Voronoi diagrams in quasi-2D hard sphere systems

Variants of the Voronoi construction, commonly applied to divide space, are analysed for quasi-two-dimensional hard sphere systems. Configurations are constructed from a polydisperse lognormal distribution of sphere radii, mimicking recent experimental investigations. In addition, experimental conditions are replicated where spheres lie on a surface such that their respective centres do not occupy a single plane. Significantly, we demonstrate that using an unweighted (no dependence on sphere size) two-dimensional Voronoi construction (in which the sphere centres are simply projected onto a single plane) is topologically equivalent to taking the lowest horizontal section through a three-dimensional construction in which the division of space is weighted in terms of sphere size. The problem is then generalised by considering the tessellations formed from horizontal sections through the three-dimensional construction at arbitrary cut height above the basal plane. This further suggests a definition of the commonly-applied packing fraction which avoids the counter-intuitive possibility of it becoming greater than unity. Key network and Voronoi cell properties (the fraction of six-membered rings, assortativity and cell height) and are analysed as a function of separation from the basal plane and the limits discussed. Finally, practical conclusions are drawn of direct relevance to on-going experimental investigations.