Chapter
2.3 Mechanical conservation and balance laws
2.3.1 Conservation of mass
2.3.2 Balance of linear momentum
2.3.3 Balance of angular momentum
2.3.4 Material form of the momentum balance equations
2.4.1 Macroscopic observables, thermodynamic equilibrium and state variables
2.4.2 Thermal equilibrium and the zeroth law of thermodynamics
2.4.3 Energy and the first law of thermodynamics
2.4.4 Thermodynamic processes
2.4.5 The second law of thermodynamics and the direction of time
2.4.6 Continuum thermodynamics
2.5 Constitutive relations
2.5.1 Constraints on constitutive relations
2.5.2 Local action and the second law of thermodynamics
2.5.3 Material frame-indifference
2.5.5 Linearized constitutive relations for anisotropic hyperelastic solids
2.6 Boundary-value problems and the principle of minimum potential energy
3 Lattices and crystal structures
3.1 Crystal history: continuum or corpuscular?
3.2 The structure of ideal crystals
3.3.1 Primitive lattice vectors and primitive unit cells
3.3.2 Voronoi tessellation and the Wigner-Seitz cell
3.3.3 Conventional unit cells
3.4.1 Point symmetry operations
3.4.2 The seven crystal systems
3.5.1 Centering in the cubic system
3.5.2 Centering in the triclinic system
3.5.3 Centering in the monoclinic system
3.5.4 Centering in the orthorhombic and tetragonal systems
3.5.5 Centering in the hexagonal and trigonal systems
3.5.6 Summary of the fourteen Bravais lattices
3.6.1 Essential and nonessential descriptions of crystals
3.6.2 Crystal structures of some common crystals
3.7 Some additional lattice concepts
3.7.1 Fourier series and the reciprocal lattice
3.7.2 The first Brillouin zone
4 Quantum mechanics of materials
4.2 A brief and selective history of quantum mechanics
4.2.1 The Hamiltonian formulation
4.3 The quantum theory of bonding
4.3.2 Electron wave functions
4.3.3 Schrodinger's equation
4.3.4 The time-independent Schrödinger equation
4.3.6 The hydrogen molecule
4.3.7 Summary of the quantum mechanics of bonding
4.4 Density functional theory (DFT)
4.4.2 Approximations necessary for computational progress
4.4.3 The choice of basis functions
4.4.4 Electrons in periodic systems
4.4.5 The essential machinery of a plane-wave DFT code
4.4.6 Energy minimization and dynamics: forces in DFT
4.5 Semi-empirical quantum mechanics: tight-binding (TB) methods
4.5.2 The Hamiltonian and overlap matrices
4.5.3 Slater-Koster parameters for two-center integrals
4.5.4 Summary of the TB formulation
4.5.5 TB molecular dynamics
4.5.6 From TB to empirical atomistic models
5 Empirical atomistic models of materials
5.1 Consequences of the Born-Oppenheimer approximation (BOA)
5.2 Treating atoms as classical particles
5.3 Sensible functional forms
5.3.1 Interatomic distances
5.3.2 Requirement of translational, rotational and parity invariance
5.4.1 Formally exact cluster potentials
5.4.3 Modeling ionic crystals: the Born-Mayer potential
5.4.4 Three- and four-body potentials
5.4.5 Modeling organic molecules: CHARMM and AMBER
5.4.6 Limitations of cluster potentials and the need for interatomic functionals
5.5.1 The generic pair functional form: the glue-EAM-EMT-FS model
5.5.2 Physical interpretations of the pair functional
5.5.3 Fitting the pair functional model
5.5.4 Comparing pair functionals to cluster potentials
5.6.1 Introduction to the bond order: the Tersoff potential
5.6.2 Bond energy and bond order in TB
5.6.4 The modified embedded atom method
5.7 Atomistic models: what can they do?
5.7.1 Speed and scaling: how many atoms over how much time?
5.7.2 Transferability: predicting behavior outside the fit
5.7.3 Classes of materials and our ability to model them
5.8 Interatomic forces in empirical atomistic models
5.8.1 Weak and strong laws of action and reaction41
5.8.2 Forces in conservative systems
5.8.3 Atomic forces for some specific interatomic models
5.8.4 Bond stiffnesses for some specific interatomic models
