Arrangements of lines of hard and soft spheres in confinement

Abstract

Chains of particles, ranging from one-dimensional to three-dimensional structures, offer an avenue of research spanning condensed matter physics to microengineering, providing insights into material behaviour across diverse length scales. In this thesis we examine a system of a line of contacting spheres, placed in a transverse confining potential, which may buckle under compression or when tilted away from the horizontal, once a critical tilt angle is exceeded. We examine this system for both the case of hard and soft spheres, through the use of computer simulations and analytical calculations. We compare our results to existing experimental work.

For the case of hard spheres, under infinitesimal compression and/or tilt, the spheres form a zig-zag pattern. These structures may be calculated using a recursion relationship based on the condition of mechanical equilibrium and geometrical constraints. For small sphere displacements, we propose a continuous formulation of these iterative equations in the form of a second-order differential equation. We explore solutions to this equation numerically, but we also find approximate analytic expressions in terms of the Jacobi, Whittaker and Airy functions.

The analysis of Whittaker functions yields exact results for the case of tilting without compression. Airy functions yield results for profiles that are both tilted and compressed. For the case of compression without tilting, we provide a detailed analysis of the relevant Jacobi functions. This analysis gives further insight into the localised nature of the buckling at small compression.

We also extend our analysis to the case of a line of contacting bubbles (soft spheres), placed in a transverse confining potential, under compression. We implement the Morse-Witten theory, which is based on a linearised version of the Laplace-Young equation, to describe bubbles under the action of applied forces. We develop a simulation method for a system of Morse-Witten bubbles which we use to find structures of bubbles in mechanical equilibrium. We find these structures correspond to the analytical and experimental work presented in this thesis.

Finally, we present an outlook for how this work may be extended in the future. We discuss how the continuum model may be generalised for a variety of boundary conditions, or how it may be used to find further mechanically stable structures. The Morse-Witten model simulations may be extended to larger systems, and without the confining potential, may be used to simulate dense packings of bubbles, i.e. foams.