Phase-field models

Phase-field models have become popular in recent years to describe a host of free-boundary problems in various areas of research. The key point of the phase-field approach is that surfaces and interfaces are implicitly described by continuous scalar fields that take constant values in the bulk phases and vary continuously but steeply across a diffuse front. In the present contribution, a distinction is made between models in which the phase field can be identified with a physical quantity (coarse-grained on a mesoscopic scale), and models in which the phase field can only be interpreted as a smoothed indicator function. Simple diffuse-interface models for the motion of magnetic domain walls, the growth of precipitates in binary alloys, and for solidification are reviewed, and it is pointed out that is such models the free energy function determines both the bulk behavior of the dynamic variable and the properties of the interface. Next, a phenomenological phase-field model for solidification is introduced, and it is shown that with a proper choice of some interpolation functions, surface and bulk properties can be adjusted independently in this model. The link between this phase-field model and the classic free-boundary formulation of solidification is established by the use of matched asymptotic analysis. The results of this analysis can then be exploited to design new phase-field models that cannot be derived by the standard variational procedure from simple free energy functionals within the thermodynamic framework. As examples for applications of this approach, the solidification of alloys and the advected field model for two-phase flow are briefly discussed.