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

journal contribution

posted on 2019-04-16, 00:00 authored by Veikko SariolaWhen
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*≈ −*k*_{1}Δ*h*or*F*≈ −*k*_{1}Δ*h*–*k*_{2}Δ*h*^{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 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.