Modeling Materials: Continuum, Atomistic and Multiscale Techniques

Modeling Meterials......Page 2
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 14
Acknowledgments......Page 17
Notation......Page 22
1.1.1 Orowan\'s pocket watch......Page 30
1.1.2 Mechanisms of plasticity......Page 32
1.1.3 Perfect crystals......Page 33
1.1.4 Planar defects: surfaces......Page 36
1.1.5 Planar defects: grain boundaries......Page 39
1.1.6 Line defects: dislocations......Page 41
1.1.7 Point defects......Page 44
1.1.8 Large-scale defects: cracks, voids and inclusions......Page 45
1.2 Materials scales: taking stock......Page 46
 Further reading......Page 47
PART I CONTINUUM MECHANICS AND THERMODYNAMICS......Page 48
2 Essential continuum mechanics and thermodynamics......Page 50
2.1.1 Tensor notation......Page 51
2.1.2 Vectors and higher-order tensors......Page 55
2.1.3 Tensor operations......Page 62
2.1.4 Properties of second-order tensors......Page 66
2.1.5 Tensor fields......Page 68
2.2.1 The continuum particle......Page 71
2.2.2 The deformation mapping......Page 72
2.2.3 Material and spatial descriptions......Page 73
2.2.4 Description of local deformation......Page 75
2.2.5 Kinematic rates......Page 78
2.3.1 Conservation of mass......Page 80
2.3.2 Balance of linear momentum......Page 82
2.3.3 Balance of angular momentum......Page 87
2.3.4 Material form of the momentum balance equations......Page 88
2.4.1 Macroscopic observables, thermodynamic equilibrium and state variables......Page 90
2.4.2 Thermal equilibrium and the zeroth law of thermodynamics......Page 94
2.4.3 Energy and the first law of thermodynamics......Page 96
2.4.4 Thermodynamic processes......Page 100
2.4.5 The second law of thermodynamics and the direction of time......Page 101
2.4.6 Continuum thermodynamics......Page 112
2.5 Constitutive relations......Page 119
2.5.1 Constraints on constitutive relations......Page 120
2.5.2 Local action and the second law of thermodynamics......Page 121
2.5.3 Material frame-indifference......Page 126
2.5.4 Material symmetry......Page 128
2.5.5 Linearized constitutive relations for anisotropic hyperelastic solids......Page 130
2.6 Boundary-value problems and the principle of minimum potential energy......Page 134
 Further reading......Page 137
 Exercises......Page 138
Part II ATOMISTICS......Page 142
3.1 Crystal history: continuum or corpuscular?......Page 144
3.3 Lattices......Page 148
3.3.1 Primitive lattice vectors and primitive unit cells......Page 149
3.3.2 Voronoi tessellation and the Wigner-Seitz cell......Page 151
3.3.3 Conventional unit cells......Page 152
3.3.4 Crystal directions......Page 153
3.4.1 Point symmetry operations......Page 154
3.4.2 The seven crystal systems......Page 158
3.5.1 Centering in the cubic system......Page 163
3.5.3 Centering in the monoclinic system......Page 166
3.5.5 Centering in the hexagonal and trigonal systems......Page 167
3.6 Crystal structure......Page 168
3.6.2 Crystal structures of some common crystals......Page 171
3.7.1 Fourier series and the reciprocal lattice......Page 175
3.7.2 The first Brillouin zone......Page 177
3.7.3 Miller indices......Page 178
 Exercises......Page 180
4.1 Introduction......Page 182
4.2 A brief and selective history of quantum mechanics......Page 183
4.2.1 The Hamiltonian formulation......Page 186
4.3.1 Dirac notation......Page 189
4.3.2 Electron wave functions......Page 192
4.3.3 Schrodinger\'s equation......Page 197
4.3.4 The time-independent Schrödinger equation......Page 200
4.3.5 The hydrogen atom......Page 201
4.3.6 The hydrogen molecule......Page 208
4.3.7 Summary of the quantum mechanics of bonding......Page 216
4.4.1 Exact formulation......Page 217
4.4.2 Approximations necessary for computational progress......Page 225
4.4.3 The choice of basis functions......Page 228
4.4.4 Electrons in periodic systems......Page 229
