Engineering Materials and Processes

Series Editor Professsor Brain Derby, Professor of Material Science Manchester Science Centre, Grosvenor Street, Manchester, Ml 7HS, UK

Bruno Predel· Michael Hoch . Monte Pool

Phase Diagrams and Heterogeneous Equilibria A Practical Introduction

With Technical Cooperation of Felicitas Predel

With 270 Figures

~ Springer

Prof. em. Dr. Dr. h. c. Bruno Predel Haugstr.26 D-70563 Stuttgart Germany predel@mf.mpg.de

Prof. em. Dr. Monte Pool University of Cincinnati Dept. of Chemical and Materials Engineering PO Box 21 00 l2 Cincinnati, OH 45221-00l2 USA monte.pool@uc.edu

With Technical Cooperation of Felicitas Predel Max-Planck-Institut fUr Metallforschung, Stuttgart

Prof. em. Dr. Michael Hoch 5300 Hamilton Av., Apt. 1706 Cincinnati, OH 45224-3165 USA niklah@fuse.net

and University of Cincinnati Dept. of Chemical and Materials Engineering P.O. Box 210012 Cincinnati, OH 45221-0012 USA

Original German Edition published by SteinkopffVerlag Darmstadt, 1992

Library of Congress Cataloging-in-Publication Data Predel, Bruno. Phase diagrams and heterogeneous equilibria: a practical introduction I Bruno Predel, Michael Hoch, Monte Pool. p. cm. - (Engineering materials and processes) Includes bibliographical references and index. ISBN 978-3-642-05727-4 ISBN 978-3-662-09276-7 (eBook) DOI 10.1007/978-3-662-09276-7

1. Phase diagrams. 2. Phase rule and equilibrium. 3. Thermodynamics. 4. Chemistry, Metallurgic. I. Hoch, M.J.R. (Michael J. R.), 1936 - II. Pool, Monte. III. Title. IV. Series. QD503.P72 2003 541' .363-dc22

ISBN 978-3-642-05727-4

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Preface

Since J.W. Gibbs in 1878 succeeded comprehensively in establishing the basic principles for an understanding of equilibria in heterogeneous systems, numer-ous books concerning constitution diagrams have been written, some of them providing a formal treatment of phase equilibria down to the small detail. The purpose of the present book is to provide an introduction to the practical ap-plications of phase diagrams. In the first instance it is intended for students of chemistry, metallurgy, mineralogy and materials science, but also for engineers and students of science and engineering disciplines concerned with materials. To facilitate the start of an involvement with heterogeneous equilibria, reactions and dynamic equilibria will be treated first, since these are familiar to chemists and metallurgists.

Of course, a description of phase equilibria is not possible without a mini-mum of formalism. The formalistic description, however, will be made lighter by clear explanations of experimental methods used to determine the constitu-tion of a system, by application examples, as well as by discussing realistic cas-es from chemistry, metallurgy, materials science and mineralogy. By this, the ne-cessity of the knowledge of phase diagrams can be shown. On the other hand a practical exercise is possible.

The physical and energetic background to phase equilibria will also be treat-ed. In so doing, the principles of thermodynamics of mixtures will be discussed and the correlation between energetics and constitution demonstrated. In this way, the more qualitative framework which often surrounds the teaching of con-stitution will be surmounted and the interested reader will be provided with a tool enabling him to make quantitative predictions also concerning phenom-enological energetics and the structural and physical factors governing an in-dividual system. It will also be possible to make predictions concerning phase equilibria for systems for which experimental results can only be obtained with difficulty.

From the standpoint of practical application, the treatment of nucleation of phase transitions, the production and stability of technologically important metastable phases and attempts to understand metallic glasses will also be dis-cussed. There is currently a large-scale technologically motivated research ef-

VI Preface

fort in this area which is providing a broader and deeper understanding. A short survey of the most important facts will be presented.

Finally, a condensed presentation of the thermodynamics and constitution of polymer systems is included.

The book "Heterogene Gleichgewichte" though printed in 1982 is still very actual. We tried to make it a thermodynamic treatment for all materials: cover ceramics, organic materials, polymers and aqueous solutions (for geologists).

