Definitions

# Phase field models

A phase field model is a mathematical model for solving interfacial problems. It has been mainly applied to solidification dynamics, but it has been also applied to other situations such as viscous fingering, fracture dynamics, vesicle dynamics, 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. The location of points where the phase field takes a given value corresponds to the position of the interface. The model is usually constructed in such a way that in the limit of small interface width (the so-called sharp interface limit) the correct interfacial dynamics is recovered. This approach permits to solve the problem by integrating a set of partial differential equations for all the 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 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 $d_o$, 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 have an explicit expression for the free energy of the system. A simple example for solidification problems is the following:

$F\left[e,phi\right]=int d\left\{mathbf r\right\} left\left[K|\left\{mathbf nabla\right\}phi|^2 + h_0f\left(phi\right) + e_0u^2 right\right]$

where $\left\{phi\right\}$ is the phase field, $u=e/e_0 + h\left(phi\right)/2$, $e$ is the local enthalpy per unit volume, $h$ is a certain polynomial function of $phi$, and $e_0=\left\{L^2\right\}/\left\{T_\left\{M\right\}c_\left\{p\right\}\right\}$ (where $L$ is the latent heat, $T_M$ is the melting temperature, and $c_\left\{p\right\}$ is the specific heat). The term with $nablaphi$ corresponds to the interfacial energy. The function $f\left(phi\right)$ is usually taken as a double-well potential describing the free energy density of the bulk of each phase, which theirself correspond to the two minima of the function $f\left(phi\right)$. The constants $K$ and $h_\left\{0\right\}$ have respectively dimensions of energy per unit length and energy per unit volume. The interface width is then given by $W=sqrt\left\{K/h_0\right\}$. The phase field model can then be obtained from the following variational relations:

$partial_\left\{t\right\} phi = -frac\left\{1\right\}\left\{tau\right\} left\left(frac\left\{delta F\right\}\left\{delta phi\right\} right\right) + \left\{eta\right\}\left(\left\{mathbf r\right\},t\right)$

$partial_\left\{t\right\} e = De_0nabla^2 left\left(frac\left\{delta F\right\}\left\{delta e\right\} right\right) - \left\{mathbf\left\{nabla\right\}\right\} cdot\left\{mathbf \left\{q\right\}\right\}_e\left(\left\{mathbf r\right\},t\right).$

where D is a diffusion coefficient for the variable $e$, and $eta$ and $mathbf \left\{q\right\}_e$ 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 $l^2/D$:

$alpha varepsilon^2partial_\left\{t\right\} phi = varepsilon^2nabla^2phi- f\text{'}\left(phi\right) - frac\left\{e_0\right\}\left\{h_0\right\} h\text{'}\left(phi\right)u+tilde eta\left(\left\{mathbf r\right\},t\right)$

$partial_\left\{t\right\}u = nabla^\left\{2\right\}u+frac\left\{1\right\}\left\{2\right\}partial_\left\{t\right\}h - \left\{mathbf nabla\right\}cdot \left\{mathbf q_\left\{u\right\}\right\}\left(\left\{mathbf r\right\},t\right)$

where $varepsilon=W/l$ is the nondimensional interface width, $alpha=\left\{Dtau\right\}/\left\{W^2h_0\right\}$, and $tildeeta\left(\left\{mathbf r\right\},t\right)$, $\left\{mathbf q_\left\{u\right\}\right\}\left(\left\{mathbf r\right\},t\right)$ are nondimensionalized noises.

#### 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 asymptoticly 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 spureous 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.