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.