Voronoi neighbor statistics of hard-disks and hard-spheres

The neighbor distribution in hard-sphere and hard-disk fluids is analyzed using Voronoi tessellation. The statistical measures analyzed are the nth neighbor coordination number (Cn), the nth neighbor distance distribution [fn(r)], and the distribution of the number of Voronoi faces (Pn). These statistics are sensitive indicators of microstructure, and they distinguish thermodynamic and annealed structures. A sharp rise in the hexagon population marks the onset of hard-disk freezing transition, and Cn decreases sharply to the hexagonal lattice values. In hard-disk random structures the pentagon and heptagon populations remain significant even at high volume fraction. In dense hard-sphere (three-dimensional) structures at the freezing transition, C1 is close to 14, instead of the value of 12 expected for a face-centered-cubic lattice. This is found to be because of a topological instability, where a slight perturbation of the positions in the centers of a pair of particles transforms a vertex in the Voronoi polyhedron into a Voronoi surface. We demonstrate that the pair distribution function and the equation-of-state obtained from Voronoi tessellation are equal to those obtained from thermodynamic calculations. In hard-sphere random structures, the dodecahedron population decreases with increasing density. To demonstrate the utility of the neighbor analysis, we estimate the effective hard-sphere diameter of the Lennard-Jones fluid by identifying the diameter of the spheres in the hard-sphere fluid which has C1 equal to that for the Lennard-Jones fluid. The estimates are within 2% deviation from the theoretical results of Barker-Henderson and Weeks-Chandler-Andersen.