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 microprotrusions. 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 W_S is derived from the observable apparent receding contact angle Θ_r^app and the density of protrusions n by W_s = γ(cos Θ_r^app + 1)/n, where γ 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 Θ_r, we calculate the energy dissipated per protrusion as W_s = πγR²[A – ln(R/κ)]f(Θ_r). Here, A = 0.60 for cylindrical micropillars and 2.9 for stacks of spheres. κ is the capillary length. f(Θ_r) is a function which depends on Θ_r and the specific geometry, f ranges from ≈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.