Fractal dimension of Apollonian packing of spherical particles

Three-dimensional Apollonian packing with random distribution of initially prepacked spheres is investigated. The computer simulation model with specially designed changing range genetic algorithm was developed to estimate the fractal dimensions of Apollonian packing of spherical particles. The fractal dimensions corresponding to the packing degree and porosity were calculated for a large range of spherical particles (in the order of millions). The result obtained provides an experimental proof of a lower bound of the fractal dimension analytically found by [3] (Aste, Phys. Rev. E, 1996; 53: 2571).     

A New Method for Prediction of Bulk Particle Packing Behavior for Arbitrary-Shaped Particles inContainers of Any Shape

This article describes the recent developments in the computer modeling of packing of complex-shaped particles and prediction of physical properties of the structures represented by the packing. The computer model DigiPac is capable of packing particles of any shapes and sizes in a container of arbitrary geometry. The ability to predict the packing structure of real particle shapes and to compute directly some structure-dependent physical properties such as liquid permeability, mechanical strength/stability, compaction and sintering, and dissolution and leaching is obviously highly desirable and has significant potential in industrial applications. Examples are presented relating to the packing of bulk and granular materials.     

A New Method for Prediction of Bulk Particle Packing Behavior for Arbitrary-Shaped Particles inContainers of Any Shape

This article describes the recent developments in the computer modeling of packing of complex-shaped particles and prediction of physical properties of the structures represented by the packing. The computer model DigiPac is capable of packing particles of any shapes and sizes in a container of arbitrary geometry. The ability to predict the packing structure of real particle shapes and to compute directly some structure-dependent physical properties such as liquid permeability, mechanical strength/stability, compaction and sintering, and dissolution and leaching is obviously highly desirable and has significant potential in industrial applications. Examples are presented relating to the packing of bulk and granular materials.     

Transport of sputtered atoms investigated by Monte Carlo method

Increasing attention has been paid to the sputtering process as a tool to deposit films and to the study of the interaction between the film properties and the deposition parameters. It is obvious that the energy and direction of these particles arriving at the substrate is in close relation with the transport process from the target to the substrate. This work deals with the computer simulation of the sputtered Ag atoms trajectories through the background gas in a diode-sputtering configuration. For that, we have developed a numerical model to simulate the transport process. We followed the three-dimensional trajectory of each sputtered atom separately and calculated the scattering angle and the energy loss if a collision took place. A statistical method, Monte Carlo simulations is used. The model predicts the flux of Ag atoms arriving at the substrate, their energies and angular distribution. The dependence of the deposition rates of Ag atoms on the gas pressure and the distance between target to substrate were investigated.     

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.     

Packing of fine particles in an electrical field

A numerical model based on the Discrete Element Method (DEM) is developed to study the packing of fine particles in an electrical field related to the dust collection in an electrostatic precipitator (ESP). The particles are deposited to form a dust cake mainly under the electrical and van der Waals forces. It is shown that for the packing formed by mono-sized charged particles, increasing either particle size or applied electrical field strength increases packing density until reaching a limit corresponding to the density of random loose packing obtained under gravity. The corresponding structural changes are analyzed in terms of coordination number, radial distribution function and other topological and metric properties generated from the Voronoi tessellation. It is shown that these properties are similar to those for the packing under gravity. Such structural similarities result from the similar changes in the competition of the cohesive forces and the driving force in the packing. In particular, it is shown that by replacing the gravity with the electrical field force, the previous correlation between packing density and the ratio of the cohesive force to the packing-driven force can be applied to the packing of fine particles in ESP.     

CFD-DEM numerical study on air impacted packing densification of equiaxed cylindrical particles

A CFD-DEM model was developed to reproduce the packing densification process of mono-sized equiaxed cylindrical particles under air impact. The effects of operating parameters on packing density were firstly studied. Then various microscopic properties of packing structures such as coordination number (CN), contact types, particle orientations, pore features were characterized and compared. And corresponding densification mechanisms were analysed based on particle motion behaviour, local structure evolution, and forces. Results indicate that the air impact can realize the packing densification of cylindrical particles under appropriate conditions. The pore size distribution in the packing of cylindrical particles shows a tail at larger pore sizes compared with that in the packing of equal spheres. Both the size and the sphericity of the pores decrease in the final dense packing; also, more surface-surface and less surface-edge contacts between two particles therein can be formed. More cylindrical particles tend to be in parallel or perpendicular contact with each other to form more stable local structures during air impact. Most particles at higher position move down (direction of gravity/air impact) with about one particle length during the densification process and most particles exhibit translational motion to realize the local rearrangement for pore filling through air impact induced inter-particle forces.     

