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 (F-Adh) required to detach a droplet from such surfaces, but analytical expressions that relate F-Adh to the surface energy, w, are still missing. In this work, we derive analytical expressions for F-Adh between droplets of radius r and super-hydrophobic surfaces on which the contact angle is greater than 150 degrees. By applying the general condition for the onset of instability in different mechanical configurations, we find F-Adh = -pi wr and F-Adh approximate to -(4/5)pi wr, 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 F-Adh on these systems.