To upgrade the book, we introduced two new solution models, which permit the calculation of enthalpy of mixing and of phase diagrams in ternary, qua-ternary, quinary and larger systems from binary data alone. This is important in practical applications, where after calculations a few experimental measure-ments are sufficient to check the results. We also deal in greater detail with sec-ond order transitions in metals and polymers. For the use of ceramicists, we especially described the phase rule in ternary systems. In the aqueous solutions we show, how solubilities of several salts in water can be calculated, again using only binary data. Last but not least, we show that the same thermodynamic for-mulas used for metals and ceramic materials can be applied to organic materi-als, polymers and aqueous solutions.

No other text to our knowledge covers all these areas. For critical review of the text, we are grateful to several experts in the field,

especially Prof. Dr. Dr. h.c. mult G. Petzow, Prof. Dr. Dr. h.c. W. Gust, Prof. Dr. F. Sommer, Prof. Dr. W. Funke, Dr. I. Arpshofen and T. Godecke. Mrs. G. Kiim-merling and Dr. I. Arpshofen have prepared drawings.

Stuttgart and Cincinnati, Summer 2003

Bruno Predel Michael Hoch, Monte Pool

Contents

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XIII

1 Fundamental Facts and Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Vaporization Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Phase Equilibria in One-Component Systems .................. 9 2.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Transformation Equilibria in the Solid State. . . . . . . . . . . . . . . . . . . . 9 2.3 Monotropic Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Phase Equilibria in Two-Component Systems Under Exclusion of the Gas Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Definition ofthe Composition................................ 17 3.2 Partial Reactions of the Solid-Liquid Transition. . . . . . . . . . . . . . . . . 18 3.3 Process of Fusion in a Two-Component System. . . . . . . . . . . . . . . . . 21 3.4 Eutectic System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 Eutectic Real Systems ....................................... 25 3.6 The Gibbs Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.7 Application of the Phase Rule ................................ 28 3.8 The Lever Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.9 Thermal Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.10 Light Microscopic and Electronmicroscopic Research Methods

to Determine Phase Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.11 X-Ray Diffraction Methods. . . . . . . .. . . . . . . .. . . . . . . . .. . . . . .. . . 40 3.12 Other Experimental Methods................................. 41 3.13 Eutectic Crystallization...................................... 41 3.14 Dendritic Crystallization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.15 Simple Phase Equilibria with Complete Solubility

in the Solid and Liquid Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

VIII

3.16 Phase Equilibria with Complete Solubility in the Solid and Liquid Phases and a Melting Point Minimum

Contents

or Melting Point Maximum .................................. 50 3.17 Real Phase Diagrams with Complete Solubility

in the Solid and Liquid Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.18 Miscibility Gap in the Solid Phase.. . . .. . . . . . . . . .. . . . . . . . . . . . .. 54 3.19 Phase Diagram with Peritectic Equilibrium. . . . . . . . . . . . . . . . . . . . 56 3.20 Miscibility Gap in the Liquid Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.21 Real Phase Diagrams with a Miscibility Gap in the Liquid Phase

and an Upper Critical Point Exclusively. . . . . . . . . . . . . . . . . . . . . . . . 62 3.22 Phases with a Superlattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.23 Systems with a Congruently Melting Compound. . . . . . . . . . . . . . . . 67 3.24 Phase Diagram with a Non-Congruently Melting Compound. . . . . 71 3.25 Phase Diagram with a Compound Forming from Two Melts . . . . . . 73 3.26 Real Diagrams with Compounds.............................. 74 3.27 Transformation Equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.28 The Iron-Carbon Phase Diagram............................. 81

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Phase Equilibria in Three-Component Systems and Four-Component Systems with Exclusion of the Gas Phase. . . . . . . . . . . . 89

4.1 The Composition Triangle................................... 89 4.2 Lever Rule in Ternary Systems.. . . . . .. . . . . . . .. . . . . . . . .. .. . . . . . 91 4.3 Compatibility Triangle ...................................... 92 4.4 Four-Phase Equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5 Representation of Ternary Phase Diagrams. . . . . . . . . . . . . . . . . . . . . 94 4.6 A Simple System with a Ternary Eutectic. . . . . . . . . . . . . . . . . . . . . . . 95 4.7 Phase Fields in a Ternary Eutectic System. . . . . . . . . . . . . . . . . . . . . . 97 4.8 Cuts at Constant Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.9 Vertical Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100 4.10 Temperature-Composition Cut through a Corner

of the Composition Triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 101 4.11 Temperature-Composition Cut Parallel to One Side

of the Composition Triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102 4.12 Simple Real Diagrams with a Ternary Eutectic .................. 103 4.13 Thermal Analysis and Structure of Simple Ternary

