated is above the recrystallization tem-perature. Its shape, however, remains as itwas after plastic deformation. If the sampleis made of a martensitic shape-memorymaterial and is plastically deformed (bent,twisted, etc.) at any temperature below Mf

and subsequently heated to temperaturesabove Af , we observe that the shape thatthe specimen had prior to the deformationstarts to recover as soon as the As tempera-ture is reached and that this restoration iscompleted at Af . This behavior is called the“shape-memory effect”, abbreviated to SME.

If the SME sample is subsequentlycooled to a temperature below Ms and itsshape remains unchanged on cooling, wetalk about the “one-way shape-memory ef-fect”. If it spontaneously deforms on cool-ing to temperatures below Ms into a shapeapproaching the shape that it had after theinitial plastic deformation, the effect iscalled the “two-way shape-memory ef-

fect”. A more visual description of thesetwo effects is given in Fig. 9-38 and a clar-ifying example is shown in Fig. 9-39,where the applicability of the one-wayshape-memory effect is given for a space-craft antenna.

The shape that has to be rememberedmust, first of course, be given to the speci-men. This is done by classical plastic de-formation by either cold or hot working.This process, however, may not involveany martensite formation. The materialmust therefore be in a special metallurgicalcondition, which may require additionalthermal treatments. In Fig. 9-40, for example, depicting a temperature-actuatedshape-memory switch, two different “re-membering” shapes are used. The initialshape may be obtained by hot extrusion orwire drawing and may or may not receivean additional cold or hot working in orderto obtain the required shape. The shapesformed must receive a heat treatment, con-sisting of a high-temperature annealing,followed by water quenching. The speci-men is now martensitic, provided that thecomposition is such that Mf is above roomtemperature. In order to induce the shapememory, the martensitic specimens arebent either to be curved or to be straightand are placed into the actuator at roomtemperature. If the temperature of the actu-ator exceeds the reverse transformation


Figure 9-38. Schematic illustration of the shape-memory effect: (a) and (e) parent phase; (b), (c) and(d) martensite phase (Otsuka and Shimizu, 1986).


temperature of the shape-memory material,the specimens recover towards their “re-membered” position. The electrical con-tacts are either closed or opened.

Special procedures for handling of theshape-memory device are needed if wewant to induce the two-way memory effect.This can be explained by again taking thetemperature-actuated switch as an exam-ple. If the specimen taken in its remem-bered position is cooled back to room tem-

perature, we do not expect further shapechanges to occur. In order to reuse thespecimens after having performed theshape-memory effect, they must be bent tobe either curved or straight again. Reheat-ing these deformed specimens for a secondtime to temperatures above Af will result inshape memory. If this cycle, bending–heat-ing–cooling, is repeated several times,gradually a two-way memory sets in. Dur-ing cooling the specimen reverts spontane-

Figure 9-39. Application ofNitinol for a shape-memoryspacecraft antenna. From “Shape Memory Alloys” byL. McDonald Shetky. Copy-right (1979) by ScientificAmerican, Inc. All rights re-served.

ously to its “deformed” positions, thusopening or closing the electrical contactson cooling. This repeated cycling, defor-mation in martensitic condition followedby a heating–cooling, is called “training”.We can thus induce two-way memory byusing a training procedure.

A further comment should be made hereconcerning the shapes that can be remem-bered. We have to distinguish three shapingprocedures: the fabrication step from rawmaterial towards, for example, a coiledwire such as for the antenna, the fabrica-tion of the “to be remembered position”,such as the additional shaping for the actu-ator, and the final deformation in the mar-tensitic condition, such as the bending ofthe actuator. The first two fabrication stepsinvolve only classical plastic deformationand, therefore, the type and degree of de-formation are in principle not limited, pro-vided that the material does not fail. Thedegree of deformation, however, is limited

in the third deformation step, because itmay not exceed the maximum strain thatcan be recovered by the phase transforma-tion itself. Because these strains are asso-ciated with the martensitic transformation,the maximum amount of recoverable strainis bound to the crystallography of the trans-formation. Exceeding this amount of de-formation in the third fabrication step willautomatically result in unrecoverable de-formation.

Many examples of shapes that can be re-membered are possible. A flat SME speci-men can elongate or shorten during heat-ing, can twist clockwise or counter-clock-wise, and can bend upwards or downwards.An SME spring can expand or contract dur-ing heating. All this depends on the secondand third fabrication steps.

What happens now if, for one reason oranother, the specimen is restrained to ex-hibit the shape-memory effect? For exam-ple, what happens if an expanded ring is


Figure 9-40. Temperature-actuated switch designed so that it opens or closes above a particular temperature.From “Shape Memory Alloys” by L. McDonald Shetky. Copyright (1979) by Scientific American, Inc. Allrights reserved.


fitted as a sleeve over a tube with an outerdiameter slightly smaller than the inner di-ameter of the expanded ring but larger thanthe inner diameter of the ring in the remem-bered position? On heating, the ring willstart to shrink as soon as the temperature As

is reached. While shrinking it will touchthe tube wall and further shrinking will behindered. From this moment, a compres-sive stress will be built up, clamping theshrinking ring around the tube. Obviously,the composition of the alloy should be suchthat the Ms temperature is below the valueat which clamping is required; in manycases, this is below room temperature. Forclamping rings it is therefore important thaton cooling back to room temperature theclamping stress is still present. This meansthat two-way memory must be avoided,which is easily achieved by choosing ashape-memory alloy that exhibits a largetemperature hysteresis.

On heating a shape-memory device,stresses can thus be built up and mechanicalwork can be done. The latter would be thecase if a compressed shape-memory springhas to lift a weight as in a shape-memoryactuated window opener (Fig. 9-41). Avery useful device is realized when ashape-memory device is used, as shown in Fig. 9-42, in combination with a biasspring made of a conventional linear elasticmaterial, both being clamped between twofixed walls and attached to each other witha plate. At temperatures below Mf , theshape-memory spring is closed and com-pressed by the bias spring. The SME springhad to be deformed, in this case com-pressed, in order to fit into the clampingunit. The clamping unit with the twosprings installed is now heated to tempera-tures above Af ; as soon as As is reached, theSME spring will start to expand and try topush back the bias spring. At Af , the shape-memory spring will not yet have regained

its original length, and further heating is re-quired to overcome the force exerted by thebias spring. At a certain temperature higherthan Af the shape-memory spring will befully recovered. This temperature will de-pend on the strength of the bias spring.During this temperature excursion, theplate that is fixed between the two springswill have moved and can, if an “engine” isattached to it, deliver work. If the clampingunit is now cooled, the bias spring will tryto compress the shape-memory spring into

Figure 9-41. A simple shape-memory windowopener made from a copper-based shape-memory al-loy. From “Shape Memory Alloys” by L. McDonaldShetky. Copyright (1979) by Scientific American,Inc. All rights reserved.

Figure 9-42. A mechanism in which a shape-mem-ory alloy (SMA) spring is used in conjunction with abias spring. From “Shape Memory Alloys” by L.McDonald Shetky. Copyright (1979) by ScientificAmerican, Inc. All rights reserved.

its deformed position. The elastic energythat has been stored in the bias spring dur-ing the heating cycle is now released, al-lowing the plate to perform work also dur-ing the cooling cycle. To describe fullysuch a working performing cycle, a ther-modynamic treatment is needed (Wollantset al., 1979). The working performing cycle can best be illustrated by taking ashape-memory spring that expands or con-tracts during heating or cooling and thatcarries a load. The working performing cy-cle can then be represented in a displace-ment– temperature, a stress– temperature,or an entropy–temperature diagram.

Although the shape-memory effect hasbeen observed in many alloy systems, onlythree systems are commercially available,

mainly because of economic factors andthe reliability of the material. The three al-loy systems are Ni–Ti, Cu–Zn–Al andCu–Al–Ni. Generally, other elements areadded in small amounts (of the order of afew weight %) in order to modify the trans-formation temperatures or to improve themechanical properties or the phase stabil-ity. In all three cases the martensite is thermoelastic. Maki and Tamura (1987) reviewed the shape-memory effect in fer-rous alloys, where a non-thermoelasticFe–Mn–Si alloy has also been found toshow a shape-memory effect, and commer-cialization is being considered. The mostimportant properties of shape-memory al-loys are summarized in Fig. 9-43, in whichthe working temperatures, the width of the


Figure 9-43. Schematic representation of the most relevant shape-memory properties (courtesy Van Hum-beeck, 1989).


hysteresis, and the maximum recoverablestrain are given. Because of the superiormechanical, chemical and shape-memoryproperties of Ni–Ti alloys, this alloy sys-tem has been applied most successfully;about 90% of the present applications usethese alloys. Owing to the continuous im-provement of the properties of Cu-basedalloys, together with their lower price, CuSME alloys have been successfully used inseveral applications.

The commercial applications of shape-memory devices can be divided into fourgroups:

1. motion: by free recovery during heat-ing and/or cooling;

2. stress: by constrained recovery duringheating and/or cooling;

3. work: by displacing a force, e.g., inactuators;

4. energy storage: by pseudoelastic load-ing of the specimen.

