Preface ...................................................... xiii
Acknowledgments ............................................... xvi
Notation ...................................................... xxi
1 Introduction ................................................. 1
1.1 Multiple scales in crystalline materials ................ 1
1.1.1 Orowan's pocket watch ............................ 1
1.1.2 Mechanisms of plasticity ......................... 3
1.1.3 Perfect crystals ................................. 4
1.1.4 Planar defects: surfaces ......................... 7
1.1.5 Planar defects: grain boundaries ................ 10
1.1.6 Line defects: dislocations ...................... 12
1.1.7 Point defects ................................... 15
1.1.8 Large-scale defects: cracks, voids and
inclusions ...................................... 16
1.2 Materials scales: taking stock ......................... 17
Further reading ........................................ 18
Part I Continuum mechanics and thermodynamics .................. 19
2 Essential continuum mechanics and thermodynamics ............ 21
2.1 Scalars, vectors, and tensors .......................... 22
2.1.1 Tensor notation ................................. 22
2.1.2 Vectors and higher-order tensors ................ 26
2.1.3 Tensor operations ............................... 33
2.1.4 Properties of second-order tensors .............. 37
2.1.5 Tensor fields ................................... 39
2.2 Kinematics of deformation .............................. 42
2.2.1 The continuum particle .......................... 42
2.2.2 The deformation mapping ......................... 43
2.2.3 Material and spatial descriptions ............... 44
2.2.4 Description of local deformation ................ 46
2.2.5 Kinematic rates ................................. 49
2.3 Mechanical conservation and balance laws ............... 51
2.3.1 Conservation of mass ............................ 51
2.3.2 Balance of linear momentum ...................... 53
2.3.3 Balance of angular momentum ..................... 58
2.3.4 Material form of the momentum balance
equations ....................................... 59
2.4 Thermodynamics ......................................... 61
2.4.1 Macroscopic observables, thermodynamic
equilibrium and state variables ................. 61
2.4.2 Thermal equilibrium and the zeroth law of
thermodynamics .................................. 65
2.4.3 Energy and the first law of thermodynamics ...... 67
2.4.4 Thermodynamic processes ......................... 71
2.4.5 The second law of thermodynamics and the
direction of time ............................... 72
2.4.6 Continuum thermodynamics ........................ 83
2.5 Constitutive relations ................................. 90
2.5.1 Constraints on constitutive relations ........... 91
2.5.2 Local action and the second law of
thermodynamics .................................. 92
2.5.3 Material frame-indifference ..................... 97
2.5.4 Material symmetry ............................... 99
2.5.5 Linearized constitutive relations for
anisotropic hyperelastic solids ................ 101
2.6 Boundary-value problems and the principle of minimum
potential energy ...................................... 105
Further reading ....................................... 108
Exercises ............................................. 109
Part II Atomistic ............................................. 113
3 Lattices and crystal structures ............................ 115
3.1 Crystal history: continuum or corpuscular? ............ 115
3.2 The structure of ideal crystals ....................... 119
3.3 Lattices .............................................. 119
3.3.1 Primitive lattice vectors and primitive unit
cells .......................................... 120
3.3.2 Voronoi tessellation and the Wigner-Seitz
cell ........................................... 122
3.3.3 Conventional unit cells ........................ 123
3.3.4 Crystal directions ............................. 124
3.4 Crystal systems ....................................... 125
3.4.1 Point symmetry operations ...................... 125
3.4.2 The seven crystal systems ...................... 129
3.5 Bravais lattices ...................................... 134
3.5.1 Centering in the cubic system .................. 134
3.5.2 Centering in the triclinic system .............. 137
3.5.3 Centering in the monoclinic system ............. 137
3.5.4 Centering in the orthorhombic and tetragonal
systems ........................................ 138
3.5.5 Centering in the hexagonal and trigonal
systems ........................................ 138
3.5.6 Summary of the fourteen Bravais lattices ....... 139
3.6 Crystal structure ..................................... 139
