Asymptotic solutions of a phase-field model or alloy solidification

One- dimensional directional solidi cation of a binary alloy is considered for the purpose of analyzing the relationship between the solution resulting from a phase-field model and that from a sharp- interface model. An asymptotic analysis based upon a small Stefan number and negligible solute diffusion in the solid phase is performed on the sharp- interface model. In the phase-field case, the small Stefan number expansion is coupled with a small interface- thickness boundary layer expansion. This approach enables us to develop analytical solutions to the phase-field model. The results show agreement at leading order between the two models for the location of the solidification front and the temperature and concentration profiles in the solid and liquid phases. However, due to the nonzero interface thickness in the phase-field model, corrections to the sharp-interface location and temperature and concentration profiles develop. These corrections result from the conduction of latent heat and from diffusion of solute across the diffuse interface. The magnitude of these corrections increases with the speed of the front, due to the corresponding increase in the release of latent heat and rejection of solute. Following Karma and Rappel [ Phys. Rev. E, 57 ( 1998), pp. 4323 4349] and Almgren [ SIAM J. Appl. Math., 59 ( 1999), pp. 2086 2107], we select the coupling between the order parameter and the temperature in the phase-field model and select the kinetic coefficient to eliminate the corrections to second order. Hence, the phase-field temperature profiles and concentration profiles agree with the sharp- interface profiles, except near the solidi cation front, where there is smoothing over the diffuse interface and no jump in the temperature gradients or concentration profile.