Shape-memory effects have also beenreported in non-martensitic system, e.g., in ferroelectric ceramics (Kimura et al.,1981), and have found applications as micro-positioning elements (Lemons andColdren, 1978). The shape change is attrib-

uted here to domain-wall motion, as shownin Fig. 9-44.

9.10.3 High Damping Capacity

The hysteresis exhibited during a pseu-doelastic loading and unloading cycle is ameasure of the damping capacity of a vi-brating device fabricated from a shape-memory material, which is cycling underextreme stress conditions exceeding thecritical stress needed to induce martensiteby stress. Vibrating fully martensitic sam-ples also exhibit high damping. A fullymartensitic sample consists of a large number of differently crystallographicallyoriented domains whose domain boundar-ies are mobile. Under the action of an ap-plied stress these boundaries move but, be-cause of friction, energy is lost during thismovement. If a cyclic stress is applied, thisforeward and backward boundary move-ment will lead to damping of the vibration.Comparing the amount of this dampingwith the damping that we observe in othernon-SME alloy systems, it is found that thedamping capacity of martensitic shape-memory alloys is one of the highest. Theshape-memory alloys are said to belong tothe high-damping materials, the so-calledHIDAMETS.

9.10.4 TRIP Effect

TRIP is the acronym for TRansforma-tion-Induced Plasticity and occurs in somehigh-strength metastable austenitic steelsexhibiting enhanced uniform ductilitywhen plastically deformed. This uniformmacroscopic strain, up to 100% elongation,accompanies the deformation-inducedmartensitic transformation and arises froma plastic accommodation process aroundthe martensite plates. This macroscopicstrain thus contrasts with that occurring in

Figure 9-44. Schematic illustration of the mecha-nism for an electronic micro-positioning (Lemonsand Coldren, 1978).

shape-memory alloys in being unrecover-able.

TRIP has been extensively studied byOlson and Cohen (1982a) and we will fol-low their approach here. They distinguishtwo modes of deformation-induced trans-formation, according to the origin of thenucleation sites for the martensite plates: “stress-assisted” and “strain-induced” trans-formation. The condition under which eachmode can operate is indicated in a tempera-ture–stress diagram as shown in Fig. 9-45.At temperatures slightly higher than Ms

s,the stress required for stress-assisted nucle-ation on the same nucleation sites followsthe line AB. At B, the yield point for slip in the parent phase is reached, defining the highest temperature Ms

s for which thetransformation can be induced solely byelastic stresses. Above this temperature,plastic flow occurs before martensite canbe induced by stress. New strain-inducednucleation sites are formed, contributing tothe kinetics of the transformation. The

stresses at which this strain-induced mar-tensite is first detected follows the curveBD. At point D, fracture occurs and thusdetermines the highest temperature Md atwhich martensite can be mechanically in-duced.

When the transformation occurs at tem-peratures below Ms

s, the plastic strain isdue entirely to transformation plasticity re-sulting from the formation of preferentialmartensite variants. The volume of the in-duced martensite is therefore linearly re-lated to the strain. The existing nucleationsites are aided mechanically by the thermo-dynamic contribution of the applied stress,reducing the chemical driving force for nucleation. Above Ms

s, the relationshipbetween strain and volume of martensitebecomes more complex, because strain is


Figure 9-46. Transformation-induced plasticity intensile tests at various temperatures (Fe–29 wt.%Ni–0.26 wt.% C) (Tamura et al., 1969).

Figure 9-45. Idealized stress-assisted and strain-in-duced regimes for mechanically-induced nucleation(Olson and Cohen, 1982a).

9.11 Recent Progress in the Understanding of Martensitic Transformations 649

then a result of plastic deformation of theparent phase and of transformation plastic-ity. Strain hardening and enhancement ofnucleation of martensite also play an essen-tial role. When martensite is formed duringtensile deformation, the strain hardeningbecomes large. Necking is then expected to be suppressed, explaining the enhanceduniform elongation. Fig. 9-46 shows, as anexample, the amount of martensite, the elongation and the ultimate strength meas-ured after tensile tests of a TRIP steel as afunction of temperature, clearly illustratingthe enhanced elongation, especially in thetemperature range between Ms

s and Md.Such a large elongation (sometimes over200%) can also be produced by subjectinga TRIP steel specimen under constant loadto thermal cycles through the transforma-tion temperature.

9.11 Recent Progress in theUnderstanding of MartensiticTransformations

We draw attention here to some recentpapers that illustrate recent progress in theunderstanding of martensite, and in which

some new approaches are also explained.Most of the information referred to in thissection was presented at the most recentICOMAT international conference on mar-tensitic transformations held in 1998 atBariloche (Ahlers et al., 1999).

New directions in martensite theory arepresented by Olson (1999). The nucleationof martensite, the growth of a single mar-tensite plate, the formation of, for example,self-accommodating groups of martensiteplates, and this within single crystals of theparent phase as well as in polycrystallinematerial, and the constraints dictated by thecomponents where martensitic materialsare only one (maybe the most important)functional element of the component, areall influenced by different interactive lev-els of structures (ranging from solute atomsto components). Nucleation is the first stepin martensite life, and a component whosefunctional properties are attributed to thoseof martensite, can be considered the finalstep. Olson (1999) constructed a flow-block diagram in which the martensitictransformation is situated in a multileveldynamic system. This new system, shownin Fig. 9-47, and the one given in Fig. 9-1,offer powerful tools for a better under-

Figure 9-47. The flow-block diagramof martensitic transformation as amultilevel dynamic system (Olson,1999).

standing of diffusionless phase transforma-tions. Its use should lead to a better designof martensitic and bainitic alloys meetingspecific requirements. Close analysis of thepapers presented at ICOMAT 98 shows thatsuch an approach can already be found inmany papers.

Special attention is given to the influ-ence of external constraints, such as hydro-static pressure, the application of a mag-netic field, and to martensite formed in thinfilms prepared by either sputtering or rapidsolidification. Kakeshita et al. (1999) stud-ied the influence of hydrostatic pressure in-stead of uniaxial stress, in order to formu-late a thermodynamic approach for a betterunderstanding of the nucleation of marten-site. The strengthening mechanisms insteel due to martensite are reviewed, thediffusion of carbon in the various states(according to the dynamic system of Fig. 9-47) of martensite is highlighted. In thiscontext, the fracture mechanism is relatedto the tempering temperature and the car-bon diffusion.

As is commonly known, the mechanismof bainite transformation is a subject withmany unresolved issues. Bhadeshia (1999)gives an overview of the transformationmechanisms proposed to explain “amongothers” the growth of bainite. The develop-ment of bainite at both high temperatures (upper bainite) and low temperatures (lower bainite) is discussed and is illustrat-ed in Fig. 9-48. According to Bhadeshia,the unresolved issues are:

the growth rate of an individual bainiteplatea theory explaining the kinetics to esti-mate the volume fraction of bainite inaustenite obtained during an isothermaltransformationthe modeling describing quantitativelythe formation of carbidesand a number of features associated withthe interaction between plastic deforma-tion and bainite formation.

The influence of carbon on the bainitictransformation is treated in great detail andis shown to be a controlling factor of themechanical properties of different multi-phase TRIP-assisted steels (Girault et al.,1999; Jacques et al., 1999).

The martensitic transformation inFe–Mn-based alloys is treated in variouspapers, showing the increasing interest indeveloping ferrous shape-memory alloys.In these alloys, austenite transforms eitherinto a h.c.p. e-phase (g Æ e) or/and into a¢-martensite (g Æ a¢).

New approaches and strategies are dis-cussed for the application of shape-mem-ory alloys in non-medical (Van Humbeeck,1999) as well as in medical applications(Duerig et al., 1999). Only two examplesare shown here. The first example (Fig. 9-49) shows that Ni–Ti superelastic alloysimprove significantly the cavitation ero-sion resistance if compared with marten-


Figure 9-48. A schematic representation of themechanism explaining the growth and developmentof bainite (Bhadeshia, 1999).

9.12 Acknowledgements 651

sitic Ni–Ti. But it should be remarked thatthis figure is only an enlargement of a fig-ure giving an overall view of the cavitationresistance of other common alloys. For ex-ample, the weight loss after 10 h is already20–30 mg for the “common” alloys incomparison with the negligible weight lossof both Ni–Ti shape-memory alloys after10 h tested under the same conditions. Avery impressive example of the applicationof Ni–Ti shape-memory alloys is given inFig. 9-50. This figure shows an atrial septalocclusion device with Nitinol (Ni–Tishape-memory alloy) wires incorporated ina sheet of polyurethane. This device allowsholes in the atrial wall of the heart to beclosed without surgery. The two umbrella-like devices are folded in two catheters,which are placed on either side of the hole. Once the two folded umbrellas arewithdrawn from their catheters, they arescrewed together in such a way that thehole is closed. Because of the flexibility ofboth materials, the heart can again beatnormally. This device illustrates the con-cept of the elastic development capacity of shape-memory alloys. Because Ni–Tishape-memory alloys have proposed to bebiocompatible (see Van Humbeeck, 1977),many applications of these Ni–Ti alloys

are currently being developed and mar-keted.