Eutectic Systems............................................ 105 4.14 Properties of Neighboring Phase Fields. . . . . . . . . . . . . . . . . . . . . . .. 108 4.15 Non-Regular Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111 4.16 Critical Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112 4.17 Schreinemakers' Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113 4.18 Ternary Systems with Unlimited Solubility in the Solid

and Liquid State, and without a Melting Point Minimum or Maximum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 114

Contents IX

4.19 Isothermal Section through a Ternary System with Unlimited Solution Formation. . . . . . . . . . . . . . . . . . . . . . . . . .. 115

4.20 Temperature-Composition Section through a Ternary System with Unlimited Solid Solution Formation. . . . . . . . . . . . . . . . . . . . .. 117

4.21 System with a Ternary Eutectic and Limited Solid Solution. . . . . .. 118 4.22 Ternary System with a Congruently Melting Binary Compound

and a Pseudobinary Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 120 4.23 Ternary System with a Binary Compound

without a Pseudobinary Section ............. . . . . . . . . . . . . . . . .. 122 4.24 Isothermal Section and Temperature-Composition Section

through a Ternary System with a Binary Compound with no Pseudo binary Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 124

4.25 Ternary System with Two Binary Compounds . . . . . . . . . . . . . . . . .. 125 4.26 Ternary Compounds with Melting Point Maxima. . . . . . . . . . . . . . .. 127 4.27 Real Ternary Systems with Binary and Ternary Compounds...... 128 4.28 Ternary System with Two Eutectic Bounding Binary Systems

with Limited Solubility in the Solid and Complete Miscibility in the Third Bounding Binary System. . . . . . . . . . . . . . . . . . . . . . . . .. 129

4.29 Ternary System with Two Peritectic Bounding Systems and Complete Solubility in the Third Bounding System . . . . . . . . .. 131

4.30 Transition between an Univariant Peritectic and an Univariant Eutectic Reaction. . . . . . . . . . . . . . . . . . . . . . . . . .. 133

4.31 Miscibility Gap in the Liquid Phase. . . . . . . . .. . . . . . . . . . . . . . . . . .. 134 4.32 Monotectic Four-Phase Reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137 4.33 Real Ternary Diagrams with Limited Solubility

in the Liquid Phase. . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . .. 138 4.34 Reaction Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 141 4.35 Four-Component Systems .................................... 143 4.36 Simple Equilibria in Four-Component Systems ................. 145 4.37 Reciprocal Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 146 4.38 Solubility of Reciprocal Salt Pairs in Water . . . . . . . . . . . . . . . . . . . .. 148 4.39 Comments to the Extent of Higher Order Systems. . . . . . . . . . . . . .. 150

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 152

5 Phase Equilibria Including a Vapor Phase ...................... 155 5.1 Vapor-Liquid Equilibrium in a One-Component System..... ..... 155 5.2 Phase Equilibria between Liquid and Vapor in Binary Systems

without a Miscibility Gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 156 5.3 Gas-Solid Equilibria in a Binary System ........................ 160 5.4 Phase Equilibria in a Binary System in which Solid, Liquid

and Gas can Appear. ... ......... ...... . ..... . ......... ...... 163 5.5 Phase Equilibria with Participation of the Gas Phase

with Limited Solubility in the Liquid Phase. . . . . . . . . . . . . . . . . . . .. 167 5.6 Vapor-Solid Equilibria with Solid Solution Formation. . . . . . . . . .. 169

X Contents

5.7 Gas-Solid Equilibria in Systems with Compounds. . . . . . . . . . . . . .. 169 5.8 Heterogeneous Equilibria at Chemical Transport Reaction ... . . .. 172

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 173

6 Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175 6.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175 6.2 Basic Thermodynamic Concepts and Definitions. . . . . . . . . . . . . . .. 176 6.3 Integral Quantities of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 178 6.4 Partial Quantities of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181 6.5 The Ideal Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 183 6.6 The Model of the Regular Solution. . . . . . . . . . . . . . . . . . . . . . . . . . .. 183 6.7 Real Solutions and Excess Functions. . . . . . . . . . . . . . . . . . . . . . . . .. 185 6.8 Analysis of Experimental Thermodynamic Data. . . . . . . . . . . . . . .. 188 6.9 Influence of the Atomic Size Difference. . . . . . . . . . . . . . . . . . . . . . .. 188 6.10 The Association Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 190 6.10.1 6.10.2 6.11 6.12 6.l3

