Computer Simulations of Dislocations

by Bulatov, Vasily; Cai, Wei

• ISBN13: 9780198526148
• ISBN10: 0198526148
• Format: Hardcover
• Copyright: 2006-12-28
• Publisher: Oxford University Press

Summary

This book presents a broad collection of models and computational methods - from atomistic to continuum - applied to crystal dislocations. Its purpose is to help students and researchers in computational materials sciences to acquire practical knowledge of relevant simulation methods.Because their behavior spans multiple length and time scales, crystal dislocations present a common ground for an in-depth discussion of a variety of computational approaches, including their relative strengths, weaknesses and inter-connections. The details of the covered methods are presented inthe form of "numerical recipes" and illustrated by case studies. A suite of simulation codes and data files is made available on the book's website to help the reader "to learn-by-doing" through solving the exercise problems offered in the book.

Author Biography

Vasily V Bulatov
Awards:
NSF Career Award (2006)
Edward Teller Fellow (2006)
Fellow of the American Physical Society (2005)
Wei Cai
Awards:
Manson Benedict Fellow, Department of Nuclear Engineering, MIT (1999)
Graduate Student Award, Materials Research Society (2000)
The Lawrence Fellowship, Lawrence Livermore National Laboratory (2001)
Presidential Early Career Award for Scientists and Engineers (2004)
Frederick E. Terman Fellowship, Stanford University (2004-2007)

