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 (η_max) 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 η_max, 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 η_max = 0.6813 ± 0.001 (in 2D case of an ensemble of hard disks) and η_max = 0.6329 ± 0.0005 (for a 3D ensemble of hard spheres).

中文翻译：

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一小组固定颗粒的结构特征和硬球体随机堆积的最大密度

摘要

考虑了硬球体的随机堆积的定义，该定义不假定短程有任何特定特征。获得的结果使（特别是）我们能够确定随机填充的最大可能密度，该随机填充没有任何类型的显式或隐藏的远程顺序。提出了新的计算机实验数据，这些数据描述了二维（2D）和三维（3D）硬球形颗粒的随机堆积的统计几何特性。研究了一小组随机选择的固定球体在不同堆积密度下的行为，以及该组球体与主要“大”集合体之间的区别（尤其是从获得的理论结果出发）。实验上发现的依赖关系与所提出的理论解决方案是一致的。让一个合奏包括N个粒子占据总体积V（堆积密度η= Nu / V，其中u是粒子体积）。球形粒子（η的无规填料的最大可能的密度_最大）由下面的几何条件中指定：一个沃罗诺伊多面体的随机紧密堆积的平均体积不能超过此包装的所有点的平均排除体积小。对于位于距球体最近的中心的距离为x处的任意一个合奏点（单位半径R = 1），排除体积w由以下关系确定。在2D空间中，我们有w（x）=π（2 - X）²在0≤ X ≤2和瓦特（X）= 0在X > 2。在3D空间中，我们有瓦特（X）=（4π/ 3）（2 - X）³在0≤ X ≤2和瓦特（X）= 0在X > 2平均化瓦特（X）为颗粒群的体积的所有点中，我们可以发现平均排除体积<瓦特>对于给定的填充密度η。我们还可以根据以下条件得出以下陈述：如果堆积密度η超过η_max，该填充在统计上不能是同质的。这项工作中使用的方法可用于计算随机密堆积的最大可能密度（上限）。此密度是η_最大= 0.6813±0.001（在硬盘的集合的2D的情况）和η_最大= 0.6329±0.0005（对于硬球的3D集合）。