Extended capillary-wave model for the liquid-vapor interface and its width in the limit of vanishing gravity

An extended capillary-wave model is derived from density functional theory and the predictions of the usual capillary-wave model are reexamined. The model is expressed in terms of the direct correlation function C(r, r') of the equilibrium fluid. A small wavevector expansion of C(r, r') reduces the extended model to the usual one but it is shown that higher-order terms also play a role in determining interfacial properties in the limit of vanishing gravity. The meaning of the root-mean-square fluctuation W of the distortions of the Gibbs dividing surface and its relation to the actual width Ω of the density profile are specified. For spatial dimensions d < 3 the predictions of the extended and the usual models agree. Explicit calculations for d = 3 show W ∼ In (Lc/a) with Lc the classical capillary length and a a length proportional to the fourth transverse moment of C(r, r'). Adoption of a scaling hypothesis near g → 0, similar to that proposed by Weeks, shows that a ∼ Lc and W does not diverge as g → 0. It is argued that this does not necessarily imply that the width Ω is also finite, although this remains a possibility. Extensions to Week's hypothesis are also examined.