Motion by curvature in generalized Cahn-Allen models

The Cahn-Allen model for the motion of phase-antiphase boundaries is generalized to account for nonlinearities in the kinetic coefficient (relaxation velocity) and the coefficient of the gradient free energy. The resulting equation is {Mathematical expression} where f is bistable. Here u is an order parameter and ? and a are physical quantities associated with the system's free energy and relaxation speed, respectively. Grain boundaries, away from triple junctions, are modeled by solutions with internal layers when e«1. The classical motion-by-curvature law for solution layers, well known when ? and a are constant, is shown by formal asymptotic analysis to be unchanged in form under this generalization, the only difference being in the value of the coefficient entering into the relation. The analysis is extended to the case when the relaxation time for the process vanishes for a set of values of u. Then a is infinite for those values. © 1994 Plenum Publishing Corporation.