5.8.5 The cutoff radius and interatomic forces
6.1 The potential energy landscape
6.2.1 Solving nonlinear problems: initial guesses
6.2.2 The generic nonlinear minimization algorithm
6.2.3 The steepest descent (SD) method
6.2.5 The conjugate gradient (CG) method
6.2.6 The condition number
6.2.7 The Newton–Raphson (NR) method
6.3 Methods for finding saddle points and transition paths
6.3.1 The nudged elastic band (NEB) method
6.4 Implementing molecular statics
6.4.2 Periodic boundary conditions (PBCs)
6.4.3 Applying stress and pressure boundary conditions
6.4.4 Boundary conditions on atoms
6.5 Application to crystals and crystalline defects
6.5.1 Cohesive energy of an infinite crystal
6.5.2 The universal binding energy relation (UBER)
6.5.3 Crystal defects: vacancies
6.5.4 Crystal defects: surfaces and interfaces
6.5.5 Crystal defects: dislocations
6.5.7 The Peierls–Nabarro model of a dislocation
6.6 Dealing with temperature and dynamics
PART III ATOMISTIC FOUNDATIONS OF CONTINUUM CONCEPTS
7 Classical equilibrium statistical mechanics
7.1 Phase space: dynamics of a system of atoms
7.1.1 Hamilton's equations
7.1.2 Macroscopic translation and rotation
7.1.3 Center of mass coordinates
7.1.4 Phase space coordinates
7.1.5 Trajectories through phase space
7.1.6 Liouville's theorem
7.2 Predicting macroscopic observables
7.2.2 The ensemble viewpoint and distribution functions
7.2.3 Why does the ensemble approach work?
7.3 The microcanonical (NVE) ensemble
7.3.1 The hypersurface and volume of an isolated Hamiltonian system
7.3.2 The microcanonical distribution function
7.3.3 Systems in weak interaction
7.3.4 Internal energy, temperature and entropy
7.3.5 Derivation of the ideal gas law
7.3.6 Equipartition and virial theorems: microcanonical derivation
7.4 The canonical (NVT) ensemble
7.4.1 The canonical distribution function
7.4.2 Internal energy and fluctuations
7.4.3 Helmholtz free energy
7.4.4 Equipartition theorem: canonical derivation
7.4.5 Helmholtz free energy in the thermodynamic limit
8 Microscopic expressions for continuum fields
8.1 Stress and elasticity in a system in thermodynamic equilibrium
8.1.1 Canonical transformations
8.1.2 Microscopic stress tensor in a finite system at zero temperature
8.1.3 Microscopic stress tensor at finite temperature: the virial stress
8.1.4 Microscopic elasticity tensor
8.2 Continuum fields as expectation values: nonequilibrium systems
8.2.1 Rate of change of expectation values
8.2.2 Definition of pointwise continuum fields
8.2.3 Continuity equation
8.2.4 Momentum balance and the pointwise stress tensor
8.2.5 Spatial averaging and macroscopic fields
8.3 Practical methods: the stress tensor
8.3.2 The virial stress tensor and atomic-level stresses
8.3.3 The Tsai traction: a planar definition for stress
8.3.4 Uniqueness of the stress tensor
8.3.5 Hardy, virial and Tsai stress expressions: numerical considerations
9.1 Brief historical introduction
9.2 The essential MD algorithm
9.3 The NVE ensemble: constant energy and constant strain
9.3.1 Integrating the NVE ensemble: the velocity-Verlet (VV) algorithm
9.3.3 Temperature initialization
9.4 The NVT ensemble: constant temperature and constant strain
9.4.2 Gauss´ principle of least constraint and the isokinetic thermostat
9.4.3 The Langevin thermostat
9.4.4 The Nosé-Hoover (NH) thermostat
9.4.5 Liouville’s equation for non-Hamiltonian systems
9.4.6 An alternative derivation of the NH thermostat
9.4.7 Integrating the NVT ensemble
9.5 The finite strain NσE ensemble: applying stress
9.5.1 A canonical transformation of variables
9.5.2 The hydrostatic stress state
9.5.3 The Parrinello-Rahman (PR) approximation
9.5.4 The zero-temperature limit: applying stress in molecular statics
9.5.5 The kinetic energy of the cell
9.6 The NσT ensemble: applying stress at a constant temperature
PART IV MULTISCALE METHODS
10 What is multiscale modeling?
