A.B. Shubin.
ABSTRACT
The definition of random packings of hard spheres, which does not imply any specific features of short-range order in an ensemble of particles, is considered. The results obtained allow us to determine the maximum possible density of random packing, in which there are no any types of explicit or hidden long-range order. New data from computer experiments are presented that describe the statistical-geometric properties of random packings of two-dimensional (2D) and three-dimensional (3D) hard spherical particles. The regularities of the behavior of a small group of randomly selected and fixed spheres at different packing densities, the differences between the properties of this group and the main “large” ensemble, which follow from the theoretical results obtained, are investigated. The regularities found experimentally are consistent with the proposed theoretical solution.
Let the ensemble consist of N particles occupying the volume V (with density η = Nu/V, where u –volume of one particle). The maximum possible density of random packing of spherical particles (ηmax) is given by the following geometric condition: the average volume of the Voronoi polyhedron in a random packing cannot be less than the average value of the excluded volume <w> for all points of this packing.
For an arbitrary point of the ensemble lying at a distance x from the center of the sphere closest to it (of unit radius, R = 1), the excluded volume w is determined by the following relations. In two-dimensional space: w(x) = p(2 – x)2 for 0 ≤ x ≤ 2; w(x) = 0 for x > 2. In three-dimensional space: w(x) = (4p/3) (2 – x)3 for 0 ≤ x ≤ 2; w(x) = 0 for x > 2. Averaging w(x) for all points of the volume of the ensemble of particles, we can find the average excluded volume <w> for given packing density (h).
We can also formulate the following statement, which follows from the condition described above: if the packing density η exceeds ηmax, then such a packing cannot be statistically homogeneous.
The approach used in the work allows us to calculate the maximum possible value (upper bound) for the density of random close packing. This value is ηmax = 0.6813 ± 0.001 (in the two-dimensional case, ensemble of hard disks) and ηmax = 0.6329 ± 0.0005 (for a three-dimensional ensemble of hard spheres).
KEY WORDS:
random close packing, average excluded volume, hard spheres, hard disks, maximum density, fixed particle, small group, random sublattice.
DOI: 10.31857/S0235010620050126