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)<<1 and xi(v)/D<<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.