Phase-field models in interfacial pattern formation out of equilibrium

The phase-field method is reviewed from the general perspective of converting a free boundary problem into a set of coupled partial differential equations. Its main advantage is that it avoids front tracking by using phase fields to locate the fronts. These fields interpolate between different constant values in each bulk phase through diffuse interfaces of finite thickness. In solidification, the phase fields can be understood as order parameters, and the model is often derived to dynamically minimise a free energy functional. However, this is not a necessary requirement, and both derivations involving a free energy (basic solidification model) and not (first viscous fingering model) are worked out. In any case, the model is required to reproduce the original free boundary problem in the limit of vanishing interface thickness. This limit and its higher order corrections, important to make quantitative contact between simulations and experiments, are discussed for both examples. Applications to fluctuations in solidification, the growth of liquid crystal mesophases, and viscous fingering in Hele-Shaw cells are presented.