## The capillary interaction between objects at liquid interfaces

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## Abstract

This thesis reports numerical and analytical results on the floatation and capillary interaction of granular–sized objects at liquid–fluid interfaces. Such objects create deformations at the liquid surfaces which result in their interaction with each other. It has been experimentally shown that this effect can be used for self-assembly of ordered structures, and there are examples in the natural world too.

The deformation created by a solid object at a liquid interface is governed by the Laplace-Young equation and appropriate boundary conditions. This is a nonlinear differential equation which is hard in general to solve analytically, and only approximate solutions exist for most of the interesting cases. We develop a new numerical solution to determine the shape of a liquid interface in the vicinity of multiple solid objects using the hp–Meshless Cloud method, which is a meshfree finite difference method. This solves the nonlinear Laplace-Young equation without any approximations.

First a system is considered where circular cylinders are immersed in a liquid. The meniscus shape is determined, and the force of interaction between a pair of cylinders is calculated as a function of the distance between them. The results are compared with previously published asymptotic solutions and experimental results. When the cylinders are sufficiently far apart, the experimental results agree with both the numerical and asymptotic results. However, as the cylinders move closer, the asymptotic solution is unable to explain the experimental results because this solution is valid only in regions with small meniscus slopes. In contrast, the numerical solution is able to accurately explain the experimental results at all distance ranges.

The numerical solution is further extended to solve for two elliptical cylinders at a liquid interface. Additionally, a new analytical solution is also developed for this problem. For the case of an isolated cylinder, this analytical solution is able to predict the same contact line shapes and meniscus profiles as the numerical solution. Both the solutions show that the force of attraction between a pair of elliptical cylinders is larger when they are in the tip–to–tip orientation, and smaller in the side–to–side orientation. The difference between the forces in the two orientations diminishes at large inter–cylinder separations. It is also shown that the meniscus far away from an elliptical cylinder is same as one created by a circular cylinder with perimeter equal to that of the elliptical cylinder.

The numerical solution is further developed to solve for multiple floating spheres. This is a complicated condition compared to the vertical cylinders be-cause the vertical locations of the spheres and the horizontal projections of the three–phase contact lines are not known a priori. A new algorithm is developed to simultaneously satisfy the force balance, Laplace–Young equation and the geometric properties of the spheres. This shows that floating and sinking of a pair of spheres can depend on their relative positions. An unexpected and new result is obtained: at an intermediate inter–particle distance range, a sphere that would sink in isolation can float as a part of a pair or a cluster. A simple and new semi–analytical solution is also developed, which also predicts the same behaviour. Additionally, the numerical solution predicts that a sphere that would float in isolation would sink as a part of a pair at very small inter-particle distances.

This numerical solution is then extended to determine the force of attraction between pairs of floating spheres. This is studied experimentally as well, by tracking the movement of particles at a liquid interface. Asymptotic solutions have previously been published for this problem. The numerical solution shows that the force deviates from the predictions of these asymptotic expressions when the density of the spheres is high. At small densities such as those used in the experiments, the asymptotic solutions correctly predict the force of attraction.