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Structural Characteristics of a Small Group of Fixed Particles and the Maximum Density of a Random Packing of Hard Spheres

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Russian Metallurgy (Metally) Aims and scope

Abstract

The definition of random packings of hard spheres, which does not assume any specific features of a short-range order, is considered. The results obtained allow (in particular) us to determine the maximum possible density of a random packing, which has no any types of explicit or hidden long-range order. New computer experiment data, which describe the statistical–geometrical properties of random packings of two-dimensional (2D) and three-dimensional (3D) hard spherical particles, are presented. The behavior of a small group of randomly chosen and fixed spheres at various packing densities and the differences between the properties of this group and the main “large” ensemble (which follow, in particular, from the theoretical results obtained) are investigated. The dependences found experimentally are consistent with the proposed theoretical solution. Let an ensemble consist of N particles occupying total volume V (at packing density η = Nu/V, where u is the particle volume). The maximum possible density of a random packing of spherical particles (ηmax) is specified by the following geometric condition: the average volume of a Voronoi polyhedron in a random close packing cannot be smaller than the average excluded volume for all points of this packing. For an arbitrary point of the ensemble lying at a distance x from the nearest center of sphere (unit radius R = 1), excluded volume w is determined by the following relations. In 2D space, we have w(x) = π(2 – x)2 at 0 ≤ x ≤ 2 and w(x) = 0 at x > 2. In 3D space, we have w(x) = (4π/3)(2 – x)3 at 0 ≤ x ≤ 2 and w(x) = 0 at x > 2. Averaging w(x) for all points of the volume of an ensemble of particles, we can find average excluded volume 〈w〉 for a given packing density η. We can also formulate the following statement, which follows from the condition given above: if packing density η exceeds ηmax, this packing cannot be statistically homogeneous. The approach used in this work can be used to calculate the maximum possible density (upper limit) of a random close packing. This density is ηmax = 0.6813 ± 0.001 (in 2D case of an ensemble of hard disks) and ηmax = 0.6329 ± 0.0005 (for a 3D ensemble of hard spheres).

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Funding

The work was carried out according to a state assignment to the Institute of Metallurgy, Ural Branch, Russian Academy of Sciences.

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  1. Institute of Metallurgy, Ural Branch, Russian Academy of Sciences, Yekaterinburg, Russia

    A. B. Shubin

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Correspondence to A. B. Shubin.

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Translated by K. Shakhlevich

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Shubin, A.B. Structural Characteristics of a Small Group of Fixed Particles and the Maximum Density of a Random Packing of Hard Spheres. Russ. Metall. 2021, 181–186 (2021). https://doi.org/10.1134/S0036029521020245

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