4.4.5 The essential machinery of a plane-wave DFT code......Page 239
4.4.6 Energy minimization and dynamics: forces in DFT......Page 250
4.5.1 LCAO......Page 252
4.5.2 The Hamiltonian and overlap matrices......Page 253
4.5.3 Slater-Koster parameters for two-center integrals......Page 256
4.5.5 TB molecular dynamics......Page 257
4.5.6 From TB to empirical atomistic models......Page 258
 Exercises......Page 264
5 Empirical atomistic models of materials......Page 266
5.1 Consequences of the Born-Oppenheimer approximation (BOA)......Page 267
5.2 Treating atoms as classical particles......Page 269
5.3 Sensible functional forms......Page 270
5.3.1 Interatomic distances......Page 271
5.3.2 Requirement of translational, rotational and parity invariance......Page 272
5.3.3 The cutoff radius......Page 274
5.4 Cluster potentials......Page 275
5.4.1 Formally exact cluster potentials......Page 276
5.4.2 Pair potentials......Page 280
5.4.3 Modeling ionic crystals: the Born-Mayer potential......Page 285
5.4.4 Three- and four-body potentials......Page 286
5.4.5 Modeling organic molecules: CHARMM and AMBER......Page 288
5.4.6 Limitations of cluster potentials and the need for interatomic functionals......Page 290
5.5 Pair functionals......Page 291
5.5.1 The generic pair functional form: the glue-EAM-EMT-FS model......Page 292
5.5.2 Physical interpretations of the pair functional......Page 293
5.5.3 Fitting the pair functional model......Page 294
5.5.4 Comparing pair functionals to cluster potentials......Page 295
5.6.1 Introduction to the bond order: the Tersoff potential......Page 297
5.6.2 Bond energy and bond order in TB......Page 300
5.6.3 ReaxFF......Page 303
5.6.4 The modified embedded atom method......Page 305
5.7.1 Speed and scaling: how many atoms over how much time?......Page 308
5.7.2 Transferability: predicting behavior outside the fit......Page 311
5.7.3 Classes of materials and our ability to model them......Page 314
5.8.1 Weak and strong laws of action and reaction41......Page 317
5.8.2 Forces in conservative systems......Page 320
5.8.3 Atomic forces for some specific interatomic models......Page 323
5.8.4 Bond stiffnesses for some specific interatomic models......Page 326
5.8.5 The cutoff radius and interatomic forces......Page 327
 Further reading......Page 328
 Exercises......Page 329
6.1 The potential energy landscape......Page 333
6.2.1 Solving nonlinear problems: initial guesses......Page 335
6.2.2 The generic nonlinear minimization algorithm......Page 336
6.2.3 The steepest descent (SD) method......Page 337
6.2.4 Line minimization......Page 339
6.2.5 The conjugate gradient (CG) method......Page 340
6.2.6 The condition number......Page 341
6.2.7 The Newton–Raphson (NR) method......Page 342
6.3 Methods for finding saddle points and transition paths......Page 344
6.3.1 The nudged elastic band (NEB) method......Page 345
6.4.1 Neighbor lists......Page 350
6.4.2 Periodic boundary conditions (PBCs)......Page 354
6.4.3 Applying stress and pressure boundary conditions......Page 357
6.4.4 Boundary conditions on atoms......Page 359
6.5 Application to crystals and crystalline defects......Page 360
6.5.1 Cohesive energy of an infinite crystal......Page 361
6.5.2 The universal binding energy relation (UBER)......Page 363
6.5.3 Crystal defects: vacancies......Page 367
6.5.4 Crystal defects: surfaces and interfaces......Page 368
6.5.5 Crystal defects: dislocations......Page 376
6.5.6 The γ-surface......Page 386
6.5.7 The Peierls–Nabarro model of a dislocation......Page 389
 Further reading......Page 400
 Exercises......Page 401
PART III ATOMISTIC FOUNDATIONS OF CONTINUUM CONCEPTS......Page 404
7 Classical equilibrium statistical mechanics......Page 406
7.1.1 Hamilton\'s equations......Page 407
7.1.2 Macroscopic translation and rotation......Page 408
7.1.3 Center of mass coordinates......Page 409
