Quantitative Phase Field Modelling of Solidification

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Contents
Authors
	Chapter 1: A Brief History of Phase Field Modelling
	Chapter 2: Overview of This Book
Section I
	Chapter 3: Recap of Grand Potential Thermodynamics
	Chapter 4: Grand Potential Phase Field Functional
		4.1. INTERACTION BETWEEN ORDER PARAMETERS
		4.2. PROPERTIES OF THE SINGLE-PHASE GRAND POTENTIAL w v(μ)
		4.3. CONCENTRATION IN A MULTI-PHASE SYSTEM IN THE GRAND POTENTIAL ENSEMBLE
	Chapter 5: Phase Field Dynamics: {O , ci} versus {O ,μi} Evolution
		5.1. TIME EVOLUTION IN THE “TRADITIONAL” FIELDS {Oa} AND {ci}
		5.2. REFORMULATION OF PHASE FIELD MODEL DYNAMICS IN TERMS OF  a AND μi
			5.2.1. Order Parameter Evolution
			5.2.2. Chemical Potential Evolution
	Chapter 6: Re-Casting Phase Field Equations for Quantitative Simulations
		6.1. NON-VARIATIONAL MODIFICATIONS TO PHASE FIELD EQUATIONS
		6.2. CHOICE OF INTERPOLATION FUNCTIONS
	Chapter 7: Equilibrium Properties of the Grand Potential Functional
		7.1. EQUILIBRIUM CONCENTRATION FIELD
		7.2. EQUILIBRIUM SOLID-LIQUID INTERFACES
		7.3. EQUILIBRIUM SOLID-SOLID INTERFACES: APPROACH I—BASIC MODEL
		7.4. EQUILIBRIUM SOLID-SOLID INTERFACES: APPROACH II—MODIFICATION OF MODEL
		7.5. SELECTING BETWEEN PHASE FIELD MODELS
	Chapter 8: Thermal Fluctuations in Phase Field Equations
		8.1. NON-DIMENSIONAL FORM OF PHASE FIELD EQUATIONS
		8.2. SIMPLIFICATION OF NOISE AMPLITUDE FOR THE ORDER PARAMETER EQUATION
		8.3. SIMPLIFICATION OF NOISE AMPLITUDE FOR THE SOLUTE EQUATION
Section II
	Chapter 9: Special Cases of the Grand Potential Phase Field Model
		9.1. POLYCRYSTALLINE MULTI-COMPONENT ALLOY SOLIDIFICATION AT LOW SUPERSATURATION
			9.1.1. Evaluating the Equilibrium Reference Chemical Potentials μ
eq i
			9.1.2. Re-Casting Differential Equations in Eq. (9.6) in Terms of Supersaturation
			9.1.3. Practical Limits of Model I: Multi-Component Version of the Model of Ofori-Opoku et al.
			9.1.4. Practical Limits of Model II: Two-Phase Binary Alloy Model of Plapp
			9.1.5. Non-Dimensional Form of the Phase Field Model Described by Eqs. (9.16) and (9.19)
			9.1.6. Thin Interface Limit of Phase Field Equations
		9.2. MULTI-PHASE BINARY ALLOY WITH QUADRATIC SOLID/LIQUID FREE ENERGIES
			9.2.1. Solid-Liquid Phase Coexistence
			9.2.2. Grand Potential Density of Phase, Multi-Phase Concentration and Susceptibility
			9.2.3. Casting the Chemical Potential and Phase Field Equations in “Supersaturation Form”
		9.3. MULTI-PHASE, MULTI-COMPONENT ALLOYS WITH QUADRATIC FREE ENERGIES
			9.3.1. Free Energy and Susceptibility of a Single Phase
			9.3.2. Vector Notation and Transformations between Concentrations and Chemical Potentials
			9.3.3. Grand Potential and Concentration of a Single Phase
			9.3.4. Multi-Phase Concentration, Susceptibility and Concentration Difference
			9.3.5. Grand Potential Driving Force for Multi-Phase Solidification