3.6.1 Essential and nonessential descriptions of
crystals ....................................... 142
3.6.2 Crystal structures of some common crystals ..... 142
3.7 Some additional lattice concepts ...................... 146
3.7.1 Fourier series and the reciprocal lattice ...... 146
3.7.2 The first Brillouin zone ....................... 148
3.7.3 Miller indices ................................. 149
Further reading ....................................... 151
Exercises ............................................. 151
4 Quantum mechanics of materials ............................. 153
4.1 Introduction .......................................... 153
4.2 A brief and selective history of quantum mechanics .... 154
4.2.1 The Hamiltonian formulation ..................... 157
4.3 The quantum theory of bonding ......................... 160
4.3.1 Dirac notation ................................. 160
4.3.2 Electron wave functions ........................ 163
4.3.3 Schrцdinger's equation ......................... 168
4.3.4 The time-independent Schrцdinger equation ...... 171
4.3.5 The hydrogen atom .............................. 172
4.3.6 The hydrogen molecule .......................... 179
4.3.7 Summary of the quantum mechanics of bonding .... 187
4.4 Density functional theory (DFT) ....................... 188
4.4.1 Exact formulation .............................. 188
4.4.2 Approximations necessary for computational
progress ....................................... 196
4.4.3 The choice of basis functions .................. 199
4.4.4 Electrons in periodic systems .................. 200
4.4.5 The essential machinery of a plane-wave DFT
code ........................................... 210
4.4.6 Energy minimization and dynamics: forces in
DFT ............................................ 221
4.5 Semi-empirical quantum mechanics: tight-binding (ТВ)
methods ............................................... 223
4.5.1 LCAO ........................................... 223
4.5.2 The Hamiltonian and overlap matrices ........... 224
4.5.3 Slater-Koster parameters for two-center
integrals ...................................... 227
4.5.4 Summary of the ТВ formulation .................. 228
4.5.5 ТВ molecular dynamics .......................... 228
4.5.6 From ТВ to empirical atomistic models .......... 229
Further reading ....................................... 235
Exercises ............................................. 235
5 Empirical atomistic models of materials .................... 237
5.1 Consequences of the Born-Oppenheimer approximation
(BOA) ................................................. 238
5.2 Treating atoms as classical particles ................. 240
5.3 Sensible functional forms ............................. 241
5.3.1 Interatomic distances .......................... 242
5.3.2 Requirement of translational, rotational and
parity invariance .............................. 243
5.3.3 The cutoff radius .............................. 245
5.4 Cluster potentials .................................... 246
5.4.1 Formally exact cluster potentials .............. 247
5.4.2 Pair potentials ................................ 251
5.4.3 Modeling ionic crystals: the Born-Mayer
potential ...................................... 256
5.4.4 Three-and four-body potentials ................. 257
5.4.5 Modeling organic molecules: CHARMM and AMBER ... 259
5.4.6 Limitations of cluster potentials and the
need for interatomic functionals ............... 261
5.5 Pair functionals ...................................... 262
5.5.1 The generic pair functional form: the
glue-EAM-EMT-FS model .......................... 263
5.5.2 Physical interpretations of the pair
functional ..................................... 264
5.5.3 Fitting the pair functional model .............. 265
5.5.4 Comparing pair functionals to cluster
potentials ..................................... 266
5.6 Cluster functionals ................................... 268
5.6.1 Introduction to the bond order: the Tersoff
potential ...................................... 268
5.6.2 Bond energy and bond order in ТВ ............... 271
5.6.3 ReaxFF ......................................... 274
5.6.4 The modified embedded atom method .............. 276
5.7 Atomistic models: what can they do? ................... 279
5.7.1 Speed and scaling: how many atoms over how
much time? ..................................... 279
5.7.2 Transferability: predicting behavior outside
the fit ........................................ 282
5.7.3 Classes of materials and our ability to model
them ........................................... 285
5.8 Interatomic forces in empirical atomistic models ...... 288
5.8.1 Weak and strong laws of action and reaction .... 288
5.8.2 Forces in conservative systems ................. 291
5.8.3 Atomic forces for some specific interatomic
