Nanoscale Limit to the Applicability of Wenzel's Equation

We examine the extent to which nanoscale geometric substrate roughness influences the contact angle droplets establish on solid surfaces. Free-energy-based Monte Carlo simulation methods are used to compute contact angles and interfacial tensions of a model Lennard-Jones fluid on substrates with regular one-dimensional heterogeneities characterized by amplitudes and periodicities in the 2-25 nm range. We focus on a relatively strong surface that facilitates the formation of Wenzel droplets. Our results enable us to probe the validity of Wenzel's model at these length scales. We find that the aforementioned model predicts the evolution of the contact angle with near-quantitative accuracy over a wide range of amplitudes for substrates with periodicities larger than approximately 20 fluid diameters, or 10 nm for an argon-like system. However, below this length scale the Wenzel model provides progressively poorer estimates of the contact angle as the periodicity of the substrate features decreases. At these relatively small length scales, the Wenzel model overestimates the influence of roughness. To complete our analysis, we introduce a means to overcome sampling difficulties that arise at intermediate densities during grand canonical simulation. Specifically, we describe a two-step process that enables us to access the free energy of a system containing a thick liquid film in contact with the substrate. The process involves high-temperature grand canonical simulation followed by temperature-expanded ensemble simulation at relatively high surface density.