Preface ..................................................... xi
1 Introduction to materials modeling and simulation ............ 1
1.1 Modeling and simulation ................................. 1
1.2 What is meant by computational materials science and
engineering? ............................................ 2
1.3 Scales in materials structure and behavior .............. 3
1.4 How to develop models ................................... 5
1.5 Summary ................................................. 7
PART ONE. SOME BASICS
2 The random-walk model ....................................... 11
2.1 Random-walk model of diffusion ......................... 11
2.2 Connection to the diffusion coefficient ................ 13
2.3 Bulk diffusion ......................................... 18
2.4 A random-walk simulation ............................... 19
2.5 Random-walk models for materials ....................... 25
2.6 Summary ................................................ 26
3 Simulation of finite systems ................................ 27
3.1 Sums of interacting pairs of objects ................... 27
3.2 Perfect crystals ....................................... 29
3.3 Cutoffs ................................................ 31
3.4 Periodic boundary conditions ........................... 32
3.5 Implementation ......................................... 34
3.6 Long-ranged potentials ................................. 35
3.7 Summary ................................................ 36
3.8 Appendix ............................................... 36
PART TWO. ATOMS AND MOLECULES
4 Electronic structure methods ................................ 45
4.1 Quantum mechanics of multielectron systems ............. 46
4.2 Early density functional theories ...................... 47
4.3 The Hohenberg-Kohn theorem ............................. 51
4.4 Kohn-Sham method ....................................... 51
4.5 The exchange-correlation functional .................... 54
4.6 Wave functions ......................................... 55
4.7 Pseudopotentials ....................................... 57
4.8 Use of density functional theory ....................... 59
4.9 Summary ................................................ 61
5 Interatomic potentials ...................................... 62
5.1 The cohesive energy .................................... 62
5.2 Interatomic potentials ................................. 63
5.3 Pair potentials ........................................ 67
5.4 Ionic materials ........................................ 76
5.5 Metals ................................................. 78
5.6 Covalent solids ........................................ 84
5.7 Systems with mixed bonding ............................. 88
5.8 What we can simulate ................................... 89
5.9 Determining parameters in potentials ................... 91
5.10 Summary ................................................ 91
5.11 Appendix ............................................... 92
6 Molecular dynamics .......................................... 96
6.1 Basics of molecular dynamics for atomic systems ........ 96
6.2 An example calculation ................................ 107
6.3 Velocity rescaling .................................... 116
6.4 Molecular dynamics in other ensembles ................. 117
6.5 Accelerated dynamics .................................. 120
6.6 Limitations of molecular dynamics ..................... 122
6.7 Molecular dynamics in materials research .............. 123
6.8 Summary ............................................... 125
6.9 Appendix .............................................. 125
7 The Monte Carlo method .................................... 131
7.1 Introduction .......................................... 131
7.2 Ensemble averages ..................................... 132
7.3 The Metropolis algorithm .............................. 134
7.4 The Ising model ....................................... 139
7.5 Monte Carlo for atomic systems ........................ 145
7.6 Other ensembles ....................................... 150
7.7 Time in a Monte Carlo simulation ...................... 154
7.8 Assessment of the Monte Carlo method .................. 155
7.9 Uses of the Monte Carlo method in materials research .. 155
7.10 Summary ............................................... 156
7.11 Appendix .............................................. 156
8 Molecular and macromolecular systems ....................... 158
8.1 Introduction .......................................... 158
8.2 Random-walk models of polymers ........................ 161
8.3 Atomistic simulations of macromolecules ............... 163
8.4 Coarse-grained methods ................................ 172
8.5 Lattice models for polymers and biomolecules .......... 175
8.6 Simulations of molecular and macromolecular
materials ............................................. 176
8.7 Summary ............................................... 177
8.8 Appendix .............................................. 178
PART THREE. MESOSCOPIC METHODS
9 Kinetic Monte Carlo ........................................ 183
9.1 The kinetic Monte Carlo method ........................ 183
9.2 Time in the kinetic Monte Carlo method ................ 187
9.3 Kinetic Monte Carlo calculations ...................... 189
9.4 Applications .......................................... 194
9.5 Summary ............................................... 195
10 Monte Carlo methods at the mesoscale ....................... 196
