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.     

Bulk and interior packing densities of random close packing of hard spheres

The packing densities of random close packing of equal hard spheres (RCPHS) are studied. The RCPHS is generated by a rearrangement algorithm with an optimization subroutine. Traditionally defined packing density, bulk density, is found to be 0.635 ± 0.002 by extrapolation to infinite number of spheres. We propose that there exist a characteristic packing density without boundary effects. This interior packing density is calculated by two methods, resulting in values without statically significant difference. Interior packing density deduced from Voronoi diagram is 0.6690 ± 0.0006. Local packing density for each sphere is defined as ratio of its volume to volume of its corresponding Voronoi cell and is sensitive to sphere's local configuration and overlapping.     

Computer Simulation of Random Sphere Packing in an Arbitrarily Shaped Container

Most simulations of random sphere packing concern a cubic or cylindric container with periodic boundary, containers of other shapes are rarely studied. In this paper, a new relaxation algorithm with pre-expanding procedure for random sphere packing in an arbitrarily shaped container is presented. Boundaries of the container are simulated by overlapping spheres which covers the boundary surface of the container. We find 0.4~0.6 of the overlap rate is a proper value for boundary spheres. The algorithm begins with a random distribution of small internal spheres. Then the expansion and relaxation procedures are performed alternately to increase the packing density. The pre-expanding procedure stops when the packing density of internal spheres reaches a preset value. Following the pre-expanding procedure, the relaxation and shrinking iterations are carried out alternately to reduce the overlaps of internal spheres. The pre-expanding procedure avoids the overflow problem and gives a uniform distribution of initial spheres. Efficiency of the algorithm is increased with the cubic cell background system and double link data structure. Examples show the packing results agree well with both computational and experimental results. Packing density about 0.63 is obtained by the algorithm for random sphere packing in containers of various shapes.     

Impact of powder composition on processing-relevant properties of pharmaceutical materials: An experimental study

In this work, we studied the influence of powder composition on packing density and other processing-relevant properties of binary mixtures, including powder flowability. Binary mixtures of pharmaceutical powders with different particle size ratios, α and varying fractions of large and small particles were analyzed systematically. Mixtures of three excipients and one API with different composition (2, 5, 10, 30, 50, 70, 90, 95 and 98 wt%) were prepared in a Turbula mixer. Powders with different properties and particle size distribution were chosen, in order to obtain three binary mixtures with different size ratios. Then, macroscopic powder properties including bulk (poured) and tapped density (BD and TD) were measured. A powder rheometer was used to measure the flow function coefficient (ffc), cohesion, compressibility and permeability of the binary mixtures. We considered experimentally three classes of binary mixtures, which are characterized by two critical ratios of particle diameter: the critical size ratio of entrance (α_c) and the critical size ratio of replacement (α_r), where α_c = 0.154 and α_r = 0.741. Below the critical size ratio of entrance (α_c), the particle asymmetry (ratio between large and small particle diameters) is high and small particles can fill the voids between larger ones. Between α_c and the critical size ratio of replacement (α_r), the smaller particles are too large to fit in the voids between larger particles (packing structure changes). Above α_r, the particles are more or less symmetric in size and overall packing structure does not change by mixing the particles. Our experiments show that there is a non-linear and non-monotonic dependence of all relevant properties on composition for powder mixtures that have an α < α_r. This non-linear behavior is even more significant for strongly asymmetric binary mixtures with α < α_c. We argue that this behavior is related to the composition dependence of random packing of particulate systems. Our results have relevance to pharmaceutical particle processing operations where constant powder mixture properties are needed to ensure quality standards are met; such operations include capsule or die filling during tableting, and the continuous feeding of powders via screw feeders. Our results suggest that for pharmaceutical particle processing operations, where constant powder mixture properties are a prerequisite for process robustness, the size ratio of API and excipient particles, α should not be smaller than α_r = 0.741.     

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.     

Effect of particle size ratio and contribution of particle size fractions on micromechanics of uniaxially compressed binary sphere mixtures

This paper is an extension of the recent work of Wi?cek (Granul Matter 18:42, 2016), wherein geometrical parameters of binary granular mixtures with various particle size ratio and contribution of the particle size fractions were investigated. In this study, a micromechanics of binary mixtures with various ratio of the diameter of small and large spheres and contribution of small particles was analyzed using discrete element simulations of confined uniaxial compression. The study addressed contact normal orientation distributions, global and partial contact force distributions and pressure distribution in packings of frictional spheres. Additionally, the effect of particle size ratio and contribution of particle size fractions on energy dissipation in granular mixtures was investigated. The particle size ratio in binary packings was chosen to prevent small particles from percolating through bedding. The bimodality of mixtures was found to have a strong effect on distribution of contact normal orientation and distribution of normal contact forces in binary mixtures. Stress transfer in binary packing was also determined by both, particle size ratio and volume fraction of small particles. Dissipation of energy was higher in mixtures with higher particle size ratios and decreased with increasing contribution of small spheres in system.     

