Asymptotic Analysis Of Phase Field Models Of Free Surfaces Driven By Surface Tension
We give an asymptotic analysis of the phase field Allen-Cahn and Cahn-Hilliard models of free surfaces with surface tension. Unlike the traditional approach to approximate the solution by the so called "matched asymptotic expansion" involving outer expansion, inner expansion and matching, our new approach uses a uniform "double asymptotic expansion" to expand the whole phase field function directly. The detailed structure of the phase field functions in the equilibrium state is given and the consistency of the phase field models with the corresponding sharp interface models is discussed, including the free surface Allen-Cahn model, Cahn-Hilliard model, and the Allen-Cahn model with volume constraint. The explicit asymptotic expansion of the phase field function reveals rich details of the phase field function structures. And it nicely explains some unusual phenomenon we observed in numerical experiments. The theory can be used to guide the future modeling and simulation of other moving boundary problems by phase field models.