Generalized Phase Field Models and Interface Problems

Problems arising from the study of an interface between two phases such as liquid and solid have been of great interest to mathematicians for centuries. Historically, these problems were modeled using an interface between the two phases (often called the Classical Stefan Model).

The phase field approach involves an interfacial region with finite thickness and a smooth transition function. This talk will review the development of this approach and the relationship between the phase field and the free boundary (or sharp interface approaches). Recent developments in collaboration with X. Chen, Ch. Eck and E. Esenturk have extended these results to include non-local interactions with anisotropy. Using differential geometry, combinatorics and asymptotic analysis, we obtain several results:

(i) a phase field model that converges more rapidly (as a function of interface thickness) to the sharp interface limiting model;

(ii) a mathematically elegant expression for the interface relation when there are anisotropic interactions;

(iii) incorporation of non-local microscopic interactions.