Phase-field model
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).
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[12] and Langer,[13] and have experienced a growing interest in solidification and other areas.
Langer,[13] had handwritten notes where he showed you could use coupled Cahn-Hilliard and Allen-Cahn equations to solve a solidification problem.
Langer felt, at the time, that the method was of no practical use since the interface thickness is so small compared to the size of a typical microstructure, so he never bothered publishing them.
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),[14] which constitutes the sharp interface 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.
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,[15] which permitted to relax this condition and has opened the way to practical quantitative simulations with phase-field models.
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.
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
The phase-field model can then be obtained from the following variational relations:[16] where D is a diffusion coefficient for the variable
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.
[17] 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,[18] 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.).
This limit is usually taken by asymptotic expansions of the fields of the model in powers of the interface width
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.
For example, this technique has permitted to cancel kinetic effects,[15] to treat cases with unequal diffusivities in the phases,[19] to model viscous fingering[2] and two-phase Navier–Stokes flows,[20] to include fluctuations in the model,[21] etc.
In multiphase-field models, microstructure is described by set of order parameters, each of which is related to a specific phase or crystallographic orientation.
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.
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.
[23] 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.
[24] A group of biological cells can self-propel in a complex way due to the consumption of Adenosine triphosphate.
