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

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 hS, which is amenable to geometry, assuming the drop was a spherical cap before snap-in. Equilibrium is reached at a distance hE < hS. At equilibrium, the normal force F = 0, and the capillary bridge is a spherical segment, amenable again to geometry. For a small normal displacement Δ h = h - hE, the normal force can be approximated with F ≈ - k1Δ h or F ≈ - k1Δ h - k2Δ h2, where k1 = -∂ F/∂ h and k2 = -1/2·∂2 F/∂ h2 are the effective linear and quadratic spring constants of the bridge, respectively. Analytical expressions for k1,2 are found using Kenmotsu's parameterization. Fixed contact angle and fixed contact radius conditions give different forms of k1,2. The expressions for k1 found here are simpler, yet equivalent to the earlier derivation by Kusumaatmaja and Lipowsky (2010). Approximate snap-in forces are obtained by setting Δ h = hS - hE. 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.