An analytical theory for the capillary bridge force between spheres

Abstract

An analytical theory has been developed for properties of a steady, axisymmetric liquid–gas capillary bridge that is present between two identical, perfectly wettable, rigid spheres. In this theory the meridional profile of the capillary bridge surface is represented by a part of an ellipse. Parameters in this geometrical description are determined from the boundary conditions at the three-phase contact circle at the sphere and at the neck (i.e. in the middle between the two spheres) and by the condition that the mean curvature be equal at the three-phase contact circle and at the neck. Thus, the current theory takes into account properties of the governing Young–Laplace equation, contrary to the often-used toroidal approximation. Expressions have been developed analytically that give the geometrical parameters of the elliptical meridional profile as a function of the capillary bridge volume and the separation between the spheres. A rupture criterion has been obtained analytically that provides the maximum separation between the spheres as a function of the capillary bridge volume. This rupture criterion agrees well with a rupture criterion from the literature that is based on many numerical solutions of the Young–Laplace equation. An expression has been formulated analytically for the capillary force as a function of the capillary bridge volume and the separation between the spheres. The theoretical predictions for the capillary force agree well with the capillary forces obtained from the numerical solutions of the Young–Laplace equation and with those according to a comprehensive fit from the literature (that is based on many numerical solutions of the Young–Laplace equation), especially for smaller capillary bridge volumes.

Generalized Shields criterion for weakly cohesive granular materials

“…tension between the liquid in a bridge and the surrounding fluid, the attractive force F cap between two particles connected by a capillary bridge can be expressed analytically. Assuming that influence of gravity on the capillary bridge can be neglected [66], the exact value of F cap depends on the shape and volume of the bridge, the radii of the two particles, and the Young-Laplace pressure, given by the surface tension γ and the local curvature [66][67][68][69][70]. The capillary force of a liquid bridge can also be approximated by the so-called "gorge method" [67,69], which gives…”

Capillary Forces between Concave Gripper and Spherical Particle for Micro-Objects Gripping

“…Geometric approximation brings a small error (<10%) with respect to the exact solution of the Young–Laplace equation because of the geometric errors [ 26 , 27 ]. Numerical solutions of the nonlinear Young–Laplace equation are effectively exact, as reported in previous works [ 25 , 28 ]. In present study, a numerical procedure was developed based on a shooting method to obtain the capillary bridge profile and solve the capillary force between a spherical concave gripper and a spherical particle.…”

Exact calculation of axisymmetric capillary bridge properties between two unequal-sized spherical particles

“…is not satisfied and equation (15) becomes redundant. The condition r 1 sin δ 1 = r 2 sin δ 2 is equivalent to y c1 = y c2 , meaning that the ordinates of the liquid bridge's triple points are different.…”

“…Many analytical approaches have investigated properties of a capillary bridge between two elastic solids, such as between a sphere and a plane [13] or between two adjacent spheres. However, most of the studies on spherical bodies are restricted to the monodisperse case 1 [7,9,14,15] and primarily focused on the convex meridional profile of the liquid bridge. The coalescence of capillary doublets between touching spheres of the same radius has also been studied theoretically by Gagneux and Millet [16].…”