Measures of fluctuations for a liquid near critical drying

We investigate density fluctuations in a liquid close to a solvophobic substrate at which a surface critical drying transition occurs. Using classical density functional theory, we determine three measures of the spatial extent and strength of the fluctuations, i.e., the local compressibility χ_{μ}(z), the local thermal susceptibility χ_{T}(z), and the reduced density χ_{*}(z); z is the distance from the substrate. While the first measure is frequently used, the second and third were introduced very recently by Eckert et al., [T. Eckert et al., Phys. Rev. Lett. 125, 268004 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.268004]. For state points in the critical drying regime, all three measures, each scaled by its bulk value, exhibit very similar forms and the ratio of χ_{T}(z) to χ_{μ}(z), for fixed z in the vapor-liquid transition region, is constant. Using a scaling treatment of surface thermodynamics, we show that such behavior is to be expected on general grounds.