(Cambridge Solid State Science Series) Ali S. Argon-The Physics of Deformation and Fracture of...

8/17/2019 (Cambridge Solid State Science Series) Ali S. Argon-The Physics of Deformation and Fracture of Polymers-Cambrid…

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The Physics of Deformation and Fracture of Polymers

Demonstrating through examples, this book presents a mechanism-based

perspective on the broad range of deformation and fracture responses of solidpolymers. It draws on the results of probing experiments and considers the similar

mechanical responses of amorphous metals and inorganic compounds to develop

advanced methodology for generating more precise forms of modeling. This, in

turn, provides better fundamental understanding of deformation and fracture

phenomena in solid polymers. Such mechanism-based constitutive response forms

have far-reaching application potential in the prediction of structural responses

and in tailoring special microstructures for tough behavior. Moreover, they can

guide the development of computational codes for deformation processing of

polymers at any level. Applications can range from large-strain industrial deform-ation texturing to production of precision micro-fluidic devices, making this book

of interest both to advanced graduate students and to practicing professionals.

A li S . A rg on is Quentin Berg Professor Emeritus in the Department of

Mechanical Engineering at Massachusetts Institute of Technology (MIT). He is

recognized world-wide as an authority on the mechanical behavior of engineering

solids, has published over 300 papers and three books, and is one of the inter-

nationally most widely cited authors in materials science. He has received a

number of honors and awards, including membership of the US National Acad-

emy of Engineering, Fellowship of the American Physical Society, DistinguishedLife Membership of the Alpha Sigma Mu (International Professional Society

of Materials and Engineering), the Nadai Medal of the American Society of

Mechanical Engineers, the Heyn Medal of the German Society for Materials

Science, and a US Senior Scientist Award of the Alexander von Humboldt

Foundation for research in Germany.


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The Physics of Deformation

and Fracture of Polymers

A . S . A R G O N

Massachusetts Institute of Technology


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c a m b r i d g e u n i v e r s i t y p r e s s

Cambridge, New York, Melbourne, Madrid, Cape Town,

Singapore, Sa ˜o Paulo, Delhi, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by

Cambridge University Press, New York

www.cambridge.org

Information on this title: www.cambridge.org/9780521821841

© A. S. Argon 2013

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without

the written permission of Cambridge University Press.

First published 2013

Printed and bound in the United Kingdom by the MPG Books Group

A catalog record for this publication is available from the British Library

Library of Congress Cataloging-in-Publication Data

Argon, Ali S.

The physics of deformation and fracture of polymers / A. S. Argon, Massachusetts Institute of

Technology.

pages cm

Includes bibliographical references and indexes.

ISBN 978-0-521-82184-1

1. Polymers–Fracture. 2. Polymers–Plastic properties. 3. Plastics. I. Title.

TA455.P58A74 2013

620.10920413–dc23

2012025871

ISBN 978-0-521-82184-1 Hardback

Cambridge University Press has no responsibility for the persistence or

accuracy of URLs for external or third-party internet websites referred toin this publication, and does not guarantee that any content on such

websites is, or will remain, accurate or appropriate.

http://www.cambridge.org/http://www.cambridge.org/9780521821841http://www.cambridge.org/http://www.cambridge.org/9780521821841http://www.cambridge.org/9780521821841http://www.cambridge.org/


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This book is dedicated to

Ian M. Ward of Leeds University for his long-term friendship

and tomy wife Xenia for her enduring support


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“An insightful exposition from one of the most influential material scientists

of our time. A must read for anybody wishing to gain a mechanician’s

(not a chemist’s!) perspective on the physics and mechanics of polymers.”Vasily Bulatov, Lawrence Livermore National Laboratory


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Contents

Preface page xv

Symbols xviii

Frequently used abbreviations xxii

1 Structure of non-polymeric glasses 1

1.1 Overview 1

1.2 Glass formability in metallic alloys 3

1.3 Atomic packing in disordered metallic solids 3

1.4 Energetic characterization of the structure of metallic glasses 7

1.4.1 The atomic site stress tensor 7

1.4.2 Calorimetry 9

1.5 Free volume 10

1.6 Viscosity of glass-forming liquids 14

1.7 Structural relaxations 16

1.7.1 A computational model 16

1.7.2 Kinetic models of structural relaxations in metallic glasses 20

1.8 The distributed character of structural relaxations and the

glass transition 21

1.9 The dependence of the glass-transition temperature on cooling rate 25

1.10 Crystallization in bulk metallic glasses 26

1.11 Deformation-induced alterations of atomic structure in sub-cooled

liquids and glasses 271.12 The range of metallic alloys that have been obtained as bulk

metallic glasses 30

1.13 The structure of amorphous silicon 30

1.14 Characterization of the structure of amorphous silicon 32

Suggested further reading on structure of non-polymeric glasses 36

References 37

2 Structure of solid polymers 40

2.1 Overview 402.2 Structure of polymers 41

2.3 Molecular architecture 46

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2.4 Molecular weight 47

2.5 Structure of amorphous polymers 49

2.5.1 Molecular-structure models of amorphous polymers 49

2.5.2 Chemically specific molecular-structure models of amorphous polymers 49

2.5.3 Chemically non-specific models of amorphous polymer

structure 53

2.5.4 Experimental means of characterization of the structure

of glassy polymers 54

2.6 Crystalline polymers 54

2.6.1 The fringed-micelle model of semi-crystalline polymers 54

2.6.2 Spherulites 55

2.6.3 Hedrites 58

2.6.4 Polymer single crystals 58

2.6.5 Crystallization from the melt and growth of spherulites 61

2.7 Defects in polymer crystals 66

2.7.1 Overview 66

2.7.2 Chain defects 67

2.7.3 Lattice defects 71

2.8 Chain-extended polymers 71

Suggested further reading on structure of solid polymers 72

References 73

3 Constitutive connections between stress and strain in polymers 77

3.1 Overview 77

3.2 Stresses and strains 77

3.2.1 Stresses 77

3.2.2 Strains 78

3.3 Linear elasticity of polymers 81

3.4 Plasticity of polymers 83

3.4.1 Generalized yield conditions 83

3.4.2 The associated-flow rule 853.5 Thermally activated deformation 87

References 89

4 Small-strain elastic response 90

4.1 Overview 90

4.2 Small-strain elasticity in crystals 91

4.2.1 The generalized Hooke’s law 91

4.2.2 Orthorhombic crystals or orthotropic solids 93

4.2.3 Hexagonal crystals 934.2.4 Cubic crystals 93

4.2.5 Isotropic materials 93

viii Contents

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4.2.6 Temperature and strain dependence of elastic response 95

4.3 Theoretical determination of elastic constants of polymers 96

4.3.1 Glassy polymers 96

4.3.2 Crystalline polymers 974.4 Elastic response of textured anisotropic polymers 102

4.5 Elastic properties of heterogeneous polymers 104

4.5.1 Methods of estimating the elastic properties of

heterogeneous polymers 104

4.5.2 The self-consistent method 105

4.5.3 The Eshelby inclusion method 106

References 109

5 Linear viscoelasticity of polymers 112

5.1 Introduction 112

5.2 Phenomenological formalisms of viscoelasticity 112

5.2.1 Uniaxial creep or stress-relaxation response 112

5.2.2 Dynamic relaxation response 116

5.2.3 Temperature dependence of viscoelastic relaxations 118

5.3 Viscoelastic relaxations in amorphous polymers 120

5.3.1 The α-relaxation 120

5.3.2 The free-volume model of the α-relaxation 122

5.3.3 Dependence of the α-relaxation on the chemical structure

of molecules 126

5.3.4 Secondary relaxations in the glassy regime 127

5.3.5 Effect of physical aging on the relaxation spectra of polymers 130

5.3.6 Secondary relaxations in polycarbonate of bisphenol-A 132

5.4 Shear relaxations in partially crystalline polymers 139

5.5 Some problems of viscoelastic-stress analysis 143

5.6 Non-linear viscoelasticity 145

Suggested further reading on linear viscoelasticity of polymers 146

References 146

6 Rubber elasticity 148

6.1 Overview 148

6.2 Molecular characteristics of rubbers 149

6.2.1 Distinctive features of rubbers 149

6.2.2 The chemical constitution of rubbers 151

6.3 Thermodynamics of rubbery behavior 151

6.4 The Gaussian statistical model of rubber elasticity 155

6.5 The non-Gaussian statistical model of rubber elasticity 159

6.5.1 The freely jointed single chain 1596.5.2 Langevin networks 161

6.5.3 Comparison of the Langevin-network model with experiments 164

ixContents

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6.6 Modes of deformation in rubber elasticity 167