考虑磁性颗粒不均匀分布的磁流变液修正微观力学模型及试验验证

为了研究磁性颗粒在磁场作用下的不均匀分布对磁流变液力学性能的影响,通过卡方分布来模拟磁性颗粒的间距分布,对现有的磁流变液微观力学模型进行修正,并通过磁流变阻尼器的力学性能试验验证了模型的有效性。在磁流变液双链微观力学模型的基础上,修正相邻磁性颗粒的间距完全相等且不随磁感应强度而变化的假设,采用卡方分布来表征磁性颗粒间距的不均匀分布,并引入分布参数来描述磁性颗粒间距随磁感应强度的变化关系,推导了考虑磁性颗粒不均匀分布的磁流变液修正微观力学模型;基于修正的微观力学模型,分析了分布参数对磁流变液剪切屈服应力的影响;将该文提出的磁流变液修正微观力学模型带入到磁流变阻尼器的准静态模型中,可以得到不同电流下的阻尼器最大出力,并与磁流变阻尼器力学性能试验数据进行对比来验证所提模型的有效性。结果表明,考虑了磁性颗粒不均匀分布的磁流变液修正微观力学模型可以更加精确地预测磁流变液在不同磁感应强度下的剪切屈服应力,尤其是在低磁感应强度情况下可以改善现有微观力学模型放大了磁流变液剪切屈服应力的缺点。     

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.     

Nearest Particle Distance and the Statistical Distribution of Agglomerates from a Model of a Finite Set of Particles

The structural analysis of a particulate composite with randomly distributed hard spheres is presented based on a model proposed earlier. The structural factors considered include the distribution of interparticle distances and the conditions for particle agglomeration. The interparticle distance was characterized by the nearest particle distance (NPD) and the distance derived from Delaunay triangulation (DT). The distances were calculated for every particle in the particle set and analyzed in the form of a cumulative distribution function (CDF). The CDF provides two parameters: the representation of particles which are in very close proximity to their neighbors and the most frequent distance between particles.     

Effects of mixing ratio and order of admixed particles with two diameters on improvement of compacted packing fraction

To improve particle flowability, a technique is used in which fine particles are admixed with the main particles. However, the effects of coating structure on the improvement in flowability are not yet fully understood. Thus, predicting the improvement resulting from this technique is difficult. In this study, we focused on the effects of the particle diameter distribution of the admixed particles on coating structures and improvement of flowability in terms of the compacted packing fraction in a particle bed. Main particles of size 397 nm with admixed particles of sizes 8 and 104 nm were used. Bimodal particle diameter distributions were adjusted by changing the mixing ratios of the two admixed particles. Furthermore, the main and admixed particles were mixed in various orders. We examined the compacted packing fractions for these different mixing ratios and orders. Scanning electron microscopy images were obtained in order to analyze the coating structures on the main particle surfaces. The results show that the main particle packing fraction was most greatly improved by pre-mixing the two admixed particles. This can be explained by a linked rigid-3-bodies model with leverage based on increasing the apparent diameter of the main particles.     

Simulation of granular packing of particles with different size distributions

The study of granular matter composed of spherical particles is of interest in manufacturing, material, and metallurgy. The viscoelastic and frictional contacts between the particles are the cause of forming the agglomeration. We present a numerical simulation for particles packing with three different kinds of size distributions: monosize, bimodal, and Gaussian, using distinct element method (DEM). The particles are initially put randomly but without any overlap, and then fall down due to the gravity force and collide with neighbor particles. Because of the dissipative factors of viscoelastic collision and frictional force, all the particles finally come together to form an agglomeration. Coordination number, porosity, radial distribution function, and force distribution are calculated for different size distributions. It is demonstrated that particle size distribution does affect the granular packing structure.     

Effect of interaction on magnetic properties of a particulate medium

Samples of magnetic particles were prepared in dry powder form, as well as in a plastic binder system. Magnetic measurements were made on the samples as function of the volumetric packing factor. Coercive force Hc, squareness, and anhysteretic magnetization measurements are correlated with the uniformity of particle dispersion. It was found that the behavior of Hc, squareness, and the initial anhysteretic susceptibility as a function of the packing factor are good indications of the degree of dispersion of the particles. It is shown that if the particles are well dispersed, Hcincreased with increased dilution, and the initial anhysteretic susceptibility increased at both high and low dilutions. A mathematical model is developed to explain the observed results. The model consists of a double distribution of interaction fields to account for the well-dispersed and the agglomerated particles.     

