Comparison of Asymptotic Solutions of a Phase-Field Model to a Sharp-Interface Model

Source abstract

A one-dimensional directional solidification problem is considered for the purpose of analyzing the relationship between the solution resulting from a phase-field model to that from a sharp-interface model. The solidification problem is posed within a finite domain, rather than an infinite extent, as in classical Stefan problems. An asymptotic analysis based upon a small Stefan number 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 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 profiles develop. These corrections result from the conduction of latent heat 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. Following Karma 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 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 agree with the sharp-interface profiles, except near the solidification front, where there is smoothing over the diffuse interface and no jump in the temperature gradients.