6.6.1 Conditions for general response 167

6.6.2 Uniaxial tension or compression 167

6.6.3 Equi-biaxial stretch 1686.6.4 Plane-strain tension and pure shear 168

6.6.5 Simple shear 169

6.6.6 Plane-strain compression flow in a channel die 171

6.7 Gaussian rubbery-type response in glassy polymers 172

References 172

7 Inelastic behavior of non-polymeric glasses 174

7.1 Overview 174

7.2 The mechanism of plasticity in non-polymeric glasses 1757.3 The kinematics of plasticity in glassy solids by shear transformations 176

7.4 Nucleation of shear transformations under stress 179

7.4.1 The elastic strain energy of a shear transformation in the

unstressed solid 179

7.4.2 The Gibbs free energy of nucleation of the shear

transformation under stress 180

7.4.3 Stages in the nucleation of the shear transformation 181

7.5 Yielding in metallic glasses 185

7.5.1 Behavior at low temperatures (T T g) 185

7.5.2 Temperature dependence of the yield stress (T T g) 187

7.5.3 Analysis of the experimental results on yield behavior of

metallic glasses at low temperatures 188

7.5.4 Yielding in metallic glasses at temperatures close to T g 189

7.5.5 Changing kinetics of plasticity near T g 193

7.6 Post-yield large-strain plastic response of glassy solids: strain

softening and strain hardening 199

7.6.1 Features of large-strain plastic flow of glassy solids 199

7.6.2 Plastic-flow-induced increase in the liquid-like

material fraction, φ 2007.6.3 Plastic-strain-induced changes in structure and the kinetics

of associated evolutions of φ 203

7.6.4 Kinetics of large-strain plastic flow of glasses at T T g 205

7.6.5 Kinetics of large-strain plastic flow of glasses at T close to T g 207

7.6.6 Multi-axial deformation: correspondences of shear, tension,

and compression at low temperatures 210

7.7 The strength-differential effect in disordered solids 213

7.8 Shear localization 216

7.8.1 The phenomenology of shear localization in metallic glasses 216

7.8.2 The mechanics of shear localization 217

x Contents

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7.8.3 Temperature rises associated with shear localization 220

7.8.4 The flow state 221

Appendix. Plastic-floor-induced structural alterations: the relation

between flow dilatations of free volume and liquid-like material 222References 224

8 Plasticity of glassy polymers 228

8.1 Overview 228

8.2 The rheology of glassy polymers 229

8.2.1 Important provisos 229

8.2.2 The phenomenology of plastic flow in glassy polymers 230

8.3 The mechanism of plastic flow in glassy polymers 234

8.3.1 Computer simulation of plastic flow 2348.3.2 Simulation results in polypropylene 236

8.3.3 Simulation results in polycarbonate 238

8.4 Temperature dependence of yield stresses of glassy polymers 243

8.5 The kinetic model of plastic yield in glassy polymers 243

8.5.1 Temperature dependence of the plastic resistance 243

8.5.2 The thermal activation parameters 247

8.5.3 A kinetic model of flow of linear-chain glassy polymers 248

8.6 Large-strain plastic flow in glassy polymers 249

8.6.1 Development of post-yield large-strain plastic flow 249

8.6.2 A model for post-yield plastic flow of glassy polymers 254

8.6.3 Stored energy and Bauschinger back strains 258

8.6.4 The strength-differential effect and the multi-axial

yield condition 259

8.7 Strain hardening in glassy polymers 262

8.8 Comparison of experiments and simulations on the yielding and

large-strain plastic flow of glassy polymers 264

References 270

9 Plasticity of semi-crystalline polymers 273

9.1 Overview 273

9.2 Mechanisms of plastic deformation 274

9.3 Plasticity of two semi-crystalline polymers: high-density polyethylene

(HDPE) and polyamide-6 (Nylon-6) 276

9.3.1 Methodology of deformation 276

9.3.2 Plastic strain-induced alterations of spherulite morphology in

Nylon-6 in uniaxial tension 277

9.3.3 Large-strain plastic flow in HDPE in plane-strain compression 280

9.3.4 Large-strain plastic flow in monoclinic Nylon-6 by plane-straincompression 291

xiContents

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9.3.5 Measurement of critical resolved shear stresses in textured

HDPE and Nylon-6 and their normal-stress dependence 292

9.4 The kinetics of plastic flow in semi-crystalline polymers 295

9.4.1 Modes of dislocation nucleation in lamellae 2989.4.2 The strain-rate expression 301

9.4.3 The dominant nucleation mode 303

9.4.4 Activation volumes 304

9.4.5 Temperature dependence of the plastic resistance 307

9.5 Simulation of plastic-strain-induced texture development in HDPE 309

9.5.1 Characteristics of the simulation 309

9.5.2 Basic assumptions of the model 309

9.5.3 Constitutive relations 311

9.5.4 Composite inclusion 315

9.5.5 Interaction law and solution procedure 315

9.5.6 Parameter selection in the model 316

9.5.7 Predicted results of the composite model and comparison

with experiments 317

Suggested further reading on plasticity of semi-crystalline polymers 321

References 321

10 Deformation instabilities in extensional plastic flow of polymers 325

10.1 Overview 325

10.2 Deformation instabilities in extensional plastic flow of polymers 325

10.3 Conditions for impending localization in extensional deformation 326

10.3.1 Basic shear response 326

10.3.2 Basic extensional response 328

10.4 Stability of extensional plastic flow 331

10.5 The effect of strain-rate sensitivity on stability in extensional

plastic flow 333

10.5.1 In the onset of necking 333

10.5.2 In the post-necking behavior 335

10.6 Plastic drawing of polymers 336References 341

11 Crazing in glassy homo- and hetero-polymers 342

11.1 Overview 342

11.2 The phenomenology of crazing in glassy homo-polymers 343

11.3 Simulation of cavitation in a glassy polymer at the atomic level 345

11.4 Craze initiation 347

11.4.1 Experimental observations 347

11.4.2 Intrinsic crazing 34911.4.3 Tension–torsion experiments 349

11.5 A craze-initiation model 353

xii Contents

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11.6 Comparison of the predictions of the craze-initiation model with

experiments 356

11.7 Craze growth 359

11.7.1 Craze stresses 35911.7.2 Craze microstructure 364

11.7.3 Craze-growth experiments 366

11.8 A craze-growth model 368

11.9 Comparison of the craze-growth model with experiments 374

11.10 Crazing in block copolymers 376

11.10.1 Morphology of diblock copolymers 376

11.10.2 Crazing experiments in PS/PB diblock copolymers 378

11.10.3 A model of craze growth in a PS/PB diblock copolymer

with spherical PB domains 381

11.10.4 Comparison of the predictions of the craze-growth model

in PS/PB diblock copolymers with experiments 385

References 387

12 Fracture of polymers 391

12.1 Overview 391

12.2 Cracks and fracture 391

12.2.1 Two complementary perspectives in crack mechanics 391

12.2.2 Cracks in LEFM 392

12.2.3 The energy-release rate GI in LEFM with crack extension 396

12.3 Cracks with plastic zones 398

12.3.1 Pervasiveness of plasticity at the crack tip 398

12.3.2 Cracks with small-scale yielding (SSY) 399

12.3.3 Crack-tip fields with contained plasticity 404

12.3.4 Crack fields in fully developed plasticity 407

12.4 Stability of crack advance 414

12.5 Intrinsic brittleness of polymers 416

12.6 Brittle-to-ductile transitions in fracture 418

12.7 Mechanisms and forms of fracture in polymers 41912.7.1 The crack-tip process zone 419

12.7.2 The role of chain scission in polymer fracture 419

12.7.3 Fracture of unoriented polymers 420

12.7.4 Cohesive separation 420

12.7.5 Fracture in glassy polymers involving crazing 422

12.7.6 Molecular-scission-controlled fracture of oriented

semi-crystalline polymers 425

12.7.7 Fracture toughnesses of a selection of polymers 428

12.8 Impact fracture of polymers 429

12.8.1 Application of fracture mechanics to impact fracture 429

12.8.2 Fracture of polymers at high strain rate 431

xiiiContents

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Suggested further reading on fracture of polymers 432

