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

Abstract

<p>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<sub>S</sub>, which is amenable to geometry, assuming the drop was a spherical cap before snap-in. Equilibrium is reached at a distance h<sub>E</sub> < h<sub>S</sub>. 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<sub>E</sub>, the normal force can be approximated with F ≈ -k<sub>1</sub>Δh or F ≈ -k<sub>1</sub>Δh - k<sub>2</sub>Δh<sup>2</sup>, where k<sub>1</sub> = -∂F/∂h and k<sub>2</sub> = -1/2·∂<sup>2</sup>F/∂h<sup>2</sup> are the effective linear and quadratic spring constants of the bridge, respectively. Analytical expressions for k<sub>1,2</sub> are found using Kenmotsu's parameterization. Fixed contact angle and fixed contact radius conditions give different forms of k<sub>1,2</sub>. The expressions for k<sub>1</sub> 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<sub>S</sub> - h<sub>E</sub>. 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.</p>