The author would like to thank M. Ah-lers, J. W. Christian, M. De Graef, R. Gott-hardt, P. Haasen, H. S. Hsu, J. Ortín, K. Ot-suka, J. Van Humbeeck and P. Wollants forsupport and advice while preparing themanuscript, and M. Van Eylen, M. Nol-mans, H. Schmidt and K. Delaey for theirassistance. The “Nationaal Fonds voor We-tenschappelijk Onderzoek” of Belgium isacknowledged for financial support (pro-ject No. 2.00.86.87). The author especiallyacknowledges the continuing interest andencouragement he received from A. De-ruyttere. For help in preparing the revisedversion, I would like to thank M. Chandra-sekaran.

Figure 9-49. The weight loss of a martensitic (NiTi–1) and a pseudoelastic (Ni Ti–2) Ni–Ti shape-memory alloy (Richman et al., 1994).

Figure 9-50. A shape-memory device for repairingdefects in the heart wall (Duerig et al., 1999).

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Wayman, C. M. (1983), in: Physical Metallurgy:Cahn, R. W., Haasen, P. (Eds.). New York: Else-vier Scientific Publishers, p. 1031.

Wayman, C. M. (1987), Proc. ICOMAT ’86. Sendai:Japanese Institute of Metals, p. 645.

Wechsler, M. S., Lieberman, D. S., Renard, T. A.(1953), Trans. AIME 197, 1503.

Williams, J. C., Thompson, A. w. (1981), in: Metal-lurgical Treatises: Tien, J. K., Elliott, J. F. (Eds.).Warrendale: Met. Soc. AIME, p. 487.

Wollants, P., De Bonte, M., Roos, J. R. (1979), Z.Metallkd. 70, 113.

Yang, J., Wayman, C. M. (1993), in: InternationalConference on Martensitic Transformation ICO-MAT ’92, Wayman, C. M., Perkins, J. (Eds.). Car-mel, CA, USA: Monterey Institute of AdvancedStudies, p. 1187.

Proceedingsof International Conferenceson Martensitic Transformations

“Physical Properties of Martensite and Bainite”(1965), Special Report 93. London: The Iron andSteel Institute.

“Shape Memory Effects in Alloys” (1975), Perkins,J. (Ed.). New York: Plenum Press.

“New Aspects of Martensitic Transformations”(1976), Suppl. to Trans. Jap. Inst. Met. 17.

“Martensitic Transformations ICOMAT ’77” (1978),Kiev: Academy of Science.

“International Conference on Martensitic Transfor-mations ICOMAT ’79” (1979), Cambridge (MA):Dep. of Mat. Science and Techn., M.I.T.

“International Conference on Martensitic Transfor-mations ICOMAT ’82” (1982), Delaey, L., Chan-drasekaran, M. (Eds.), Journal de Physique 43,Conf. C-4.

“International Conference on Shape Memory Al-loys” (1986), Youyi Chu, Hsu, T. Y., Ko. T. (Eds.).Beijing: China Academic Publishers.

“International Conference on Martensitic Transfor-mations ICOMAT ’86” (1987), Tamura, I. (Ed.).Sendai: The Japan Inst. of Metals.

“The Martensitic Transformation in Science andTechnology” (1989), Hornbogen, E., Jost, N.(Eds.). Oberursel: DGM-InformationsgesellschaftVerlag.

“Shape Memory Materials” (1989), Otsuka, K., Shi-mizu, K. (Eds.), Proceedings of the MRS Interna-tional Meeting on Advanced Materials, vol. 9.Pittsburgh: Mat. Research Soc.

“International Conference on Martensitic Transfor-mations ICOMAT ’89” (1990), Morton, A. J.,Dunne, B., Kelly, P. M., Kennon, N. (Eds.) to bepublished.

“International Conference on Martensitic Transfor-mations ICOMAT ’92” (1993), Wayman, C. M.,Perkins, J. (Eds.). Carmel, CA, USA: Monterey In-stitute of Advanced Studies.

“International Symposium and Exhibition on ShapeMemory Materials (SME ’99)” (2000). Zurich:Trans. Tech. Publications.

“International Conference on Shape Memory andSuperelastic Technologies (SMST ’97)” (1997),Pelton, A., Hodgson, D., Russell, S., Duerig, T.(Eds.). Santa Clara, CA, USA: SMST Proceedings.

“Displacive Phase Transformations and Their Appli-cations in Materials Engineering” (1998), Inoue,K., Mukherjee, K., Otsuka, K., Chen, H. (Eds.).Warrendale, OH, USA: TMS Publications.

“European Symposium on Martensitic Transforma-tions ESOMAT ’97” (1997), Beyer, J., Böttger, A.,Mulder, J. H. (Eds.). EDP-Sciences Press.

“European Symposium on Martensitic Transforma-tions ESOMAT ’94” (1995), Planes, A., Ortin, J.,Lluis Manosa. Les Ulis, France: Les Editions dePhysique.

“International Conference on Martensitic Transfor-mations ICOMAT ’98” (1999), Ahlers, M., Kos-torz, G., Sade, M. (Eds.). Amsterdam: Elsevier.

“Martensitic Transformation and Martensite” (1980),Hsu, T. Y. (Xu Zuyao) (Ed.). Shanghai: SciencePress (in Chinese).

“Pseudoelasticity and Stress-Induced MartensiticTransformations” (1977), Otsuka, K., Wayman, C.M. (Eds.), in: “Reviews on the Deformation Be-havior of Materials”.


10 High Pressure Phase Transformations

Martin Kunz

ETH Zürich, Labor für Kristallographie, Zürich, Switzerland

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 65710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65910.2 Pressure-Driven Phase Transitions . . . . . . . . . . . . . . . . . . . . 66010.2.1 Framework flexion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66110.2.2 Increase in coordination number . . . . . . . . . . . . . . . . . . . . . . . 66310.2.3 Pressure-induced ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 66410.3 Generating High Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 66610.3.1 Dynamic pressure generation . . . . . . . . . . . . . . . . . . . . . . . . 66710.3.2 Static pressure devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66810.3.2.1 Large-volume presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66810.3.2.2 Diamond anvil cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67110.4 Probing Phase Transformations in Materials at High Pressure . . . . . 67310.4.1 Volumetric techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67410.4.2 Spectroscopic techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 67410.4.2.1 Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67410.4.2.2 Raman and infrared spectroscopy . . . . . . . . . . . . . . . . . . . . . . 67410.4.2.3 Mössbauer spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 67510.4.2.3 X-Ray absorption spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 67610.4.2.5 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67610.4.3 Ultrasonic sound velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 67710.4.4 Diffraction techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67810.4.4.1 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67810.4.4.2 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67910.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67910.5.1 Zincblende-type semiconductors . . . . . . . . . . . . . . . . . . . . . . 67910.5.1.1 Si and Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68010.5.1.2 GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68110.5.1.3 InSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68210.5.2 Materials in the B-C-N system . . . . . . . . . . . . . . . . . . . . . . . . 68310.5.2.1 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68410.5.2.2 B–N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68510.5.2.3 C–N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68610.5.2.4 B–C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68610.5.2.5 B–C–N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68610.5.3 H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68710.5.3.1 Ice Ih, XI and Ic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688


10.5.3.2 Ice II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68910.5.3.3 Ice III and Ice IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68910.5.3.4 Ice V, Ice IV and Ice XII . . . . . . . . . . . . . . . . . . . . . . . . . . . 69010.5.3.5 Ice VI, Ice VII, Ice VIII, Ice X . . . . . . . . . . . . . . . . . . . . . . . 69010.5.3.6 Amorphous ice at high pressure . . . . . . . . . . . . . . . . . . . . . . . 69110.6 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69210.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692

656 10 High Pressure Phase Transformations



a, b, c lattice parametersE0 internal energy at zero pressure in shock experimentEH Hugoniot internal energy in a shock experimentG Gibbs energyH enthalpyl wavelengthm chemical potentialn number of molesP pressureP0 ambient pressure PH Hugoniot pressureQ heatr0 density at zero pressureS entropyT temperatureTc critical temperature in superconductorsT0 temperature at zero pressureTH Hugoniot temperatureU internal energyUP sample velocityUs shock velocityV molar volumeV0 molar volume at zero pressureVH molar volume in the shocked Hugoniot stateDVdis difference in molar volume due to an order/disorder processVdisordered molar volume of the disordered stateVordered molar volume of the ordered stateW work

ADX angle dispersive X-ray diffractionAX compound semiconductors of 1 :1 stoichiometrybcc body centered cubicccp cubic closest packing (= fcc)ct caratDAC diamond anvil celldhcp hexagonal closest packing with doubled c-axisEDX energy dispersive X-ray diffractionEXAFS extended X-ray absorption fine struturefcc face centered cubic (=ccp)hcp hexagonal closest packingHDA high density amorphous phase of H2OHEL Hugoniot elastic limitHP high pressure

HT high temperatureIR infraredLDA low-density amorphous phase of H2ONMR nuclear magnetic resonanceRT room temperatureXAS X-ray absorption spectroscopy



10.1 Introduction

The understanding of condensed matterrequires a knowledge of the relationshipbetween temperature, pressure and chemi-cal environment on the one hand and vol-ume, bonding and electronic and magneticstructure on the other. In gases, pressure,temperature and volume largely follow asimple and universal relationship, but thesituation in condensed matter is much morecomplicated. This is mainly due to inter-atomic interactions, which are muchstronger in condensed material than ingases and cause structural, electronic andmagnetic correlations. Pelton (2001, Chap. 1of this book) explores the interdependencebetween chemical composition and tem-perature in great detail. In this chapter, wewill mainly focus on the effects of pressureon the state of solid material.

