# Correlation function structure in square-gradient models of the liquid-gas interface: Exact results and reliable approximations

## TL;DR

This paper analyzes the structure of density correlations at liquid-gas interfaces using square-gradient models, demonstrating that certain approximations are highly accurate across different wave-vectors and models, including near tricritical points.

## Contribution

The study provides exact results and reliable approximations for correlation functions in square-gradient models, validated against new analytically solvable cases and numerical solutions.

## Key findings

- Approximations accurately describe correlation functions across all wave-vectors.
- Some models yield exact correlation functions with these approximations.
- Near tricritical points, approximations closely match numerical solutions.

## Abstract

In a recent article, we described how the microscopic structure of density-density correlations in the fluid interfacial region, for systems with short-ranged forces, can be understood by considering the resonances of the local structure factor occurring at specific parallel wave-vectors $q$. Here, we investigate this further by comparing approximations for the local structure factor and correlation function against three new examples of analytically solvable models within square-gradient theory. Our analysis further demonstrates that these approximations describe the correlation function and structure factor across the whole spectrum of wave-vectors, encapsulating the cross-over from the Goldstone mode divergence (at small $q$) to bulk-like behaviour (at larger $q$). As shown, these approximations are exact for some square-gradient model potentials, and never more than a few percent inaccurate for the others. Additionally, we show that they very accurately describe the correlation function structure for a model describing an interface near a tricritical point. In this case, there are no analytical solutions for the correlation functions, but the approximations are near indistinguishable from the numerical solutions of the Ornstein-Zernike equation.