Asymptotic decay of liquid structure: oscillatory liquid-vapour density profiles and the Fisher-Widom line

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Abstract
Recent work has highlighted the existence of a unified theory for the asymptotic decay of the density profile ρ(r) of an inhomogeneous fluid and of the bulk radial distribution function g(r). For a given short-ranged interatomic potential ρ(r) decays into bulk in the same fashion as g(r), i.e. with the same exponential decay length (α0/-1) and, for sufficiently high bulk density (ρb) and/or temperature (T), oscillatory wavelength (2π/α1). The quantities α0 and α1 are determined by a linear stability analysis of the bulk fluid; they depend on only the bulk direct correlation function. In this paper we reintroduce the concept of the Fisher-Widom (FW) line. This line was originally introduced, in say the (ρb, T plane, as that which separates pure exponential from exponentially damped oscillatory decay of g(r). We explore the relevance of the FW line for the form of the density profile at a liquid-vapour interface. Using a weighted density approximation (WDA) density functional theory we locate the FW line for the square-well model of an atomic fluid. We find that this line crosses the liquid branch of the liquid-vapour coexistence curve at T/T c ≈ 0·9, where T c is the critical temperature. Accordingly, for T≲0·9T c very general statistical mechanical theory predicts damped oscillatory decay of the liquid-vapour density profile into the bulk liquid. Since the amplitude of the oscillations is not determined by the linear analysis we have calculated explicit nonlinear numerical solutions of our WDA theory, using a high quality finite element method. Our results show that in a mean-field treatment the amplitude of the oscillatory profile in the saturated liquid tail is about 2% of ρb at temperatures approaching the triple point and decreases rapidly as T increases towards the FW line. The predictions of the asymptotic profile decay theory are confirmed by our explicit results and the unified nature of the phenomena is illustrated by comparing results for the liquid-vapour profile with profiles calculated for attractive wall-liquid interfaces at the same bulk liquid state point. The effects of capillary-wave fluctuations on the oscillatory nature of liquid-vapour profiles, above the FW line, are discussed, and we argue that while incorporating such fluctuations should lead to a significant reduction in the amplitude of oscillations, in d = 3, at least, there should be no change to the period and decay length for the profile in the liquid tail. The implications of our results for other interfacial properties, for computer simulations of the liquid-vapour interface, for studies of wetting transitions and for the nature of the solvation force that arises when a fluid is confined between two planar walls are considered.

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