7.1.4 Phase space coordinates......Page 410
7.1.5 Trajectories through phase space......Page 411
7.1.6 Liouville\'s theorem......Page 413
7.2.1 Time averages......Page 416
7.2.2 The ensemble viewpoint and distribution functions......Page 418
7.2.3 Why does the ensemble approach work?......Page 421
7.3.1 The hypersurface and volume of an isolated Hamiltonian system......Page 432
7.3.2 The microcanonical distribution function......Page 435
7.3.3 Systems in weak interaction......Page 438
7.3.4 Internal energy, temperature and entropy......Page 441
7.3.5 Derivation of the ideal gas law......Page 447
7.3.6 Equipartition and virial theorems: microcanonical derivation......Page 449
7.4 The canonical (NVT) ensemble......Page 452
7.4.1 The canonical distribution function......Page 453
7.4.2 Internal energy and fluctuations......Page 457
7.4.3 Helmholtz free energy......Page 458
7.4.4 Equipartition theorem: canonical derivation......Page 460
7.4.5 Helmholtz free energy in the thermodynamic limit......Page 461
 Further reading......Page 466
 Exercises......Page 467
8 Microscopic expressions for continuum fields......Page 469
8.1.1 Canonical transformations......Page 471
8.1.2 Microscopic stress tensor in a finite system at zero temperature......Page 476
8.1.3 Microscopic stress tensor at finite temperature: the virial stress......Page 479
8.1.4 Microscopic elasticity tensor......Page 489
8.2 Continuum fields as expectation values: nonequilibrium systems......Page 494
8.2.1 Rate of change of expectation values......Page 495
8.2.2 Definition of pointwise continuum fields......Page 496
8.2.4 Momentum balance and the pointwise stress tensor......Page 498
8.2.5 Spatial averaging and macroscopic fields......Page 504
8.3 Practical methods: the stress tensor......Page 508
8.3.1 The Hardy stress......Page 509
8.3.2 The virial stress tensor and atomic-level stresses......Page 510
8.3.3 The Tsai traction: a planar definition for stress......Page 511
8.3.4 Uniqueness of the stress tensor......Page 516
8.3.5 Hardy, virial and Tsai stress expressions: numerical considerations......Page 517
 Exercises......Page 518
9.1 Brief historical introduction......Page 521
9.2 The essential MD algorithm......Page 524
9.3.1 Integrating the NVE ensemble: the velocity-Verlet (VV) algorithm......Page 526
9.3.3 Temperature initialization......Page 533
9.4 The NVT ensemble: constant temperature and constant strain......Page 536
9.4.1 Velocity rescaling......Page 537
9.4.2 Gauss´ principle of least constraint and the isokinetic thermostat......Page 538
9.4.3 The Langevin thermostat......Page 540
9.4.4 The Nosé-Hoover (NH) thermostat......Page 542
9.4.5 Liouville’s equation for non-Hamiltonian systems......Page 545
9.4.6 An alternative derivation of the NH thermostat......Page 546
9.4.7 Integrating the NVT ensemble......Page 547
9.5 The finite strain NσE ensemble: applying stress......Page 549
9.5.1 A canonical transformation of variables......Page 550
9.5.2 The hydrostatic stress state......Page 556
9.5.3 The Parrinello-Rahman (PR) approximation......Page 557
9.5.4 The zero-temperature limit: applying stress in molecular statics......Page 559
9.6 The NσT ensemble: applying stress at a constant temperature......Page 562
 Exercises......Page 563
PART IV MULTISCALE METHODS......Page 566
10.1 Multiscale modeling: what is in a name?......Page 568
10.2 Sequential multiscale models......Page 570
10.3 Concurrent multiscale models......Page 572
10.3.1 Hierarchical methods......Page 573
10.3.2 Partitioned-domain methods......Page 575
10.4 Spanning time scales......Page 576
 Further reading......Page 578
11 Atomistic constitutive relations for multilattice crystals......Page 579
11.1.1 Restricted ensembles......Page 583
11.1.2 Properties of a metastable state from a restricted canonical ensemble......Page 585
11.2.1 Multilattice crystals and mean positions......Page 587
