A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag

We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.