References 433

13 Toughening of polymers 435

13.1 Overview 435

13.2 Strategies of toughening of polymers 436

13.3 Different manifestations of toughness in polymers 437

13.4 The generic fracture response of polymers in uniaxial tension 438

13.5 Toughening of crazable glassy polymers by compliant particles 440

13.5.1 Types of compliant composite particles 440

13.5.2 Brittleness of glassy homo-polymers and alleviating it

through craze plasticity 443

13.5.3 The mechanism of toughening in particle-modified crazableglassy polymers 445

13.5.4 Elasticity of compliant particles 447

13.5.5 Craze initiation from compliant particles and the craze-flow

stress 449

13.5.6 The role of compliant-particle size in toughening

glassy polymers 449

13.5.7 A model for the craze-flow stress of particle-toughened

polystyrene 452

13.5.8 Special HIPS blends prepared to evaluate the

toughening model 454

13.5.9 Comparison of the behavior of special HIPS blends

with model predictions 457

13.6 Diluent-induced toughening of glassy polymers 459

13.6.1 Different manifestations of toughening with diluents 459

13.6.2 Factors affecting diluent toughening of PS 462

13.6.3 A model of diluent-induced toughening of glassy polymers 465

13.6.4 Comparison of the diluent-induced-toughening model with

experiments 472

13.7 Toughening of semi-crystalline polymers 47513.7.1 Toughness of unmodified HDPE and polyamides of

Nylon-6 and -66 475

13.7.2 Toughening semi-crystalline polymers by particle

modification 477

13.8 Toughening of brittle thermosetting polymers 492

References 497

Author index 501

Subject index 507

xiv Contents

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Preface

The chemistry and physics of polymers, and their molecular microstructure,

morphology, and larger-scale organization have been extensively studied and

described in many treatises.

In comparison the plastic deformation and fracture processes, both in the

laboratory and in industrial practice, have largely been dealt with at a phenom-

enological level, and often separately for different polymers and blends, rather

than from a unified and comprehensive mechanistic perspective. This has left the

mechanisms governing the deformation and fracture resistance of polymers far

less well understood.

On the other hand, fundamental developments in polymer physics and polymer

materials science in the recent past are now making it possible to consider broad

ranges of their deformation and fracture from a mechanistic point of view at an

appropriate molecular and morphological level. Moreover, insight gained fromstudies of corresponding responses of amorphous metals and semiconductors,

reinforced by computational simulations and mechanistic modeling, has also

broadened the perspective.

The purpose of this book is to present a coherent picture of the inelastic

deformation and fracture of polymers from a mechanistic point of view, addressed

to graduate students of material science and mechanical engineering and to

professional practitioners in the field.

The book concentrates heavily on research conducted at the Massachusetts

Institute of Technology from the mid 1980s to the mid 2000s by the author and

a group of collaborators. It reports on extensive experimental studies and related

computational simulations. In the latter there is much emphasis on development

of mechanistic models ranging from unit plastic relaxation events to the evolution

of deformation textures in channel die compression flow to large plastic strains. At

every level the experimental results are compared in detail with predictions from

the models.

The core of the book is devoted to subjects starting with anelastic behavior of

polymers and rubber elasticity, but proceeds with greater emphasis in following

chapters to mechanisms of plastic relaxations in glassy polymers and semi-

crystalline polymers with initial spherulitic morphology. Other chapters concen-trate on craze plasticity in homo-polymers and block copolymers, culminating

with a chapter on toughening mechanisms in brittle polymers. To make the


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main chapters on plastic flow and toughening tractable to the reader, the book

starts with a brief tutorial chapter devoted to the structure of polymers from the

chain molecular levels to morphological aggregation of crystalline lamellae and

their further aggregation into spherulites. Since unit plastic relaxations arecomplex phenomena in glassy polymers, which, however, exhibit parallel phe-

nomena that can be followed more transparently in amorphous metals and

amorphous silicon, a chapter is also included at the start on the atomic structure

of such simpler elemental glasses. Finally, since fracture involves propagation of

cracks emanating from notches, with crack tips being modified by plastic zones

of various levels of pervasiveness, a relatively comprehensive chapter on fracture

mechanisms and mechanics is included to precede the chapter on toughening

mechanisms.

Each chapter starts with an overview laying out the topics to be presented to

give an overall perspective. Copious references are provided at the ends of chap-

ters, often supplemented with lists of additional references that develop some

topics in greater depth.

It is assumed that the reader has had an introductory course on materials

science and perhaps on polymers such as e.g. An Introduction to the Mechanical

Properties of Solid Polymers by I. M. Ward and J. Sweeney, John Wiley & Sons,

second edition 2004.

Clearly, the present book covers in depth only a narrow subject area on the

mechanical response of polymers; thus, as such, it is not intended as a review. The

informed reader will note that much work of other investigators falling outsidethe main scope has not been included. This omission is intentional, in order to

preserve a coherent central perspective.

Many colleagues at various levels contributed significantly to the conduct of

the research discussed in the book and the overall development of the subject of

this book, either in experimentation or in computational modeling. These

include, in order of depth of involvement, R. Cohen, U. Suter, A. Gałęski,

Z. Bartczak, E. Pio ´ rkowska, H. Brown, D. Parks, O. Gebizlioglu, S. Ahzi,

M. Hutnik, P. Mott, O. Muratoglu, B. Lee, J. Vancso, J. Qin, and G. Dagli.

In private discussions on many aspects of polymer research, G. Rutledge con-

tributed some important perspective. M. Weinberg of DuPont supplied specially

pedigreed polymer samples and blends for the experimental studies. P. Geil,

B. Wunderlich, E. Kramer, and E. Ma generously furnished electronic files of

some key micrographs and computer-generated images. The text was prepared,

and numerous modifications were implemented, always cheerfully, by Doris

Elsemiller. The illustrations were ably produced by Andrew Standeven. All this

would not have been possible without funds provided by Deans T. Magnanti

and S. Suresh and Department Heads R. Abeyaratne and M. Boyce. Finally,

the very thorough copy-editing by Dr. Steven Holt on behalf of Cambridge

University Press that uncovered a number of inconsistencies in referencingbetween the text and the lists of references at the ends of chapters is also

gratefully acknowledged.

xvi Preface


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The serious entry of the author into the field of deformation and fracture of

polymers started in 1971 during a sabbatical leave at Leeds University in Britain

with Professor Ian Ward. The friendly association with Ward has continued until

the present. For this reason the book is dedicated first of all to him in appreciationof his long-term friendship. Secondly, however, the book is dedicated in equal

measure to my wife Xenia for her enduring support.

xviiPreface


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Symbols

All mathematical symbols are fully defined in the text where they are introduced.

Some material parameters have been referred to by different symbols, preserving

their usage in the literature.