			9.3.6. Casting the Driving Force in Terms of Supersaturation
			9.3.7. Final Form of Phase Field Equations in Terms of Supersaturation Driving Forces
	Chapter 10: Application: Phase Field Modelling of Ternary Alloys
		10.1. THERMAL SPRAY COATING DEPOSITION OF WC-Co
		10.2. REPRESENTATION OF THERMODYNAMIC PHASES
		10.3. TABULATED TIELINES IN THE LOW SUPERSATURATION LIMIT
		10.4. MOBILITY AND DIFFUSION COEFFICIENTS IN TERNARY SYSTEMS
		10.5. LOW SUPERSATURATION LIMIT OF A TERNARY ALLOY IN THE GRAND POTENTIAL PHASE FIELD MODEL
		10.6. PHASE FIELD PARAMETERS FOR EMULATING THE SHARP INTERFACE LIMIT
		10.7. SIMULATIONS OF CARBIDE DISSOLUTION
Section III
	Chapter 11: Interpreting Asymptotic Analyses of Phase Field Models
		11.1. WHAT IS AN ASYMPTOTIC ANALYSIS OF A PHASE FIELD MODEL ABOUT?
		11.2. UNDERSTANDING THE ROLE OF - AS AN ASYMPTOTIC CONVERGENCE PARAMETER
		11.3. INTERPRETING THE ROLE OF - IN ASYMPTOTIC ANALYSIS AND THE NOISE AMPLITUDE
Section IV
	Chapter 12: The Regime of Rapid Solidification
	Chapter 13: Modelling Continuous Growth Kinetics in the Diffuse Interface Limit of Grand Potential Phase Field Equations
		13.1. REVIEW OF THE CONTINUOUS GROWTH MODEL OF RAPID SOLIDIFICATION
			13.1.1. Kinetic Undercooling of the Interface in Henrian Solutions
		13.2. CONTINUOUS GROWTH MODEL LIMIT OF THE GRAND POTENTIAL PHASE FIELD MODEL
			13.2.1. Specializing Eq. (13.20) into the CGM Model of Eq. (13.6): Full Drag Case
			13.2.2. Specializing Eq. (13.20) into the CGM Model with Zero Drag
			13.2.3. Relating 1/vPF c to Interface Kinetic Coefficient B for the Case of Ideal Binary Alloys
		13.3. NON-EQUILIBRIUM PARTITION COEFFICIENT k(V0) AND CHOICE OF ANTI-TRAPPING
			13.3.1. Chemical Potential Jump at the Interface
			13.3.2. Evaluation of 
 ¯ F and an Implicit Equation for k(v0) from Eq. (13.31)
			13.3.3. Computing k(v0) for an Ideal Binary Alloy
	Chapter 14: Application: Phase Field Simulations of Rapid Solidification of a Binary Alloy
Section V
	Appendix A: Incorporating Temperature in the Grand Potential Phase Field Model
	Appendix B: Asymptotic Analysis of the Grand Potential Phase Field Equations
		B.1. LENGTH AND TIME SCALES
		B.2. PHASE FIELD EQUATIONS WRITTEN IN PERTURBATION VARIABLES
			B.2.1. Convenient Notations and Definitions
		B.3. FIELD EXPANSIONS AND MATCHING CONDITIONS OF OUTER/INNER SOLUTIONS
		B.4. OUTER EQUATIONS SATISFIED BY PHASE FIELD EQUATIONS
		B.5. INNER EQUATIONS SATISFIED BY PHASE FIELD EQUATIONS
			B.5.1. Phase Field Equation
			B.5.2. Chemical Potential Equation
			B.5.3. Constitutive Relation between c and μ
		B.6. ANALYSIS OF INNER EQUATIONS AND MATCHING TO THEIR OUTER FIELDS
			B.6.1. O(1) Phase Field Equation (B.24)
			B.6.2. O(1) Diffusion Equation (B.27)
			B.6.3. O(E) Phase Field Equation (B.25)
			B.6.4. O(E) Diffusion Equation (B.29)
			B.6.5. O(E2) Phase Field Equation (B.26)
			B.6.6. O(E2) Diffusion Equation (B.29)
Bibliography
Index