Nonlinear aspects of the Cahn-Hilliard equation

This paper treats phase separation within the context of the phenomenological Cahn-Hilliard equation, c_t = · M(c)(∂f/∂c - K²c)], where c is the concentration, M(c) is the mobility, and f(c) is homogeneou s free energy, which is assumed here to be a fourth degree polynomial. Natural boundary conditions are introduced. The full set of equilibrium solutions is specified. A comparison theorem for stability criteria which was postulated by Langer is proved here within the framework of the natural boundary conditions. Energy methods are used to define and estimate the limit of monotonic global stability. It is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution, there may still exist some internal “excitable” region in which the homogeneous solution possesses growing fluctuations. Furthermore a periodic instability is shown to exist in the metastable region in addition to the well-known nucleation instability.