models ......................................... 294
5.8.4 Bond stiffnesses for some specific
interatomic models ............................. 297
5.8.5 The cutoff radius and interatomic forces ....... 298
Further reading ............................................ 299
Exercises .................................................. 300
6 Molecular statics .......................................... 304
6.1 The potential energy landscape ........................ 304
6.2 Energy minimization ................................... 306
6.2.1 Solving nonlinear problems: initial guesses .... 306
6.2.2 The generic nonlinear minimization algorithm ... 307
6.2.3 The steepest descent (SD) method ............... 308
6.2.4 Line minimization .............................. 310
6.2.5 The conjugate gradient (CG) method ............. 311
6.2.6 The condition number ........................... 312
6.2.7 The Newton-Raphson (NR) method ................. 313
6.3 Methods for finding saddle points and transition
paths ................................................. 315
6.3.1 The nudged elastic band (NEB) method ............ 316
6.4 Implementing molecular statics ........................ 321
6.4.1 Neighbor lists ................................. 321
6.4.2 Periodic boundary conditions (PBCs) ............ 325
6.4.3 Applying stress and pressure boundary
conditions ..................................... 328
6.4.4 Boundary conditions on atoms ................... 330
6.5 Application to crystals and crystalline defects ....... 331
6.5.1 Cohesive energy of an infinite crystal ......... 332
6.5.2 The universal binding energy relation (UBER) ... 334
6.5.3 Crystal defects: vacancies ..................... 338
6.5.4 Crystal defects: surfaces and interfaces ....... 339
6.5.5 Crystal defects: dislocations .................. 347
6.5.6 The γ-surface .................................. 357
6.5.7 The Peierls-Nabarro model of a dislocation ..... 360
6.6 Dealing with temperature and dynamics ................. 371
Further reading ............................................ 371
Exercises .................................................. 372
Part III Atomistic foundations of continuum concepts .......... 375
7 Classical equilibrium statistical mechanics ................ 377
7.1 Phase space: dynamics of a system of atoms ............ 378
7.1.1 Hamilton's equations ........................... 378
7.1.2 Macroscopic translation and rotation ........... 379
7.1.3 Center of mass coordinates ..................... 380
7.1.4 Phase space coordinates ........................ 381
7.1.5 Trajectories through phase space ............... 382
7.1.6 Liouville's theorem ............................ 384
7.2 Predicting macroscopic observables .................... 387
7.2.1 Time averages .................................. 387
7.2.2 The ensemble viewpoint and distribution
functions ...................................... 389
7.2.3 Why does the ensemble approach work? ........... 392
7.3 The microcanonical (NVE) ensemble ..................... 403
7.3.1 The hypersurface and volume of an isolated
Hamiltonian system ............................. 403
7.3.2 The microcanonical distribution function ....... 406
7.3.3 Systems in weak interaction .................... 409
7.3.4 Internal energy, temperature and entropy ....... 412
7.3.5 Derivation of the ideal gas law ................ 418
7.3.6 Equipartition and virial theorems:
microcanonical derivation ...................... 420
7.4 The canonical (NVT) ensemble .......................... 423
7.4.1 The canonical distribution function ............ 424
7.4.2 Internal energy and fluctuations ............... 428
7.4.3 Helmholtz free energy .......................... 429
7.4.4 Equipartition theorem: canonical derivation .... 431
7.4.5 Helmholtz free energy in the thermodynamic
limit .......................................... 432
Further reading ............................................ 437
Exercises .................................................. 438
8 Microscopic expressions for continuum fields ............... 440
8.1 Stress and elasticity in a system in thermodynamic
equilibrium ........................................... 442
8.1.1 Canonical transformations ...................... 442
8.1.2 Microscopic stress tensor in a finite system
at zero temperature ............................ 447
8.1.3 Microscopic stress tensor at finite
temperature: the virial stress ................. 450
8.1.4 Microscopic elasticity tensor .................. 460
8.2 Continuum fields as expectation values:
nonequilibrium systems ................................ 465
8.2.1 Rate of change of expectation values ........... 466
8.2.2 Definition of pointwise continuum fields ....... 467
8.2.3 Continuity equation ............................ 469
