Microstructures Defined Microstructure Evolution Why Simulate Microstructure Evolution? Further Reading On Microstructures and Their Evolution from a Noncomputational Point of View On What Is Not Treated in This Book Thermodynamic Basis of Phase Transformations Reversible and Irreversible Thermodynamics The First Law of Thermodynamics The Gibbs Energy Molar Quantities and the Chemical Potential Entropy Production and the Second Law of Thermodynamics Driving Force for Internal Processes Conditions for Thermodynamic Equilibrium Solution Thermodynamics Entropy of Mixing The Ideał Solution Regular Solutions General Solutions in Multiphase Eąuilibrium The Dilute Solution Limit—Henry’s and Raoult’s Law The Chemical Driving Force Influence of Curvature and Pressure General Solutions and the CALPHAD Formalism Practical Evaluation of Multicomponent Thermodynamic Eąuilibrium Monte Carlo Potts Model Two-State Potts Model (Ising Model) Hamiltonians Dynamics (Probability Transition Functions) Lattice Type Boundary Conditions The Vanilla Algorithm Motion by Curvature The Dynamics of Kinks and Ledges Temperaturę Boundary Anisotropy 3.3Q-State Potts Model Uniform Energies and Mobilities Self-Ordering Behavior Boundary Energy Boundary Mobility Pinning Systems Stored Energy 3.4Speed-Up Algorithms The Boundary-Site Algorithm The A-Fold Way Algorithm Parallel Algorithm Applications of the Potts Model Grain Growth Incorporating Realistic Textures and Misorientation Distributions Incorporating Realistic Energies and Mobilities Validating the Energy and Mobility Implementations Anisotropic Grain Growth Abnormal Grain Growth Recrystallization Zener Pinning 3.7Finał Remarks Cellular Automata 4.2A One-Dimensional Introduction One-Dimensional Recrystallization Before Moving to Higher Dimensions 4.3CA Modeling of Recrystallization CA-Neighborhood Definitions in Two Dimensions The Interface Discretization Problem 4.4+2D CA Modeling of Grain Growth

Approximating Curvature in a Cellular Automaton Grid A Mathematical Formulation of Cellular Automata Irregular and Shapeless Cellular Automata Irregular Shapeless Cellular Automata for Grain Growth In the Presence of Additional Driving Forces Hybrid Cellular Automata Modeling PrincipleLattice Gas Cellular Automata Principle—BooleanLGCA BooleanLGCA—Example of Application Network Cellular Automata—A Development for the Future Combined Network Cellular Automata CNCA for Microstructure Evolution Modeling Further ReadingModelingSolid-State Diffusion Diffusion Mechanisms in Crystalline Solids Microscopic Diffusion The Principle of Time Reversal A Random Walk Treatment Einstein’s Eąuation Macroscopic Diffusion Phenomenological Laws of Diffusion Solutions to Fick’s Second Law Diffusion Forces and Atomie Mobility Interdiffusion and the Kirkendall Effect Multicomponent Diffusion Numerical Solution of the Diffusion Eąuation g Modeling Precipitation as a Sharp-Interface Phase Transformation Statistical Theory of Phase Transformation The Extended Volume Approach—KJM A Kinetics Solid-State Nucleation Macroscopic Treatment of Nucleation—Classical Nucleation Theory Transient Nucleation Multicomponent NucleationTreatment of Interfacial Energies Diffusion-Controlled Precipitate Growth Zener’s Approach for Planar Interfaces Quasi-static Approach for Spherical Precipitates Moving Boundary Solution for Spherical Symmetry Multiparticle Precipitation Kinetics The Numerical Kampmann-Wagner Model The SFFK Model—A Mean-Field Approach for Complex Systems Comparing the Growth Kinetics of Different Models Phase-Field Modeling 7.2Phase-Field Model for Pure Substances Anisotropy Formulation Materiał and Model Parameters Application to Dendritic Growth Phase-Field Eąuation Finite Difference Discretization Boundary Values Stability Condition Structure of the Code Main Computation Parameter File MatLab Visualization 7.4Model for Multiple Components and Phases Model Formulation Entropy Density Contributions Evolution Eąuations Nondimensionalization Finite Difference Discretization and Staggered Grid Optimization of the Computational Algorithm Parallelization Adaptive Finite Element Method Simulations of Phase Transitions and Microstructure Evolution Introduction to Discrete Dislocations Statics and Dynamics Basics of Discrete Plasticity Models 8.2Linear Elasticity Theory for Plasticity Fundamentals of Elasticity Theory Eąuilibrium Eąuations Compatibility Eąuations Hooke’s Law—The Linear Relationship between Stress and Strain Elastic Energy Green’s Tensor Function in Elasticity Theory The Airy Stress Function in Elasticity Theory Dislocation Statics Two-Dimensional Field Eąuations for Infinite Dislocations in an Isotropic Linear Elastic Medium Two-Dimensional Field Eąuations for Infinite Dislocations in an Anisotropic Linear Elastic Medium Three-Dimensional Field Eąuations for Dislocation Segments in an Isotropic Linear Elastic Medium Three-Dimensional Field Eąuations for Dislocation Segments in an Anisotropic Linear Elastic Medium Dislocation Dynamics Newtonian Dislocation Dynamics Viscous and Viscoplastic Dislocation Dynamics Kinematics of Discrete Dislocation Dynamics Dislocation Reactions and Annihilation Finite Elements for Microstructure Evolution Solution of Partial Differential EquationsIntroduction to the Finite Element Method Finite Element Methods at the Meso- and Macroscale The Eąuilibrium Eąuation in FE Simulations Finite Elements and Shape Functions Assemblage of the Stiffness Matrix Solid-State Kinematics for Mechanical Problems Conjugate Stress-Strain Measures