Foundations and Extensions of the Cahn-Hilliard-Cook Theory

Abstract

Abstract In this paper we examine the foundations of the Cahn-Hilliard-Cook (CHC) theory of phase separation, which has been used routinely in recent years to interpret dynamic scattering experiments on binary polymer blends, and discuss its extension to multi-component mixtures. The CHC-theory is based on the following nonlinear Langevin equation1: where Φ(q,t) is the Fourier transform of the local volume fraction Φr,t) of one of the components of the binary mixture which is assumed to be incompressible; λ(q) is the q-dependent onsager coefficient; μ(q,t) is the Fourier transform of the local chemical potential difference μ(r,t); and η(q,t) is the random force which is added to account for the thermal fluctuations. The latter is assumed to be a stationary white noise process with zero mean and an autocovariance (η(q,t) q(-q,t′)) = λ(q) q2 δ(t - t′), which is determined from fluctuation-dissipation theorem. The use of the Langevin equation method to describe fluctuations in nonlinear macroscopic systems, in the way done in the CHC theory, was questioned by van Kampen2,3 for reasons which we discuss in this paper. The proper way of using the Langevin equation method in nonlinear systems was provided by Akcasu4 in 1977 through van Kampen's system size expansion, which we sketch next.