Comparison of asymptotic solutions of a phase-field model to a sharp-interface model (Q2783709)

A one-dimensional directional solidification problem is studied. The relationship between the solution resulting from a phase-field model and the solution from a sharp-interface model is analyzed. The solidification problem is posed within a finite domain (not within an infinite extent). In the asymptotic analysis of the phase-field case a small Stephan number expansion is coupled with a small interface-thickness boundary layer expansion. An agreement is obtained for the location of the solidification front and the temperature profile by the two models. Corrections are obtained from the latent heat across the diffuse interface. Coupling between the order parameter and the temperature in the phase-field model and the kinetic coefficient are selected to eliminate corrections to second order. Phase-field temperature profiles agree with sharp-interface profiles, except near the solidification front.