A area of bar A0 area of perfect bar B(v, β ) energy factor in STC proportionality factor

D diameter of craze fibril; diameter of particle Dc critical particle diameter for craze initiation D0 mean spacing of craze fibrils E Young’s modulus F Helmholtz free energy

F0 self Helmholtz free energy of ST Fint interaction (Helmholtz) energy with σ m of STΔ F0 ¼ F0þ FintΔ Fv activation free energy for viscous flowFe edge-dislocation line energyFs screw-dislocation line energyGI mode I energy-release rateGIC critical mode I energy-release rate for crack advanceΔG* Gibbs free energy of activationΔ H * activation enthalpy

I n normalization factor for stresses in J integral field J I J integral non-linear crack-tip energy-release rate J IC critical J integral crack driving force J U unrelaxed creep compliance J R relaxed creep compliance K I mode I stress intensity factor K IC critical mode I stress intensity factor for crack growth in plane strain K S critical mode I stress intensity factor for crack growth for plane stress K C mode I stress intensity factor for growth of cracks between plane stress

and plane strain: K S> K C> K IC L load on deforming barL Langevin function

M e entanglement molecular weight in rubbers

M n number-average molecular weight M w weight-average molecular weight M w/ M n polydispersity ratio


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N strain exponentQ Heat; dQ, heat increment

R rate; universal gas constant

RSD strength differential ratioS entropy; dS, change in entropyT temperatureT 0 (¼ΔG

*/k )T BD brittle-to-ductile transition temperatureT g glass-transition temperatureT m melting temperatureU internal energy; dU , change in internal energyV volumeW p plastic work; dW p, increment of plastic workY tensile uniaxial yield strength

Y c intrinsic craze yield stressY 0 athermal tensile yield strengthY C yield strength in compressionY T yield strength in tensiona crack length; Cartesian coordinate axisa0 molecular diameteraT viscoelastic shift factorb Burgers vector; Cartesian coordinate axisc Cartesian coordinate axis; volume fractioncf fraction; free-volume fractioncij Voigt elastic constant elementcijkl tensor elastic constant elementeij Voigt deviatoric strain element

f fraction f a amorphous fraction f c crystalline fractiong( λ) (¼ λ2 1/ λ) Gaussian orientation hardening functionk Boltzmann’s constant; yield strength in sheark r rate constantl monomer link length; generic lengthl_ elongation ratem (¼ dln γ_/dln σ ) phenomenological stress exponentmT Taylor factor in polycrystalline aggregates

p pressureq cooling rater radial coordinates applied simple shear stress, deviatoric shear stresss0 athermal shear resistancesij Voigt elastic compliancesijkl tensor compliance elementt timet f time to fractureur radial displacementuθ angular displacement

uz axial displacementvf volume fractionz polar coordinate axis

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x, y, z Cartesian axesF fluidityΔ process-zone length

χ interface energy, crystallinityΛ matrix ligament thicknessΛc critical matrix ligament thickness where a toughness jump occursO atomic volumeOf volume of ST clusterOmon monomer volumeα proportionality constantα (¼ τ ̂/ μ(0)) normalized threshold shear resistanceαb (¼ vcb/vc) proportionality factor between craze-border velocity and

craze-tip velocity β (¼ ε T/γT) activation dilatancy

β level of porosity, secondary relaxation β e activation-energy attenuation factorγ tangential shear strainγp plastic shear strain (deviatoric)γT transformation shear strain_γ shear strain rate_γp plastic shear strain rate_γe elastic shear strain rate_γ0 frequency factor in thermal activationδ crack-tip opening displacementδij Kronecker deltaε normal strainε T free-standing transformation strain tensor

ε C constrained transformation strain tensorε T activation dilatation (¼ βγT)ε equivalent total normal straindε

pij plastic normal strain increment

dε equivalent total strain incrementε TC craze strain as dilatational transformation strainε _ nominal strain rateε e equivalent strain (deviatoric)ε

pf plastic strain at fractureε y normal strain at yield (¼ σ y/ E)

φ fraction, liquid-like-material fractionφs liquid-like-material fraction at the flow state λ1, λ2, λ3 principal extension ratios in a rubber λc chain-extension ratio in Langevin rubbery response λL locking stretch λe stress-attenuation factor in activation energy λn natural draw ratio between onset of instability and regaining of stabil-

ity in fiber drawing, also in craze matter fibril strain μ shear modulus μ0 storage modulus in viscoelasticity μ00 loss modulus in viscoelasticity

μu unrelaxed modulus in viscoelasticity μr relaxed modulus in viscoelasticity, friction factorn Poisson’s ratio

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vD Debye frequencyvG pre-exponential frequency factor, an eigenfrequency ρ material density

ρm mobile dislocation densityσ generally an applied shear stress, sometimes normal stress (deviatoric)σ B brittle strengthσ 1, σ 2, σ 3 principal normal stressesσ c craze-border tractionσ C flow stress in compressionσ e uniaxial Mises equivalent axial stressσ S flow stress in shearσ T flow stress in tensionσ m mean normal stress (¼ σ n)σ TH thermal misfit negative pressure in particle

σ y (¼

Y ) uniaxial yield strengthσ von Mises equivalent stress (¼ σ e)σ ̂ ideal cavitation strength in UBER modelσ ∞

applied tensile stress promoting craze growthθ angular coordinateθ (¼ σ / μ(T )) reduced shear stress normalized with shear modulusθ (¼ T /T g) reduced temperature normalized with the glass transition

temperatureτ stress tensorτ time periodτ a shear resistance of amorphous component in HDPEτ c shear resistance of a crystalline component in HDPEτ ̂ threshold plastic shear resistance at T ¼ 0 Kτ ̂C threshold uniaxial plastic resistance in compression

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Frequently used abbreviations

CD constraint directionCN center-notchedCR compression ratio

DAM dry as moldedDEN double-edge-notchedDGEBA diglycidyl ether of bisphenol-A, a common epoxy resinFD free directionHDPE high-density polyethyleneHRR Hutchinson–Rice–Rosengren (model)KRO-1 a diblock resinLD loading directionPB polybutadienePMMA polymethyl methacrylatePS polystyrene

QSC quasi-single-crystalline (deformation texture)RH relative humidityRVE representative volume elementSANS small-angle neutron scatteringSAXS small-angle X-ray scatteringSEN single-edge-notchedST shear transformationTEM transmission electron microscopyWAXS wide-angle X-ray scattering


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1 Structure of non-polymeric glasses

1.1 Overview

The principal assignment of this book is to present the physics of inelastic

deformation and fracture of polymers, incorporating microstructural forms

ranging from fully disordered glassy polymers to semi-crystalline morphologies

of quite considerable crystalline perfection. While the semi-crystalline polymers

have few, if any, parallels in morphology among other solids, the glassy polymers

have such parallels in their atomic packing forms and morphologies in metallic

glasses and space-network glasses, which exhibit most of the forms of structural

relaxation, inelastic response, and fracture behavior of glassy polymers, albeit

often in modified forms and on somewhat different scales. Since these non-

polymeric glasses are free of the severe molecular-segmental-level topological

constraints, they exhibit the corresponding forms of mechanical response in afar simpler context, which is amenable to more precise analysis. For this reason we

start our assignment in this chapter by considering in some depth the hierarchical

details of atomic-packing forms of metallic glasses and those of amorphous silicon

as a surrogate for a space-network glass before we deal with the molecular

structure of glassy polymers and semi-crystalline polymers in Chapter 2. The

atomic structure of amorphous silicon, in particular, makes contact with other

directionally bonded covalent glasses and acts as a bridge between the densely

packed amorphous metals with close-to-isotropic atomic interaction and high

levels of atomic coordination and the structures of randomly snaking chain

molecules of polymer glasses.

In both cases, namely for amorphous metals and for space-network glasses, in

this chapter we develop important concepts such as free volume or liquid-like

atomic environments that both serve to promote structural rearrangements and

also play crucial roles in triggering shear relaxations under stress that can range

from few-atom clusters to far-reaching avalanches of plastic events. In every case,

however, the presentation of the quantitative details of the topology and kinetics

of such relaxations will be deferred to later chapters, where they are discussed

together with the corresponding phenomena in glassy polymers, using the simpler

processes in amorphous metals as guides to the more complex processes inpolymers. For metallic glasses, in which crystallization is suppressed and replaced

by disorder, or at best only by some short-to-medium-range order, the

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characteristic atomic packing can be reached operationally in a variety of ways.

These include, e.g., direct condensation from a vapor into a solid and irradiation

of a crystalline solid by energetic particles at relatively high fluences at low

temperatures at which reordering of knock-on atoms is largely suppressed. How-ever, the most widely used route for obtaining a metallic glass is by rapid cooling

of a complex alloy melt that is fast enough to override crystallization. It is this

route that will be of exclusive interest to us.

