Phase-field model
A phase-field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fingering, fracture mechanics, hydrogen embrittlement, and vesicle dynamics. 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 and Langer, and have experienced a growing interest in solidification and other areas. 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 function  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 function 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 formula_1, 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. Variational formulations. A model for a phase field can be constructed by physical arguments if one has an explicit expression for the free energy of the system. A simple example for solidification problems is the following: where formula_3 is the phase field, formula_4, formula_5 is the local enthalpy per unit volume, formula_6  is a certain polynomial function of formula_3, and formula_8 (where formula_9 is the latent heat, formula_10 is the melting temperature, and formula_11 is the specific heat). The term with formula_12 corresponds to the interfacial energy. The function formula_13 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 formula_13. The constants formula_15 and formula_16 have respectively dimensions of energy per unit length and energy per unit volume. The interface width is then given by formula_17. The phase-field model can then  be obtained from the following variational relations: where "D" is a diffusion coefficient for the variable formula_5, and formula_21 and formula_22 are stochastic terms accounting for thermal fluctuations (and whose statistical properties can be obtained from the fluctuation dissipation theorem). The first equation gives an equation for the evolution of the phase field, whereas the second one is a diffusion equation, which usually is rewritten for the temperature or for the concentration (in the case of an alloy). These equations are, scaling space with formula_23 and times with formula_24: where formula_27 is the nondimensional interface width, formula_28, and formula_29, formula_30  are nondimensionalized noises. Alternative energy-density functions. The choice of free energy function, formula_13, can have a significant effect on the physical behaviour of the interface, and should be selected with care. The double-well function represents an approximation of the Van der Waals equation of state near the critical point, and has historically been used for its simplicity of implementation when the phase-field model is employed solely for interface tracking purposes. But this has led to the frequently observed spontaneous drop shrinkage phenomenon, whereby the high phase miscibility predicted by an Equation of State near the critical point  allows significant interpenetration of the phases and can eventually lead to the complete disappearance of a droplet whose radius is below some critical value. Minimizing perceived continuity losses over the duration of a simulation requires limits on the Mobility parameter, resulting in a delicate balance between interfacial smearing due to convection, interfacial reconstruction due to free energy minimization (i.e. mobility-based diffusion), and phase interpenetration, also dependent on the mobility. A recent review of alternative energy density functions for interface tracking applications has proposed a modified form of the double-obstacle function which avoids the spontaneous drop shrinkage phenomena and  limits on mobility, with comparative results provide for a number of benchmark simulations using the double-well function and the volume-of-fluid sharp interface technique. The proposed implementation has a computational complexity only slightly greater than that of the double-well function, and may prove useful for interface tracking applications of the phase-field model where the duration/nature of the simulated phenomena introduces phase continuity concerns (i.e. small droplets, extended simulations, multiple interfaces, etc.). Sharp interface limit of the phase-field equations. A phase-field model can be constructed to purposely reproduce a given interfacial dynamics as represented  by a sharp interface model. In such a case the sharp interface limit (i.e. the limit when the interface width goes to zero) of the proposed set of phase-field equations should be performed. This limit is usually taken by asymptotic expansions of the fields of the model in powers of the interface width formula_32. These expansions are performed both in the interfacial region (inner expansion) and in the bulk (outer expansion), and then are asymptotically matched order by order. The result gives a partial differential equation for the diffusive field and a series of boundary conditions at the interface , which should correspond to the sharp interface model and whose comparison with it provides the values of the parameters of the phase-field model. Whereas such expansions were in early phase-field models performed up to the lower order in formula_32 only, more recent models use higher order asymptotics (thin interface limits) in order to cancel undesired spurious effects or to include new physics in the model. For example, this technique has permitted to cancel kinetic effects, to treat cases with unequal diffusivities in the phases, to model viscous fingering and two-phase Navier–Stokes flows, to include fluctuations in  the model, etc. Multiphase-field models. In multiphase-field models, microstructure is described by set of order parameters, each of which is related to a specific phase or crystallographic orientation. This model is mostly used for solid-state phase transformations where multiple grains evolve (e.g. grain growth, recrystallization or first-order transformation like austenite to ferrite in ferrous alloys). Besides allowing the description of multiple grains in a microstructure, multiphase-field models especially allow for consideration of multiple thermodynamic phases occurring e.g. in technical alloy grades. Phase-field models on graphs. Many of the results for continuum phase-field models  have discrete analogues for graphs, just replacing calculus with calculus on graphs. Phase Field Modeling in Fracture Mechanics. Fracture in solids is often numerically analyzed within a finite element context using either discrete or diffuse crack representations. Approaches using a finite element representation often make use of strong discontinuities embedded at the intra-element level and often require additional criteria based on, e.g., stresses, strain energy densities or energy release rates or other special treatments such as virtual crack closure techniques and remeshing to determine crack paths. In contrast, approaches using a diffuse crack representation retain the continuity of the  displacement field, such as continuum damage models and phase-field fracture theories. The latter traces back to the reformulation of Griffith’s principle in a variational form and has similarities to gradient-enhanced damage-type models. Perhaps the most attractive characteristic of phase-field approaches to fracture is that crack initiation and crack paths are automatically obtained from a minimization problem that couples the elastic and fracture energies. In many situations, crack nucleation can be properly accounted for by following branches of critical points associated with elastic solutions until they lose stability. In particular, phase-field models of fracture can  allow nucleation even when the elastic strain energy density is spatially constant. A limitation of this approach is that nucleation is based on strain energy density and not stress. An alternative view based on introducing a nucleation driving force seeks to address this issue. Phase Field Models for Collective Cell Migration. A group of biological cells can self-propel in a complex way due to the consumption of Adenosine triphosphate. Interactions between cells like cohesion or several chemical cues can produce movement in a coordinated manner, this phenomenon is called "Collective cell migration". A theoretical model for these phenomena is  the phase-field model and incorporates a phase field for each cell species and additional field variables like chemotactic agent concentration. Such a model can be used for phenomena like cancer, wound healing, morphogenesis and ectoplasm phenomena.