Phase-field model
Mathematical model / From Wikipedia, the free encyclopedia
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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.