The first report of a metallic glass was that by Duwez and co-workers concern-

ing an Au–Si alloy (Klement et al. 1960). Since that time the science and technol-

ogy of the production of the metallic glasses has progressed from a scientific

curiosity to a very active area of materials science, leading to the development

of a myriad of increasingly more stable glasses with wide-ranging potential for

product applications.

To understand the thermodynamics and kinetics of formation of metallic

glasses through rapid cooling of an alloy melt of complex composition by overrid-

ing crystallization, a number of interrelated subjects need to be understood in

considerable detail. These include the evolving changes in atomic packing in sub-

cooled melts, the kinetics of inter-diffusion of the constituent atom species that are

part of the kinetics of atomic relaxations in the sub-cooled melts entering the glass

transition range, and the kinetics of the competing crystallization processes.

In the following sections we discuss first the atomic packing in sub-cooled alloy

melts near a glass transition, referring to results obtained from recent combin-

ations of modeling studies and associated experiments for some successful alloycompositions of metallic glasses. We follow this by considering the kinetics of

structural relaxations in some metallic glass compositions supported by actual

inter-diffusion experiments on constituent atom species. We contrast these

observations with competing forms and kinetics of polymorphic crystallization

processes in these compositions to arrive at classical time–temperature–

transformation (TTT) diagrams. Following these considerations of the structure

of sub-cooled alloy melts, we consider the all-important process of glass transition

from a point of view of cessation of percolation of unit structural relaxation

elements of atomic clusters possessing liquid-like character. Finally, employing

mechanistic considerations and their kinetics, we examine some very successful

metallic glass compositions that now permit one to obtain metallic glasses in

relatively bulky form with sufficient stability in the sub-cooled melt to permit

increasingly complex processing paths.

We follow the discussion of metallic glass alloys with a brief parallel consider-

ation of the behavior of amorphous silicon that is based primarily on computer

simulations that have not only introduced the corresponding behavior of space-

network glasses but also permitted a much deeper mechanistic understanding

both of structural relaxations, and, more importantly, of the nature of plastic

shear relaxations by ubiquitous shear transformations in glassy solids of alltypes, particularly in glassy polymers. These are developed in detail later in

Chapters 7 and 8.


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1.2 Glass formability in metallic alloys

The requirements for glass formability in metallic alloy melts by rapid quenching

have received much attention since the first report by Duwez and co-workers in

1960 on obtaining a metallic glass in an Au–Si alloy composition. Both laboratory

experiments and computer studies have established that melts of pure metals tend

to crystallize at such high rates that they cannot be quenched rapidly enough to

obtain a glass. Thus, obtaining an alloy glass requires satisfying a number of

interrelated conditions that stifle crystallization. Success in the early investigations

with binary metal–metalloid compositions of, e.g., Au–Si, Pd–Si, Fe–B, etc. with

atom number ratios of 4:1 between metal and metalloid ions of substantial atomic-

size difference already demonstrated the importance of atomic-size difference

between constituents to stabilize the melt and retard crystallization. Other relatedfactors that emerged as essential for glass formability include the presence of a

deep eutectic in the alloy composition that is beneficial in shortening the path

between the melt and the glass; a high viscosity of the sub-cooled alloy melt at the

liquidus range; and well-chosen alloy constituents requiring complex polymorphic

crystallization involving coupled, sluggish diffusive atom fluxes among alloy

constituents. Such fluxes produce topological and chemical short-range order that

minimizes free-energy differences between the sub-cooled melt and the crystalline

phase, and results in low levels of free volume in the sub-cooled melt at the glass-

transition range. These have all proved to be important factors for glass form-

ability. Detailed studies up to the present have demonstrated that many of these

requirements are not independent but emanate from a need for efficient atomic

packing in the sub-cooled melt, in which an important factor is the atomic-size

mismatch among the alloy constituents.

1.3 Atomic packing in disordered metallic solids

The atomic packing in disordered solids was investigated first by Bernal (1964),

who considered the problem in the context of a model of a simple liquid that

consisted of randomly close-packed hard spheres of uniform size and described

the structure as a distribution of five different canonical polyhedra with well-

defined volume fractions.

A more realistic computer model of a disordered solid considering both attract-

ive and repulsive atom interactions, carried out by Finney (1970), gave very

similar results, establishing that the hard-sphere repulsive interactions did indeed

play a dominant role in the dense random packing of atoms. While these pioneer-

ing models for liquids transforming into disordered solids gave reasonable agree-

ment between the structure of the models and the radial distribution functions(RDFs) of atom positions of simple liquids (Bernal 1964), they severely under-

predicted the densities of liquids at melting (or by extension, the densities of

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glasses) in comparison with face-centered cubic (fcc) crystals, at a level of a density

reduction of around 13.5% (Miracle et al. 2003). In comparison the density

reduction of fcc crystals upon melting is only, on average, 4.5% (Brandes 1983).

To explain the large density difference between the models of dense randomly

packed uniform-sized spheres and the actual density of metallic glasses a number

of factors for more efficient packing of spheres were considered. Since the early

metallic-glass compositions were of metal–metalloid type such as Au–Si, Pd–Si,

and Fe–B with number ratios of 4:1 between metal and metalloid atoms at a size

ratio R of around 0.7, Polk (1972) proposed that the smaller solute metalloid

atoms might more nearly fit into the interstitial spaces of the metal ions, thereby

achieving a higher density. However, it was soon recognized that the interstitial

volumes between metal atoms in the glass are far too small to accommodate the

metalloid atoms without a large misfit strain. A number of more complete packing

exercises for spheres of different size ratios R between solute (metalloid) and

solvent (metal) atoms in binary systems (Visscher and Bolsterli 1972; Zheng

et al. 1995; Lee et al. 2003) showed conclusively that atomic-size differences

between constituents in the framework of dense random packing of hard spheres

could not by themselves account for the larger density difference between actualmetallic glasses and models. This is well demonstrated in Fig. 1.1, showing that the

packing density of 0.64 of the models of dense randomly packed uniform-sized

fcc structure

98% of fcc structure

Number Fraction of Smaller Spheres

P a c k i n g D e n s i t y

00.60

0.65

0.70

0.80

0.75

0.2 0.4

R = 0.60

R = 0.40

R = 0.20

R = 0.80

0.80.6 1

Fig. 1.1 Relative atomic packing density of a binary mixture of spheres with radius

ratios R ranging from 0.2 to 0.8, based on developments of Zheng et al. (1995)

(from Miracle et al. (2003): courtesy of Taylor and Francis).


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spheres remains well below the density of 0.74 of an fcc crystal, for all reasonable

number fractions of smaller solute atoms in binary alloys for all atom ratios R

down to 0.4 (Miracle et al. 2003). In a comprehensive study Egami and Waseda(1984) calculated critical solute-concentration limits that were based on determin-

ation of excess enthalpies of binary systems through evaluation of the elastic misfit

interactions between solute and solvent atoms before phase separation occurs.

However, this led to no further improvement in accounting for the density

disparity. The failure of these considerations resulted, in time, in a recognition

that the dense random-packing models of spheres do not represent the atom

packing in metallic glasses even when actual flexibility of atoms is considered

and that there must be quite considerable short-to-medium-range packing order of

atoms that results in the relatively high actual densities of metallic glasses. Thus,

from combined modeling and experimental structural studies of Miracle et al.

(2003), Miracle (2004a, 2004b, 2006), and Ma and co-workers (Sheng et al. 2006),

among others, it has emerged that a high degree of short-to-medium-range atomic

order exists in metallic-glass alloys. In alloys with a primary solvent component

and one or more solute components the principal packing order is in the form of

solute-centered polyhedra for all solute-to-solvent radius ratios R in the range

0.7–1.3. An excellent example of this is shown in Fig. 1.2, namely an Ni80P20binary glass where the principal solute (P) appears as the small black spheres

surrounded by solvent Ni atoms (dark gray) in the first icosahedral-type shells.

