Second-Order Phase Field Asymptotics for Unequal Conductivities

We extend Karma and Rappel’s improved asymptotic analysis of the phase field model to different diffusivities in solid and liquid. We consider both second-order “classical” asymptotics, in which the interface thickness is taken much smaller than the capillary length, and the new “isothermal” asymptotics, in which the two lengths are considered comparable. In the first case, if the phase field model is required to be gradient flow for an entropy functional, then for unequal diffusivities it is impossible to construct a phase equation with finite kinetics which converges with second-order accuracy to a Gibbs–Thomson equilibrium condition with infinitely fast kinetics. In the second case, some error terms are pushed to higher orders, and it is easy to eliminate the remaining errors with finite phase kinetics.