Finite-size corrections for the static structure factor of a liquid slab with open boundaries

TL;DR

This paper derives an exact expression for the static structure factor of a liquid slab with open boundaries, revealing finite-size effects and their implications for small-system thermodynamics and experimental interpretation.

Contribution

It provides a novel, exact formula for the slab structure factor with open boundaries, linking finite-size corrections to thermodynamic properties and particle fluctuations.

Findings

Finite-size corrections significantly affect the structure factor at small wavenumbers.

The slab can be modeled as an open system with a well-defined chemical potential.

Simulation data confirms the theoretical predictions for Lennard-Jones liquids.

Abstract

The presence of a confining boundary can modify the local structure of a liquid markedly. In addition, small samples of finite size are known to exhibit systematic deviations of thermodynamic quantities relative to their bulk values. Here, we consider the static structure factor of a liquid sample in slab geometry with open boundaries at the surfaces, which can be thought of as virtually cutting out the sample from a macroscopically large, homogeneous fluid. This situation is a relevant limit for the interpretation of grazing-incidence diffraction experiments at liquid interfaces and films. We derive an exact, closed expression for the slab structure factor, with the bulk structure factor as the only input. This shows that such free boundary conditions cause significant differences between the two structure factors, in particular at small wavenumbers. An asymptotic analysis of this result yields the scaling exponent and an accurate, useful approximation of these finite-size corrections. Furthermore, the open boundaries permit the interpretation of the slab as an open system, supporting particle exchange with a reservoir. We relate the slab structure factor to the particle number fluctuations and discuss conditions under which the subvolume of the slab represents a grand canonical ensemble with chemical potential $\mu$ and temperature $T$. Thus, the open slab serves as a test-bed for the small-system thermodynamics in a $\mu T$ reservoir. We provide a microscopically justified and exact result for the size dependence of the isothermal compressibility. Our findings are corroborated by simulation data for Lennard-Jones liquids at two representative temperatures.