The exploration of the physics andchemistry of material under high pressureis of a remarkably inter-disciplinary inter-est. This is reflected in the variety of scien-tific problems that are the focus of modernhigh-pressure research. The understandingof the interdependence between structuraldistortions and electronic and magneticproperties is a typical example of a physi-cal problem that benefits from experimentsperformed at high pressure. Another topicwithin physics that relies on high-pressureexperiments is the calibration and refine-ment of theoretical models describing theinteraction of atoms. High-pressure experi-ments on very simple covalent systemssuch as solid hydrogen or helium providevaluable data, which help to formulate fun-damental concepts on the nature of matter.A fascinating problem in condensed-matterchemistry is the structural change of a solid(or liquid) and its phase transitions as a re-action on an external pressure. Its under-standing offers chemists vital insights into

the thermodynamics of solid or liquid material (see also the Chapter by Binder,2001). In a more practical way, materialsscientists are interested in adding pressureto the variables temperature and chemicalcomposition in order to stabilize new mate-rials with technically interesting proper-ties. A prominent example of a class of ma-terials whose synthesis and exploration de-pend on the application of high pressure isthe family of novel abrasives and super-hard materials. In recent years, high-pres-sure experiments are even used in the lifesciences. A popular key issue in this area is the question on the origin of life (e.g.,Pedersen, 1997). A more applied interest of biology in high pressure is the search for commercially viable alternative waysof food sterilization (e.g., Ondrey and Ka-miya, 2000; Thakur and Nelson, 1998;Mermelstein, 1999).

Forcing a given set of atoms into a smallvolume by applying high pressure in-creases the interatomic interaction. This inturn leads to structural phase transitionsand changes in physical properties. Thedriving forces for high pressure-inducedphase transitions are obviously linked tothe need to optimize the volume occupiedby the atoms for a given pressure. In Sec.10.2, we will discuss the most importantmechanisms, which are the immediatecause of pressure-driven phase transitions.Extracting information on solid or liquidmaterial at high pressure (“high” pressurein this chapter refers to pressures in therange 0.1 –100 GPa) is an experimentalchallenge and was first tackled by PercyWilliam Bridgman in the first half of thiscentury (e.g., Bridgman, 1946). He wasawarded the Nobel Prize for physics in1946 for his ground-breaking achieve-ments in this important field. Since thetime of Bridgman the technology necessaryto perform experiments at high pressure

has experienced extensive development.Section 10.3 will briefly summarize thecurrent state of the art of high-pressure ex-perimentation. Although third-generationsynchrotron sources produced a minor rev-olution in high-pressure crystallography,X-ray diffraction is by no means the onlyway of investigating material when sub-jected to pressure. In Sec. 10.4, we will re-view the most common probes used forphysical and chemical investigations athigh pressure. Section 10.5 treats in detailthe high-pressure phase transitions for a se-lected sample of materials, which are ofspecial interest within materials science.

10.2 Pressure-Driven PhaseTransitions

The physical quantities that define thethermodynamic state of a system can bedifferentiated into intensive quantities (i.e.,temperature T, pressure P, chemical poten-tial µ) and extensive quantities (i.e., en-tropy S, volume V, number of moles n).Conjugate quantities are pairs of an inten-sive and an extensive variable (i.e., T, S orP, V ). Their products have the dimensionsof energy or a volume-normalized energy,which can be used to describe the relativestability of a given system. The thermody-namic stability is determined by the mini-mum of the internal energy U, which is de-fined as a sum of a heat term Q and a workterm W,

U = Q + W (10-1)

where Q is a function of temperature andentropy and W is a function of pressure andvolume. On an atomistic level, Q can beviewed as the vibrational energy of the at-oms oscillating around their equilibriumposition. W can be visualized as the sum ofthe potential energies from the inter atomic

interactions between the ensemble of con-stituting atoms.

The absolute value of the internal energycannot be measured. However, the differ-ence between two states is independent ofthe path and mechanism of the change ofthe system (Hess’s law). This makes thedifference in the internal energy dU a veryimportant quantity in comparing a systemat two different states:

dU = T dS – P dV (10-2)

Entropy S and volume V are two indepen-dent variables that are very difficult to con-trol in an experiment. However, two suc-cessive Legendre transformations of Utransform the internal energy first into theenthalpy H

H = U + PV (10-3)

and then into the Gibbs energy G

G = H – TS = U + PV – TS (10-4)

The total exact differential of G yields

dG = – S dT + VdP (10-5)

which has the desirable property that theindependent variables T and P are easilymodified and controlled in an experiment.This makes G the critical quantity, whichhas to be considered when comparing asystem at different states. While Pelton(2001) focuses on the variation of G as afunction of the chemical composition andtemperature, we will here investigate thechange in G upon increasing pressure.

If a system is subject to changing pres-sure at constant temperature, Eq. (10-5) re-duces to

dG = VdP (10-6)

The change in the Gibbs energy on chang-ing pressure is thus

(∂G/∂P)T = + V (10-7)


10.2 Pressure-Driven Phase Transitions 661

Equation (10-7) states an intuitive trivial-ity, i.e., that in order to minimize the Gibbsenergy upon increasing pressure, a givensystem will reduce its volume. Volume re-duction is thus the ultimate driving force ofstructural change and phase transformationas a reaction on increasing pressure. How-ever, the way a volume is reduced can bedifferent, depending on the initial structureand configurational entropy of the systemand also on the amount of pressure applied.In the following we will focus on the threemost important mechanisms responsiblefor structural changes at high pressure.These are framework flexion, increase incoordination number and pressure-inducedordering.

10.2.1 Framework flexion

A number of technologically importantoxide materials (e.g., zeolites, perovskites)can be viewed as being built of relativelyrigid corner-linked polyhedra forming arather flexible framework. These materialsare known to react on changing tempera-ture and/or pressure mainly by flexion atthe polyhedral joints rather than by polyhe-dral compression, i.e., bond length reduc-tion (e.g., Hazen and Finger, 1978; Veldeand Besson, 1982; Hemley et al., 1994).Silica (SiO2) can be viewed as a typical ex-ample of the behavior of framework struc-tures under pressure. It has a rich phase di-agram in P–T space. The ambient condi-tion phase is a-quartz, which consists ofinterconnected spirals of corner-linkedSiO4 tetrahedra. Silica has two high-pres-sure polymorphs, of which only coesite canbe viewed as a tetrahedral frameworkstructure. Its framework is characterized byfour-fold rings of tetrahedra connected tochains, which in turn are packed via sharedtetrahedral corners to the densest knowntetrahedral framework. At ambient condi-

tions (coesite can be quenched to roompressure), both structures exhibit nearlyideal Si–O–Si angles of between 143° and144°. The phase transition is thus structu-rally characterized by building the tetrahe-dral framework in coesite more denselythan in a-quartz. This can be demonstratedby looking at the oxygen surroundings inboth structures: In a-quartz, each oxygenhas three oxygen neighbors at ~ 2.6 Å,which represent the tetrahedral oxygenneighbors. The closest oxygen atoms fromany other tetrahedron are found at a dis-tance of ~ 3.5 Å. In coesite on the otherhand, there are also three oxygen neighborsat 2.6 Å around each O, indicating the stiffbehavior of the SiO4 tetrahedrals. The nextnearest oxygen neighbors in coesite, how-ever, are observed at a distance of only3.0 – 3.2 Å (room pressure). Both structuretypes react to applied pressure with a verylarge decrease in the Si–O–Si angles (Fig.10-1), while the Si–O bond lengths remainmore or less constant. Densification withinthe stability field of quartz is thus achievedby flexion of the framework through rigidrotations of the SiO4 tetrahedra, resultingin a decrease of the Si–O–Si angle from144° to 125°. If this reaches a limit, denserpacking of the polyhedra is achieved byframework reconstruction. Further densifi-cation, however, eventually involves an in-crease in the coordination number (see Sec.10.2.2). Thus, a pure compression at RT ora rapid (shock) compression of a-quartzleads to an amorphization involving bothframework collapse and an increase in co-ordination number (Hemley et al., 1994).Pressure-induced amorphization seems tobe a common feature of many frameworkstructures, indicating a volume reductionby a collapse of the framework. This de-stroys the long-range order while stilllargely maintaining the short-range orderof the primary building blocks. A similar

effect is also thought to be responsible forthe high-density form of amorphization ofice (Mishima et al., 1984; see also Sec.10.5.3.6).

The most open frameworks known in in-organic chemistry are adopted by the zeolitestructure family. These structures exhibitsome peculiar effects upon compression.Zeolite frameworks are characterized bylarge cages and channels, which can be oc-cupied by extra-framework cations or mole-cules. As in silica, volume reduction on zeo-lite can be most easily achieved by a reduc-tion of the framework channels, which isachieved through rigid body rotations of theframework tetrahedra and thus bending ofthe Si–O–Si angles. A study of the naturalzeolite natrolite for example showed thatcompression leads to a continuous reductionof the unit-cell volume without any structu-ral phase transition up to ~ 7 GPa (Belitskyet al., 1992). Above 7 GPa, the investigatedsamples underwent amorphization, similarto the transition observed for silica. Thisamorphization again indicates the collapse

of the tetrahedral framework into a glassytetrahedral arrangement.