10.1 Modeling Grain Growth ................................. 196
10.2 The Monte Carlo Potts model ........................... 198
10.3 The N-fold way ........................................ 202
10.4 Example applications of the Potts model ............... 205
10.5 Applications in materials science and engineering ..... 208
10.6 Summary ............................................... 210
11 Cellular automata .......................................... 211
11.1 Basics of cellular automata ........................... 211
11.2 Examples of cellular automata in two dimensions ....... 215
11.3 Lattice-gas methods ................................... 218
11.4 Examples of cellular automata in materials research ... 219
11.5 Relation to Monte Carlo ............................... 227
11.6 Summary ............................................... 227
12 Phase-field methods ........................................ 229
12.1 Conserved and non-conserved order parameters .......... 229
12.2 Governing equations ................................... 230
12.3 A one-dimensional phase-field calculation ............. 233
12.4 Free energy of an interface ........................... 237
12.5 Local free-energy functions ........................... 238
12.6 Two examples .......................................... 241
12.7 Other applications in materials research .............. 244
12.8 Summary ............................................... 244
12.9 Appendix .............................................. 244
13 Mesoscale dynamics ......................................... 249
13.1 Damped dynamics ....................................... 249
13.2 Langevin dynamics ..................................... 251
13.3 Simulation "entities" at the mesoscale ................ 252
13.4 Dynamic models of grain growth ........................ 253
13.5 Discrete dislocation dynamics simulations ............. 256
13.6 Summary ............................................... 265
PART FOUR. SOME FINAL WORDS
14 Materials selection and design ............................. 269
14.1 Integrated computational materials engineering ........ 269
14.2 Concurrent materials design ........................... 271
14.3 Methods ............................................... 273
14.4 Materials informatics ................................. 275
14.5 Summary ............................................... 278
PART FIVE. APPENDICES
A Energy units, fundamental constants, and conversions ....... 281
A.l Fundamental constants ................................. 281
A.2 Units and energy conversions .......................... 281
В A brief introduction to materials .......................... 283
B.l Introduction .......................................... 283
B.2 Crystallography ....................................... 284
B.3 Defects ............................................... 291
B.4 Point defects ......................................... 291
B.5 Dislocations .......................................... 292
B.6 Polycrystalline materials ............................. 302
B.7 Diffusion ............................................. 306
С Mathematical background .................................... 310
C.l Vectors and tensors ................................... 310
C.2 Taylor series ......................................... 314
C.3 Complex numbers ....................................... 315
C.4 Probability ........................................... 317
C.5 Common functions ...................................... 318
C.6 Functionals ........................................... 321
D A brief summary of classical mechanics ..................... 324
D.l Newton's equations .................................... 324
D.2 The Hamiltonian ....................................... 326
D.3 Example: the harmonic oscillator ...................... 327
D.4 Central-force potentials .............................. 328
E Electrostatics ............................................. 330
E.l The force ............................................. 330
E.2 Electrostatic potentials and energies ................. 330
E.3 Distribution of charges: the multipole expansion ...... 331
F Elements of quantum mechanics .............................. 334
F.l History ............................................... 334
F.2 Wave functions ........................................ 335
F.3 The Schrödinger equation .............................. 336
E.4 Observables ........................................... 336
F.5 Some solved problems .................................. 337
F.6 Atoms with more than one electron ..................... 342
F.7 Eigenvalues and eigenvectors .......................... 346
F.8 Multielectron systems ................................. 348
F.9 Quantum mechanics of periodic systems ................. 349
F.10 Summary ............................................... 349
G Statistical thermodynamics and kinetics .................... 351
G.l Basic thermodynamic quantities ........................ 351
G.2 Introduction to statistical thermodynamics ............ 352
G.3 Macrostates versus microstates ........................ 352
G.4 Phase space and time averages ......................... 353
G.5 Ensembles ............................................. 355
G.6 Fluctuations .......................................... 362
G.7 Correlation functions ................................. 364
G.8 Kinetic rate theory ................................... 368
G.9 Summary ............................................... 374
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