# Phase field models

Phase field models

A phase field model is a mathematical model for solving interfacial problems. It has been mainly applied to solidification dynamics, [ [http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.matsci.32.101901.155803 WJ. Boettinger et al. Annual Review of Materials Research Vol. 32: 163-194 (2002)] ] but it has been also applied to other situations such as viscous fingering, [http://prola.aps.org/abstract/PRE/v60/i2/p1734_1 R. Folch et al. Phys. Rev. E 60, 1734 - 1740 (1999)] ] fracture dynamics, [ [http://prola.aps.org/abstract/PRL/v87/i4/e045501 A. Karma et. al. Phys. Rev. Lett. 87, 045501 (2001)] ] vesicle dynamics, [ [http://link.aps.org/doi/10.1103/PhysRevE.72.041921 T. Biben et al. Phys. Rev. E 72, 041921 (2005)] ] 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 [G.J. Fix, in Free Boundary Problems: Theory and Applications, Ed. A.Fasano and M. Primicerio, p. 580, Pitman (Boston, 1983).] and Langer, [J.S. Langer, Models of pattern formation in ﬁrst–order phase transitions,in Directions in Condensed Matter Physics p. 165, Ed. G. Grinstein and G.Mazenko, World Scientiﬁc, Singapore, (1986).] 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), [ [http://prola.aps.org/abstract/RMP/v52/i1/p1_1 J.S. Langer, Rev. Mod. Phys. 52, 1 (1980)] ] 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, [http://prola.aps.org/abstract/PRE/v57/i4/p4323_1 A. Karma and W.J. Rappel Phys. Rev. E 57, 4323 - 4349 (1998)] ] 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:

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

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 $e_0=\left\{L^2\right\}/\left\{T_\left\{M\right\}c_\left\{p$ (where $L$ is the latent heat, $T_M$ is the melting temperature, and $c_\left\{p\right\}$ is the specific heat). The term with $ablaphi$ 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: [ [http://prola.aps.org/abstract/RMP/v49/i3/p435_1 P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977)] ]

$partial_\left\{t\right\} phi = -frac\left\{1\right\}\left\{ au\right\}left\left(frac\left\{delta F\right\}\left\{delta phi\right\} ight\right) + \left\{eta\right\}\left(\left\{mathbfr\right\},t\right)$

$partial_\left\{t\right\} e =De_0 abla^2 left\left( frac\left\{delta F\right\}\left\{delta e\right\} ight\right) -\left\{mathbf\left\{ abla cdot\left\{mathbf \left\{q_e\left(\left\{mathbf r\right\},t\right).$

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 $l^2/D$:

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

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

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

harp 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, [ [http://dx.doi.org/10.1016/S0167-2789(00)00064-6 G. B. McFadden et al., Physica D 144, 154-168 (2000)] ] to model viscous fingering and two-phase Navier–Stokes flows, [ [http://www.ingentaconnect.com/content/ap/cp/1999/00000155/00000001/art06332 D. Jacqmin, J. Comput. Phys. 155,96-127 (1999)] ] to include fluctuations in the model, [ [http://dx.doi.org/10.1103/PhysRevE.71.061603 R. Benítez and L. Ramírez-Piscina Phys. Rev. E 71, 061603 (2005)] ] etc.

* [http://arxiv.org/abs/cond-mat/0305058 R. Gonzalez-Cinca et al., in Advances in Condensed Matter and Statistical Mechanics, ed. by E. Korucheva and R. Cuerno, Nova Science Publishers (2004)] a review on phase field models.
* [http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.matsci.32.112001.132041?journalCode=matsci L-Q Chen, Annual Review of Materials Research, Vol. 32: 113-140 (2002)] Phase field models in solidification

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Phase contrast microscopy — Phase contrast image of a cheek epithelial cell Phase contrast microscopy is an optical microscopy illumination technique of great importance to biologists in which small (invisible to the human eye) phase shifts in the light passing through a… …   Wikipedia

• Field electron emission — It is requested that a diagram or diagrams be included in this article to improve its quality. For more information, refer to discussion on this page and/or the listing at Wikipedia:Requested images. Field emission (FE) (also known as field… …   Wikipedia

• Phase-type distribution — Probability distribution name =Phase type type =density pdf cdf parameters =S,; m imes m subgenerator matrixoldsymbol{alpha}, probability row vector support =x in [0; infty)! pdf =oldsymbol{alpha}e^{xS}oldsymbol{S}^{0} See article for details… …   Wikipedia

• Phase transition — This diagram shows the nomenclature for the different phase transitions. A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another. A phase of a thermodynamic system and the states of matter… …   Wikipedia

• Phase plane — A phase plane is a visual display of certain characteristics of certain kinds of differential equations.Certain systems of differential equations can be written in the form::dx/dt = Cx where C may be any combination of constants in order to… …   Wikipedia

• Field emission display — A field emission display (FED) is a display technology that incorporates flat panel display technology that uses large area field electron emission sources to provide electrons that strike colored phosphor to produce a color image as a electronic …   Wikipedia

• Quantum field theory — In quantum field theory (QFT) the forces between particles are mediated by other particles. For instance, the electromagnetic force between two electrons is caused by an exchange of photons. But quantum field theory applies to all fundamental… …   Wikipedia

• Magnetic field — This article is about a scientific description of the magnetic influence of an electric current or magnetic material. For the physics of magnetic materials, see magnetism. For information about objects that create magnetic fields, see magnet. For …   Wikipedia

• Force field (chemistry) — In the context of molecular mechanics, a force field (also called a forcefield) refers to the functional form and parameter sets used to describe the potential energy of a system of particles (typically but not necessarily atoms). Force field… …   Wikipedia

• Quantitative models of the action potential — In neurophysiology, several mathematical models of the action potential have been developed, which fall into two basic types. The first type seeks to model the experimental data quantitatively, i.e., to reproduce the measurements of current and… …   Wikipedia