The role of capillary wave fluctuations in determining the liquid-vapour interface

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Abstract
We investigate the role played by capillary-wave like fluctuations in the determination of the equilibrium density profile and the intermolecular correlation functions of various models of a planar liquid-vapour interface in an external gravitational field. We show that the interfacial profile calculated from the van der Waals model, which is the simplest (squaregradient) approximation in a density functional theory of inhomogeneous fluids, is stable against capillary-wave fluctuations and therefore cannot be used to describe the ‘bare’ interface in a capillary wave model. The density-density correlation function of the van der Waals model exhibits long-ranged correlations in the interface. These are of precisely the same form as those predicted by Wertheim and can be associated with capillary-wave like fluctuations of the Gibbs dividing surface. These fluctuations do not lead to a ‘diffuse’ interfacial profile in the limits of infinite interfacial area and zero gravity; the interfacial thickness remains finite. In order to determine the equilibrium profile it is necessary to include the complete spectrum of density fluctuations, not just the capillary wave contributions. We contrast our results with those of field theoretical treatments (capillary wave models) which predict that the capillary wave fluctuations lead to a logarithmic divergence of the equilibrium interfacial thickness in the appropriate limits. The relative merits of the density functional and field theoretical approaches are discussed.