Geometry Dependence of the Receding Angle of a Droplet on a Solid Cylindrical Surface

Droplets that partially wet solid surfaces exhibit hysteresis in their contact angle. The values of the minimum (receding) and maximum (advancing) angles, θ_R and θ_A, are empirically well-defined and thought to be unique for a given set of materials. We measured the contact angles of water droplets hanging from hydrophobic, PDMS-functionalized glass and found that the value of θ_R varies with the curvature of the glass. The effect is substantial: θ_R changes from 86.0 ± 1.9° on a flat plate to 95.6 ± 1.9° on a 2 mm diameter rod of the same material. The measured values of θ_A were independent of geometry (θ_A = 103.2 ± 0.9°). We found a consistent trend among PDMS-functionalized glass cylinders with diameters ranging from 2 to 12.7 mm. We also measured the speed at which the contact line moved just after receding; these results showed a receding speed ∝ cos(θ_R) - cos(θ_E) and a consistent equilibrium contact angle, θ_E = 103.4 ± 2.3°. Finally, we measured the sliding of water droplets as rods were tilted. The larger θ_R (and thus smaller hysteresis) for a 2 mm-diameter rod led to droplets sliding at a tilt angle of just 21° from horizontal, compared to the 48° minimum tilt for a 7 mm rod. The results show that hysteresis arises from an energy barrier that depends on the shape of the droplet and contact line, both of which change with substrate curvature. The results may lead to designing surfaces that better trap water droplets or shed them for self-cleaning or water-harvesting applications.