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 unmi xing is considered in a thin-film geometry confined between two parallel wa lls, 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 th e film for an average concentration of c = 1/2. We consider how the concent ration profile c(z) across the thin film depends on the film thickness D. B y Monte Carlo simulation of a lattice model for a polymer mixture it is sho wn that for relatively small D the width of the interface scales like w pro portional 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 e xtension of the Cahn-Hilliard "gradient-square"-type theory, while the beha vior for large D can be traced back to the behavior of capillary waves expo sed to a short-range potential by the wails. Corrections due to fast concen tration variations, as they occur in the strong-segregation limit of a poly mer mixture, can be accounted for by self-consistent field theory. Subtle p roblems occur, however, with respect to the proper combination of these the ories with the capillary wave approximation, particularly at intermediate v alues of D.