1 Introduction 1.1 Microstructures Defined 1.2 Microstructure Evolution 1.3 Why simulate Microstructure evolution? 1.4 Further Reading 1.4.1 On Microstructures and their evolution from a noncomputationalpoint of view 1.4.2 On what is not treated in this book 2 Basic Thermodynamics 2.1 Reversible and Irreversible Thermodynamics 2.1.1 The first law of thermodynamics 2.1.2 The Gibbs energy 2.1.3 Molar quantities and the chemical potential 2.1.4 Entropy production and the Second Law of Thermodynamics2.1.5 Driving force for internal processes 2.1.6 Conditions for thermodynamic equilibrium 2.2 Solution thermodynamics 2.2.1 Entropy of mixing 2.2.2 The ideal solution 2.2.3 Regular solutions 2.2.4 General solutions in multi-phase equilibrium 2.2.5 The dilute solution limit – Henry’s and Raoult’s law 2.2.6 The chemical driving force 2.2.7 Influence of curvature and pressure 2.2.8 General solutions and the CALPHAD formalism 2.2.9 Practical evaluation of multi-component thermodynamicequilibrium 56 CONTENTS3 Monte Carlo Potts Model 3.1 Introduction 3.2 Two state Potts model (Ising model) 3.2.1 Hamiltonians 3.2.2 Dynamics (Probability Transition Functions) 3.2.3 Lattice Type 3.2.4 Boundary Conditions 3.2.5 The Vanilla Algorithm 3.2.6 Motion by Curvature 3.2.7 The Dynamics of Kinks and Ledges 3.2.8 Temperature 3.2.9 Boundary Anisotropy 3.2.10 Summary 3.3 Q-State Potts Model 3.3.1 Uniform Energies and Mobilities 3.3.2 Self-Ordering Behaviour 3.3.3 Boundary Energy 3.3.4 Boundary Mobility 3.3.5 Pinning Systems 3.3.6 Stored Energy 3.3.7 Summary 3.4 Speed-Up Algorithms 3.4.1 The Boundary-Site Algorithm 3.4.2 The N-Fold Way Algorithm 3.4.3 Parallel Algorithm 3.4.4 Summary 3.5 Applications of the Potts Model 3.5.1 Grain Growth 3.5.2 Incorporating Realistic Textures and MisorientationDistributions 3.5.3 Incorporating Realistic Energies and Mobilities 3.5.4 Validating the Energy and Mobility Implementations 3.5.5 Anisotropic Grain Growth 3.5.6 Abnormal Grain Growth 3.5.7 Recrystallisation 3.5.8 Zener Pinning 3.6 Summary 3.7 Final Remarks 3.8 Acknowledgements CONTENTS 74 Cellular Automata 4.1 A Definition 4.2 A One Dimensional Introduction 4.2.1 One Dimensional Recrystallization 4.2.2 Before moving to higher dimensions 4.3 +2D CA Modeling of recrystallization 4.3.1 CA-Neighborhood Definitions in 2D 4.3.2 The Interface Discretization Problem 4.4 +2D CA Modeling of Grain Growth 4.4.1 Approximating Curvature in a Cellular Automaton Grid4.5 Mathematics of Cellular Automata 4.6 Irregular and Shapeless Cellular Automata 4.6.1 Irregular Shapeless Cellular Automata for Grain Growth4.6.2 In the Presence of Additional Driving Forces 4.7 Hybrid Cellular Automata Modeling 4.7.1 Principle 4.7.2 Case example 4.8 Lattice Gas Cellular Automata 4.8.1 Principle – Boolean lgca 4.8.2 Boolean lgca – Example of Application 4.9 Network Cellular Automata 4.9.1 Combined Network Cellular Automata 4.9.2 CNCA for Microstructure Evolution Modeling 4.10 Further Reading 5 Solid-state diffusion 5.1 Diffusion mechanisms in crystalline solids 5.2 Microscopic diffusion 5.2.1 The principle of time reversal 5.2.2 A random walk treatment 5.2.3 Einstein’s equation 5.3 Macroscopic diffusion 5.3.1 Phenomenological laws of diffusion 5.3.2 Solutions to Fick’s second law 5.3.3 Diffusion forces