Force of adhesion between droplets and super-hydrophobic surfaces: Closed-form analytical expressions

Super-hydrophobic and liquid-repellent surfaces can be characterized experimentally in tensile adhesion experiments with the force (FAdh) required to detach a droplet from such surfaces, but analytical expressions that relate FAdh to the surface energy, w, are still missing. In this work, we derive analytical expressions for FAdh between droplets or radius r and super-hydrophobic surfaces on which the contact angle is greater than 150°. By applying the general condition for the onset of instability in different mechanical configurations, we find FAdh = -πwr and FAdh ≈ -(4/5)πwr, for the fixed-force and fixed-grip configurations, respectively, as well as other expressions that depend on the ratio of the spring constant of a generic force measuring apparatus to the surface tension of the liquid composing the droplet. These expressions open the possibility of retrieving w or, equivalently, the receding contact angle, by measuring FAdh on these systems.