11.2.2 Cauchy-Born kinematics......Page 588
11.2.3 Centrosymmetric crystals and the Cauchy-Born rule......Page 590
11.2.4 Extensions and failures of the Cauchy-Born rule......Page 591
11.3.1 Periodic supercell of a multilattice crystal......Page 592
11.3.2 Helmholtz free energy density of a multilattice crystal......Page 595
11.3.3 Determination of the reference configuration......Page 596
11.3.4 Uniform deformation and the macroscopic stress tensor......Page 599
11.3.5 Elasticity tensor......Page 604
11.4.1 Quasiharmonic Helmholtz free energy......Page 607
11.4.2 Determination of the quasiharmonic reference configuration......Page 611
11.4.3 Quasiharmonic stress and elasticity tensors......Page 615
11.4.4 Strict harmonic approximation......Page 619
11.5.1 General expressions for the stress and elasticity tensors......Page 621
11.5.2 Stress and elasticity tensors for some specific interatomic models......Page 622
11.5.3 Crystal symmetries and the Cauchy relations......Page 624
 Exercises......Page 627
12.1 Finite elements and the Cauchy–Born rule......Page 630
12.2 The essential components of a coupled model......Page 633
12.3.1 Total energy functional......Page 637
12.3.2 The quasi-continuum (QC) method......Page 639
12.3.3 The coupling of length scales (CLS) method......Page 642
12.3.4 The bridging domain (BD) method......Page 643
12.3.5 The bridging scale method (BSM)......Page 645
12.3.6 CACM: iterative minimization of two energy functionals......Page 646
12.3.7 Cluster-based quasicontinuum (CQC-E)......Page 647
12.4 Ghost forces in energy-based methods......Page 649
12.4.1 A one-dimensional Lennard-Jones chain of atoms......Page 651
12.4.3 Ghost forces in a generic energy-based model of the chain......Page 652
12.4.4 Ghost forces in the cluster-based quasicontinuum (CQC-E)......Page 656
12.4.5 Ghost force correction methods......Page 659
12.5.1 Forces without an energy functional......Page 660
12.5.2 FEAt and CADD......Page 662
12.5.4 The atomistic-to-continuum (AtC) method......Page 663
12.5.6 Spurious forces in force-based methods......Page 665
12.6.1 A simple example:shearing a twin boundary......Page 667
12.6.2 Setting up the model......Page 669
12.6.3 Solution procedure......Page 671
12.6.4 Twin boundary migration......Page 673
12.6.5 Automatic model adaption......Page 674
12.7 Quantitative comparison between the methods......Page 676
12.7.1 The test problem......Page 677
12.7.2 Comparing the accuracy of multiscale methods......Page 679
12.7.3 Quantifying the speed of multiscale methods......Page 683
12.7.4 Summary of the relative accuracy and speed of multiscale methods......Page 684
 Exercises......Page 685
13 Atomistic–continuum coupling: finite temperature and dynamics......Page 687
13.1 Dynamic finite elements......Page 688
13.2 Equilibrium finite temperature multiscale methods......Page 690
13.2.1 Effective Hamiltonian for the atomistic region......Page 691
13.2.2 Finite temperature QC framework......Page 696
13.2.3 Hot-QC-static:atomistic dynamics embedded in a static continuum......Page 699
13.2.4 Hot-QC-dynamic: atomistic and continuum dynamics......Page 701
13.2.5 Demonstrative examples: thermal expansion and nanoindentation......Page 704
13.3 Nonequilibrium multiscale methods......Page 706
13.3.2 Wave reflections......Page 707
13.3.3 Generalized Langevin equations......Page 712
13.3.4 Damping bands......Page 716
 Exercises......Page 718
A: Mathematical representation of interatomic potentials......Page 719
A.1 Interatomic distances and invariance......Page 720
A.2 Distance geometry: constraints between interatomic distances......Page 722
A.3 Continuously differentiable extensions of Vint(s)......Page 725
A.4 Alternative potential energy extensions and the effect on atomic forces......Page 727
References......Page 731
Index......Page 775