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Droplets on Lubricated Surfaces: The slow dynamics of skirt formation

A key question in the interaction of droplets with lubricated and liquid-infused surfaces is what determines the apparent contact angle of droplets. Previous work has determined this using measured values of the geometry of the `skirt' -- the meniscus-like deformation that forms around the base of the deposited droplet. Here, we consider theoretically the equilibrium of a droplet on a smooth, impermeable lubricant-coated surface, and argue that the small effect of gravity within the skirt and the size of the substrate are important for determining the final equilibrium. However, we also show that the evolution of the skirt towards this ultimate equilibrium is extremely slow (on the order of days for typical experimental parameter values). We therefore suggest that previous experiments on smooth lubricated surfaces may have observed only slowly-evolving transients, rather than `true' equilibria, potentially explaining why a wide range of skirt sizes have been reported.