Common API Reference

This page documents the common types and functions shared across all phase field models.

Types

Interface Parameters

PhaseFields.InterfaceParams — Type

InterfaceParams

Parameters for solid-liquid interface modeling.

Fields

W₀::Float64: Interface width parameter [m]
σ₀::Float64: Interface energy [J/m²]
τ::Float64: Relaxation time [s]
δ::Float64: Anisotropy strength [-]
n_fold::Int: Anisotropy symmetry (4=FCC, 6=HCP)

Example

# FCC metal with 4-fold anisotropy
params = InterfaceParams(
    W₀ = 1e-6,    # 1 μm interface width
    σ₀ = 0.3,     # 0.3 J/m² interface energy
    τ = 3e-8,     # 30 ns relaxation time
    δ = 0.04,     # 4% anisotropy
    n_fold = 4    # 4-fold cubic symmetry
)

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Diffusion Parameters

PhaseFields.DiffusionParams — Type

DiffusionParams

Temperature-dependent diffusion parameters.

Fields

D_liquid::Float64: Pre-exponential factor for liquid [m²/s]
D_solid::Float64: Pre-exponential factor for solid [m²/s]
Q_liquid::Float64: Activation energy for liquid [J/mol]
Q_solid::Float64: Activation energy for solid [J/mol]

Example

params = DiffusionParams(
    D_liquid = 1e-9,  # m²/s
    D_solid = 1e-13,  # m²/s
    Q_liquid = 40e3,  # 40 kJ/mol
    Q_solid = 80e3    # 80 kJ/mol
)

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Material Parameters

PhaseFields.MaterialParams — Type

MaterialParams

General material properties.

Fields

Vm::Float64: Molar volume [m³/mol]
L::Float64: Latent heat [J/mol]
Cp::Float64: Heat capacity [J/(mol·K)]
k::Float64: Thermal conductivity [W/(m·K)]

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Interpolation Functions

Phase field models use interpolation functions h(φ) and double-well potentials g(φ) to smoothly transition between phases.

Interpolation h(φ)

Properties:

h(0) = 0, h(1) = 1
Smooth transition in interface region

using PhaseFields

φ = 0.5

# Polynomial: h = 3φ² - 2φ³
h = h_polynomial(φ)

# Sinusoidal: h = (1 - cos(πφ))/2
h = h_sin(φ)

# Derivative
dh = h_prime(φ)

PhaseFields.h_polynomial — Function

h_polynomial(φ::T) where T<:Real

Polynomial interpolation function h(φ) = 3φ² - 2φ³.

Most commonly used interpolation for phase field models. Properties: h(0) = 0, h(1) = 1, h'(0) = h'(1) = 0.

Arguments

φ: Order parameter ∈ [0, 1]

Returns

Interpolated value ∈ [0, 1]

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PhaseFields.h_sin — Function

h_sin(φ::T) where T<:Real

Sinusoidal interpolation function h(φ) = (1 - cos(πφ))/2.

Alternative interpolation with similar properties to polynomial. Properties: h(0) = 0, h(1) = 1, h'(0) = h'(1) = 0.

Arguments

φ: Order parameter ∈ [0, 1]

Returns

Interpolated value ∈ [0, 1]

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PhaseFields.h_prime — Function

h_prime(φ::T) where T<:Real

Derivative of polynomial interpolation: h'(φ) = 6φ(1-φ).

Analytical form for better performance than AD.

Arguments

φ: Order parameter ∈ [0, 1]

Returns

Derivative value

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Double-Well g(φ)

Properties:

g(0) = g(1) = 0
Maximum at φ = 0.5
Provides energy barrier

# Standard: g = φ²(1-φ)²
g = g_standard(φ)

# Obstacle: g = |φ(1-φ)|
g = g_obstacle(φ)

# Derivatives
dg = g_prime(φ)
d2g = g_double_prime(φ)

PhaseFields.g_standard — Function

g_standard(φ::T) where T<:Real

Standard double-well potential g(φ) = φ²(1-φ)².

Properties: g(0) = g(1) = 0, minimum at φ = 0 and φ = 1. Maximum at φ = 0.5 with g(0.5) = 1/16.

