Phase-field model

A phase-field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics,^[1] but it has also been applied to other situations such as viscous fingering,^[2] fracture mechanics,^[3][4][5][6] hydrogen embrittlement,^[7] and vesicle dynamics.^{[8][9][10][11]}

The method substitutes boundary conditions at the interface by a partial differential equation for the evolution of an auxiliary field (the phase field) that takes the role of an order parameter. This phase field takes two distinct values (for instance +1 and −1) in each of the phases, with a smooth change between both values in the zone around the interface, which is then diffuse with a finite width. A discrete location of the interface may be defined as the collection of all points where the phase field takes a certain value (e.g., 0).

A phase-field model is usually constructed in such a way that in the limit of an infinitesimal interface width (the so-called sharp interface limit) the correct interfacial dynamics are recovered. This approach permits to solve the problem by integrating a set of partial differential equations for the whole system, thus avoiding the explicit treatment of the boundary conditions at the interface.

Phase-field models were first introduced by Fix^[12] and Langer,^[13] and have experienced a growing interest in solidification and other areas. Langer,^[13] had handwritten notes where he showed you could use coupled Cahn-Hilliard and Allen-Cahn equations to solve a solidification problem. George Fix worked on programing problem. Langer felt, at the time, that the method was of no practical use since the interface thickness is so small compared to the size of a typical microstructure, so he never bothered publishing them.