8.2.4 Momentum balance and the pointwise stress
tensor ......................................... 469
8.2.5 Spatial averaging and macroscopic fields ....... 475
8.3 Practical methods: the stress tenser .................. 479
8.3.1 The Hardy stress ............................... 480
8.3.2 The virial stress tensor and atomic-level
stresses ....................................... 481
8.3.3 The Tsai traction: a planar definition for
stress ......................................... 482
8.3.4 Uniqueness of the stress tensor ................ 487
8.3.5 Hardy, virial and Tsai stress expressions:
numerical considerations ....................... 488
Exercises .................................................. 489
9 Molecular dynamics ......................................... 492
9.1 Brief historical introduction ......................... 492
9.2 The essential MD algorithm ............................ 495
9.3 The NVE ensemble: constant energy and constant
strain ................................................ 497
9.3.1 Integrating the NVE ensemble: the velocity-
Verlet (VV) algorithm .......................... 497
9.3.2 Quenched dynamics .............................. 504
9.3.3 Temperature initialization ..................... 504
9.3.4 Equilibration time ............................. 507
9.4 The NVT ensemble: constant temperature and constant
strain ................................................ 507
9.4.1 Velocity rescaling ............................. 508
9.4.2 Gauss'principle of least constraint and the
isokinetic thermostat .......................... 509
9.4.3 The Langevin thermostat ........................ 511
9.4.4 The Nosé-Hoover (NH) thermostat ................ 513
9.4.5 Liouville's equation for non-Hamiltonian
systems ........................................ 516
9.4.6 An alternative derivation of the NH
thermostat ..................................... 517
9.4.7 Integrating the NVT ensemble ................... 518
9.5 The finite strain NσE ensemble: applying stress ....... 520
9.5.1 A canonical transformation of variables ........ 521
9.5.2 The hydrostatic stress state ................... 527
9.5.3 The Parrinello-Rahman (PR) approximation ....... 528
9.5.4 The zero-temperature limit: applying stress
in molecular statics ........................... 530
9.5.5 The kinetic energy of the cell ................. 533
9.6 The NσT ensemble: applying stress at a constant
temperature ........................................... 533
Further reading ............................................ 534
Exercises .................................................. 534
Part IV Multiscale methods .................................... 537
10 What is multiscale modeling? ............................... 539
10.1 Multiscale modeling: what is in a name? ............... 539
10.2 Sequential multiscale models .......................... 541
10.3 Concurrent multiscale models .......................... 543
10.3.1 Hierarchical methods ........................... 544
10.3.2 Partitioned-domain methods ..................... 546
10.4 Spanning time scales .................................. 547
Further reading ....................................... 549
11 Atomistic constitutive relations for multilattice
crystals ................................................... 550
11.1 Statistical mechanics of systems in metastable
equilibrium ........................................... 554
11.1.1 Restricted ensembles ........................... 554
11.1.2 Properties of a metastable state from
a restricted canonical ensemble ................ 556
11.2 Relating mean positions to applied deformation: the
Cauchy-Born rule ...................................... 558
11.2.1 Multilattice crystals and mean positions ....... 558
11.2.2 Cauchy-Born kinematics ......................... 559
11.2.3 Centrosymmetric crystals and the Cauchy-Born
rule ........................................... 561
11.2.4 Extensions and failures of the Cauchy-Born
rule ........................................... 562
11.3 Finite temperature constitutive relations for
multilattice crystals ................................. 563
11.3.1 Periodic supercell of a multilattice crystal ... 563
11.3.2 Helmholtz free energy density of
a multilattice crystal ......................... 566
11.3.3 Determination of the reference configuration ... 567
11.3.4 Uniform deformation and the macroscopic
stress tensor .................................. 570
11.3.5 Elasticity tensor .............................. 575
11.4 Quasiharmonic approximation ........................... 578
11.4.1 Quasiharmonic Helmholtz free energy ............ 578
11.4.2 Determination of the quasiharmonic reference
configuration .................................. 582
11.4.3 Quasiharmonic stress and elasticity tensors .... 586