If the channels and holes in a frameworkare large enough, we can even obtain theseemingly paradox result of a negativecompressibility. This happens if the molec-ular size of the pressure medium is smallenough to be squeezed into the channels.Because the pressure medium seems to bepacked more densely within the zeolitestructure than in the fluid, the total volumeof the system (crystal plus pressure me-dium) decreases. The volume of the crystalalone, however, increases with appliedpressure. In a similar way, Hazen (1983)observed compressional anomalies for zeo-lite 4a, which exhibits different phase tran-sitions depending on the pressure mediumused. All of the observed high-pressurephases showed higher compressibilities athigh pressure.

A more frequent but still unusual phe-nomenon observed upon compressingframework structures is a negative linear orareal compressibility. Materials with these


Figure 10-1. Si–O–Si angles (solid symbols) and average Si–O bond lengths (open symbols) vs. pressure forquartz (squares) and coesite (diamonds). Note the (within experimental errors) constant ·Si–OÒ distances with in-creasing pressure, while the Si–O–Si angles show a significant negative correlation. This demonstrates that frame-work structures (such as tetrahedral SiO2 polymorphs) adapt their volume through framework flexion. Data fromLevien et al. (1980) and Levien and Prewitt (1981).


properties will expand in one or two di-mensions upon hydrostatic compression.As shown by Baughman et al. (1998), thisalso implies a negative Poisson’s ratio, i.e.,lateral contraction upon uniaxial compres-sion. These authors also demonstrate thatproperties such as a negative Poisson’s ra-tio are in most cases linked to geometricconstraints in hinged framework structures.Because such behavior can result in in-creasing surface area upon increasing hy-drostatic pressure, it is of potential interestto material scientists. In principle it is con-ceivable to manufacture these compoundsinto porous composite material with zeroor negative volume compressibilities.

Another example of a structure familywhose phase transitions are characterizedby framework flexion are the perovskites.This material, with the general formulaABX3, can be viewed as a stuffed deriva-tive of WO3-type structures. WO3 (BX3) isbuilt up of a three-dimensional network ofcorner-linked WO6 octahedra, the W (B)cations occupying the corners of the cube-shaped unit cell. In perovskite the center ofthis cube is occupied by the A cations. Inthe ideal structure, the B–X–B angle at theconnecting octahedral corners is 180°. Asshown by Glazer (1972), this octahedralframework is susceptible to a number ofdistortions which are characterized by rigidoctahedral rotations, thus pure frameworkflexions. These distortions can even be ob-served at ambient conditions depending onthe nature of the A and B cations. At ambi-ent conditions, the geometry is controlledby the relative size and thus the bonding re-quirements of the cations involved. In asimilar way the effect of compression de-pends on the relative size at ambient condi-tions and the relative compressibilities ofthe A and B cations. In most cases the Acation is the more compressible unit, thusleading to increasing polyhedral tilting

(i.e., framework flexion) with increasingpressure. An example of this is the increas-ing orthorhombic distortion of MgSiO3

with increasing pressure (Fiquet et al.,2000).

10.2.2 Increase in coordination number

Many crystal structures are best de-scribed by a closest packing of anions withsome of the interstitial sites (two tetrahedraand one octahedron per anion) occupied bya cation. In this simple, but in many casesvery successful picture, the large and softanions are in contact with each other andthe small, more rigid cations are isolatedfrom each other and in contact only withtheir surrounding anions. The number ofanions surrounding any given cation is de-termined by the ratio of the “sizes” of thecations and anions (for the problem of de-fining the “size” of an atom see e.g., Rossand Price (1997)). In such a close-packedarray of anions, rigid polyhedral rotation isusually not able to accommodate a volumereduction imposed by increasing pressure.The only way to optimize the volume uponincreasing pressure is through more effi-cient packing of the anions involved, thusreducing the anion–anion distances withinthe structure. This reduces the ‘size’ of theanions without affecting the respective sizeof the cations, which leads to a higher cat-ion/anion size ratio. As a consequence, thecoordination number of the cations in-creases. Such a coordination increase isgenerally accompanied by a lengthening ofthe cation–anion distances, which at firstglance may appear a surprising effect for ahigh-pressure phase transition. However,the lengthening of the first coordinationsphere is compensated by a shortening ofthe second shell. It is therefore worthwhilelooking not only at the nearest neighbors ofa cation. When looking at both the first and

second coordination shells of a cation, acoordination increase can be visualized asmaking the bonding environment of a cat-ion less distorted. Keeping this in mind, theincrease in coordination number can bequantitatively rationalized by the distortiontheorem (Brown, 1992). The compressionof a close-packed array of anions leads firstto a reduction in the anion–anion distancewithout rearranging the actual packing.This reduces all cation–anion distances,which leads to a strengthening of the indi-vidual bonds. In the framework of Brown’sbond valence approach, a general strength-ening of cation–anion bonds around agiven atom induces an ‘overbonding’ ofthe atoms, which in turn destabilizes thestructure. Because of the exponential rela-tionship between bond strength and bondlength (Fig. 10-2), a set of equal bondlengths will have a lower bond valence sumthan the same number of bonds with thesame average value, but different individ-ual bond lengths (distorted arrangement).An increase in coordination number thusreduces the overbonding of the atoms inthe crystal by making the bonding environ-ment around the atoms less distorted. Avery instructive example of this effect isthe aforementioned phase transition in sil-

ica (SiO2) from coesite (4-coordinated Si,2-coordinated O) to stishovite (6-coordi-nated Si, 3-coordinated O). While the co-esite structure is described by a very densepacking of corner-linked SiO4 tetrahedra,stishovite adopts the rutile structure, char-acterized by chains of edge-sharing SiO6

octahedra. Other prominent examples ofincreasing coordination numbers at in-creasing pressure are the well-known phasetransitions from the NaCl (6-fold) typestructure to the CsCl (8-fold) structure inalkali chlorides or the high-pressure transi-tions from zincblende (4-fold) to NaCl orb-Sn (6-fold) in many AX semiconductors(see Sec. 10.5.1).

10.2.3 Pressure-induced ordering

The potential effect of high pressure onorder–disorder phase transitions has onlyrecently been fully realized. The interestedreader is referred to the excellent andthorough review by Hazen and Navrotsky(1996). In this section we only give a briefsummary of the main features of pressure-induced order–disorder phenomena.

Atoms on a given crystallographic sitecan order with respect to their chemicalspecies, exact position, magnetic moment


Figure 10-2. The exponential rela-tionship between bond valence andbond length leads to more regulargeometries at high pressure; Highpressure tends to destabilize struc-tures through overbonding. For agiven average bond length, the aver-age bond valence is higher (= moreoverbonding) the more the individ-ual bond lengths are different fromeach other (= irregular geometry).


or electronic state. In the following, disor-der with respect to chemical species is re-ferred to as ‘substitutional disorder’,whereas disorder on two neighboring sitesis called ‘structural disorder’. Changes inany of these order parameters across aphase transition are known to have an ef-fect on the molar volume of the material(e.g., Owen and Liu, 1947). Depending onwhether the molar volume is smaller for anordered or disordered state, pressure canthus – in principle – promote ordering ordisordering effects. Nevertheless, such ef-fects are very often hampered by reduceddiffusion rates at high pressure. It is there-fore fair to say that at moderate tempera-tures (1500 K), high pressure does notnecessarily induce order–disorder transi-tions, but certainly supports such reactions.The fast kinetics above ~1500 K, however,allow rapid equilibration with respect to or-dering and even in many cases inhibitquenching of an ordering pattern stable athigh pressure and high temperature (Hazenand Navrotsky, 1996).

Based on present data, the volumechange DVdis = Vdisordered – Vordered of struc-tural or substitutional disorder tends to bepositive, suggesting that high pressure fa-

vors an ordered arrangement (Hazen andNavrotsky, 1996). This can be understoodby the fact that an ordered arrangement oftwo atomic species of different ‘sizes’leads to an alternation of ‘small’ and ‘large’ layers or rods (Fig. 10-3a). A disor-dered arrangement, on the other hand,forces each of the disordered sites to havethe apparent size of the largest atom shar-ing this site (Fig. 10-3b). An ordering de-pendence of the molar volume (and there-fore a pressure dependence of the ordering)can also be observed in flexible frameworkstructures such as feldspars. At first glance,the molar volume of such structures shouldnot be critically correlated with the size ofthe cations, because in these structuresmost of the volume change induced by or-dering or pressure can be accommodatedby the intra-polyhedral angles (Sec.10.2.1). The observed volume changes,however, can be understood on the basis ofthe variation of intra-polyhedral angles, de-pending on the cation species occupyingthe respective polyhedra (e.g., Geisinger etal., 1985). Simple geometric considera-tions show that varying distributions of agiven set of angles in space leads to differ-ent enclosed volumes (Fig. 10-4).

Figure 10-3. Schematic drawingto illustrate a possible mecha-nism of pressure-induced order-ing. Two sets of balls of differentsizes occupy (a) a smaller vol-ume in an ordered arrangementthan (b) the same number ofballs in a disordered distribution.

