Resumen
One-dimensional
directional solidification 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 solidification
front, where there is smoothing over the diffuse interface and no
jump in the temperature gradients or concentration profile.