单分散圆盘堆积的Voronoi统计模型

Statistical Voronoi Model for Monodisperse Disk Packings

研究随机堆积的几何结构和特征对于理解颗粒物质、泡沫、胶体等无序体系的宏观物理性质有重要意义。结合实验与计算机模拟,人们对不同形状、不同维度的颗粒形成的随机堆积进行了探究。理论方面,人们提出了一些基于统计几何、平均场近似或随机过程的模型分析处理随机堆积的体积分数、平均配位数等问题。不过由于堆积结构的约束条件复杂、无序的程度难以严格定义等原因,即使是单分散圆盘堆积也很难进行严格地分析和计算。对于随机密堆积的体积分数,不同的研究中给出了不同的结果。本文提出Voronoi统计模型对单分散圆盘堆积的几何特征进行理论研究。采用Voronoi网描述堆积的几何构形,导出一般情况下二维Voronoi网的面积公式,利用排斥圆和定向Voronoi圆给出了几个确定刚性圆盘间Voronoi近邻关系的定理。然后讨论平衡稳定堆积,利用接触关系对Voronoi元胞的影响,导出对称Voronoi元胞的体积分数与接触数的关系、Voronoi元胞面积及几何配位数关于接触线夹角的公式。最后,利用Voronoi网的统计分析导出平均几何配位数、平均约化自由体积关于接触线夹角概率分布的积分公式。具体的理论计算结果表明,Voronoi元胞的体积分数随其对称程度的升高而增大,随着接触数的增多也在增大;随机密堆积的平均接触数是4,平均体积分数是π2/12;这些结果可用于理解无摩擦圆盘体系形成的堆积结构和特征。

It is of great significance to study the geometric structure and characteristics of random packings for understanding the macroscopic physical properties of disordered systems such as granular matter, foam, colloid, etc.Combining with experiments and computer simulations, researchers have explored the random packings of particles with different shapes and dimensions. In theory, some models based on statistical geometry, mean field approximation or stochastic process were proposed to investigate the volume fraction and average coordination number of random packings. However, due to the complex constraints of packing structure, the difficulty of setting a criterion for the disorder, etc., it is difficult to perform rigorous analysis and calculation even for monodisperse disk packings.For the volume fraction of the random closed packing, different studies provided different results. In this paper,a statistical Voronoi model was proposed for the theoretical research of the geometric properties of the monodisperse disk packings. The Voronoi network was used to describe the configuration of a packing and an area formula of the Voronoi network was deduced for general case. Based on the concepts of excluded circle and Voronoi circle, several theorems were given for determining the Voronoi nearest neighbor relationship between rigid disks. For balanced and stable disk packings, based on the relationships between the features of a Voronoi cell and the contact structure of nearest neighbor disks, this paper derived several formulae such as the volume fraction of a symmetric Voronoi cell varying with the contact number, the area of a Voronoi cell and the geometric coordination number varying with angles of neighbor contact lines. Finally, this paper derived the integral formulae for the average geometric coordination number and the average reduced free volume with respect to the probability distribution of the contact line angle by using the statistical analysis of Voronoi network. Theoretical calculation results show that the volume fraction of a Voronoi cell increases both with the increase of its symmetry and its contact number, the average contact number is 4 and the average volume fraction is π~2/12 for the random closed packing. These results can be used to understand the geometric structure and feature of the frictionless disk packing.