The largest volume changes are ob-served for substitutional ordering pro-cesses. A4+B2

2+O4 spinels, for example,show a difference in molar volume of up to3.5% between the normal (fully ordered)and the inverse (disordered on the octahe-dral site) modifications (Hazen and Yang,1999). The pressure behavior of the spinelfamily is especially interesting, becausethey show both pressure-induced disorder-ing as well as pressure-induced orderingfor one and the same structure type de-pending on the chemical species involved(Wittlinger et al., 1998; Hazen and Yang,1999).

The rather scarce data on pressure de-pendence of charge distribution seems toindicate that charge ordering tends to besuppressed by high pressure. NaV2O5 forexample shows a charge-ordering transi-tion around 35 K at ambient pressure. Thistransition seems to be shifted to lower tem-peratures at higher pressures and disap-pears completely around 1 GPa (Ohwada etal., 1999). This suggests that in the case ofNaV2O5, high pressure induces a charge-disorder phase transition. In a similar way,the application of pressure to Sm4Bi3 shifts

the charge-ordering transition to lowerpressures and eventually even induces aniso-structural phase transition where themixed valence compound Sm3

2+Sm3+Bi33–

changes to a purely 3-valent materialSm4

3+Bi34– (Ochiai et al., 1985).

Another technically interesting phenom-enon that is connected to the pressure de-pendence of ordering and correlation ofcharge carriers is the well-documentedpressure dependence of Tc in certain ce-ramic superconductors (e.g., Acha et al.,1997; Han et al., 1997). Although neitherthe superconductivity nor its relationshipto high pressure is fully understood in ox-ide materials, it is justified to assume thatthe observed strong shift in Tc with increas-ing pressure is connected to a subtle inter-play between pressure-induced structuraldistortions, orbital overlaps and charge-carrier distribution.

10.3 Generating High Pressure

The technology of pressure cells com-patible with in situ experiments for mate-rial characterization has experienced tre-


Figure 10-4. Schematic two-dimensional sketch to demon-strate how the distribution of agiven set of rigid triangles(polyhedra in 3-d) affects thearea (volume in 3-d) enclosed.The area between the trianglesin (a) is 2.450, while the trian-gles in (b) enclose an area of2.414 (after Hazen and Navrot-sky, 1996).

10.3 Generating High Pressure 667

mendous development since the ground-breaking work of Bridgman. Consequently,there is a very comprehensive and vast lit-erature on this subject to which the morecommitted reader is referred (Miletich etal., 2000; Holzapfel, 1997; Eremets, 1996;Ahrens, 1987). Only a brief overview willbe given in this section.

There is a huge variety of different tech-niques for generating pressures. They caninitially be divided into static methods anddynamic techniques.

10.3.1 Dynamic pressure generation

The highest pressures (102–103 GPa)can be obtained using dynamic shock-wavegeneration. This is achieved by means ofexplosives or by a projectile that is acceler-ated toward the target with a gas gun (Ah-rens, 1980, 1987). The shock-wave tech-nique was originally developed in the mid-1950s at Los Alamos, USA, in the courseof the development of atomic bombs (Walsh and Christian, 1955). In its simplestcase, the impact of a projectile on the targetproduces a uniaxial shock wave. The shockwave passes through the sample at shockvelocity Us . The sample itself is acceler-ated to the sample velocity Up, and Us andUp, together with the temperature, are thequantities measured during a shock-waveexperiment. The velocities are usually de-termined by measuring entrance and exittimes of the shock wave. For samples of a few millimeters in length, the time to be measured is in the range of 10–1 to ~ <101 µs. Ignoring the yield strength ofthe solid (which is justified at the highshock pressures encountered during ashock-wave experiment), the material be-haves as a fluid. In such an experiment, thevolume decreases from V0 to VH, the tem-perature increases from T0 to TH and thepressure from P0 to PH, while the internal

energy rises from E0 to EH. The Rankine–Hugoniot relations combine these quanti-ties to (r0 = density at ambient conditions):

VH = V0 (US – UP)/US (10-8)

PH = r0 US UP (10-9)

EH – E0 = (V0 – VH) PH/2 (10-10)

The measurements of US and UP give thequantities on the left of Eqs. (10-8) to (10-10) for one experiment. Various experi-ments at different strengths of explosion ordifferent velocities of the projectile pro-duce different points, forming the Hugon-iot curve describing the Hugoniot equa-tions of state VH (PH). In order to reduce theHugoniot equations of state into an isother-mal equation of state, careful thermody-namic corrections have to be applied (Poirier, 1991). A material passed by ashock wave usually displays variousstages. Up to a pressure of the Hugoniotelastic limit (HEL) (0.2 to 20 GPa), thesample behaves elastically, correspondingto the propagation of the longitudinalshock wave. Above the HEL, plastic defor-mation of the material occurs, giving riseto the fluid-like behavior that creates theHugoniot curves. If a material undergoes aphase transition, this is readily observed aschanges of slope in the Hugoniot curves,separated by a mixed-phase regime (Fig.10-5).

In this sense, in a shock-wave experi-ment the material can only be investigatedby the difference in UP and US, which bothdepend on the volume reduction of thesample. The accuracy of these measure-ments becomes critical once the volume re-ductions are small at very high pressures.Third-generation synchrotron radiationsources or even free-electron lasers of thefuture may offer improved possibilities.The increase in X-ray flux of these X-raysources may allow for stroboscopic X-ray

diffraction in the microsecond regime. Thiswould enable structural information to beextracted of a material under extreme dynamic pressure. Today, shock-wave ex-periments are highly complementary tostatic high-pressure experiments. Undoubt-edly, one of their most crucial roles is thecreation of an equation of state up to ex-treme pressures without any additionalpressure standard. This provides invaluableanchor points for all pressure scales cur-rently used in static high-pressure experi-ments (Mao et al., 1978; Jamieson et al.,1980).

10.3.2 Static pressure devices

Devices to generate static pressures thatare sustainable for an a priori indefinitetime can be divided into ‘large-volumepresses’ and ‘diamond anvil cells’. The pri-mary difference between these two fami-lies of devices lies in the volume of mate-rial subjected to high pressure and conse-quently the maximum pressure attainable.For large-volume devices the compressedvolume lies between 1 mm3 and 1 cm3.This allows maximum pressures of around

20 GPa to be obtained. Diamond anvilcells, in contrast, enclose a volume of <10–3 mm3, i.e., only a few pico-liters.These devices are capable of maximalpressures up to 200 to 500 GPa.

103.2.1 Large-volume presses

Historically, the first large-volume cellswere hydraulic presses as built by Bridg-man. He was able to compress fluids up to10 GPa. In modern high-pressure research,hydraulic devices are limited to maximalpressures between 1 and 2 GPa and willtherefore not be discussed any further inthis chapter. Presses operating with solidpressure media up to 50 GPa are the mostimportant tools in modern materials sci-ence for the synthesis of materials at simul-taneously high pressure and high tempera-ture. Increasingly, large-volume devices arealso used for in situ studies in combinationwith synchrotron and neutron radiation.

Piston-Cylinder systems are the simplestdesign for compressing material. They relyon a cylinder acting as the sample chamber.The sample is compressed by a piston fit-ted within the cylinder. Closed-end cylin-


Figure 10-5. Hugoniot curve of ahypothetical material undergoing aphase transformation at high pres-sure. The low-pressure phase ap-proaches its hydrostatic behavior af-ter passing the Hugoniot elastic limit(HEL). The existence of a shock-in-duced high-pressure phase is seenby the onset of the mixed-phase re-gime. The hydrostat of the high-pressure phase can be recognized inthe high-pressure phase regime (af-ter Ahrens, 1980).


ders are closed on one side and the sampleis compressed with one piston from theother side. Open-end cylinders rely on twoopposite pistons. The maximal pressure of5 GPa attained by piston-cylinder devicesis generally limited by the yield strength ofthe cylinder. Various techniques exist to in-crease this yield strength. In a techniquecalled frettaging the cylinder is preparedsuch that in its default state, the outer partretains a residual tensile stress and the in-ner part a compressive stress. Adding thework stress leads to a partial cancellationof these stresses, thus allowing for the ap-plication of higher loads to the sample.Pre-loading the cylinder is another way toincrease the yield strength of the cylinderto higher values. Pre-load can be achievedby using a soft but incompressible material (lead) as the inner cylinder material and by initially overstraining it into its flow re-gime. Another method relies on the appli-cation of an external load on the cylindersimultaneously with the application of theload on the piston. Alternatively, cylinderscan be reinforced by winding a strong wireunder tension around the cylinder. The

most straightforward way to reinforce acylinder is by increasing the ratio of its di-ameter to length. If this is pursued conse-quently, we arrive at two conical pistonscompressing a sample contained in a girdle(Fig. 10-6a). A modification of the girdledesign is obtained by optimizing the coni-cal shape of the anvils. A cycloid shape ofthe pistons, as shown in Fig. 10-6b, en-sures the optimal compromise ensuringsufficient cylinder support at low pressurewhile maintaining a reasonable stroke inthe high-pressure regime. This ‘beltdesign’ is by far the most widely used high-pressure apparatus for materials synthesisand is frequently applied in industry for thesynthesis of super-hard material. The max-imum pressure achieved by belt devices isaround 10 GPa. Such devices have alsobeen optimized with respect to sample vol-ume; a flat belt apparatus constructed byFukunaga et al. (1987) was capable ofcompressing a sample of 125 ml up to pres-sures of 5 – 7 GPa.

