Interfacial profiles between coexisting phases in thin films:: Cahn-Hilliard treatment versus capillary waves

A symmetric binary mixture (A,B) below its critical temperature T(c) of unmixing is considered in a thin-film geometry confined between two parallel walls, where it is assumed that one wall prefers A and the other wall prefers B. Then an interface between the coexisting unmixed phases is stabilized, which (above the wetting transition temperature) occurs in the center of the film for an average concentration of c = 1/2. We consider how the concentration profile c(z) across the thin film depends on the film thickness D. By Monte Carlo simulation of a lattice model for a polymer mixture it is shown that for relatively small D the width of the interface scales like w proportional to D, while for larger D a crossover to a behavior w proportional to root D occurs. This behavior is explained by phenomenological theories: it is shown that the behavior at small D can be understood by a suitable extension of the Cahn-Hilliard "gradient-square"-type theory, while the behavior for large D can be traced back to the behavior of capillary waves exposed to a short-range potential by the wails. Corrections due to fast concentration variations, as they occur in the strong-segregation limit of a polymer mixture, can be accounted for by self-consistent field theory. Subtle problems occur, however, with respect to the proper combination of these theories with the capillary wave approximation, particularly at intermediate values of D.