Sharp interface limits of phase-field models

The use of continuum phase-field models to describe the motion of well-defined interfaces is discussed for a class of phenomena, that includes order/disorder transitions, spinodal decomposition and Ostwald ripening, dendritic growth, and the solidification of eutectic alloys. The projection operator method is used to extract the ``sharp interface limit’’ from phase field models which have interfaces that are diffuse on a length scale $\xi$. In particular,phase-field equations are mapped onto sharp interface equations in the limits $\xi \kappa \ll 1$ and $\xi v\mathrm{/}D\ll 1$, where $\kappa$ and $v$ are respectively the interface curvature and velocity and $D$ is the diffusion constant in the bulk. The calculations provide one general set of sharp interface equations that incorporate the Gibbs-Thomson condition, the Allen-Cahn equation and the Kardar-Parisi-Zhang equation.