The disadvantage of piston-cylinder as-semblies is the intrinsic opacity of both pis-ton and cylinder, thus inhibiting most in

Figure 10-6. Schematic drawing of (a) a girdle-type press and (b) a belt apparatus. The shaded area corre-sponds to gasket material. S shows the sample position. Note how consequent cutting of stress amplifying cor-ners on the girdle and decrease of length-to-width ratio on the piston lead from a girdle press to a belt design.

situ observations. This problem is allevi-ated by opposed anvil systems which oper-ate with a gasketing system instead of acylinder. The very first example of thistype was built by Bridgman (1952). A log-ical extension of the Bridgman design areprofiled Bridgman anvils as first proposedby Ivanov and coworkers in the 1960s(e.g., Ivanov et al., 1991) leading to tor-oidal anvils (Fig. 10-7). This principle wasoptimized most consequently in theParis–Edinburgh cell design (Besson et al.,1992). The advantage of X-ray and neutrontransparent gasketing has been exploited tothe maximum in this cell. This is the reasonwhy this cell design has advanced to thestate-of-the-art model for in situ high-pres-sure neutron diffraction and is also em-ployed for X-ray experiments. A develop-ment of the Los Alamos neutron grouppushes the limits of P and T attainable withtoroidal anvils up to 50 GPa and 3000 K,thus enabling in situ neutron studies up tothese conditions.

A quite different approach to compress-ing large volumes up to pressures of30 GPa was first developed by researchersin Japan using multi-anvil designs (Aki-moto, 1987). A very successful product isthe DIA-type cubic anvil press. In this de-vice, six tungsten carbide (WC) or sintereddiamond anvils are arranged parallel to thefaces of a cube that encloses the cell as-sembly. A ram exerts an axial force, which

is also translated into an equatorial com-pression through wedges attached to thefour equatorial anvils. Although initiallydesigned for synthesis experiments, vari-ous models of this type have been used in recent years at synchrotron radiationsources for in situ X-ray diffraction studiesin Japan (Photon factory and Spring-8),Brookhaven (NSLS) and Hamburg (DESY). This is possible because the smallgaps between the individual anvils allowfor entry and exit of X-rays, and the cell as-sembly containing the sample is made of X-ray transparent material (BN, epoxy). Withtapered anvils made of sintered diamond,maximal pressures of 20 – 25 GPa at tem-peratures around 1000 °C can be attained.

A rather different geometry in multi-an-vil technology is the split-sphere design(Liebermann and Wang, 1992) pioneered atOkayama University by Ito. It consists of auniaxial ram applying force on a splitsphere (first stage) that contains a cube-shaped cavity with the body diagonal alongthe axis of the ram. The cavity is occupiedby a cube built of eight WC cubes withtruncated corners. The truncated corners inturn form an octahedral cavity that hoststhe cell assembly. The cell assembly is usu-ally a MgO octahedron with a Pt capsulecontaining the sample in its center. A vari-ation of this design is the split-cylinder de-sign in which the sphere of the first stage isreplaced by a cylinder. Multi-anvil presses


Figure 10-7. The principle ofthe toroidal anvil. The cut iscircular symmetric around thevertical axis of the plot. Thesample (S) is surrounded by awasher within a toroidal beltcontaining gasket material. Thisgeometry prevents the samplefrom being extruded, thus creat-ing a hydrostatic pressure de-spite the uniaxial force applied.


based on the split-sphere approach are alsoused as in situ devices at the ‘SynchrotronRadiation Source’ in Daresbury (UK) andthe ‘Advanced Photon Source’ at ArgonneNational Laboratory (USA).

However, the determination of pressureis not straightforward. In principle, pres-sure can be calculated by dividing the forceby the area on which it is acting. In practicethis is not applicable, mainly because ofthe unpredictable friction losses and illde-fined compressibilities of the cell assem-bly. Therefore, pressure has to be deter-mined through a calibration procedure.When in situ diffraction techniques are im-possible, this can be done by determiningphase transitions revealed through changesin resistivity of metals and semiconductors(e.g., Bi, Ba, ZnS, GaAs, GaP). The re-spective high-pressure phase transitionsare determined in hydraulic pressure de-vices whose pressure can be directly meas-ured by a pressure gauge. If in situ observa-tion is possible, the equations of state ofmaterials such as NaCl, Cu, Mo, Ag andPd, which have been determined up to veryhigh pressures through shock-wave experi-ments, can be applied to calibrate the press.Because the reproducibility for a givenpress and cell assembly is very high, pres-sure of subsequent experiments can be de-termined from the force applied on the cellassembly.

The strength of large-volume deviceslies in their potential for material synthesisat simultaneously high temperature andhigh pressure. The relatively large size ofthe compressed volume (~ 0.1–1 cm3) al-lows inclusion of a heater (usually cylindri-cal graphite or LaCrO3 resistance heaters)as well as thermocouples, which enablepressure and temperature to be combinedin a very controlled way. Due to the largevolume, these devices are limited to maxi-mal pressures of 50 GPa at the very best.

10.3.2.2 Diamond anvil cells

Experiments at extreme pressures, whichare of interest not only to geophysicists andplanetologists, but also to physicists andchemists studying, for example, solidifica-tion and metallization of ‘gases’, can beachieved using diamond anvil cells (DAC).Diamond anvil cells are in principle verysmall opposed-anvil devices of Bridgmantype. The anvils are made of diamond sin-gle crystals, shaped in the brilliant cut withthe bottom tip truncated to form the anvilsurface (culet). This simple design allowsthe special properties of diamonds to beused in two ways. First, the extreme hard-ness of diamonds allows very high pressureto be generated. Pressures attained dependof course on the size of the culet. Maximalpressures of 500 GPa have been reported(Xu et al., 1986) and pressures between100 and 200 GPa can be reliably repro-duced. The second advantage of using single crystalline diamonds as pressure anvils is the high transparency of diamondfor almost the entire electromagnetic spec-trum. This allows us not only to easily observe samples under high pressure, butalso to probe them with spectroscopicmethods as well as X-ray diffraction(Sec. 10.4). The strength of DACs is thustheir huge range of pressure combined withthe ease of performing in situ experiments,and this all with a device of the size of afist, which is also easy to operate by non-specialists.

The very first DAC was constructed us-ing a big (8 ct) gem-quality diamond (takenfrom smugglers by the US government anddonated to the US National Bureau of Stan-dards) in which a hole was drilled (Jamie-son, 1957; Lawson and Tang, 1950). Pres-sure on the sample in the hole was appliedvia a piston (piano string) pressed on thesample. The limited pressure range obtain-

able with this approach was painfully real-ized when the 8 ct diamond was crushedduring an experiment. As a consequence,the opposed-anvil geometry was developed(Weir et al., 1959; Jamieson et al., 1959),first by simply squeezing a powdered sam-ple between the culets of two opposed an-vils and later by introducing the gasketingtechnique (Van Valkenburg, 1964). Its prin-ciple has remained unchanged since its in-vention and is as simple as it is efficient(Fig. 10-8). A hole about 100 to 200 mm indiameter is drilled in a metal foil (Fe, W,Re). This hole serves as sample chamberand is filled with the sample (powder orsingle crystal) and a pressure medium (al-cohol, liquid gas). The pressure medium iscompressed through an axial force exertedby the diamonds on the gasket. The gasketseals the sample chamber and at the sametime transforms part of the axial pressureinto an equatorial pressure through theplastic deformation of the gasket material.In this way the pressure medium is com-pressed isotropically and therefore trans-mits a hydrostatic pressure on the sample.This design was popularized for materials

research by Bassett (Merrill and Bassett,1974) and Mao (Mao and Bell, 1975). Fur-ther details on the technology of DACs andalso on its various modifications and devel-opments are given in excellent reviews byHazen and Finger (1982) and Miletich etal. (2000).

As for the large volume devices, the ac-curate determination of pressure in a dia-mond anvil cell is a difficult issue. Themost accurate values are obtained by add-ing an internal standard to the samplewhose equation of state is known with suf-ficient precision to relate its diffractionpattern to a pressure value (e.g., Angel etal., 1997). A very convenient and popularalternative, albeit not quite as accurate, isthe exploitation of the pressure shift of flu-orescence lines. The most frequently usedfluorescence is the R1 line of ruby (Mao etal., 1978). This method again benefits fromthe transparency of the diamond high-pres-sure windows. A ruby, which is packed to-gether with the sample into the gasket hole,is illuminated with a green or blue laserthat induces a red fluorescence line. Thewavelength of this line depends on pres-


Figure 10-8. Sketch illustrating the prin-ciple of the diamond anvil cell. The axialforce applied to the diamonds is partlytranslated into a plastic deformation ofthe gasket, which results in a circularsymmetric equatorial force. This ensuresquasi-hydrostatic conditions at the sample(S), which is embedded in a pressure me-dium.

