In the talk we introduce the notion of phases, and of macroscopic and microscopic models of phase transitions.
When modeling flow and transport phenomena, it is clearly important to recognize the phase of the fluids. For example, water can exist in liquid, gaseous, and solid (ice) phase, depending on the temperature and pressure conditions. Partial differential equation (PDE) models can describe phase transitions by tracking the primary unknowns such as temperature. For example, for water, if the temperature T is less than 0'C, we recognize the phase as ice, and if it is greater than 100'C, we recognize the phase as vapor. When water changes phase at 0' (or 100') C, the phase transition is associated with additional energy exchange (and enthalpy difference). This approach works well in macroscopic models.
At the scale of interfaces and at the porescale, a more refined description is needed. Phase field models or diffuse interface models are complicated evolution PDEs which were originally designed to describe phase transitions in fluids, but more recently have been also used in other context than fluids, e.g. for modeling human migration (aggregation and disaggregation), predicting crystal formation in materials, crack propagation in rock and materials, and in other applications. The variables in phase field models attempt to find minimum of the free energy functional, which has the stabilizing (convex) diffusive part and the destabilizing (concave) nonlinear part. These two parts of the model compete with each other, and produce an interface with a (small) width dependent on the parameters of the model. The numerical approximation of phase field models is challenging and exciting, due to the scales, sharp gradients, and nonlinearities involved. Some techniques, e.g., time-stepping, address explicitly the convexity and concavity of the energy. New challenges involve modeling and numerics for realistic phase field models for mixtures.