Analytical Expressions for Spring Constants of Capillary Bridges and Snap-in Forces of Hydrophobic Surfaces

[Image: see text] When a force probe with a small liquid drop adhered to its tip makes contact with a substrate of interest, the normal force right after contact is called the snap-in force. This snap-in force is related to the advancing contact angle or the contact radius at the substrate. Measuring snap-in forces has been proposed as an alternative to measure the advancing contact angles of surfaces. The snap-in occurs when the distance between the probe surface and the substrate is h(S), which is amenable to geometry, assuming the drop was a spherical cap before snap-in. Equilibrium is reached at a distance h(E) < h(S). At equilibrium, the normal force F = 0, and the capillary bridge is a spherical segment, amenable again to geometry. For a small normal displacement Δh = h – h(E), the normal force can be approximated with F ≈ −k(1)Δh or F ≈ −k(1)Δh – k(2)Δh(2), where k(1) = −∂F/∂h and k(2) = −1/2·∂(2)F/∂h(2) are the effective linear and quadratic spring constants of the bridge, respectively. Analytical expressions for k(1,2) are found using Kenmotsu’s parameterization. Fixed contact angle and fixed contact radius conditions give different forms of k(1,2). The expressions for k(1) found here are simpler, yet equivalent to the earlier derivation by Kusumaatmaja and Lipowsky (2010). Approximate snap-in forces are obtained by setting Δh = h(S) – h(E). These approximate analytical snap-in forces agree with the experimental data from Liimatainen et al. (2017) and a numerical method based on solving the shape of the interface. In particular, the approximations are most accurate for super liquid-repellent surfaces. For such surfaces, readers may find this new analytical method more convenient than solving the shape of the interface numerically.