10.1 Multiscale modeling: what is in a name?
10.2 Sequential multiscale models
10.3 Concurrent multiscale models
10.3.1 Hierarchical methods
10.3.2 Partitioned-domain methods
10.4 Spanning time scales
11 Atomistic constitutive relations for multilattice crystals
11.1 Statistical mechanics of systems in metastable equilibrium
11.1.1 Restricted ensembles
11.1.2 Properties of a metastable state from a restricted canonical ensemble
11.2 Relating mean positions to applied deformation: the Cauchy-Born rule
11.2.1 Multilattice crystals and mean positions
11.2.2 Cauchy-Born kinematics
11.2.3 Centrosymmetric crystals and the Cauchy-Born rule
11.2.4 Extensions and failures of the Cauchy-Born rule
11.3 Finite temperature constitutive relations for multilattice crystals
11.3.1 Periodic supercell of a multilattice crystal
11.3.2 Helmholtz free energy density of a multilattice crystal
11.3.3 Determination of the reference configuration
11.3.4 Uniform deformation and the macroscopic stress tensor
11.4 Quasiharmonic approximation
11.4.1 Quasiharmonic Helmholtz free energy
11.4.2 Determination of the quasiharmonic reference configuration
11.4.3 Quasiharmonic stress and elasticity tensors
11.4.4 Strict harmonic approximation
11.5 Zero-temperature constitutive relations
11.5.1 General expressions for the stress and elasticity tensors
11.5.2 Stress and elasticity tensors for some specific interatomic models
11.5.3 Crystal symmetries and the Cauchy relations
12 Atomistic–continuum coupling: static methods
12.1 Finite elements and the Cauchy–Born rule
12.2 The essential components of a coupled model
12.3 Energy-based formulations
12.3.1 Total energy functional
12.3.2 The quasi-continuum (QC) method
12.3.3 The coupling of length scales (CLS) method
12.3.4 The bridging domain (BD) method
12.3.5 The bridging scale method (BSM)
12.3.6 CACM: iterative minimization of two energy functionals
12.3.7 Cluster-based quasicontinuum (CQC-E)
12.4 Ghost forces in energy-based methods
12.4.1 A one-dimensional Lennard-Jones chain of atoms
12.4.2 A continuum constitutive law for the Lennard-Jones chain
12.4.3 Ghost forces in a generic energy-based model of the chain
12.4.4 Ghost forces in the cluster-based quasicontinuum (CQC-E)
12.4.5 Ghost force correction methods
12.5 Force-based formulations
12.5.1 Forces without an energy functional
12.5.3 The hybrid simulation method (HSM)
12.5.4 The atomistic-to-continuum (AtC) method
12.5.5 Cluster-based quasicontinuum (CQC-F)
12.5.6 Spurious forces in force-based methods
12.6 Implementation and use of the static QC method
12.6.1 A simple example:shearing a twin boundary
12.6.2 Setting up the model
12.6.3 Solution procedure
12.6.4 Twin boundary migration
12.6.5 Automatic model adaption
12.7 Quantitative comparison between the methods
12.7.2 Comparing the accuracy of multiscale methods
12.7.3 Quantifying the speed of multiscale methods
12.7.4 Summary of the relative accuracy and speed of multiscale methods
13 Atomistic–continuum coupling: finite temperature and dynamics
13.1 Dynamic finite elements
13.2 Equilibrium finite temperature multiscale methods
13.2.1 Effective Hamiltonian for the atomistic region
13.2.2 Finite temperature QC framework
13.2.3 Hot-QC-static:atomistic dynamics embedded in a static continuum
13.2.4 Hot-QC-dynamic: atomistic and continuum dynamics
13.2.5 Demonstrative examples: thermal expansion and nanoindentation
13.3 Nonequilibrium multiscale methods
13.3.1 A naÏve starting point
13.3.3 Generalized Langevin equations
A: Mathematical representation of interatomic potentials
A.1 Interatomic distances and invariance
A.2 Distance geometry: constraints between interatomic distances
A.3 Continuously differentiable extensions of Vint(s)
A.4 Alternative potential energy extensions and the effect on atomic forces
Modeling Materials
：Continuum, Atomistic and Multiscale Techniques
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