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.