10.4 Probing Phase Transformations in Materials at High Pressure 673

sure (dl/dP = 0.37 nm GPa–1) and can thusbe used to determine the pressure withinthe sample chamber. More recent develop-ments of this approach use different fluo-rescence lines with various pressure andtemperature dependence in order to simul-taneously determine pressure and tempera-ture in the sample chamber (e.g., Datchi etal., 1997).

Because of the very small volumes of thesample chamber (~ 0.001 mm3), combiningpressure and temperature in a diamond an-vil cell is a difficult task. In principle thereare two different approaches, namely exter-nal and internal heating. With externalheating, the whole sample chamber, includ-ing diamonds, is enclosed in a resistanceheater (e.g., Hazen and Finger, 1982; Bas-sett et al., 1993) that heats the entire assem-bly consisting of gasket, diamond anvilsand sample. Temperature is measured viathermocouples attached to the outer dia-mond facets, assuming that the high ther-mal conductivity of the diamonds allowsfor only a very small temperature gradientbetween sample and outer diamond facets.This technique has consequently been opti-mized by using the gasket material itself asa resistance heater, thus minimizing theheated volume and thermal gradients (Du-brovinsky et al., 1997). An alternative ap-proach is the use of an infrared laser beam,which is focused through the (IR-lasertransparent) diamonds onto the sample,where it is absorbed and thus transformedinto heat. This technique was again pio-neered by Bassett (e.g., Bassett and Ming,1972) and then further developed and opti-mized by Boehler (Boehler and Chopelas,1991) and Fiquet and Andrault (Fiquet etal., 1994). Measuring temperature with thistechnique is even more difficult because itexplicitly assumes that only the sample isheated up. The only way to obtain a quanti-tative estimate of the sample temperature

is by measuring the black-body radiation of the glowing sample and fitting it toPlanck‘s spectral function (e.g., Bassettand Weathers, 1987). When doing HP-HTexperiments with laser heating we shouldbear in mind the possibility of thermalpressure, i.e., the increase of pressure inthe heated area, while the pressure deter-mined by a ruby chip outside the hot-spotremains constant (Andrault et al., 1996).

10.4 Probing Phase Transformations in Materials at High Pressure

A very important aspect when doing ex-periments on high-pressure phase transfor-mations is the possibility of investigatingthe phase transformation in situ at condi-tions of high pressure and possibly simulta-neously high temperature. As mentionedabove, this is much easier to do in static ex-periments than with dynamic shock-wavetechniques. This is mainly due to the veryshort time that is available in a shock-waveexperiment. We will therefore focus onstatic experiments in the following. Amongthe static experiments, the transparency ofthe diamond pressure windows in a DACallows for much more versatile experimen-tal techniques compared with large-volumepresses. For large-volume experiments, di-rect observation of the sample is limited toeither transport properties (e.g., electric re-sistance) or, if using some sort of electro-magnetic radiation as a probe, severe com-promises in signal-to-background and ac-cessible space have to be accepted. Never-theless, the range of in situ techniques ap-plicable to both DACs and large-volumedevices has grown considerably and con-tinues to expand. There is an extensive spe-cialized literature on each of the varioustechniques. We will give here a brief over-

view on and introduction to the availabletechniques, with references to the more de-tailed literature. Many techniques can beused to measure different physical proper-ties and vice versa. In the following we willdifferentiate by the technique rather thanby the measured property.

10.4.1 Volumetric techniques

The first in situ observations in large-volume cells were pioneered by Bridgman(e.g., Bridgman, 1940) on piston-cylinderdevices. He made volumetric measure-ments by simply measuring the stroke ofthe piston as a function of force (pressure)applied to the sample. This technique obvi-ously assumes that deformation of the pis-ton and cylinder, as well as leakage, can beneglected. The accuracy in V/V0 obtainedthrough this method can be fine-tuned toabout 1 in 1000 (Anderson and Swenson,1984). In general, the accuracy is limited tolower values by deformation of the pistonand the cylinder. Compressibilities aretherefore most often and much more accu-rately determined using diffraction tech-niques (see Sec. 10.4.4). Prior to the avail-ability of in situ diffraction techniques,however, volumetric measurements werethe only way of measuring compressibil-ities as a function of pressure, and they pro-vided extremely valuable data.

10.4.2 Spectroscopic techniques

There are a variety of spectroscopictechniques making use of various segmentsof the electromagnetic spectrum. Althoughthe transparency of diamonds makes DACsa very obvious tool for spectroscopic tech-niques, they are by no means limited tothem, but – due to the advent of brilliantsynchrotron radiation sources – are in-creasingly also applied in combinationwith large-volume presses.

10.4.2.1 Microscopy

A rather cheap, but in many cases highlyefficient ‘spectroscopic’ device is the hu-man eye. Visual observation of a sampleunder high pressure can be a very usefuland sensitive tool for observing a phasetransformation and pinpointing it in P–Tspace. An example of this has been de-scribed by Arlt et al. (2000), where thepressure dependence of a high-temperaturephase transition in Mn-pyroxenes turnedout to be difficult to determine using dif-fraction techniques, but could be opticallyobserved through the discontinuous changein birefringence.

10.4.2.2 Raman and infrared spectroscopy

The most popular spectroscopic tech-niques applied in high-pressure studies areRaman and infrared (IR) spectroscopies.They both probe the vibrational propertiesof the material under investigation. Be-cause lattice vibrations strongly depend onthe topology of the chemical bonds of asubstance, these spectroscopic techniquesare very sensitive to phase transformations.Vibrational modes involving a dipolechange can be excited by absorbing an in-frared photon giving rise to an absorptionband in the infrared. The energies of theselattice modes are thus of the same magni-tude as the energy of the IR photons. If thephoton energy is much higher, the interac-tion of the photon with the lattice can in-duce a vibrational mode, where the energyof the lattice mode is transferred from thephoton to the lattice. This decreases the en-ergy of the photon and therefore induces awavelength shift on the scattered photons(Raman spectroscopy). Because energy canbe transferred in both ways (to and fromthe lattice), the respective wavelength shift


10.4 Probing Phase Transformations in Materials at High Pressure 675

can be positive or negative. By measuringthe wavelength shifts of transmitted laserlight of a wavelength around 500 nm, acharacteristic fingerprint of the vibrationalproperties of a given substance is obtained.Again, this fingerprint critically dependson bond strengths and the structural config-uration and is therefore ideal for detectingphase transformations. While in generalRaman shifts are measured for the opticalbranches of the vibrational spectrum, verycarefully designed experiments are able todetermine the minute frequency shiftscaused by the acoustic branches, too. SuchBrillouin scattering experiments are veryinteresting, because they allow the fullelastic tensor to be measured as a functionof pressure (e.g., Sinogeikin and Bass,1999). The elastic tensor is a material prop-erty that is useful in its own right. Its strongdependence on structural parametersmakes it a very sensitive probe for detect-ing and investigating phase transforma-tions (Carpenter and Salje, 1998). A veryinstructive and comprehensive overview ofhigh-pressure specific problems and appli-cations of IR and Raman spectroscopy isgiven by Gillet et al. (1998).

10.4.2.3 Mössbauer spectroscopy

Mössbauer spectroscopy has become in-creasingly popular in materials science toprobe site-dependent distributions ofcharge and magnetic moment. Its applica-tion to high pressure is limited to DAC ex-periments (Pasternak and Taylor, 1996;McCammon, 2000). In a classical Möss-bauer experiment, the radiation emitted by the g-source lifts the sample nuclei intoan excited state through an absorptionevent. During re-emission, a fraction of theg-quanta is emitted without recoil on thelattice (recoil-free fraction) and can thus bereabsorbed by a nucleus in the same struc-

tural environment (resonant absorption)while lifting it into an excited state. Theenergy of the excited state of a given nu-cleus is a function of its structural environ-ment, and therefore varies between indi-vidual substances. This energy shift (iso-mershift) is another characteristic finger-print for a given structural state as well asfor the electronic charge of the nucleus.The isomer shift can be measured by alter-ing the relative energy of the g-quantathrough the Doppler effect caused by rela-tive movements of sample and source. Theg-quanta of the most popular Mössbauernuclei (57Fe26) are in the range of 14.4 keVand are thus strongly absorbed even by di-amonds. It is for this reason that the firsthigh-pressure Mössbauer experimentswere performed using more exoticMössbauer nuclei such as 153Eu, 129I and170Yb, which have higher g-energies.These experiments helped answer someinteresting questions about the physics ofmagnetic materials (e.g., Pasternak et al.,1986; Abd-Elmeguid et al., 1980). The de-velopment of especially miniaturized dia-mond anvil cells (e.g., Pasternak and Tay-lor, 1990, 1996) together with advances indetector technology also allowed success-ful Mössbauer experiments to be per-formed on 57Fe nuclei (e.g., McCammon etal., 1998). While traditional Mössbauer ex-periments make use of the energy structureof the emitted g-rays, modern synchrotronsources with their time-pulsed radiationalso allow us to exploit the time structure ofa Mössbauer event. As mentioned above, a Mössbauer event involves the absorptionof a g-quantum by lifting the Mössbauernucleus in an excited state. This state lastsfor a time span in the range of 100 ns be-fore it decays and emits the scattered radia-tion. Because individual synchrotron X-raybursts can be gated into a time interval inthe range of 100 ps with 1 ms–1 repetition

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