Correlation between superhydrophobicity and the power spectral density of randomly rough surfaces

We show experimentally and analytically that for single-valued, isotropic, homogeneous, randomly rough surfaces consisting of bumps randomly protruding over a continuous background, superhydrophobicity is related to the power spectral density of the surface height, which can be derived from microscopy measurements. More precisely, superhydrophobicity correlates with the third moment of the power spectral density, which is directly related to the notion of Wenzel roughness (i.e., the ratio between the real area of the surface and its projected area). In addition, we explain why randomly rough surfaces with identical root-mean-square roughness values may behave differently with respect to water repellence and why roughness components with wavelength larger than 10 μm are not likely to be of importance or, stated otherwise, why superhydrophobicity often requires a contribution from submicrometer-scale components such as nanoparticles. The analysis developed here also shows that the simple thermodynamic arguments relating superhydrophobicity to an increase in the sample area are valid for this type of surface, and we hope that it will help researchers to fabricate efficient superhydrophobic surfaces based on the rational design of their power spectral density.