A three dimensionally periodic,eleven coordinated,dense packing of symmetry equivalent spheres

Symmetry equivalent equal size spheres can be packed in an array filling 71.87% of space. Each sphere in this packing has eleven neighbors at unit distance. The density of this tetragonal close packing is second only to closest packing with twelve coordination. Anions in numerous compounds, including some which are solid electrolytes, arrange themselves in this pattern. It occurs among others in rutiles, β-BeO, Li₄GeO₄ and CaSO₄.     

Computer simulation of morphology and packing behaviour of irregular particles, for predicting apparent powder densities

This paper describes a methodology for prediction of powder packing densities which employs a new approach, designated as random sphere construction (RSC), for modelling the shape of irregular particles such as those produced by water atomization of iron. The approach involves modelling an irregular particle as a sphere which incorporates smaller corner spheres located randomly at its surface. The RSC modelling technique has been combined with a previously developed particle packing algorithm (the random build algorithm), to provide a computer simulation of irregular particle packings. Analysis of the simulation output data has allowed relationships to be established between the particle modelling parameters employed by the RSC algorithm, and the density of the simulated packings. One such parameter is η, which is the number of corner spheres per particle. A relationship was established between η (which was found to have a profound influence on packing density), and the fractional density of the packing, fd. Vision system techniques were used to measure the irregularity of the simulated particles, and this was also related to η. These two relationships were then combined to provide a plot of fractional density for a simulated packing against irregularity of the simulated particles. A comparison was made of these simulated packing densities and observed particle packing densities for irregular particles, and a correlation coefficient of 0.96 was obtained. This relatively good correlation indicates that the models developed are able to realistically simulate packing densities for irregular particles. There are a considerable number of potential applications for such a model in powder metallurgy (PM), process control. In combination with on-line particle image analysis, the model could be used to automatically predict powder densities from particle morphology.     

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.     

SIZE DISTRIBUTION OF CLUSTERS INHERENT IN RANDOM DISPERSIONS OF EQUAL SPHERES WITH A COAGULATION RADIUS

Abstract

The spatial structure of a computer-simulated random dispersion of equal spheres is investigated. The sphere of influence, i.e. the coagulation radius, is assumed for each particle without any particle movement. If the distance between sphere centers lies within the coagulation radius, the spheres are regarded as being connected to each other to form a cluster. Various sizes of clusters exist inherently in the random dispersion. The effects of the coagulation radius and the bulk-mean particle volume fraction on the size distribution of clusters are discussed theoretically and experimentally.     

SIZE DISTRIBUTION OF CLUSTERS INHERENT IN RANDOM DISPERSIONS OF EQUAL SPHERES WITH A COAGULATION RADIUS

The spatial structure of a computer-simulated random dispersion of equal spheres is investigated. The sphere of influence, i.e. the coagulation radius, is assumed for each particle without any particle movement. If the distance between sphere centers lies within the coagulation radius, the spheres are regarded as being connected to each other to form a cluster. Various sizes of clusters exist inherently in the random dispersion. The effects of the coagulation radius and the bulk-mean particle volume fraction on the size distribution of clusters are discussed theoretically and experimentally.     

Simulation of random close packed discs and spheres

Random packing is a phenomenon which is observed on a regular basis but which currently lacks any constructive mathematical formalism. Consequently, much of our understanding of packing structures comes from experimental physics and mathematical modeling. Using Monte Carlo simulations as a model of real sphere aggregates, estimates of contact distributions are obtained for various binary mixtures of spheres. We find that the number of contacts can be described by a single algebraic function of two variables: the ratio of radii, and the concentration of small spheres.     

Predicting porosity of binary mixtures made out of irregular nonspherical particles: Application to natural sediments

Predicting porosity or packing density of sediments made of coarse and fine components of arbitrary geometry is critical to many science and engineering applications. Well-established analytical models for packing of spheres express porosity of the binary mixture as a function of fine-to-coarse particle size ratio. Nevertheless, the applicability of such models to natural granular materials is limited given the nonspherical and irregular nature of the particles whose packing depends on both particle size and shape. The objective of this study is to develop a model that predicts the porosity of binary mixtures made up of irregular nonspherical particles. We modified a previously developed linear sphere-packing model so that it takes into account the effect of both the particle size and shape. As an input, the modified model uses the coarse-to-fine particles specific surface area ratio instead of using the particle size ratio required by the sphere-packing model. We tested the modified model by predicting the porosities of a binary mixture composed of coarse and fine calcite aggregates. We further validate the model by using published data on the porosity of binary mixtures made of synthesized, cubical and cylindrical particles. Our model predictions show good agreement with the measured porosity.     

