Morphological instability in phase-field models of solidification

We analyze the linear stability of a planar solidification front with sharp-interface and phase-field models in two physical situations: (1) an isothermal system at the melting point in the unperturbed state, and (2) constant-speed growth of a crystal into its hypercooled melt. The parameters in the phase-field models are chosen to scale with the nondimensional interface thickness so that in the limit of vanishing interface thickness, the sharp-interface model is recovered. Comparison of the results from the two models shows the following trends as the interface between the melt and solid is made thicker. (1) Perturbations to the plane front are stabilized as if the surface energy of the interface was increased. (2) The planar front and its perturbations behave as if the interfacial attachment kinetics was made faster, as long as the interface is significantly smaller than the capillary length. If the interface thickness is on the order of the capillary length, then the attachment kinetics may appear either slower or faster than for sharp-interface models. Stability results under ``heat trapping'' conditions are computed, and only planar fronts whose speed increases with undercooling are found to be stable.