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

A one-dimensional directional solidi cation problem is considered for the p urpose of analyzing the relationship between the solution resulting from a phase-field model to that from a sharp-interface model. The solidi cation p roblem is posed within a finite domain, rather than an infinite extent, as in classical Stefan problems. An asymptotic analysis based upon a small Ste fan number is performed on the sharp-interface model. In the phase-field ca se, the small Stefan number expansion is coupled with a small interface-thi ckness boundary layer expansion. This approach enables us to develop analyt ical solutions to the phase-field model. The results show agreement at lead ing order between the tw models for the location of the solidi cation front and the temperature pro les in the solid and liquid phases. However, due t o the nonzero interface thickness in the phase-field model, corrections to the sharp-interface location and temperature pro les develop. These correct ions result from the conduction of latent heat across the di use interface. The magnitude of these corrections increases with the speed of the front d ue to the corresponding increase in the release of latent heat. Following K arma and Rappel [ Phys. Rev. E (3), 57 (1998), pp. 4323-4349] and Almgren [ SIAM J. Appl. Math., 59 (1999), pp. 2086-2107], we select the coupling betw een the order parameter and the temperature in the phase-field model and se lect the kinetic coefficient to eliminate the corrections to second order. Hence the phase-field temperature pro les agree with the sharp-interface pr o les, except near the solidi cation front, where there is smoothing ver th e di use interface and no jump in the temperature gradients.