Attempts to extend the capillary-wave theory of fluid interfacial fluctuations to microscopic wavelengths, by introducing an effective wave-vector $\left(q\right)$ -dependent surface tension ${\sigma }_{\mathrm{eff}}\left(q\right)$, have encountered difficulties. There is no consensus as to even the shape of ${\sigma }_{\mathrm{eff}}\left(q\right)$. By analyzing a simple density functional model of the liquid-gas interface, we identify different schemes for separating microscopic observables into background and interfacial contributions. In order for the backgrounds of the density-density correlation function and local structure factor to have a consistent and physically meaningful interpretation in terms of weighted bulk gas and liquid contributions, the background of the total structure factor must be characterized by a microscopic $q$-dependent length $\zeta \left(q\right)$ not identified previously. The necessity of including the $q$ dependence of $\zeta \left(q\right)$ is illustrated explicitly in our model and has wider implications; i.e., in typical experimental and simulation studies, an indeterminacy in $\zeta \left(q\right)$ will always be present, reminiscent of the cutoff used in capillary-wave theory. This leads inevitably to a large uncertainty in the $q$ dependence of ${\sigma }_{\mathrm{eff}}\left(q\right)$.

• Received 21 November 2014
• Revised 28 January 2015

DOI:https://doi.org/10.1103/PhysRevE.91.030401