6.14 6.15

6.16 6.17 6.18 6.19 6.20 6.21

6.22

6.23 6.24 6.25 6.26 6.27

6.28 6.29

6.30 6.31

Basic Formulae of the Association Model ..................... . Application to Liquid Binary and Ternary Alloys ............... . The Hoch-Arpshofen ModeL ................................ . Difference in Heat Capacity between Liquid and Solid Cp(L-S) ... . Calculation of Thermodynamic Functions in Multi-Component Systems ................................ . The Thermodynamic Activity ............................... . General Comments about Experimental Methods to Determine Thermodynamic Mixing Properties .............. . The High-Temperature Calorimeter .......................... . Partial Vapor Pressure Measurements ........................ . Activity Determination from the EMF of Galvanic Cells ......... . Fusion Equilibrium in a One-Component System .............. . Fusion Equilibria in Binary Systems .......................... . Equilibrium between a Binary Liquid and the Crystal of one Component in the Ideal System ........................ . Equilibrium between a Binary Liquid and a Solid Solution in an Ideal System ......................................... . Fusion Equilibrium in a Regular System ...................... . Fusion Equilibrium in a Real System ......................... . Melting Point Minimum .................................... . Phase Equilibrium During Demixing ......................... . Calculation of the Miscibility Gap Based on the Regular Solution Model ............................................ . Evaluation of Solubility Equilibria ........................... . Evaluation of a Fusion Equilibrium with Small Liquidus Concentrations ............................................ . Critical Demixing Temperature in Real Solutions ............... . The Spinodal .............................................. .

191 193 198 203

206 213

217 217 218 221 222 224

230

233 234 236 236 238

240 243

246 248 249

Contents XI

6.32 Calculation of a Simple Ordering Reaction in Solid Solutions . . . .. 249 6.33 Degree of Order in a Superlattice as a Function of Temperature . .. 255 6.34 Comments on the Character of Phase Transformations .......... 257 6.35 Thermodynamic Properties of Ternary Alloys . . . . . . . . . . . . . . . . .. 258 6.36 Calculation of Fusion Equilibria in Ternary Systems. . . . . . . . . . . .. 263

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267

7 Nucleation During Phase Transitions. . . . . . . . . . . . . . . . . . . . . . . . .. 269 7.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 269 7.2 Homogeneous Nucleation Without Change of Composition. . . . .. 269 7.3 Heterogeneous Nucleation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 272 7.4 Homogeneous Nucleation with Change in Composition. . . . . . . . .. 275 7.5 Spinodal Decomposition ..................................... 278

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 281

8 Metastable Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 283 8.1 Energetics ofthe Nucleation of Metastable Crystalline Phases. . . .. 283 8.2 Guinier-Preston Zones in AI-Cu Alloys. . . . . .. . . . . . . . . . . . . . . . . .. 285 8.3 Short Range Order in Liquid Solutions and Their Effect

on Solid-Liquid Equilibria. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. 288 8.3.1 Some Empirical Findings about Short Range Order

in Liquid Alloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 288 8.3.2 The Model of Homogeneous Equilibria for the Quantitative

Description of the Short Range Order in Liquid Alloys. . . . . . . . . .. 294 8.3.3 The Effect of Homogeneous Equilibria on Fusion Equilibria. . . . .. 296 8.3.3.1 Liquidus Lines in the Vicinity of the Melting Point

of a Congruently Melting Compound. . . . . . . . . . . . . . . . . . . . . . . . .. 296 8.3.3.2 Miscibility Gap in a Binary Liquid Alloy with a Strong

Compound Forming Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 298 8.4 Metallic Glasses ............................................ 302 8.5 Metastable, Non-Crystalline Metallic Phases. . . . . . . . . . . . . . . . . . .. 304 8.6 Methods to Obtain Extremely High Cooling Rates ............... 307 8.7 Crystallization of Glasslike Alloys. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 309

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 311

9 Effect of Diffusion on Phase Transformations . . . . . . . . . . . . . . . . .. 313 9.1 Distribution of Components During Solidification of Liquids. . . .. 313 9.2 Constitutional Undercooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 317 9.3 Effects ofthe Constitutional Undercooling . . . . . . . . . . . . . . . . . . . .. 319 9.4 Purification by Zone Melting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 322 9.5 Precipitation Reactions in the Solid State. . . . . . . . . . . . . . . . . . . . . .. 324

