Analysis of the packing structure of wet spheres by Voronoi–Delaunay tessellation

The structure of a packing of narrowly sized wet spheres with packing density 0.435 is analysed against the well-established random close packing with packing density 0.64 by means of the Voronoi and Delaunay tessellation. The topological and metric properties of Voronoi polyhedra, such as the number of faces, perimeter, area and volume of a polyhedron, the number of edges, perimeter and area of a polyhedron face, have been quantified. Compared to the well established random close packing, the distributions become wider and more asymmetric with a long tail at the higher values. The volume and sphericity of each Delaunay cell have also been quantified. Their distributions are shown to be wider and more asymmetric than those for the random close packing, but the peaks are almost the same. For the wet particle packing, the correlations between Voronoi polyhedron size and shape and between Delaunay cell size and shape are more scattered. The topological and metric results are also shown to be consistent with those obtained for the packing of fine particles, although the dominant forces in forming a packing differ. The results should be useful to the quantitative understanding of the structure of loosely packed particles.     

DEM simulation of binary sphere packing densification under vertical vibration

The densification of random binary sphere packings subjected to vertical vibration was modeled by using the discrete element method (DEM). The influences of operating parameters such as the vibration conditions, sphere size ratio (diameter ratio of larger versus small spheres), and composition (volume fraction of large spheres) of the binary mixture on the fractional packing density φ (defined as the volume of spheres divided by the volume of container) were studied. Two packing states, i.e., random loose packing (RLP) and random close packing (RCP), were reproduced and their micro properties such as the coordination number (CN), radial distribution function (RDF), and force structure were characterized and compared. The results indicate that properly controlling vibration conditions can realize the transition of binary packing structure from the RLP to RCP state when the sphere size ratio and composition are fixed, and the fractional packing density for RCP after vibration can reach φ_RCP?≈?0.86. Different packing characteristics from RLP and RCP identify that RCP shows much denser and more uniform structure than RLP. The current modeling results are in good agreement with those obtained from both the physical experiments and the proposed empirical models in the literature.     

Effect of vibration direction on the packing of sphero-cylinders

Particle packing is widely encountered when coping with granular materials, while mechanical vibration is usually used for packing densification. Vibration direction has been proven to be crucial for the ordered packing of spherical particles, but there are few reports for non-spherical ones in this regard. In this study, the effect of vibration direction on the macroscopic and microscopic packing parameters of sphero-cylinders are systematically examined using discrete element method (DEM). Due to the anisotropic shapes of sphero-cylinders, their packing characteristics are much richer and also more complex than those of spheres. It is found that vibration direction affects both the packing density and the packing structure of sphero-cylinders through tuning their orientation distributions and contact modes. Moreover, vibration direction plays a significant role in determining the optimal vibration intensities for dense packing. When the sphericity of Voronoi cell decreases and/or the density increases, the Nematic order parameter increases accordingly. Besides, no obvious relationship between the packing density and the average contact number is observed.     

Quasi-universality in the packing of uniform spheres under gravity

A hypothesis that packing fraction alone can be used to characterize the structure of a sphere packing, known as the quasi-universality in the literature, is tested. The analysis, conducted in terms of coordination number, radial distribution function, and structural properties from the Voronoi/Delaunay tessellation, is based on the packing results generated under different conditions, covering a wide packing fraction range. The results show strong similarities in these properties for a given packing fraction, indicating that although not generally valid, the quasi-universality approximately holds for the packing of spheres formed when the gravity is the driving force. The usefulness of this finding is also demonstrated through representative examples.     

Chord length distribution of Voronoi diagram in Laguerre geometry with lognormal-like volume distribution

The Voronoi diagram in the Laguerre geometry based on random close packing of spheres (RCP-LV diagram) has been found to be a better representation of polycrystalline structure than the conventional Poisson–Voronoi diagram. Stereology of the RCP-LV diagram with lognormal-like volume distribution has been investigated by the classical intercept count method. An improved five-parameter gamma distribution function is proposed to integrate the fact that probability density of the chord length remains non-zero as the chord length approaches zero. The proportional coefficient between the average grain size and the average chord length varies from 1.60 to 1.14 as the coefficient of variation of grain volume increases from 0.4 to 2.2. It is shown that there exists the possibility that not only the average grain size but also the grain size distribution can be estimated if the chord distribution characterization is fully explored with the aid of RCP-LV diagram simulation.     

