Phasenfeld

Phasenfeld, or phase-field, is a computational framework used to simulate microstructure evolution in materials. It replaces sharp interfaces with diffuse, continuous field variables that distinguish phases by their values. Interfaces emerge from dynamics rather than being tracked explicitly.

A phase field ϕ(x,t) varies between phases. Dynamics derive from a free-energy functional F[ϕ], with a local potential and a gradient term that penalizes sharp variations. For non-conserved order parameters, the Allen–Cahn equation applies; for conserved ones, the Cahn–Hilliard equation applies.

In multicomponent systems, several order parameters ϕ_i are used with the constraint sum_i ϕ_i = 1. The free-energy density includes interfacial energy terms and phase interactions. The framework can couple to mechanics, diffusion, and thermal fields.

Numerical implementation relies on discretization methods such as finite differences, finite elements, or spectral schemes. Accurate resolution of interfaces and appropriate mobility choices are essential for reliable results.

Applications include solidification and dendritic growth, grain growth and recrystallization, phase transformations in metals and alloys, polymers, and ceramics, as well as fracture, damage, and battery microstructure evolution.

The method builds on diffuse-interface theory, notably the Cahn–Hilliard framework, and has matured into a standard tool in materials science and related fields.