Identities for droplets with circular footprint on tilted surfaces

Abstract

Exact mathematical identities are presented between the relevant parameters of droplets displaying circular contact boundary based on flat tilted surfaces. Two of the identities are derived from the force balance, and one from the torque balance. The tilt surfaces cover the full range of inclinations for sessile or pendant drops, including the intermediate case of droplets on a wall (vertical surface). The identities are put under test both by the available solutions of a linear response approximation at small Bond numbers as well as the ones obtained from numerical solutions, making use of the Surface Evolver software. The subtleties to obtain certain angle-averages appearing in identities by the numerical solutions are discussed in detail. It is argued how the identities are useful in two respects. First is to replace some unknown values in the Young–Laplace equation by their expressions obtained from the identities. Second is to use the identities to estimate the error for approximate analytical or numerical solutions without any reference to an exact solution.

New insights into the capillary retention force and the work of adhesion

“…For example, the force necessary to detach a sessile drop is different than for a pendant drop, because weight opposes (helps) the detachment of a sessile (pendant) drop. In addition, even in the absence of gravity, the Laplace pressure [14,[28][29][30][31] produces a force in the normal direction that needs to be taken into account. However, even though f ⊥ is different from f d , in this paper we will refer to f ⊥ as the normal force of adhesion for two reasons.…”

“…First, f ⊥ is the only force that is exerted by the solid on the triple line and that tries to anchor the drop on the solid in the normal direction. The force produced by the pressure at the solid-liquid contact area points away from the surface [14,28], and therefore it seems natural to take f ⊥ to be the force that truly anchors a drop to a solid surface in the direction perpendicular to the surface. Second, the conceptual comparison of Eqs.…”

“…(2.11) has been studied (with a different notation and for different situations) only in Refs. [14,[28][29][30][31]. Our approach is essentially the same as that of Refs.…”

“…Our approach is essentially the same as that of Refs. [14,[28][29][30][31], the only real difference being that for us the normal force of adhesion does not take the (Laplace) pressure into account, whereas in Refs. [28][29][30][31] the normal force of adhesion is implicitly taken to be the same as the detachment force, which does include the contribution of the force produced by pressure.…”

“…Although the lateral retention force has been studied extensively (see Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein), its normal counterpart f ⊥ has not received nearly as much attention [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], in spite of the fact that it plays an essential role in many situations. A familiar situation is a pendant drop, which must be anchored to the surface by a force that is normal to the surface.…”