Equations of The Phase Field Model
Phase field models are usually constructed in order to reproduce a given interfacial dynamics. For instance, in solidification problems the front dynamics is given by a diffusion equation for either concentration or temperature in the bulk and some boundary conditions at the interface (a local equilibrium condition and a conservation law), which constitutes the sharp interface model.
A number of formulations of the phase field model are based on a free energy functional depending on an order parameter (the phase field) and a diffusive field (variational formulations). Equations of the model are then obtained by using general relations of Statistical Physics. Such a functional is constructed from physical considerations, but contains a parameter or combination of parameters related to the interface width. Parameters of the model are then chosen by studying the limit of the model with this width going to zero, in such a way that one can identify this limit with the intended sharp interface model.
Other formulations start by writing directly the phase field equations, without referring to any thermodynamical functional (non-variational formulations). In this case the only reference is the sharp interface model, in the sense that it should be recovered when performing the small interface width limit of the phase field model.
Phase field equations in principle reproduce the interfacial dynamics when the interface width is small compared with the smallest length scale in the problem. In solidification this scale is the capillary length, which is a microscopic scale. From a computational point of view integration of partial differential equations resolving such a small scale is prohibitive. However, Karma and Rappel introduced the thin interface limit, which permitted to relax this condition and has opened the way to practical quantitative simulations with phase field models. With the increasing power of computers and the theoretical progress in phase field modelling, phase field models have become a useful tool for the numerical simulation of interfacial problems.
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