Non-Conservative Phase-Field: Allen-Cahn

Goal

To simulate the morphological evolution of structural transformations (like solidification, melting, or curvature-driven grain growth) using the Allen-Cahn (time-dependent Ginzburg-Landau) equation. This tracks a non-conservative order parameter $\phi$ which distinguishes between phases (e.g., solid vs. liquid).

Instructions

1. Mathematical Formulation

The Allen-Cahn equation describes the evolution of a non-conserved order parameter $\phi$ down a free energy gradient: $$ \frac{\partial \phi}{\partial t} = -M \frac{\delta F}{\delta \phi} = M \left( \epsilon^2 \nabla^2 \phi - \frac{\partial f(\phi)}{\partial \phi} \right) $$ Where $M$ is the mobility, $\epsilon$ is the gradient energy coefficient controlling the interface thickness, and $f(\phi) = W \phi^2(1-\phi)^2$ is the double-well potential barrier between the two phases ($\phi=0$ and $\phi=1$).

Unlike Cahn-Hilliard, Allen-Cahn does not conserve the integral of $\phi$. It naturally drives systems to reduce their total interfacial area, resulting in curvature-driven boundary migration.

2. Running Curvature-Driven Grain Growth

Use the provided script to set up a 2D grid containing a circular solid grain in a liquid matrix and observe its capillarity-driven shrinkage.

# Env: phasefield-agent
python .agents/skills/mat-phase-field-non-conservative/scripts/run_grain_growth.py \
    --grid-size 100 \
    --radius 30 \
    --steps 200 \
    --dt 0.1 \
    --output grain_growth.gif

Parameters:

• --grid-size: Number of grid points per dimension (e.g., 100 for a 100x100 2D grid).
• --radius: Initial radius of the circular grain in grid units.
• --steps: Total number of time steps to run.
• --dt: Time step size.
• --output: Filepath to save the resulting .gif animation or .png.

Examples

Classic Shrinking Circular Grain

A universal mathematical benchmark for the Allen-Cahn equation is proving that a circular domain shrinks at a rate proportional to its curvature (the $v = M \gamma K$ law). The area of the circle must decrease linearly with time.

# Env: phasefield-agent
python .agents/skills/mat-phase-field-non-conservative/scripts/run_grain_growth.py \
    --grid-size 100 \
    --radius 35 \
    --steps 300 \
    --dt 0.5 \
    --output examples/benchmark-grain/classic_shrinking_grain.gif

See the examples/benchmark-grain/README.md for the expected output.

Constraints

• Environments: Scripts require the phasefield-agent Conda environment. Each code block MUST specify the environment.
• Interface Thickness: The spatial resolution dx must be small enough to resolve the diffuse interface (typically requiring at least 4-5 grid points across the interface controlled by $\epsilon$).

References

• Allen, S. M., & Cahn, J. W., "A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening", Acta Metallurgica, 1979. DOI

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Author: Bowen Deng