Phase-field Modeling of Phase Change Phenomena

The phase-field method has become a popular numerical tool for moving boundary problems in recent years. In this method, the interface is intrinsically diffuse and stores a mixing energy that is equivalent to surface tension. The major advantage of this method is its energy formulation which makes it easy to incorporate different physics. Meanwhile, the energy decay property can be used to guide the design of energy stable numerical schemes.

In this dissertation, we investigate the application of the Allen-Cahn model, a member of the phase-field family, in the simulation of phase change problems. Because phase change is usually accompanied with latent heat, heat transfer also needs to be considered. Firstly, we go through different theoretical aspects of the Allen-Cahn model for nonconserved interfacial dynamics. We derive the equilibrium interface profile and the connection between surface tension and mixing energy. We also discuss the well-known convex splitting algorithm, which is linear and unconditionally energy stable. Secondly, by modifying the free energy functional, we give the Allen-Cahn model for isothermal phase transformation. In particular, we explain how the Gibbs-Thomson effect and the kinetic effect are recovered. Thirdly, we couple the Allen-Chan and heat transfer equations in a way that the whole system has the energy decay property. We also propose a convex-splitting-based numerical scheme that satisfies a similar discrete energy law. The equations are solved by a finite-element method using the deal.ii library. Finally, we present numerical results on the evolution of a liquid drop in isothermal and non-isothermal settings. The numerical results agree well with theoretical analysis.