Shapes of Delaunay Simplexes and Structural Analysis of Hard Sphere Packings

In this chapter we apply a computational geometry technique to investigate the structure of packings of hard spheres. The hard sphere model is the base for understanding the structure of many physical matters: liquids, solids, colloids and granular materials. The structure analysis is based on the concept of the Voronoi Diagram (Voronoi-Delaunay tessellation), which is well known in mathematics and physics. The Delaunay simplexes are used as the main instrument for this work. They define the simplest structural elements in the three-dimensional space. A challenging problem is to relate geometrical characteristics of the simplexes (e.g. their shape) with structural properties of the packing. In this chapter we review our recent results related to this problem. The presented outcome may be of interest to both mathematicians and physicists. The idea of structural analysis of atomic systems, which was first proposed in computational physics, is a subject for further mathematical development. On the other hand, physicists, chemists and material scientists, who are still using traditional methods for structure characterization, have an opportunity to learn more about this new technique and its implementation. We present the analysis of hard sphere packings with different densities. Our method permits to tackle a renowned physical problem: to reveal a geometrical principle of disordered packings. The proposed analysis of Delaunay simplexes can also be applied to structural investigation of other molecular systems.

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