phasefield

Phase-field methods are mathematical and computational techniques for simulating the evolution of microstructures in materials. They describe interfaces between phases with an order parameter phi that varies smoothly between values representing different phases, creating a diffuse interface of finite thickness rather than a sharp boundary. The evolution is driven by a free energy functional F[phi] that typically contains a local potential with two or more minima corresponding to stable phases and a gradient term that penalizes spatial variations. A common choice is F[phi] = ∫ (f0(phi) + (kappa/2) |∇phi|^2) dV, where f0 is a double-well potential.

The chemical potential is μ = δF/δphi = f0'(phi) - kappa ∇^2 phi, and the dynamics follow either the Allen-Cahn equation ∂phi/∂t = -L μ for non-conserved order parameters, or the Cahn-Hilliard equation ∂phi/∂t = ∇·(M ∇μ) for conserved order parameters, with L or M a mobility coefficient (possibly phi-dependent). In practice, phase-field models couple phi to other fields such as temperature, composition, or displacement to simulate phase transformations, diffusion, and elastic effects. The interface thickness is governed by kappa and f0.

Applications include solidification and dendritic growth, spinodal decomposition, grain growth, precipitation, and phase-field fracture models. The method avoids explicit front tracking and naturally handles complex morphologies and topological changes. It requires resolving the diffuse interface numerically, often via finite element, finite difference, or spectral methods, and can be extended to multicomponent and multiphysics problems. The phase-field approach has a long history, evolving from diffuse-interface concepts introduced by Cahn and Hilliard in 1958 and the Allen–Cahn equation in 1979, and has since become a standard tool in computational materials science.