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

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 $q$ [Nat. Phys. 15, 287 (2019)]. Here we investigate this further by comparing approximations for the local structure factor and pair correlation function against three new examples of analytically solvable models within square-gradient theory. Our analysis further demonstrates that these approximations describe the pair correlation function and structure factor across the whole spectrum of wave vectors, encapsulating the crossover from the Goldstone mode divergence (at small $q$) to bulklike behavior (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 describe very accurately 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 nearly indistinguishable from the numerical solutions of the Ornstein-Zernike equation.

Figure 1

Potentials $\mathrm{\Delta }\varphi \left(\rho \right)$ for the five models considered here: Landau quartic potential [Eq. (27)], single-resonance model for $n=2$ (SRM) [Eq. (43)], trigonometric model [Eq. (64)], double-cubic (DC) model [Eq. (71)], and tricritical or Fisk-Widom model [Eq. (94)].

Figure 2

Comparison of the exact expressions (57) and (56) for $S(0;q)$ and $G(0,0;q)$ with the general approximations (25) and (26), respectively, for a model with a single resonance at $q=\sqrt{3}\kappa$ [Eq. (43)]. Percentage errors are shown in the insets. For the local structure factor, $S(0;q)$, comparison is also made with the improved approximation (87). See text for details.

Figure 3

Comparison of the exact expression (70) for $S(0;q)$ with the general approximation (25) for the trigonometric model [Eq. (64)]. Percentage errors are shown in the inset. For this model, the approximation (26) for $G(0,0;q)$ recovers the exact analytical result.

Figure 4

Comparison of the exact expressions (57) for $S(0;q)$ and (56) for $G(0,0;q)$ with the general approximations (25) and (26) for the double-cubic model [Eq. (71)]. Percentage errors are shown in the insets.

Figure 5

Comparison of the general approximations (25) and (26) for $S(0;q)$ and $G(0,0;q)$ with the results obtained from solving numerically the OZ equations (14) and (12) for the tricritical model [Eq. (94)]. Percentage errors are shown in the insets.