phase-field modeling.doc

Phase field modelsFrom Wikipedia, the free encyclopediaJump to: navigation, search 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 dynamics, [3] vesicle dynamics,[4] etc.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[5] and Langer,[6] and have experienced a growing interest in solidification and other areas.Contents 1 Equations of the Phase field model o 1.1 Variational formulations o 1.2 Sharp interface limit of the Phase field equations  2 Multi Phase Field Models  3 Software  4 Further reading  5 References [edit] Equations of the Phase field modelPhase 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),[7] which constitutes the sharp interface model.A two phase microstructure and the order parameter φ profile is shown on a line across the domain. Gradual change of order parameter from one phase to another shows diffuse nature of the interface.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 do, 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,[8] 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.[edit] Variational formulationsA model for a phase field can be constructed by physical arguments if one have an explicit expression for the free energy of the system. A simple example for solidification problems is the following:where φ is the phase field, u = e / e0 + h(φ) / 2, e is the local enthalpy per unit volume, h is a certain polynomial function of φ, and e0 = L2 / TMcp (where L is the latent heat, TM is the melting temperature, and cp is the specific heat). The term with corresponds to the interfacial energy. The function f(φ) is usually taken as a double-well potential describing the free energy density of the bulk of each phase, which themselves correspond to the two minima of the function f(φ). The constants K and h0 have respectively dimensions of energy per unit length and energy per unit volume. The interface width is then given by . The phase field model can then be obtained from the following variational relations:[9]where D is a diffusion coefficient for the variable e, and 。