Consistency of Capillary Wave Theory in Three Dimensions: Divergence of the Interface Width and Agreement with Density Functional Theory

We compare the predictions of the capillary wave model for fluid interfaces with known exact expressions, due to Triezenberg, Zwanzig, and Wertheim, that relate the surface tension to the interfacial structure, and find agreement in all dimensions. It is shown that contrary to the case d3, the interfacial pair correlation function does not obey scaling in d=3 dimensions on length scales less than the capillary length L sub c. Nevertheless, once the sensitive dependence on the gravitational field and the short distance cutoff are properly taken into account, we find no evidence for recent allegations that the capillary wave model leads to inherently inconsistent results for the direct correlation function in d=3. Several issues regarding the connection with density functional theory and the interpretation of the Triezenberg-Zwanzig formula, on which these results touch, are discussed.