Table of Contents

1 Introduction to Crystal Dislocations	1
1.1 Perfect Crystal Structures	2
1.1.1 Lattices and Bases	2
1.1.2 Miller Indices	5
Summary	5
Problems	6
1.2 The Concept of Crystal Dislocations	6
1.2.1 How to Make a Dislocation	6
1.2.2 The Burgers Vector	10
Summary	13
Problems	14
1.3 Motion of a Crystal Dislocation	15
1.3.1 Driving Forces for Dislocation Motion	16
1.3.2 Conservative versus Non-conservative Motion	18
1.3.3 Atomistic Mechanisms of Dislocation Motion	20
Summary	23
I ATOMISTIC MODELS
2 Fundamentals of Atomistic Simulations	27
2.1 Interatomic Interactions	27
2.1.1 Interatomic Potential Models	28
2.1.2 Locality of Interatomic Interactions	32
2.1.3 Computational Cost of Interatomic Interaction Models	33
Summary	35
Problem	36
2.2 Equilibrium Distribution	36
Summary	38
Problems	39
2.3 Energy Minimization	39
2.3.1 The Steepest-descent Method	40
2.3.2 Conjugate Gradient Relaxation	41
2.3.3 Global Minimization	42
Summary	43
2.4 Monte Carlo	44
2.4.1 Average over the Configurational Space	44
2.4.2 Designing a Stochastic Monte Carlo Process	45
2.4.3 Metropolis Algorithm	46
Summary	47
Problems	47
2.5 Molecular Dynamics	48
2.5.1 The Verlet Algorithm	49
2.5.2 The Velocity Verlet Algorithm	49
2.5.3 Energy Conservation	50
Summary	51
Problems	51
3 Case Study of Static Simulation	53
3.1 Setting up an Initial Configuration	53
Summary	57
Problem	57
3.2 Boundary Conditions	58
3.2.1 Periodic Boundary Conditions	59
Summary	62
Problem	62
3.3 Data Analysis and Visualization	63
Summary	67
Problems	68
4 Case Study of Dynamic Simulation	69
4.1 Setting up an Initial Configuration	69
Summary	71
4.2 Initializing Atomic Velocities	71
Summary	74
Problem	74
4.3 Stress Control	75
Summary	77
4.4 Temperature Control	78
4.4.1 Ensembles and Extended Systems	78
4.4.2 Nose—Hoover Thermostat	79
Summary	81
Problem	81
4.5 Extracting Dislocation Velocity	81
Summary	83
Problem	83
5 More about Periodic Boundary Conditions	84
5.1 Setting up an Initial Configuration	85
5.1.1 A Naive Approach	85
5.1.2 Conditional Convergence and the Linear Error Field	87
5.1.3 Adjusting the Shape of the Supercell	89
Summary	91
Problem	92
5.2 Dislocation Core Energy	92
Summary	95
Problems	96
5.3 Peierls Stress	96
Summary	98
Problems	98
6 Free-energy Calculations	101
6.1 Introduction to Free Energy	101
6.1.1 Free Energies of Configurational States	102
Summary	104
6.2 Harmonic Approximation	104
6.2.1 Free Energy of a Vacancy in the Dislocation Core	106
6.2.2 Bookkeeping of the Free Energies	107
Summary	108
Problems	109
6.3 Beyond Harmonic Approximation	109
6.3.1 Free Energy of a Core Vacancy Revisited	113
Summary	116
Problems	116
7 Finding Transition Pathways	117
7.1 The Rare Event Problem	117
Summary	119
7.2 Transition State Theory	119
Summary	121
7.3 Local Path Optimization	122
7.3.1 Constrained Minimization	122
7.3.2 The Chain of States Method	124
7.3.3 Case Study: Kink Migration in Silicon	126
Summary	128
7.4 Global Path Optimization	128
7.4.1 Case Study I: The Two-dimensional Potential	130
7.4.2 Case Study II: Kink Migration in Silicon Revisited	131
Summary	132
7.5 Temperature-accelerated Sampling	132
7.5.1 Case Study: Kink Migration in Silicon Yet Again	134
Summary	136
II CONTINUUM MODELS
8 Peierls–Nabarro Model of Dislocations	139
8.1 Model Formulation	140
8.1.1 Volterra Model Versus Peierls–Nabarro Model of Dislocations	140
8.1.2 Elastic Energy of the PN Dislocation	142
8.1.3 The Misfit Energy	143
8.1.4 The Analytic Solution	145
Summary	146
Problems	146
8.2 Numerical Solutions	147
8.2.1 The Sinusoid Misfit Potential	148
8.2.2 Misfit Potential with Intermediate Minima	149
Summary	152
8.3 Extension to Two Displacement Components	152
8.3.1 Numerical Solutions	156
8.3.2 Stress Effects	158
Summary	160
Problems	160
8.4 Further Extensions	162
8.4.1 Lattice Resistance to Dislocation Motion	162
8.4.2 Non-locality of Dislocation Energy	163
8.4.3 More Complex Dislocation Geometries	163
Summary	163
Problems	164
9 Kinetic Monte Carlo Method	166
9.1 Non-interacting Model	167
9.1.1 Model Formulation	169
9.1.2 Markov Chains	171
9.1.3 Kinetic Monte Carlo Algorithm	171
Summary	174
Problems	174
9.2 Dealing with Elastic Interactions	176
9.2.1 Model Formulation	176
9.2.2 Expressions for the Elastic Energy of a Kinked Dislocation	176
9.2.3 Energy of a Kink Pair	179
9.2.4 Computing Energy Changes due to Segment Motion	180
9.2.5 Kinetic Monte Carlo Simulation Results	182
Summary	183
9.3 Kink Pair Nucleation as a Rare Event	183
9.3.1 Sustainable Kink Pair Nucleation	184
9.3.2 Survival Probability	185
9.3.3 Average Time of First Arrival	188
9.3.4 Probability Distribution for the First Arrival Time	189
9.3.5 An Enhanced kMC Simulation	192
Summary	194
Problems	195
10 Line Dislocation Dynamics	196
10.1 Nodal Representation and Forces	197
10.1.1 Nodal Representation of Dislocation Networks	197
10.1.2 Energy and Forces	199
10.1.3 Elastic Energy and Force Contributions	199
10.1.4 Core Energy and Force Contributions	203
10.1.5 Periodic Boundary Conditions	204
Summary	205
Problems	205
10.2 Nodal Mobility Functions	208
10.2.1 A Linear Mobility Model	209
10.2.2 A Mobility Model for FCC Metals	211
10.2.3 A Mobility Model for BCC Metals	213
Summary	214
Problems	214
10.3 Time Integrators	214
10.3.1 An Example: Frank—Read Source	216
Summary	218
10.4 Topological Changes	218
10.4.1 Remeshing	219
10.4.2 Split and Merge Operators	221
10.4.3 When and How to Use merge	223
10.4.4 When and How to split a Node	224
10.4.5 A Complete Line DD Algorithm	226
10.4.6 Frank—Read Source Revisited	226
Summary	227
Problems	228
10.5 Parallel Simulations	229
10.5.1 Scalability	230
10.5.2 Spatial Domain Decomposition	231
10.5.3 Dynamic Load Balance	233
10.5.4 Single-Processor Performance	235
Summary	236
Problem	236
10.6 A Virtual Straining Test	236
10.6.1 Mobility Function	237
10.6.2 Boundary and Initial Conditions	237
10.6.3 Loading Condition	237
10.6.4 Results	239
Summary	240
11 Phase Field Method	241
11.1 General Phase Field Approach	242
11.1.1 Case A (Relaxational or Ginzburg–Landau)	243
11.1.2 Case B (Diffusional or Cahn–Hilliard)	248
Summary	250
Problems	251
11.2 Dislocations as Phase-Field Objects	251
11.2.1 The Elastic Energy	254
11.2.2 The Lattice Energy	254
11.2.3 The Gradient Energy	255
Summary	256
11.3 Elastic Energy of Eigenstrain Fields	256
11.3.1 Useful Expressions in the Fourier Space	259
11.3.2 Stress Expressions in Two Dimensions	260
Summary	261
Problem	261
11.4 A Two-Dimensional Example	262
Summary	264
Problems	265
11.5 Dislocation–Alloy Interaction	266
Summary	269
Problems	269
11.6 PFM or Line DD?	271
Summary	274
Bibliography	275
Subject Index	281