Numerical prediction of the ground state of constrained two-dimensional monodisperse and bidisperse foams

Abstract

For bubbles of equal area tiling the plane, the hexagonal honeycomb has been proved to be the optimal arrangement with the least perimeter and therefore the lowest surface energy. For two-dimensional foams with a finite number of bubbles, a number of questions about the lowest surface energy (ground state) structure for constrained cases have not been answered. In particular, we seek the ground state of a foam consisting of N bubbles of equal area (monodisperse) confined within a circle or on a sphere, and also the ground state of a foam consisting of N bubbles of two different areas (bidisperse) under the same constraints. To investigate these problems, a numerical approach based on the use of Surface Evolver software is employed. We systematically developed optimisation algorithms suitable for foam systems and capable of operating in Surface Evolver, combined with a suitable choice of initial structure, which greatly enhances the capability and reliability of the overall numerical approach. We present the ground state conjectures of all constrainted monodisperse foams for N up to 100. Conjectures for the ground state of bidisperse foams with up to 72 bubbles are also proposed under specific conditions, i.e., a composition ratio of 1:1 and an area ratio of 1.4 for bubbles of different areas in the foam. We analyse the features of the ground state structures under different constraints in order to provide a priori predictions of the ground state for larger systems. For monodisperse foams confined within a circle, when N ≳ 91 bubbles close to the boundary circle form a shell-like structure, which is distinguished from the hexagonal grid structure in the bulk of the foam cluster. We provide analytical curves or best-fit curves to describe the changes to the number of bubbles in each shell of the structure. For monodisperse foams confined on a sphere, the difference between structures is reflected in how the 12 topological defects with a charge of +1 are distributed, where structures with high symmetry (in particular icosahedral symmetry) are more energetically stable. A low degree of bubble area dispersity introduces additional defects in planar foams, but has little effect on foams confined on a sphere. However, in either system, the preferential occupation of disclinations with a charge of +1 by small bubbles, as well as the segregation between bubbles, is energetically preferred. Organised structures in nature also tend to follow the principle of minimising energy, and the ground state structures mentioned in this thesis (especially on the sphere) can provide guidance on equilibrium structures in other systems, such as the selforganised structures of colloids, virus capsid structures, and cellular structures. This may then provide insights into the design of nanomaterials or deepen the understanding of biological systems.
Finally, we also present two conjectures on the ground states of monodisperse foams confined on a sphere. First, we conjecture that for the case N = 42, there exist two structurally different ground states, one of which has D5h symmetry, while the other has icosahedral (Ih) symmetry. Secondly, we conjecture that a lower bound on the perimeter of monodisperse foams on a sphere can be given by an imaginary hexagonal grid wrapping on a sphere.

Date of Award	2024
Original language	English
Awarding Institution	Aberystwyth University
Supervisor	Simon Cox (Supervisor) & Tudur Davies (Supervisor)