Energy Dissipation of Moving Drops on Superhydrophobic and Superoleophobic Surfaces

A water drop moving on a superhydrophobic surface or an oil drop moving on a superoleophobic surface dissipates energy by pinning/depinning at nano- and micro-protrusions. Here, we calculate the work required to form, extend, and rupture capillary bridges between the protrusions and the drop. The energy dissipated at one protrusion Ws is derived from the observable apparent receding contact angle Theta(app)(r) and the density of protrusions n by W-s = gamma(cos Theta(app)(r) + 1)/n, where gamma is the surface tension of the liquid. To derive an expression for W-s that links the microscopic structure of the surface to apparent contact angles, two models are considered: A superhydrophobic array of cylindrical micropillars and a superoleophobic array of stacks of microspheres. For a radius of a protrusion R and a receding materials contact angle Theta(r), we calculate the energy dissipated per protrusion as W-s = pi gamma R-2[A - ln(R/K)]f(Theta(r)). Here, A = 0.60 for cylindrical micropillars and 2.9 for stacks of spheres. xis the capillary length. f(Or) is a function which depends on Theta(r) and the specific geometry, f ranges from approximate to 0.25 to 0.96. Combining both equations above, we can correlate the macroscopically observed apparent receding contact angle with the microscopic structure of the surface and its material properties.