A geometric algorithm based on tetrahedral meshes to generate a dense polydisperse sphere packing

In the discrete element method, the packing generation of polydisperse spheres with a high packing density value is a major concern. Among the methods already developed, few algorithms can generate sphere packing with a high density value. The aim of this paper is to present a new geometric algorithm based on tetrahedral meshes to generate dense isotropic arrangements of non-overlapping spheres. The method consists of first filling in every tetrahedron with spheres in contact (i.e., hard-sphere clusters). Then, the algorithm increases the packing density value by detecting the large empty spaces and filling them with new spheres. This new geometric algorithm can also generate a complex shape structure.     

A hierarchical approach to simulate the packing density of particle mixtures on a computer

In many fields of materials science it is important to know how densely a particle mixture can be packed. The “packing density” is the ratio of the particle volume and the volume of the surrounding container needed for a random close packing of the particles. We present a method for estimating the packing density for spherical particles based on computer simulations only, i.e. without the need for additional experiments. Our method is particularly suited for particle mixtures with an extremely wide range of particle diameters as they occur e.g. in modern concrete mixtures. A single representative sample from such mixtures would be much larger than can be handled on present standard computers. In our hierarchical approach the diameter range is therefore divided into smaller intervals. Samples from these limited diameter intervals are drawn and their packing density is estimated from a simulated packing. The results are used to “fill” the interstices in the sample from the next larger particle interval. To account for the interaction between particles of different sizes we include larger particles into the sample of smaller ones. The larger ones act as part of the boundary during the packing. Thus we obtain more realistic estimates of how dense a fraction of particles can be packed within the whole mixture. The focus of this paper is on the divide-and-conquer approach and on how the simulation results from the fractions can be collected into an overall estimate of the packing density. We do not go into details of the simulation technique for the single packing. We compare our results to some experimental data to show that our method works at least as good as the classical analytical models like CPM without the need for any experiments.     

Experimental study on the packing densification of mixtures of spherical and cylindrical particles subjected to 3D vibrations

To identify the dense packing of cylinder–sphere binary mixtures (spheres as filling objects), the densification process of such binary mixtures subjected to three-dimensional (3D) mechanical vibrations was experimentally studied. Various influential factors including vibration parameters (such as vibration time t, vibration amplitude A, frequency ω, vibration acceleration Γ) as well as particle size ratio r (small sphere vs. large cylinder), composition of the binary mixtures X_L (volume fraction of cylinders), and container size D (container diameter) on the packing density ρ were systematically investigated. The results show that the optimal vibration parameters for different binary cylinder–sphere mixtures are different. The smaller the size ratio, the less vibration acceleration is needed to form a stable dense packing. For each binary mixture, high packing density can be obtained when the volume fraction of large cylindrical particles is dominant. Meanwhile, increasing the container size can decrease the container wall effect and get higher packing density. The proposed analytical model has been proved to be valid in predicting the packing densification of current cylinder–sphere binary mixtures.     

Concerning the geometric limit of the density of a loose medium modeled by identical spherical particles

Based on the introduced system of definitions, the statistical-geometric property of a random close packing (RCP) of identical solid spheres (SS) is found that determines the geometric limit of the packing density. The results of calculations using computer models of SS packings show that the magnitude of the geometric limit virtually coincides with a real limiting density of the RCP of SS.Notation _max limiting density of packing in an RCP of SS - V ₀ volume of packing before placing trial spheres - V volume of packing after placing trial spheres - N number of spheres in a packing - k quantity of trial spheres - density of packing - x distance from a given point to the nearest center of SS - inaccessible volume - x _max the largest value ofx in a packing - _g model estimate of the limiting density of packing Ural State Technical University, Ekaterinburg, Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 564–568, July–August, 1995.     

Isothermal fluid flow in a packing of spheres

Simultaneous measurements of the velocity profiles inside and behind a packing made of spheres are used to establish the pattern of isothermal flow of a fluid inside the packing.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 49, No. 5, pp. 827–833, November, 1985.