11.4.4 Strict harmonic approximation .................. 590
11.5 Zero-temperature constitutive relations ............... 592
11.5.1 General expressions for the stress and
elasticity tensors ............................. 592
11.5.2 Stress and elasticity tensors for some
specific interatomic models .................... 593
11.5.3 Crystal symmetries and the Cauchy relations .... 595
Further reading ............................................ 598
Exercises .................................................. 598
12 Atomistic-continuum coupling: static methods ............... 601
12.1 Finite elements and the Cauchy-Born rule .............. 601
12.2 The essential components of a coupled model ........... 604
12.3 Energy-based formulations ............................. 608
12.3.1 Total energy functional ........................ 608
12.3.2 The quasi-continuum (QC) method ................ 610
12.3.3 The coupling of length scales (CLS) method ..... 613
12.3.4 The bridging domain (BD) method ................ 614
12.3.5 The bridging scale method (BSM) ................ 616
12.3.6 CACM: iterative minimization of two energy
functionals .................................... 617
12.3.7 Cluster-based quasicontinuum (CQC-E) ........... 618
12.4 Ghost forces in energy-based methods .................. 620
12.4.1 A one-dimensional Lennard-Jones chain of
atoms .......................................... 622
12.4.2 A continuum constitutive law for the Lennard-
Jones chain .................................... 623
12.4.3 Ghost forces in a generic energy-based model
of the chain ................................... 623
12.4.4 Ghost forces in the cluster-based
quasicontinuum (CQC-E) ......................... 627
12.4.5 Ghost force correction methods ................. 630
12.5 Force-based formulations .............................. 631
12.5.1 Forces without an energy functional ............ 631
12.5.2 FEAt and CADD .................................. 633
12.5.3 The hybrid simulation method (HSM) ............. 634
12.5.4 The atomistic-to-continuum (AtC) method ........ 634
12.5.5 Cluster-based quasicontinuum (CQC-F) ........... 636
12.5.6 Spurious forces in force-based methods ......... 636
12.6 Implementation and use of the static QC method ........ 638
12.6.1 A simple example: shearing a twin boundary ..... 638
12.6.2 Setting up the model ........................... 640
12.6.3 Solution procedure ............................. 642
12.6.4 Twin boundary migration ........................ 644
12.6.5 Automatic model adaption ....................... 645
12.7 Quantitative comparison between the methods ........... 647
12.7.1 The test problem ............................... 648
12.7.2 Comparing the accuracy of multiscale methods ... 650
12.7.3 Quantifying the speed of multiscale methods .... 654
12.7.4 Summary of the relative accuracy and speed of
multiscale methods ............................. 655
Exercises .................................................. 656
13 Atomistic-continuum coupling: finite temperature and
dynamics ................................................... 658
13.1 Dynamic finite elements ............................... 659
13.2 Equilibrium finite temperature multiscale methods ..... 661
13.2.1 Effective Hamiltonian for the atomistic
region ......................................... 662
13.2.2 Finite temperature QC framework ................ 667
13.2.3 Hot-QC-static: atomistic dynamics embedded in
a static continuum ............................. 670
13.2.4 Hot-QC-dynamic: atomistic and continuum
dynamics ....................................... 672
13.2.5 Demonstrative examples: thermal expansion and
nanoindentation ................................ 675
13.3 Nonequilibrium multiscale methods ..................... 677
13.3.1 A naive starting point ......................... 678
13.3.2 Wave reflections ............................... 678
13.3.3 Generalized Langevin equations ................. 683
13.3.4 Damping bands .................................. 687
13.4 Concluding remarks .................................... 689
Exercises .................................................. 689
Appendix A Mathematical representation of interatomic
potentials ................................................. 690
A.l Interatomic distances and invariance ................... 691
A.2 Distance geometry: constraints between interatomic
distances .............................................. 693
A.3 Continuously differentiable extensions of  int(s) ...... 696
A.4 Alternative potential energy extensions and the
effect on atomic forces ................................ 698
References .................................................... 702
Index ......................................................... 746
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