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Contents
1 Introduction
1.1 The Role of Microstructure Materials Science
1.2 Free Boundary Problems and Microstructure Evolution
1.3 Continuum versus Sharp Interface Descriptions
2 Thermodynamics Primer
3 Mean Field Theory of Phase Transformations
3.1 Simple Lattice Models
3.2 Introduction to Landau Theory
4 Spatial Variations and Interfaces
4.1 The Ginzburg-Landau Free Energy Functional
4.2 Equilibrium Interfaces and Surface Tension
5 Nonequilibrium Dynamics
5.1 Driving Forces and Fluxes
5.2 The Diffusion Equation
5.3 Dynamics of Conserved Order Parameters: Model B
5.4 Dynamics of Nonconserved Order Parameters: Model A
5.5 Generic Features of Models A and B
5.6 Equilibrium Fluctuations of Order Parameters
5.7 Stability and the Formation of Second Phases
5.8 Interface Dynamics of Phase Field Models
5.9 Numerical Methods
6 Introduction to Phase Field Modeling: Solidification of Pure Materials
6.1 Solid Order Parameters
6.2 Free Energy Functional for Solidification
6.3 Single Order Parameter Theory of Solidification
6.4 Solidification Dynamics
6.5 Sharp and Thin Interface Limits of Phase Field Models
6.6 Case Study: Thin Interface Analysis
6.7 Numerical Simulations of Model C
6.8 Properties of Dendritic Solidification in Pure Materials
7 Phase Field Modeling of Solidification in Binary Alloys
7.1 Alloys and Phase Diagrams: A Quick Review
7.2 Microstructure Evolution in Alloys
7.3 Phase Field Model of a Binary Alloy
7.4 Equilibrium Properties of Free Energy Functional
7.5 Phase Field Dynamics
7.6 Thin Interface Limits of Alloy Phase Field Models
7.7 Case Study: Detailed Analysis of a Dilute Binary Alloy Model
7.8 Numerical Simulations of Dilute Alloy Phase Field Model
7.9 Deriving Alloy Phase field Models in the Grand Potential Ensemble
7.10 Properties of Dendritic Solidification in Binary Alloys
8 Multiple Phase Fields and Order Parameters
8.1 Multiorder Parameter Models
8.2 Multiphase Field Models
8.3 Phase Field Models with Vector Order Parameters
8.4 Orientational Order Parameter for Polycrystalline Modeling
9. Introductory Classical Density Functional Theory of Freezing
10 Field Models with Periodic Order Parameters
10.1 Generic Properties of Periodic Systems
10.2 Periodic Free Energies and the Swift-Hohenberg Equation
10.3 The Phase Field Crystal Model
10.4 Deriving Phase Field Crystal Models from Classical Density Functional Theory
10.5 Equilibrium Properties in a One-Mode Approximation
10.6 Elastic Constants of PFC Model
10.7 Structural phase field Crystal (XPFC) models
10.8 Multiscale Modeling: Amplitude Expansions
11 Phase Field Crystal Modeling of Binary Alloys
11.1 A Two-Component PFC Model for Alloys
11.2 Simplification of Binary Model
11.3 PFC Alloy Dynamics
11.4 Structural phase field Crystal (XPFC) modelling of Alloys
11.5 Applications of the Alloy PFC Models
Appendix A Thin Interface Limit of a Binary Alloy Phase Field Model
A.1 Phase Field Model
A.2 Curvilinear Coordinate Transformations
A.3 Length and Timescales
A.4 Matching Conditions between Outer and Inner Solutions
A.5 Outer Equations Satisfied by Phase Field Model
A.6 Inner Expansion of Phase Field Equation
A.7 Analysis of Inner Equations and Matching to Outer Fields
A.8 Summary of Results of Sections A.2-A.7
A.9 Elimination of Thin Interface Correction Terms
Appendix B Basic Numerical Algorithms for Phase Field Equations
B.1 Explicit Finite Difference Method for Model A
B.2 Explicit Finite Volume Method for Model B
B.3 Stability of Time Marching Schemes
B.4 Semi-Implicit Fourier Space Method
B.5 Finite Element Method
Appendix C Miscellaneous Derivations
C.1 Structure Factor
C.2 Transformations from Cartesian to Curvilinear Coordinates
C.3 Newton's Method for Nonlinear Algebraic Equations
