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COMPUTATIONAL
MATERIALS ENGINEERING
An Introduction to Microstructure Evolution
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COMPUTATIONAL
MATERIALS ENGINEERING
An Introduction to Microstructure Evolution
KOENRAAD G. F. JANSSENS
DIERK RAABE
ERNST KOZESCHNIK
MARK A. MIODOWNIK
BRITTA NESTLER
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier
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Library of Congress Cataloging-in-Publication Data
Computational materials engineering: an introduction to microstructure evolution/editors Koenraad G. F.
Janssens ... [et al.].
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-12-369468-3 (alk. paper)
ISBN-10: 0-12-369468-X (alk. paper)
1. Crystals–Mathematical models. 2. Microstructure–Mathematical models. 3. Polycrystals–Mathematical
models. I. Janssens, Koenraad G. F., 1968-
TA418.9.C7C66 2007
548
.7–dc22
2007004697
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
ISBN 13: 978-0-12-369468-3
ISBN 10: 0-12-369468-X
For all information on all Elsevier Academic Press publications
visit our Web site at www.books.elsevier.com
Printed in the United States of America
07 08 09 10 11 12 10 9 8 7 6 5 4 3 2 1
to Su
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Table of Contents
Preface xiii
1 Introduction 1
1.1 Microstructures Defined 1
1.2 Microstructure Evolution 2
1.3 Why Simulate Microstructure Evolution? 4
1.4 Further Reading 5
1.4.1 On Microstructures and Their Evolution from
a Noncomputational Point of View 5
1.4.2 On What Is Not Treated in This Book 6
2 Thermodynamic Basis of Phase Transformations 7
2.1 Reversible and Irreversible Thermodynamics 8
2.1.1 The First Law of Thermodynamics 8
2.1.2 The Gibbs Energy 11
2.1.3 Molar Quantities and the Chemical Potential 11
2.1.4 Entropy Production and the Second Law of
Thermodynamics 12
2.1.5 Driving Force for Internal Processes 15
2.1.6 Conditions for Thermodynamic Equilibrium 16
2.2 Solution Thermodynamics 18
2.2.1 Entropy of Mixing 19
2.2.2 The Ideal Solution 21
2.2.3 Regular Solutions 22
2.2.4 General Solutions in Multiphase Equilibrium 25
2.2.5 The Dilute Solution Limit—Henry’s and Raoult’s Law 27
2.2.6 The Chemical Driving Force 28
2.2.7 Influence of Curvature and Pressure 30
2.2.8 General Solutions and the CALPHAD Formalism 33
2.2.9 Practical Evaluation of Multicomponent Thermodynamic
Equilibrium 40
vii
3 Monte Carlo Potts Model 47
3.1 Introduction 47
3.2 Two-State Potts Model (Ising Model) 48
3.2.1 Hamiltonians 48
3.2.2 Dynamics (Probability Transition Functions) 49
3.2.3 Lattice Type 50
3.2.4 Boundary Conditions 51
3.2.5 The Vanilla Algorithm 53
3.2.6 Motion by Curvature 54
3.2.7 The Dynamics of Kinks and Ledges 57
3.2.8 Temperature 61
3.2.9 Boundary Anisotropy 62
3.2.10 Summary 64
3.3 Q-State Potts Model 64
3.3.1 Uniform Energies and Mobilities 65
3.3.2 Self-Ordering Behavior 67
3.3.3 Boundary Energy 68
3.3.4 Boundary Mobility 72
3.3.5 Pinning Systems 75
3.3.6 Stored Energy 77
3.3.7 Summary 80
3.4 Speed-Up Algorithms 80
3.4.1 The Boundary-Site Algorithm 81
3.4.2 The N-Fold Way Algorithm 82
3.4.3 Parallel Algorithm 84
3.4.4 Summary 87
3.5 Applications of the Potts Model 87
3.5.1 Grain Growth 87
3.5.2 Incorporating Realistic Textures and Misorientation
Distributions 89
3.5.3 Incorporating Realistic Energies and Mobilities 92
3.5.4 Validating the Energy and Mobility Implementations 93
3.5.5 Anisotropic Grain Growth 95
3.5.6 Abnormal Grain Growth 98
3.5.7 Recrystallization 102
3.5.8 Zener Pinning 103
3.6 Summary 107
3.7 Final Remarks 107
3.8 Acknowledgments 108