Highly filled particulate thermoplastic composites: Part I Packing density of irregularly shaped particles

The packing density of irregular shaped particles greatly affects the properties of highly filled particulate composite materials. The effects of particle size distribution parameters on the packing density of fused silica powder and cristobalite flour powder of different size ranges is reported. Various size distributions, according to the log-normal function, were prepared by sieving and characterized by light scattering, using a Malvern 2600 light scattering instrument. The apparent and tap density of the various powders was used to characterize the packing density. The size distribution width was found to have a major effect on the packing density. In addition, the particle size was found to affect the packing density however, its significance depends on the size range and shape of the particles. Mixtures of powders, each having a different size distribution, behave differently. This revised version was published online in November 2006 with corrections to the Cover Date.     

Structural properties of wet granular beds subjected to tapping

Based on a simple numerical model for wet granular beds, we study the structural properties of wet particles subjected to tapping in terms of the global anisotropy and angular distribution presented by their contacts. The model allows to generate 2-dimensional packings of disks that can form capillary bridges due to the presence of interstitial liquid. A pseudodynamic simulation of adhesive hard disks has been implemented. The bed is subjected to a tapping-like excitation and we study the evolution of structural anisotropy of the packing with the number of taps. We also analyse the behavior of the angular distribution of contacts and anisotropy as a function of tapping intensity and liquid content. Present results help to better understand the behavior found in a previous work for the packing fraction of these systems. They also demonstrate that anisotropy alone not always helps to completely understand the behavior of the structural properties of wet particles.     

粉体堆积密度的理论计算

考察了紧密堆积时多级理想球形颗粒混合粉体的堆积密度的理论值与粒级组分数的关系,紧密堆积时颗粒的粒度分布特征,以及堆积密度的理论值与单一粒径粉体空隙体积分数、颗粒干扰宽度的相关关系。研究表明,原始堆积密度越大,多粒级理想球形颗粒混合粉体达到相同的堆积密度所需的粒级数越少,各粒级的体积含量随着粒级的增加呈指数下降,且原始堆积密度越大,下降速度越快;其粒度分布符合对数正态分布;堆积密度的理论值与单一粉体空隙体积分数、颗粒干扰宽度的符合二元二次非线性关系。     

Optimization of a Computer Simulation Model for Packing of Concrete Aggregates

A simulation algorithm was developed for modeling the dense packing of large assemblies of particulate materials (in the order of millions). These assemblies represent the real aggregate systems of portland cement concrete. Two variations of the algorithm are proposed: sequential packing model and particle suspension model. A developed multicell packing procedure as well as fine adjustment of the algorithm's parameters were useful to optimize the computational resources (i.e., to realize the trade-off between the memory and packing time). Some options to speed up the algorithm and to pack very large volumes of spherical entities (up to 10 million) are discussed. The described procedure resulted in a quick method for packing of large assemblies of particulate materials. The influence of model variables on the degree of packing and the corresponding distribution of particles was analyzed. Based on the simulation results, different particle size distributions of particulate materials are correlated to their packing degree. The developed algorithm generates and visualizes dense packings corresponding to concrete aggregates. These packings show a good agreement with the standard requirements and available research data. The results of the research can be applied to the optimal proportioning of concrete mixtures.     

Sequential sphere packing by trilateration equations

Specimen generation by packing of particles is the initial step in most numerical simulations for discrete media. In many cases, especially for virtual soil compaction, filtration, or penetration tests, this preparation work is essential for the success of the tests because the discrete particles are not able to be redistributed during the simulation. This paper presents a novel sphere sequential packing method for specimen generation using the trilateration method and its relevant equations. The method is developed on an assumption that spherical particles must be in contact with at least three neighbour particles to be kept in balance. This semi-analytical method has three advantages: quick generation speed, adjustable porosity, and good control over the spatial distribution of particles at local scale. This paper is a study on two typical spatial distributions: (1) layer-wise, where particles with similar sizes have priority of being positioned next to each other; and (2) discrete, where small particles are located preferentially in between large particles.     

Microdynamic analysis of solid flow in a shear cell

Granular flow in a model shear cell under conditions relevant to those in an annular cell is investigated based on the results obtained by means of the discrete element method. The spatial and statistical distributions of microdynamic variables such as velocity, porosity, coordination number and contact force are established, and the dependence of these variables on some key physical and operational parameters of particles and the cell is studied. It is shown that the normal pressure, shear velocity of the cell, particle friction coefficient and rolling friction coefficient have noticeable influences on these microdynamic variables. However, the effects of wall friction coefficient and damping coefficient are negligible. There is a linear relationship between overall coordination number and packing density, when the coordination number ranges between 5 and 6.5. The deviation from the relation derives from the cases where the normal pressure is varied as a result of the significant change in the normal contact forces between particles.