The light-gray spheres represent, in turn, Ni atoms shared by neighboring soluteatoms lying in shells outside those depicted in the figure (Sheng et al. 2006).

In alloys with other solute components, the latter also either form additional

Fig. 1.2 Model of solute-centered icosahedral type atom packing in an Ni80P20 binary glass

obtained through Monte Carlo modeling (from Sheng et al. (2006): courtesy of Nature).

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polyhedral shells in which these solutes are surrounded by other solvent Ni atoms,

or the additional solute atoms are preferentially accommodated in the interstitial

spaces of the solute-centered polyhedral shells (Miracle 2006). While there is very

good evidence for this icosahedral-type packing order (Sheng et al. 2006), it is not

based only on purely geometrical effects of fit of atoms of different size ratios R

but also arises because the free energy of the alloy is governed importantly by the

concentrations of the misfit-produced elastic strain energies of atoms in the

ordered polyhedral shells. The latter effects have been considered by Egami and

Waseda (1984) to lead to estimates of limits to the composition of glasses provided

by specific constituents. It is clear that the short-to-medium-range order that is

present in the sub-cooled melt is accentuated with decreasing temperature as the

elastic misfits are systematically reduced as much as possible by diffusional

exchanges of atoms. The existing evidence suggests that, e.g., in the most stable

Zr-based bulk metallic glass alloys the atomic ordering results in an increase in

density and a decrease in free volume in the alloy to a fractional concentration of a

mere 1%–2% just prior to reaching the glass transition where the kinetics of

atomic ordering decreases below a critical low level (Busch 2000). It is this form

of ordering that results in the very significant decrease in atomic mobility and

increase in viscosity of the sub-cooled melt that suppress crystallization in thesealloys. This is demonstrated well in Fig. 1.3 with the classical TTT diagram of the

Zr-based alloy Vitreloy-1 (Zr41.2Ti13.8Cu12.5Ni10Be22.5). Figure 1.3 introduces a

glass

supercooledliquid

crystalline

Log time (s)

600

700

800

900

1000

1100

T g

T liq

1 2 3 4

T e m p e r a t u r e , K

Fig. 1.3 A time–temperature–transformation diagram for the Vitreloy 1 glass-forming

liquid (▴, obtained using electrostatic levitation; , obtained using carbon crucibles) (fromBusch (2000): courtesy of TMS).


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number of important kinetic concepts, which we develop further in more detail in

subsequent sections. These include the equilibrium liquidus temperature T liqbelow which, depending on time or cooling rate, a number of different scen-

arios can develop. For short periods of time or higher cooling rates a super-cooled liquid is maintained in which the viscosity increases monotonically as

the temperature decreases. For longer periods of time or lower cooling rates

polymorphic crystallization sets in. The border between the supercooled liquid

and initiation of crystallization is given by the characteristic “C”-shaped curve

which has a critical nose at a location of 895 K and 60 s. Cooling rates faster

than those that merely graze the nose of the curve maintain the supercooled

liquid to lower temperatures, albeit with continued monotonic increase in

viscosity. Finally, depending on the cooling rate, the atomic mobility in the

supercooled liquid becomes too low to permit further structural relaxation

and atomic compaction at the given rate of cooling. Then, the supercooled

liquid undergoes a glass transition at T g that is higher the higher the cooling

rate, below which the excess volume per atom, the free volume, decreases only

very sluggishly. The rather long transformation time window of 60 s at the nose

permits comfortable cooling rates in the range of 1.0 K/s to avoid incipient

crystallization for many processing histories for such stable glasses. The tempera-

ture dependence of the viscosity of this alloy in its supercooled liquid region is

shown in Fig. 1.4 (Masuhr et al. 1999). The figure also shows the viscosities of

many pure metals at their melting points near the bottom. These viscosities are

typically three orders of magnitude lower than that of the Vitreloy 1 alloy,demonstrating why crystallization is extremely rapid in pure metals upon quench-

ing and the extreme difficulty for them to undergo a successful glass transition.

The vertical, upward-directed arrow in Fig. 1.4 shows where the glass transition

occurs, at a viscosity of 1012 Pa s, but that some fluid-like behavior still persists at

lower temperatures and higher viscosities.

1.4 Energetic characterization of the structure of metallic glasses

1.4.1 The atomic site stress tensor

A very important form of characterization of the structural state of disorder in a

glass is through the atomic site stress tensor introduced by Egami and Vitek

(1983). While all atoms in a crystal are in mechanical equilibrium in their orderly

arrangement that results in a low internal energy, in a glass all atoms are also in

mechanical equilibrium but experience very large misfit-induced forces of inter-

action with their neighbors. This permits one to define an atomic site stress tensor

τ resulting from the very substantial local interaction forces between atoms. Two

scalar invariants of this stress tensor, defined for every atomic site, namely theatomic site pressure, p, and the atomic site deviatoric stress, σ , are of most interest.

They are defined as

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p τ ð Þ ¼ 1=3ð Þtr τ ð Þ ð1:1Þ

and

σ τ ð Þ ¼jτ 1=3ð Þtr τ ð Þ I j ð1:2Þ

where tr stands for the trace of the tensor τ and Ι stands for the identity tensor

(Demkowicz and Argon 2005a). These two quantities furnish directionless scalar

measures of the size misfit and distortional misfit, respectively, of atomic sites.

Figures 1.5(a) and (b) show the distributions of the atomic site pressure and

deviatoric stress calculated by Egami and Vitek (1983) from a three-dimensional

(3D) computational model of a well-relaxed glass. The pressure distribution is

nearly symmetric by virtue of overall traction equilibrium since there are both

dilated and compacted domains. However, the overall volume fraction of the

dilatation somewhat dominates over the compaction because of the unsymmet-

rical character of the atomic binding potential around zero stress. The deviatoric

stress distribution, however, is always positive by definition and by virtue of non-directionality. We note that the tail end of the pressure distribution on the

negative side (i.e., positive mean normal stress) borders on levels of de-cohesion,

Temperature, K

1000/ T (1/K)

T (K) V i s c o s i t y ( P a s ) v

f / v m ( % )

Ti NiZr

Be Cu

T g

T liq

0.5 1.0

500 1000

0.5

0.0

1.0

4000 2000 1000 600

10 –2

100

102

104

106

108

1010

1012

1014

1016

10 –41.5

Fig. 1.4 The temperature dependence of the thermal equilibrium viscosity of liquid Vitreloy1 compared with experimental data from viscosity experiments (○), and beam-bending

experiments (□). The inset gives the temperature dependence of the free volume of this

alloy. The viscosities of many pure metals (▵), including Zr, Ti, Ni, Be, and Cu, are shown

close to the temperature axis (from Masuhr et al. (1999): courtesy of the APS).


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albeit over only atomic dimensions, while the high-end tail of the deviatoric stress

distribution is a large fraction of the shear modulus. These characteristics of

atomic sites in glasses emphasize the very important fact that glassy solids store

a very substantial excess enthalpy of disorder, which has its origin in the elastic

strain energies associated with the atomic site structural misfit.

1.4.2 Calorimetry

A precise way of monitoring the thermodynamic properties of a glassy metal is

accomplished through differential scanning calorimetry (DSC), in which the

amount of heat required to increase the temperature of a sample is measured in

comparison with that of a reference sample with well-known heat capacity. The

technique supplies very useful information on the onset of property changes such

as the glass transition and phase changes such as crystallization or melting as well

as distinguishing thermodynamic-property differences such as levels of excess

enthalpy of disorder and specific heat associated with different thermal and

mechanical treatments. For example, Fig. 1.6 shows three DSC scans for a

Pd40Ni40P20 metallic-glass alloy heated at a rate of 20 K/min. The solid line

represents the gradually increasing heat flow into an initially quenched sample

as it transitions from a glassy solid into an under-cooled liquid at a glass-transition

temperature of around 585 K. The dotted and dashed lines, on the other hand,

show the similar endothermic transitions in samples pre-annealed at 540 K for

1.0 h and 50 h, respectively. Clearly, the much more stabilized sample with the 50 h

of annealing required considerably more heat input before undergoing the transi-

tion. If crystallization had set in above the glass transition with a strong negative

heat flow, as an exothermic process, a significant dip would have occurred in thescan. Alternatively, the occurrence of melting would produce a substantial upward

peak (Duine et al. 1992).

