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. In particular, phase-field equations are mapped onto sharp-interface equations in the limits xi kappa<<1 and xi nu /D << 1, where kappa and nu 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.