4 Cellular Automata 109
4.1 A Definition 109
4.2 A One-Dimensional Introduction 109
4.2.1 One-Dimensional Recrystallization 111
4.2.2 Before Moving to Higher Dimensions 111
4.3 +2D CA Modeling of Recrystallization 116
4.3.1 CA-Neighborhood Definitions in Two Dimensions 116
4.3.2 The Interface Discretization Problem 118
4.4 +2D CA Modeling of Grain Growth 123
viii TABLE OF CONTENTS
4.4.1 Approximating Curvature in a Cellular Automaton Grid 124
4.5 A Mathematical Formulation of Cellular Automata 128
4.6 Irregular and Shapeless Cellular Automata 129
4.6.1 Irregular Shapeless Cellular Automata for Grain Growth 131
4.6.2 In the Presence of Additional Driving Forces 135
4.7 Hybrid Cellular Automata Modeling 136
4.7.1 Principle 136
4.7.2 Case Example 137
4.8 Lattice Gas Cellular Automata 140
4.8.1 Principle—Boolean LGCA 140
4.8.2 Boolean LGCA—Example of Application 142
4.9 Network Cellular Automata—A Development for the Future? 144
4.9.1 Combined Network Cellular Automata 144
4.9.2 CNCA for Microstructure Evolution Modeling 145
4.10 Further Reading 147
5 Modeling Solid-State Diffusion 151
5.1 Diffusion Mechanisms in Crystalline Solids 151
5.2 Microscopic Diffusion 154
5.2.1 The Principle of Time Reversal 154
5.2.2 A Random Walk Treatment 155
5.2.3 Einstein’s Equation 157
5.3 Macroscopic Diffusion 160
5.3.1 Phenomenological Laws of Diffusion 160
5.3.2 Solutions to Fick’s Second Law 162
5.3.3 Diffusion Forces and Atomic Mobility 164
5.3.4 Interdiffusion and the Kirkendall Effect 168
5.3.5 Multicomponent Diffusion 171
5.4 Numerical Solution of the Diffusion Equation 174
6 Modeling Precipitation as a Sharp-Interface Phase Transformation 179
6.1 Statistical Theory of Phase Transformation 181
6.1.1 The Extended Volume Approach—KJMA Kinetics 181
6.2 Solid-State Nucleation 185
6.2.1 Introduction 185
6.2.2 Macroscopic Treatment of Nucleation—Classical Nucleation
Theory 186
6.2.3 Transient Nucleation 189
6.2.4 Multicomponent Nucleation 191
6.2.5 Treatment of Interfacial Energies 194
6.3 Diffusion-Controlled Precipitate Growth 197
6.3.1 Problem Definition 199
6.3.2 Zener’s Approach for Planar Interfaces 202
6.3.3 Quasi-static Approach for Spherical Precipitates 203
6.3.4 Moving Boundary Solution for Spherical Symmetry 205
6.4 Multiparticle Precipitation Kinetics 206
6.4.1 The Numerical Kampmann–Wagner Model 206
6.4.2 The SFFK Model—A Mean-Field Approach for Complex
Systems 209
6.5 Comparing the Growth Kinetics of Different Models 215
TABLE OF CONTENTS ix
7 Phase-Field Modeling 219
7.1 A Short Overview 220
7.2 Phase-Field Model for Pure Substances 222
7.2.1 Anisotropy Formulation 224
7.2.2 Material and Model Parameters 226
7.2.3 Application to Dendritic Growth 226
7.3 Case Study 228
7.3.1 Phase-Field Equation 229
7.3.2 Finite Difference Discretization 229
7.3.3 Boundary Values 231
7.3.4 Stability Condition 232
7.3.5 Structure of the Code 232
7.3.6 Main Computation 233
7.3.7 Parameter File 236
7.3.8 MatLab Visualization 237
7.3.9 Examples 238
7.4 Model for Multiple Components and Phases 241
7.4.1 Model Formulation 241
7.4.2 Entropy Density Contributions 242
7.4.3 Evolution Equations 245
7.4.4 Nondimensionalization 248
7.4.5 Finite Difference Discretization and Staggered Grid 249
7.4.6 Optimization of the Computational Algorithm 252
7.4.7 Parallelization 253
7.4.8 Adaptive Finite Element Method 253
7.4.9 Simulations of Phase Transitions and Microstructure
Evolution 253
7.5 Acknowledgments 265
8 Introduction to Discrete Dislocations Statics and Dynamics 267
8.1 Basics of Discrete Plasticity Models 267
8.2 Linear Elasticity Theory for Plasticity 268
8.2.1 Introduction 268