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 329

XII Contents

10 Organic and Polymeric Materials ............................. 331 10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 331 10.2 Organic Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 331 10.3 Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 333 lOA Polymer Blends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 334

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 339

Subject Index ..................................................... 341

List of Symbols

latin letters at-% A atomic percent of component A aA thermodynamic activity of component A Cp molar heat capacity at constant pressure Cv molar heat capacity at constant volume c concentration, common v vapor e binary eutectic point E ternary eutectic point EQ quaternary eutectic point F free energy per g-atom (molar free energy) G Gibbs energy per g-atom (molar Gibbs energy) H heat content per g-atom (molar enthalpy) L liquid, melt 1 lamellar spacing in an eutectic or in a lamellar precipitation MA atomic weight of component A n number of moles, common p pressure, common; in T-x phase diagrams: binary peritectic point; ter-

nary peritectic point p A partial vapor pressure of component A p A 0 vapor pressure of pure component A PK critical pressure Q molar heat per g-atom QF activation energy of the freezing reaction QM activation energy of the melting reaction R gas constant RA rate of reaction; common RA rate of precipitation RE rate of the eutectic reaction RK net rate of crystallization RF rate of freezing RM rate of melting S a) entropy per g-atom (molar entropy)

b) used as an index: solid

XIV

T TA TAO

Te, TE TF Tg TG U

V VA wt-% A xA Xe,xE Z

temperature in K; common melting temperature of component A boiling temperature of component A eutectic temperature freezing temperature; common temperature of equilibrium temperature of glass transformation a) internal energy per g-atom; (molar energy) b) in phase diagrams: ternary transition equilibrium molar volume partial volume of component A in g-atom weight percent of component A mol fraction (atomic fraction) of component A eutectic concentration coordination number

Greek Letters a, B, y, E, 11 in phase diagrams: solid phases vL oscillation frequency of atoms in a liquid vS oscillation frequency of atoms in a solid a specific grain boundary energy

Differential Quantities

List of Symbols

ilF variation of the molar Gibbs energy due to formation of a mixed phase (integral molar Gibbs energy of mixing)

ilG changes in the molar Gibbs energy a) due to formation of a mixed phase (integral molar Gibbs energy of mix-ing) b) by a phase transformation

ilG A F GAL - GAS = molar Gibbs energy offreezing of component A GAL, GAS Gibbs energy of liquid, respectively solid A

ilGex Integral molar excess Gibbs energy ilGA Partial molar Gibbs energy of mixing of component A ilGi Ideal integral molar Gibbs Energy of mixing ilG A(i) Ideal partial molar Gibbs Energy of mixing of component A ilH change in molar enthalpy due to the formation of a mixed phase (integral

molar enthalpy of mixing or of formation) ilHAF molar enthalpy of melting of component A ilHu molar enthalpy of transformation ilHV molar enthalpy of evaporation ilS change in molar entropy due to the formation of a mixed phase (integral

molar entropy of mixing) ilSAF molar entropy of melting of component A ilSex Integral molar excess entropy

List of Symbols xv

LlSi Ideal integral entropy of mixing LlS A Partial molar entropy of mixing of component A LlS A(i) Ideal Partial molar entropy of mixing of component A Ll T difference in temperature, general Ll V change in molar volume due to the formation of a mixed phase (integral

molar volume of mixing) Ll VF change of molar volume due to melting in cm3/g-atom (molar volume of

melting) LlVu change of volume due to transformation (molar transformation vol-

ume) Ll VA Partial molar volume of mixing of component A

Annotation If not mentioned otherwise, the extensive state functions were referred to 1 mol, for instance F, G, H, S and all differential values marked with Ll (except LlT). For mixtures with common content, these values are identical with those related to 1 g-atom.

Care should be taken, if molecules or molecule like species were considered, which have an exactly defined stoichiometry. In such cases the values related to 1 mol have to be converted into values related to 1 g-atom. This is done by dividing the molar value by the number of atoms in the formula unit, for, as well known, 1 mol consists of 6.023 * 1023 molecules, whereas 1 g-atom contains 6.023 * 1023 atoms.

As an example: Melting enthalpy of water is LlH~20 = 6.007 kJ/mol. This cor-responds to LlH~20 = 2002 kJ/g-atom.