–20 20 20 301000

100

200

100

200

300

0.0

p (GPa)

N (p )(a) (b)

N (s /√3)

s /√3 (GPa)

Fig. 1.5 Histogram of smoothed Gaussian distributions of (a) atomic site pressure p and

(b) atomic site deviatoric stress σ , calculated from a model of amorphous Fe (from Egami

and Vitek (1983): courtesy of the Metallurgical Society of AIME).

91.4 Energetic characterization

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1.5 Free volume

A concept of critical importance in understanding atomic mobility in glass-

forming liquids and even glasses, that is referred to as free volume, was introduced

by Fox and Flory (1950), and all subsequent theoretical developments on atomic

mobility in disordered structures and their structural relaxation processes have

been based on this concept. In a disordered structure like a dense liquid, for an

atom to migrate, room must be provided in its immediate neighborhood for it tomove into. In such structures atoms occupy, on average, volumes v equal to or

larger than v0, the van der Waals volume of the atom, or its size in an ordered

reference structure. When the actual size of the volume the atom occupies in the

structure exceeds a critical value vc (vc> v0 ), locally the excess can be considered as

free volume. Then atomic transport occurs only when momentary voids of some

critical size v* approximately equal to the atomic volume v0 appear by redistri-

bution of the local free volume as a result of fluctuations. In a liquid such

redistribution of free volume is considered not to require overcoming an energy

barrier. Fox and Flory (1950) defined the local free volume as

vf ¼ v v0 ð1:3Þ

using v0 rather than vc, which needs a more precise definition that will be given below.

600 625575550525 –1

Temperature, K

H e a t f l o w ( m W )

0

1

2

3

4

Fig. 1.6 DSC scans of thermal effects in Pd40Ni40P20 with a heating rate of 20 K/min: the

solid line is for a quenched sample; the dotted line and dashed lines are for samples

pre-annealed at 540 K for 1.0 h and 50 h, respectively (from Duine et al. (1992): courtesy

of Pergamon Press).


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Using the above ideas, Doolittle (1951) proposed an expression for the viscosity

η of a liquid given by

η ¼ η0

exp bv0=vf ð Þ ð1:4Þ

where both η0 and b were considered adjustable parameters and vf represents the

volume-average free volume. This equation was quite successful in representing

the viscosities of simple hydrocarbon fluids. Earlier Vogel (1921), Fulcher (1925),

and Tamman and Hesse (1926) quite independently proposed a different-

appearing expression for the viscosity of molten-oxide glasses that had the form

ln η ¼ A þ B

T T 0ð1:5Þ

with A, B, and T 0 being often considered as adjustable constants, but having

distinct physical meanings. The constant B, e.g., having the dimension of tempera-

ture, is the reciprocal of the difference between the volumetric coefficient of

thermal expansion of the liquid and that of the solid below the glass-transition

temperature. This so-called Vogel–Fulcher–Tamman (or VFT) equation, which

was arrived at entirely empirically, has proved to be very successful in representing

the equilibrium viscosities of many sub-cooled liquids, including metal alloys and

some molten-oxide glasses at high temperature in the melt region. A quick exam-

ination shows that the VFT equation is identical with the Doolittle equation if it is

recognized that

A ¼ ln η0 ð1:6aÞ

and

vf ¼ v0 b= Bð Þ T T 0ð Þ ð1:6bÞ

where the ratio b/ B must be a constant having the dimension of reciprocal

temperature. With this comparison, the free volume is seen to be linearly

temperature-dependent, representing a form of thermal expansion of the liquid

that is always substantially larger than that of its solid form. We note that on this

comparison the free volume vanishes at T ¼ T 0 and that the viscosity according to

Eq. (1.4) becomes unbounded, being based on trends having no experimental

support, requiring further refinement of the concept.

Cohen and Turnbull (1959) clarified the physical significance of the Doolittle

equation by demonstrating that the probability P(v*) of finding a hole of size v* or

larger in a field of free volume can be expressed as

P vð Þ ¼ exp γv

vf

ð1:7Þ

and that, following the arguments of Fox and Flory, this would lead to a diffusion

constant

D ¼ gau exp γv=vf ð Þ ð1:8Þ

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where a* is the molecular diameter, g a geometrical constant of order unity, and u

the gas kinetic velocity. Moreover, since the Stokes–Einstein relation for dilute

systems, which gives

D ¼ kT = 3pað Þð ÞF ð1:9Þ

that relates the fluidity, F ¼ 1/η, to the diffusion constant D for dilute systems, is

also largely applicable to the liquids of interest here, an expression for the shear

viscosity η follows directly as

η ¼ η0 exp γv

vf

ð1:10Þ

which is the Doolittle equation, where η0 ¼ kT =3pðaÞ2u

and has the dimensions

of Pa s, if g is taken as 1.0.

Finally, the average free volume, given by

vf ¼ αv0 T T 0ð Þ ð1:11Þ

where α is the volumetric thermal coefficient of expansion, provides the physical

basis of the VFT relation for the shear viscosity. The form of eq. ( 1.11) with

T 0 vc, in which atoms are more weakly bound to




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their neighbors and are surrounded by an excess of volume, were referred to as liquid-

like (LL). In this picture the local free volume, vf , is redefined more precisely as

vf ¼ v vc ð1:12Þ

Since the volume per atom v is temperature-dependent and will vary with fluctu-

ations, the free volume will also fluctuate and in a liquid will wander around, not

necessarily being associated with specific atoms. In the Cohen and Grest model

the probability distribution of v, P(v), and the local free-energy function, f (v),sketched in Figs. 1.7(a) and (b) permit the determination of a specific expression

for the average free volume vf and its temperature dependence given by

0

v 0 v c v

v

f (v )

(b)

0

v 0 v 1

(a)f (v )

Fig. 1.7 A schematic representation of the free-energy function f (v), which is dependent

primarily on the atomic volume at an atom site: (a) the binding-energy plot showing

equilibrium volume v0 and inflexion point v1; (b) division of f (v) into two parts, consisting

of a central, strongly bonded harmonic part with v < vc and a linear part with v > vc used todefine solid-like, SL, and liquid-like, LL, atomic environments (from Cohen and Grest

(1979): courtesy of the APS).

131.5 Free volume

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vf ¼ k

A T T 0ð Þ þ T T 0ð Þ

2 þ 2 AvaT

k

1=2( ) ð1:13Þ

where k is Boltzmann’s constant and other parameters, such as A representing ashape parameter of the free-energy function having the dimensions of modulus

(Pa), and va and T 0, with dimensions of volume and temperature, respectively, are

all derived from the free-volume model and have similar meanings to those used

above, but can be treated as adjustable. Unlike the free-volume expression of

eq. (1.3) or eq. (1.11), which vanishes at T ¼ T 0, the expression of eq. (1.13)

vanishes only when T → 0 and removes a critical flaw in eqs. (1.6b) and (1.11).

When used in the Doolittle equation, the free-volume expression of eq. (1.13)

provides the best fit to the thermal-equilibrium viscosity of sub-cooled melts and

dense liquids over the widest range of temperature. In Fig. 1.4 it gives the solid

curve capturing the entire range of experimental measurements of the viscosity of

Vitreloy 1, while the temperature dependence of the free-volume expression vf of

eq. (1.13) is shown in the inset of this figure, as normalized by vm(¼v0), the van der

Waals volume in the alloy melt (Masuhr et al. 1999).

1.6 Viscosity of glass-forming liquids

To better understand the viscosity of sub-cooled liquids, the formation of metallicglasses, and the kinetics of their glass transition, as well as the kinetics of the

competing processes of crystallization, it is useful to view their viscosity in the

context of a broader collection of other potentially glass-forming liquids. Angell

(1995), who has considered this comparison in quite considerable detail, has

introduced an insightful classification grading liquids in a range from strong to

fragile depending on the form of the temperature dependence of their viscosity.