8.2.2 Fundamentals of Elasticity Theory 269
8.2.3 Equilibrium Equations 273
8.2.4 Compatibility Equations 274
8.2.5 Hooke’s Law—The Linear Relationship between Stress and
Strain 275
8.2.6 Elastic Energy 280
8.2.7 Green’s Tensor Function in Elasticity Theory 280
8.2.8 The Airy Stress Function in Elasticity Theory 283
8.3 Dislocation Statics 284
8.3.1 Introduction 284
8.3.2 Two-Dimensional Field Equations for Infinite Dislocations
in an Isotropic Linear Elastic Medium 285
8.3.3 Two-Dimensional Field Equations for Infinite Dislocations
in an Anisotropic Linear Elastic Medium 287
8.3.4 Three-Dimensional Field Equations for Dislocation Segments
in an Isotropic Linear Elastic Medium 289
x TABLE OF CONTENTS
8.3.5 Three-Dimensional Field Equations for Dislocation Segments
in an Anisotropic Linear Elastic Medium 292
8.4 Dislocation Dynamics 298
8.4.1 Introduction 298
8.4.2 Newtonian Dislocation Dynamics 299
8.4.3 Viscous and Viscoplastic Dislocation Dynamics 307
8.5 Kinematics of Discrete Dislocation Dynamics 310
8.6 Dislocation Reactions and Annihilation 311
9 Finite Elements for Microstructure Evolution 317
9.1 Fundamentals of Differential Equations 317
9.1.1 Introduction to Differential Equations 317
9.1.2 Solution of Partial Differential Equations 320
9.2 Introduction to the Finite Element Method 321
9.3 Finite Element Methods at the Meso- and Macroscale 322
9.3.1 Introduction and Fundamentals 322
9.3.2 The Equilibrium Equation in FE Simulations 324
9.3.3 Finite Elements and Shape Functions 324
9.3.4 Assemblage of the Stiffness Matrix 327
9.3.5 Solid-State Kinematics for Mechanical Problems 329
9.3.6 Conjugate Stress–Strain Measures 331
Index 335
TABLE OF CONTENTS xi
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Preface
Being actively involved since 1991 in different research projects that belong under the field of
computational materials science, I always wondered why there is no book on the market which
introduces the topic to the beginning student. In 2005 I was approached by Joel Stein to write
a book on this topic, and I took the opportunity to attempt to do so myself. It was immediately
clear to me that such a task transcends a mere copy and paste operation, as writing for experts is
not the same as writing for the novice. Therefore I decided to invite a respectable collection of
renowned researchers to join me on the endeavor. Given the specific nature of my own research,
I chose to focus the topic on different aspects of computational microstructure evolution. This
book is the result of five extremely busy and active researchers taking a substantial amount of
their (free) time to put their expertise down in an understandable, self-explaining manner. I am
very grateful for their efforts, and hope the reader will profit. Even if my original goals were
not completely met (atomistic methods are missing and there was not enough time do things as
perfectly as I desired), I am satisfied with—and a bit proud of—the result.
Most chapters in this book can be pretty much considered as stand-alones. Chapters 1 and 2
are included for those who are at the very beginning of an education in materials science; the
others can be read in any order you like.
Finally, I consider this book a work in progress. Any questions, comments, corrections, and
ideas for future and extended editions are welcome at [email protected]. You may also
want to visit the web site accompanying this book at http:// books.elsevier.com/companions/
9780123694683.
Koen Janssens,
Linn, Switzerland,
December 2006
xiii
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