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 d_o, 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: : F[e,\varphi]=\int d{\mathbf r} \left[ K|{\mathbf\nabla}\varphi|^2 + h_0f(\varphi) + e_0u(\varphi)^2 \right] where \varphi is the phase field, u(\varphi)=e/e_0 + h(\varphi)/2, e is the local
enthalpy per unit volume, h is a certain polynomial function of \varphi, and e_0={L^2}/{T_M c_p} (where L is the
latent heat, T_M is the melting temperature, and c_{p} is the specific heat). The term with \nabla\varphi corresponds to the interfacial energy. The function f(\varphi) 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(\varphi). The constants K and h_{0} have respectively dimensions of energy per unit length and energy per unit volume. The interface width is then given by W=\sqrt{K/h_0}. The phase-field model can then be obtained from the following variational relations: : \partial_t \varphi = -\frac{1}{\tau} \left(\frac{\delta F}{\delta \varphi} \right) + \eta({\mathbf r},t) : \partial_t e = De_0\nabla^2 \left( \frac{\delta F}{\delta e} \right) - {\mathbf{\nabla}} \cdot{\mathbf q}_e(\mathbf r,t). where
D is a diffusion coefficient for the variable e, and \eta and \mathbf q_e 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 l and times with l^2/D: : \alpha \varepsilon^2\partial_t \varphi = \varepsilon^2\nabla^2\varphi- f'(\varphi) - \frac{e_0}{h_0} h'(\varphi)u+\tilde \eta({\mathbf r},t) : \partial_t u = \nabla^2 u+\frac{1}{2}h'(\varphi) \partial_t \varphi - \mathbf \nabla\cdot \mathbf q_u(\mathbf r,t) where \varepsilon=W/l is the nondimensional interface width, \alpha={D\tau}/{W^2h_0}, and \tilde\eta({\mathbf r},t), \mathbf q_u(\mathbf r,t) are nondimensionalized noises.
Alternative energy-density functions The choice of free energy function, f(\varphi), 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 \varepsilon. 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 \varepsilon 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 model viscous fingering to include fluctuations in the model, etc. == Multiphase-field models==