Figure 1.8 shows an Angell plot of a limited set of glass-forming liquids chosen for

the purpose of a comparison of the alloys that form bulk metallic glasses (BMGs)

with some other liquids. Since these liquids have vastly different relaxation pro-

cesses, the temperature scale is normalized with the glass-transition temperature

T g of the liquids considered. For this purpose the glass transition is defined as

occurring when the viscosity of the liquid reaches 1012 Pas.

At one limit, that of the strong liquids, are liquids of covalent and strongly

directionally bonded types such as SiO2, GeO2, and molten Si, all of which

maintain their directionally bonded character in the liquid state. The temperature

dependence of these liquids is of Arrhenius type, which, in the context of the VFT

framework of viscosities represented by eq. (1.5), means that T 0 vanishes and

B ¼Δ Fv/ R, where Δ Fv is the activation energy of viscous flow and R is the universal

gas constant. The Arrhenius dependence is shown by the straight line for SiO 2 inFig. 1.8. In the context of distributed structural relaxation processes, rather than

mono-energetic types, which we discuss in Section 1.8, Δ Fv refers to the terminal


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activation energy of a characteristic spectrum of activation energies for flow of a

disordered medium, in which the low end of the spectrum represents those relax-

ations of low volume fraction that are readily accomplished, while Δ Fv represents

the most sluggish background relaxation that governs global flow. In these strong

liquids diffusion and structural relaxations are thermally assisted directly rather

than being governed indirectly by free-volume fluctuations as in the VFT model of

diffusion and fluidity that reflects the behavior of liquids bound by non-directional

metallic bonds or by van der Waals interactions.

At the other limit of the gradation of fluidity response are the fragile liquids that

exhibit viscosities with a VFT type of temperature dependence reflecting atomic

mobility governed indirectly by the volume fluctuation as discussed in Section 1.5.

In Fig. 1.8, glycerol and KCa(NO3) are two such fragile liquids. To account for

the differences in behavior of fragile liquids in the VFT framework, Angell (1995)

introduced a so-called fragility constant D into the VFT viscosity relation, giving

η ¼ η0 exp DB= T T 0ð Þð Þ ð1:14Þ

where D is of the order of unity, i.e., 1–3, etc., for the fragile liquids, whereas for

the behavior of strong liquids D is in the range of 100 or so. Clearly, apart from

the presence of T 0, the increasing fragility constant reflects an increase of theactivation energy for flow for strong liquids. In Fig. 1.8 the behaviors of

the BMG-forming liquids of Vitreloy 1 and 4 and the metallic alloy melt of

fragile

strong

glycerol

T g* / T

T gSiO2 = 1410 K

T gV1 = 606 K

T mV1 = 1030 K

K+Ca2+(NO3)3–

Na2O2˙SiO2

Zr41.2Ti13.8 Cu12.8Ni10Be22.5(V1)

Zr46.75Ti8.25Cu7.5Ni10Be27.5(V4)

Mg65

Cu25

Y10

103

107

1011

10 –5

10 –1

0.4 0.6 0.8 1.0

V i s c o s i t y ( P a s )

SiO2

Fig 1.8 An Angell plot of the dependences of the viscosities of a series of liquids on their

reduced temperature T =T g, defining strong (Arrhenian) and fragile (VFT) liquids (fromBusch (2000): courtesy of TMS).

151.6 Viscosity of glass-forming liquids

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Mg65Cu25Y10 are intermediate between those of fragile and strong liquids and

demonstrate their success in retarding crystallization.

1.7 Structural relaxations

1.7.1 A computational model

The free-energy function sketched out in Fig. 1.7, from which free volume is

defined, indicates that only LL cells with v> vc possess free volume, while those

with v< vc that are SL cells provide negligible free volume. In this characteriza-

tion, the SL cells are expected to be stiff, whereas LL cells are much more

compliant both to dilatation and to distortion. Thus, clusters of cells with large

LL fractions of atom environments would be expected to play a dominant role in

structural relaxations. It would then follow that in a liquid or sub-cooled melt that

readily changes shape LL environments must percolate through the structure at all

times on the average. This condition that exists at the melting point of an alloy

must then continue to hold also in the sub-cooled melt region. As the temperature

decreases and structural relaxations continue steadily, albeit at a lower rate, the

LL material fraction must decrease systematically, resulting in progressive, large

increases in viscosity as, e.g., the trend in Fig. 1.4 shows. In all cases, however, the

LL fraction of atom environments must continue to percolate through the struc-

ture to maintain fluid behavior. At a certain temperature T g at which the percola-tion of the LL environments ceases for a characteristic period of observation, the

SL fraction will form a topologically continuous stiff background and the alloy

begins to exhibit solid behavior. This represents a glass transition at T g, below

which some LL environments may still be present but in frozen-in and isolated

form. The systematic decrease of the LL environments in the sub-cooled melt

between the melting point and the glass-transition temperature has been studied in

computer simulations. For example, in a two-dimensional (2D) molecular-

dynamics simulation of the structure of a generic Cu50Zr50 alloy melt and sub-

cooled liquid Deng et al. (1989a), using a 4–8 Lennard-Jones-type atomic potential

to represent the interactions of the Cu and Zr atoms, carried out limited studies of

melting and quenching in small mats of atoms under periodic boundary condi-

tions. With such models they explored both the topological features and the

kinetics of structural relaxations (Deng et al. 1989b, 1989c). Figure 1.9 shows

the dimensionless volume per atom v* as a function of dimensionless temperature

T * in the melting and quenching simulation, the details of which, including the

definitions of dimensionless quantities, can be found in Deng et al. (1989a). On

tessellating the 2D field of atoms in the melt and the sub-cooled liquid between T mand T g by Voronoi polygons they noted first that the field contained primarily two

distinct families of atom environments. One family was in the form of distortedhexagons of roughly similar size. These environments were labeled as solid-like,

SL. While the second family of environments consisted primarily of pentagons




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and heptagons, also occasionally present were a few squares and octagons. The

average area of these different types of polygons increased monotonically with

increasing number of sides as shown in Fig. 1.10. Of the non-hexagonal family of

polygons, the pentagons and heptagons showed a strong association into 5–7-

sided structural dipoles. Figure 1.10 demonstrates that the area (volume) of each

structural dipole was, on average, 7% larger than that of an average pair of

hexagons. These dipolar environments were labeled as liquid-like, LL, and were

considered as principal carriers of free volume. Separate simulations of plastic

deformation in such 2D mats demonstrated that the LL atom environments did

indeed act as the principal facilitator of plastic flow, confirming their anticipated

role in providing atomic mobility (Deng et al. 1989d). Figures 1.11(a) –(c) show

three 2D atom mats with a Voronoi tessellation at temperatures of (a) T ¼ 1:2T m,

(b) T g


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showed that LL environments moved over atom sites rapidly without muchdiscernible actual exchange of atoms, indicating that in general environments

are transient and are not necessarily locked onto specific atoms.

It is useful to consider the fluctuating disorder in the liquid in Fig. 1.11(a) as a

graphical representation of the configurational entropy, Sc, attributable to the

presence of the LL component. This can be defined along conventional lines as

Sc ¼ k ln N =2ð Þ!= N =2 nð Þ!n!½ ð1:15Þ

where n is the number of dipolar LL sites and N represents the total number of

sites, or N /2 the total number of pairs of either 5–7-sided LL polygons or the 6–6-

sided SL environments. The maximum of Sc occurs at n ¼ N /4, or when n repre-

sents half of the total of pair sites or an LL site concentration φ ¼ 0.5. We take this

as a characteristic of the flow state.

The graphical representation of percolation of LL environments above T g and

the absence of such percolation below T g are in broad agreement with the Cohen

and Grest (1979) model of the structure of simple atomic liquids and glasses

discussed in Section 1.5. Two-dimensional simulations of the type discussed above

have been carried out also by others and have largely supported the picture

sketched above (Perrera and Harrowell 1999; Hentschel et al. 2007).

The relatively simple simulations of the topology of structural disorder andassociated atom mobility hav

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