Mixing narrow coarse and fine coal fractions – The maximum volume fraction of suspensions

A main objective of coal-water slurry fuel (CWSF) preparation is to achieve maximum loading of coal. In the absence of a strong colloidal attractive force, the maximum loading is determined by the packing density of the particles which in turn is a function of the particle size distribution. In this study, coal fractions of different mean sizes with a narrow distribution were separated by sieving. Mixtures of different mean coarse to fine size ratios were then prepared. For each size ratio, different amounts of coarse particle contents were prepared. With these different mixtures, water was added to produce the CWSF. The maximum volume packing density, φ_max, for each mixture was determined using a rheological vane yield stress technique. The determination of φ_max involving the direct yield stress measurements of extremely high concentration suspensions is an entirely new and accurate approach. It was found that the highest φ_max was obtained when the coarse to fine ratio was ～10 located at a coarse coal content of 70 wt%. This result is consistent with that obtained by theoretical modelling of bimodal mixing of monodisperse size spheres with a size ratio of 10. At lower size ratio, φ_max obtained at the optimum coarse coal content was lower.     

Inclusion interaction and effective material properties in a particle-filled composite material system

This work presents a methodology implementing random packing of spheres combined with commercial finite element method (FEM) software to optimize the material properties, such as Young’s modulus, Poisson’s ratio, and coefficient of thermal expansion (CTE) of two-phase materials used in electronic packaging. The methodology includes an implementation of a numerical algorithm of random packing of spheres and a technique for creating conformal FEM mesh of a large aggregate of particles embedded in a medium. We explored the random packing of spheres with different diameters using particle generation algorithms coded in MATLAB. The FEM meshes were generated using software MATLAB and TETGEN. After importing the databases of the nodes and elements into commercial FEM software ANSYS, the composite materials with spherical fillers and the polymer matrix were modeled using ANSYS. The effective Young’s modulus, Poisson’s ratio, and CTE along different axes were calculated using ANSYS by applying proper loading and boundary conditions. It was found that the composite material was virtually isotropic. The Young’s modulus and Poisson’s ratio calculated by FEM models were compared to a number of analytical solutions in the literature. For low volume fraction of filler content, the FEM results and analytical solutions agree well. However, for high volume fraction of filler content, there is some discrepancy between FEM and analytical models and also among the analytical models themselves. The discrepancy is attributed to the multi-body interaction effect of the filler particles when they are getting close.     

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.     

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.     

Mono-sized sphere packing algorithm development using optimized Monte Carlo technique

In this paper, fuel cell catalyst layer was developed using the optimized sphere packing algorithm. An optimization technique named adaptive random search technique (ARSET) was employed in this packing algorithm. The ARSET algorithm will generate the initial location of spheres and allow them to move in the random direction with the variable moving distance, randomly selected from the sampling range (α), based on the Lennard–Jones potential and Morse potential of the current and new configuration. The solid fraction values obtained from this developed algorithm are in the range of 0.610–0.624 while the actual processing time can significantly be reduced by 5.58–34% based on the number of spheres. The initial random number sampling range (α) was investigated and the appropriate α value is equal to 0.5.     

Synthesis of hollow lead sulfide microspheres

Lead sulfide (PbS) uniform hollow spheres have been successfully synthesized by γ-irradiating PMMA–CS₂–ethanol aqueous solution that contains Pb(CH₃COO)₂·3H₂O at room temperature. X-ray diffraction (XRD), field emission scanning electron microscope (FE-SEM), transmission electron micrograph (TEM), and high-resolution transmission electron micrograph (HRTEM) experimental results show that the diameter of PbS hollow spheres, the thickness of sphere shell, and the size of these crystallites are about 500, 20, and 10 nm, respectively. Room temperature UV–vis absorption spectrum of the PbS hollow spheres gives its peak centered at around 238 nm and a weak shoulder peak centered at about 322 nm. The obvious blue shift of the absorption peak may be attributed to the small dimension of the PbS bricks of the spheres wall. A possible growth mechanism of PbS hollow sphere is also presented. The successful preparation of PbS hollow spheres in large scale under mild conditions could be of interest for both applications and fundamental studies.