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 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 Δ h = h - h_E, 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 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 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 = 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.