KKS Model (Kim-Kim-Suzuki)

Overview

The KKS model handles multi-phase, multi-component systems with local thermodynamic equilibrium at the interface. It avoids the artificial interface energy problem by partitioning the composition between phases.

Key feature: Local equilibrium constraint ensures equal diffusion potentials between phases at each point.

Mathematical Formulation

Phase Field Equation

\[\tau \frac{\partial \phi}{\partial t} = W^2 \nabla^2 \phi - g'(\phi) + h'(\phi) \cdot \Delta\omega\]

Concentration Equation

\[\frac{\partial c}{\partial t} = \nabla \cdot \left( M(\phi) \nabla \mu \right)\]

Local Equilibrium Constraints

At each point, the phase compositions (cs, cl) satisfy:

\[c = h(\phi) c_s + (1 - h(\phi)) c_l \quad \text{(mass conservation)}\]

\[\mu_s(c_s) = \mu_l(c_l) \quad \text{(equal chemical potential)}\]

Grand Potential Difference

\[\Delta\omega = f_s(c_s) - f_l(c_l) - \mu (c_s - c_l)\]

Quick Example

Basic Usage with Parabolic Free Energy

using PhaseFields

# Create model
model = KKSModel(τ=1.0, W=1.0, m=1.0, M_s=1.0, M_l=10.0)

# Parabolic free energies for testing
f_s = ParabolicFreeEnergy(A=1.0, c_eq=0.2)  # Solid
f_l = ParabolicFreeEnergy(A=1.0, c_eq=0.8)  # Liquid

# Partition composition at interface
c_avg = 0.5  # Average composition
φ = 0.5      # At interface

c_s, c_l, μ, converged = kks_partition(c_avg, φ, f_s, f_l)
println("Solid: c_s = $c_s, Liquid: c_l = $c_l")

# Grand potential difference
Δω = kks_grand_potential_diff(f_s, f_l, c_s, c_l, μ)

With CALPHAD Thermodynamics

using PhaseFields
using OpenCALPHAD

# Load database
db = read_tdb("AgCu.TDB")

# Create CALPHAD-coupled KKS model
T = 1000.0  # Temperature [K]
model, f_s, f_l = create_calphad_kks_model(db, T, "FCC_A1", "LIQUID")

# Partition with real thermodynamics
c_s, c_l, μ, converged = kks_partition(c_avg, φ, f_s, f_l)

API Reference

PhaseFields.KKSModel — Type

KKSModel

Kim-Kim-Suzuki model for solidification with local equilibrium.

The KKS model introduces phase concentrations (cs, cl) that satisfy:

Mass conservation: h(φ)·cs + (1-h(φ))·cl = c
Equal diffusion potential: μs(cs, T) = μl(cl, T)

Evolution equations:

Phase field:  τ·∂φ/∂t = W²∇²φ - g'(φ) + h'(φ)·Δω
Concentration: ∂c/∂t = ∇·(M(φ)·∇μ)

where Δω = fs(cs) - fl(cl) - μ·(cs - cl) is the grand potential difference.

Fields

τ::Float64: Relaxation time [s]
W::Float64: Interface width parameter [m]
m::Float64: Driving force scale (typically 1.0 for KKS)
M_s::Float64: Mobility in solid phase [m²/(J·s)]
M_l::Float64: Mobility in liquid phase [m²/(J·s)]

Example

model = KKSModel(τ=1.0, W=1.0, m=1.0, M_s=1.0, M_l=10.0)

Reference

Kim, Kim, Suzuki, Phys. Rev. E 60 (1999) 7186

source

PhaseFields.ParabolicFreeEnergy — Type

ParabolicFreeEnergy

Simple parabolic free energy for testing KKS model.

f(c) = A·(c - c_eq)²

Fields

A::Float64: Curvature (must be positive for stability)
c_eq::Float64: Equilibrium concentration

source

PhaseFields.free_energy — Function

free_energy(f::ParabolicFreeEnergy, c)

Parabolic free energy density: f(c) = A·(c - c_eq)²

source

PhaseFields.chemical_potential — Function

chemical_potential(f::ParabolicFreeEnergy, c)

Chemical potential (first derivative): df/dc = 2A·(c - c_eq)

source

PhaseFields.d2f_dc2 — Function

d2f_dc2(f::ParabolicFreeEnergy, c)

Second derivative of free energy: d²f/dc² = 2A

source

PhaseFields.kks_partition — Function

kks_partition(c_avg, φ, f_s, f_l; maxiter=50, tol=1e-12, φ_min=1e-8)

Solve the KKS local equilibrium problem using Newton iteration.

Given average concentration c and phase fraction φ, find phase concentrations (cs, cl) that satisfy:

h(φ)·cs + (1-h(φ))·cl = c (mass conservation)
μs(cs) = μl(cl) (equal chemical potential)

Arguments

c_avg: Average concentration at this point
φ: Phase field value (0=liquid, 1=solid)
f_s: Solid phase free energy (ParabolicFreeEnergy or similar)
f_l: Liquid phase free energy

Keyword Arguments

maxiter: Maximum Newton iterations (default: 50)
tol: Convergence tolerance (default: 1e-12)
φ_min: Minimum φ for Newton iteration (default: 1e-8)

Returns

(c_s, c_l, μ, converged): Phase concentrations, chemical potential, convergence flag

Example

f_s = ParabolicFreeEnergy(A=1000.0, c_eq=0.1)  # Solid equilibrium at c=0.1
f_l = ParabolicFreeEnergy(A=1000.0, c_eq=0.9)  # Liquid equilibrium at c=0.9

c_s, c_l, μ, converged = kks_partition(0.5, 0.5, f_s, f_l)

source

PhaseFields.kks_grand_potential_diff — Function

kks_grand_potential_diff(f_s, f_l, c_s, c_l, μ)

Compute the grand potential difference (driving force for phase field).

Δω = f_s(c_s) - f_l(c_l) - μ·(c_s - c_l)

Arguments

f_s, f_l: Solid and liquid free energy functions
c_s, c_l: Phase concentrations
μ: Chemical potential (equal in both phases)

Returns

Grand potential difference Δω [J/mol]

source

PhaseFields.kks_phase_rhs — Function

kks_phase_rhs(model, φ, ∇²φ, Δω)

Compute the right-hand side of the KKS phase field equation.

τ·∂φ/∂t = W²∇²φ - g'(φ) + m·h'(φ)·Δω
⟹ ∂φ/∂t = [W²∇²φ - g'(φ) + m·h'(φ)·Δω] / τ

Arguments

model: KKSModel parameters
φ: Phase field value
∇²φ: Laplacian of phase field
Δω: Grand potential difference from kksgrandpotential_diff

Returns

Time derivative ∂φ/∂t

source

PhaseFields.kks_concentration_rhs — Function

kks_concentration_rhs(model, φ, ∇²μ)

Compute the right-hand side of the KKS concentration equation.

∂c/∂t = ∇·(M(φ)·∇μ) ≈ M(φ)·∇²μ  (for constant mobility in each phase)

Arguments

model: KKSModel parameters
φ: Phase field value (for mobility interpolation)
∇²μ: Laplacian of chemical potential

Returns

Time derivative ∂c/∂t

source

PhaseFields.kks_mobility — Function

kks_mobility(model, φ)

Compute effective mobility using interpolation.

M(φ) = h(φ)·M_s + (1-h(φ))·M_l

Returns

Effective mobility at this phase field value

source

PhaseFields.kks_interface_width — Function

kks_interface_width(model)

Get the interface width parameter W.

source