2 Second - Order Phase Field Asymptotics

We extend Karma and Rappel's improved asymptotic analysis of the phase eld model to diierent diiusivities in solid and liquid. We consider both second-order \classical" asymp-totics, 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 rst case, if the phase eld model is required to be gradient ow for an entropy functional, then for unequal diiusivities it is impossible to construct a phase equation with nite kinetics which converges with second-order accuracy to a Gibbs-Thomson equilibrium condition with innnitely fast kinetics. In the second case, some error terms are pushed to higher orders, and it is easy to eliminate the remaining errors with nite phase kinetics.