and atomic mobility 5.3.4 Interdiffusion and the Kirkendall effect 5.3.5 Multi-component diffusion 5.4 Numerical solution of the diffusion equation 8 CONTENTS6 Modelling precipitation 6.1 Statistical theory of phase transformation 6.1.1 The extended volume approach - KJMA kinetics 6.2 Solid-state Nucleation 6.2.1 Introduction 6.2.2 Macroscopic treatment of nucleation - CNT 6.2.3 Transient nucleation 6.2.4 Multi-component nucleation 6.2.5 Treatment of interfacial energies 6.3 Diffusion-controlled precipitate growth 6.3.1 Problem definition 6.3.2 Zener’s approach for planar interfaces 6.3.3 Quasi-static approach for spherical precipitates 6.3.4 Moving boundary solution for spherical symmetry 6.4 Multi-particle precipitation kinetics 6.4.1 The numerical Kampmann-Wagner model 6.4.2 The SFFK model - A mean-field approach for complexsystems 6.5 Comparing the growth kinetics of different models 7 Phase-field Modelling 7.1 A Short Overview 7.2 Phase-field Model for Pure Substances 7.2.1 Anisotropy formulation 7.2.2 Material and Model Parameters 7.2.3 Application to Dendritic Growth 7.3 Case Study 7.3.1 Phase-field Equation 7.3.2 Finite Difference Discretization 7.3.3 Boundary Values 7.3.4 Stability Condition 7.3.5 Structure of the Code 7.3.6 Main computation 7.3.7 Parameter file 7.3.8 MatLab visualization 7.3.9 Examples 7.4 Model for Multiple Components and Phases 7.4.1 Model formulation 7.4.2 Entropy Density Contributions 7.4.3 Evolution Equations CONTENTS 97.4.4 Non-dimensionalization 7.4.5 FD discretization and staggered grid 7.4.6 Optimization of the Computational Algorithm 7.4.7 Parallelization 7.4.8 Adaptive Finite Element Method 7.4.9 Simulations of phase transitions and microstructureevolution 7.5 Acknowledgements 8 Discrete Dislocations 8.1 Basics of Discrete Plasticity Models 8.2 Linear Elasticity Theory for Plasticity 8.2.1 Introduction 8.2.2 Fundamentals of Elasticity Theory 8.2.3 Equilibrium Equations 8.2.4 Compatibility Equations 8.2.5 Hooke’s Law—the Linear Relationship between Stressand Strain 8.2.6 Elastic Energy 8.2.7 Green’s Tensor Function in Elasticity Theory 8.2.8 The Airy Stress Function in Elasticity Theory 8.3 Dislocation Statics 8.3.1 Introduction 8.3.2 2D Field Equations for Infinite Dislocations in an IsotropicLinear Elastic Medium 8.3.3 2D Field Equations for Infinite Dislocations in an AnisotropicLinear Elastic Medium 8.3.4 3D Field Equations for Dislocation Segments in anIsotropic Linear Elastic Medium 8.3.5 3D Field Equations for Dislocation Segments in anAnisotropicLinear Elastic Medium 8.4 Dislocation Dynamics 8.4.1 Introduction 8.4.2 Newtonian Dislocation Dynamics 8.4.3 Viscous and Viscoplastic Dislocation Dynamics 8.5 Kinematics of Discrete Dislocation Dynamics 8.6 Dislocation Reactions and Annihilation 10 CONTENTS9 FEM 9.1 Fundamentals of Differential Equations 9.1.1 Introduction to Differential Equations 9.1.2 Solution of Partial Differential Equations 9.2 Introduction to the finite element method 9.3 Finite Element Methods at the Meso–Macroscale 9.3.1 Introduction and Fundamentals 9.3.2 The Equilibrium Equation in FE Simulations 9.3.3 Finite Elements and Shape Functions 9.3.4 Assemblage of the Stiffness Matrix 9.3.5 Solid-State Kinematics for Mechanical Problems 9.3.6 Conjugate Stress–Strain Measures