Arguments

φ: Order parameter ∈ [0, 1]

Returns

Potential value ≥ 0

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PhaseFields.g_obstacle — Function

g_obstacle(φ::T) where T<:Real

Obstacle double-well potential g(φ) = φ(1-φ).

Simpler form than standard, but with same zeros. Used in some thin-interface models.

Arguments

φ: Order parameter ∈ [0, 1]

Returns

Potential value ≥ 0

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PhaseFields.g_prime — Function

g_prime(φ::T) where T<:Real

Derivative of standard double-well: g'(φ) = 2φ(1-φ)(1-2φ).

Analytical form for better performance than AD.

Arguments

φ: Order parameter ∈ [0, 1]

Returns

Derivative value

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PhaseFields.g_double_prime — Function

g_double_prime(φ::T) where T<:Real

Second derivative of standard double-well: g''(φ) = 2(1 - 6φ + 6φ²).

Arguments

φ: Order parameter ∈ [0, 1]

Returns

Second derivative value

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Anisotropy Functions

For dendritic growth, crystallographic anisotropy modifies the interface energy and kinetics.

Cubic Anisotropy (4-fold)

For FCC and BCC crystals:

θ = π/4  # Angle

# 4-fold anisotropy: a(θ) = 1 + ε cos(4θ)
a = anisotropy_cubic(θ; ε=0.05)

PhaseFields.anisotropy_cubic — Function

anisotropy_cubic(θ::T; δ::Real=0.04) where T<:Real

4-fold anisotropic interface energy for FCC/BCC crystals.

σ(θ) = 1 + δ·cos(4θ)

Arguments

θ: Interface normal angle [rad]
δ: Anisotropy strength (default: 0.04 = 4%)

Returns

Normalized interface energy (base value = 1)

Example

# Interface energy at θ = 0 (⟨100⟩ direction)
σ = anisotropy_cubic(0.0, δ=0.04)  # = 1.04

# Interface energy at θ = π/4 (⟨110⟩ direction)
σ = anisotropy_cubic(π/4, δ=0.04)  # = 0.96

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HCP Anisotropy (6-fold)

For hexagonal crystals:

# 6-fold anisotropy: a(θ) = 1 + ε cos(6θ)
a = anisotropy_hcp(θ; ε=0.05)

PhaseFields.anisotropy_hcp — Function

anisotropy_hcp(θ::T; δ::Real=0.02) where T<:Real

6-fold anisotropic interface energy for HCP crystals.

σ(θ) = 1 + δ·cos(6θ)

Arguments

θ: Interface normal angle [rad]
δ: Anisotropy strength (default: 0.02 = 2%)

Returns

Normalized interface energy (base value = 1)

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Custom Anisotropy

# n-fold anisotropy
a = anisotropy_custom(θ; ε=0.05, n=8)

PhaseFields.anisotropy_custom — Function

anisotropy_custom(θ::T; n::Int, δ::Real) where T<:Real

n-fold anisotropic interface energy.

σ(θ) = 1 + δ·cos(n·θ)

Arguments

θ: Interface normal angle [rad]
n: Symmetry order (4 for cubic, 6 for hexagonal)
δ: Anisotropy strength

Returns

Normalized interface energy (base value = 1)

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AD Configuration

PhaseFields.jl uses DifferentiationInterface.jl for automatic differentiation.

PhaseFields.DEFAULT_AD_BACKEND — Constant

Default AD backend (ForwardDiff). Can be changed via set_ad_backend!.

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PhaseFields.set_ad_backend! — Function

set_ad_backend!(backend)

Set the default AD backend for PhaseFields.jl.

Arguments

backend: A DifferentiationInterface backend (e.g., DI.AutoForwardDiff())

Example

using DifferentiationInterface
set_ad_backend!(DI.AutoForwardDiff())  # Default
# set_ad_backend!(DI.AutoEnzyme())     # For high performance (future)

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Module

PhaseFields — Module

PhaseFields.jl

Phase Field method simulation package for materials science. Provides domain-specific functionality for solidification modeling with CALPHAD thermodynamic coupling.

Features

Interface modeling (interpolation, double-well, anisotropy)
Phase field models (Allen-Cahn, Cahn-Hilliard, KKS, WBM)
CALPHAD coupling via OpenCALPHAD.jl (optional extension)
AD-compatible design using DifferentiationInterface.jl

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