The capillary interaction between two vertical cylinders

Abstract

Particles floating at the surface of a liquid generally deform the liquid surface. Minimizing the energetic cost of these deformations results in an inter-particle force which is usually attractive and causes floating particles to aggregate and form surface clusters. Here we present a numerical method for determining the three-dimensional meniscus around a pair of vertical circular cylinders. This involves the numerical solution of the fully nonlinear Laplace-Young equation using a mesh-free finite difference method. Inter-particle force-separation curves for pairs of vertical cylinders are then calculated for different radii and contact angles. These results are compared with previously published asymptotic and experimental results. For large inter-particle separations and conditions such that the meniscus slope remains small everywhere, good agreement is found between all three approaches (numerical, asymptotic and experimental). This is as expected since the asymptotic results were derived using the linearized Laplace-Young equation. For steeper menisci and smaller inter-particle separations, however, the numerical simulation resolves discrepancies between existing asymptotic and experimental results, demonstrating that this discrepancy was due to the nonlinearity of the Laplace-Young equation.

Self-assembly of repulsive interfacial particles via collective sinking

“…To make our models more realistic would require detailed simulations of the meniscus around an array of spherical particles. While this would be an involved procedure, we believe that it may soon be feasible computationally 35,40 and, further, may yield new insight beyond existing mean-field theories [25][26][27][28] . In particular, these mean-field theories use a linear superposition of the far-field, small deflection meniscus around an axisymmetric object, h(r) ∼ K 0 (r/ c ), even though close to small axisymmetric objects a subtly different meniscus form is more appropriate 41 .…”

Deformation of a free interface pierced by a tilted cylinder: Variation of the contact angle

“…In future work it will be interesting to change the shape of the object that pierces the interface [21,22], and to characterize the interaction between several objects [20], again varying systematically both the contact angle and the angle of inclination. In principle, the method could be applied to a wide range of systems in which gravity does not play a role.…”

“…These include mesh-free finite difference methods [20]; boundary-element simulations [21]; Surface Evolver simulations coupled to finitedifference simulations [22]; and an ordinary differential equation solver (MATLAB) [3].…”

The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young’s and Young-Laplace equations

“…A widely used approach to calculate a minimum energy surface is by means of the Surface Evolver program. 42 But several other approaches, both theoretical and numerical, have been used for studying the fluid-fluid interface shape in different physical problems, e.g., menisci shapes and capillary interactions, [43][44][45][46][47][48][49][50][51][52] droplet shapes, [53][54][55][56][57] diffuse interfaces, [58][59][60] or fluid-fluid interfaces in contact with deformable solids. [61][62][63] In this article, we introduce a new numerical method to obtain the minimum-energy shape of a fluid-fluid interface.…”