The microstructures of cold dense systems as informed by hard sphere models and optimal packings

Abstract

Sphere packings, or arrangements of "billiard balls" of various sizes that never overlap, are especially informative and broadly applicable models. In particular, a hard sphere model describes the important foundational case where potential energy due to attractive and repulsive forces is not present, meaning that entropy dominates the system's free energy. Sphere packings have been widely employed in chemistry, materials science, physics and biology to model a vast range of materials including concrete, rocket fuel, proteins, liquids and solid metals, to name but a few.

Despite their richness and broad applicability, many questions about fundamental sphere packings remain unanswered. For example, what are the densest packings of identical three-dimensional spheres within certain defined containers? What are the densest packings of binary spheres (spheres of two different sizes) in three-dimensional Euclidean space [special characters omitted]? The answers to these two questions are important in condensed matter physics and solid-state chemistry. The former is important to the theory of nucleation in supercooled liquids and the latter in terms of studying the structure and stability of atomic and molecular alloys. The answers to both questions are useful when studying the targeted self-assembly of colloidal nanostructures.

In this dissertation, putatively optimal answers to both of these questions are provided, and the applications of these findings are discussed. The methods developed to provide these answers, novel algorithms combining sequential linear and nonlinear programming techniques with targeted stochastic searches of conguration space, are also discussed. In addition, connections between the realizability of pair correlation functions and optimal sphere packings are studied, and mathematical proofs are presented concerning the characteristics of both locally and globally maximally dense structures in arbitrary dimension d. Finally, surprising and unexpected findings are presented concerning structural signatures inherent to nonequilibrium glassy states of matter, as modeled using a prototypical glass of 1,000,000 identical spheres.