Random Loose Packing Research Articles

Computer simulation of random packing of hard spheres

A computer simulation which can randomly pack hard spheres for gravitationally stable lattices ranging from random loose packing to random close packing is described. The algorithm tests a model for random packing which proposes that the characteristics of a lattice are determined by its local interstices and that the shape of these interstices is formed by bridging and the angular separation of contacting spheres. The model predicts that random packings form a quadrilateral in the packing density/mean coordination plane with packing densities ranging from 0.509 to 0.638 and mean coordination numbers ranging from 4.4 to 5.9.

Microstructural descriptors characterizing granular deposits

AbstractQuantitative results are presented on mean coordination number and coordination number distribution, contact normal distribution and fabric tensor of simulated anisotropic granular deposits with resulting solid fractions between ca. 15% for ballistic deposits and 58%, corresponding to a random loose packing. The deposits, generated by the capture of uniform size spherical particles arriving normal to a target, were simulated using a simple algorithmic model.We focus on microstructural quantities which explicitly take into account the discrete nature of the granules comprising the deposit. Such measures are important in determining the heat transport properties of the deposits for Fourier conduction through the solid phase, as well as their mechanical and sintering properties. The variation of mean coordination number with deposit solid fraction was successfully correlated using a unit‐cell model (Eq. 13). This correlation, in conjunction with entropy maximization arguments (after Nayak and Tien, 1978) has been further used to predict the coordination number distribution. The usefulness of the results reported here is illustrated by computing an upper bound to the deposit effective thermal conductivity (Jagota and Hui, 1990) and comparing it to both ‘exact’ simulation results (Tassopoulos and Rosner, 1991b) and experimental data (Koh, 1971).

Monte carlo simulations of structural properties of packed beds

Monte Carlo simulations are used to obtain void fraction (Φ) and particle—particle contacts radial profiles for nondeformable spheres contained within cylinders with impernetrable walls. Simulated packings of monosize and multisize spheres are created by their sequential random placement within a prescribed enclosure using packing rules consistent with loading procedures in packed bed reactors. The resulting random-loose packings (Φ = 0.42) differ significantly in local and global properties and in the extent of ordering from random-close packings (Φ = 0.36). Void fraction and particle—particle contact radial profiles in monosize packings show a heavily damped decay behavior near the wall, extending only about two sphere diameters away from it. The damped oscillatory behavior characteristic of random-close packings is not observed. Similar trends are observed for multisize sphere distributions. Void fraction profiles for multisize spheres prescribed by continuous distributions are accurately described by a simple linear combination of the profiles of the individual size components. In bidisperse size distributions, similar superimposition rules also describe the qualitative details of the void fraction profiles, but do not account for compaction effects caused by penetration of small spheres into the interstices between larger spheres. Void fraction and contacts in segregated packings, in which monosize spheres of different size are dropped into annular and core regions in the cylindrical enclosure, are also described. These simulations provide the basis for our current studies of heat transport properties in packed beds, and for the generation of realistic pore structure models of overlapping randomly-arranged spheres.

Random loose packings of uniform spheres and the dilatancy onset.

The random-loose-packing fraction of uniform spheres at the limit of zero gravitational force is determined to be 0.555\ifmmode\pm\else\textpm\fi{}0.005. This structure corresponds to a sphere packing at its rigidity-percolation threshold. The onset of dilatancy was also measured to be at approximately the same packing fraction. Preliminary evidence also indicates that shear thickening of suspensions with macroscopic spheres and a negative pore liquid pressure develops at the rigidity threshold. These provide further evidence that a hard-sphere glass transition could exist in the packing fraction range of 0.555 to 0.645.

Packing of sands—A review

The minimum and maximum void ratios, corresponding to states of dense and loose random packing, are the most meaningful measures of packing behaviour. Size distribution, sphericity, roundness and surface roughness of sand particles, all affect the packing and compressibility of sands. Size affects packing, only when particles are less than 50 μm. Maximum and minimum void ratios decrease with increasing roundness and sphericity. For narrowly graded sands, compaction is almost independent of moisture content except near air-dryness and near saturation, where it is greater. The influence of both sand-particle size distribution and particle characteristics on compaction and plow-pan formation in sandy soils is discussed.

Statistical mechanical considerations on the random packing of granular materials

The historical development on the random packing of granular materials is briefly discussed. The geometrical probability space for granular packings is then introduced and the significance and relevance of characteristic microelements, i.e. ‘Voronoi polyhedra’, is elaborated upon. The classical statistical mechanical theory based on Boltzmann's postulate is remodeled and applied to a collection of ‘Voronoi cells’ in the form of routine ensemble phase averages. The exact probability density functions for the distribution of void ratios in a random aggregate of granular materials are then found to be exponential and the associated partition functions are obtained. The statistical entropy is shown to be a global maximum for the ‘loose random packing’ state which is also equivalent to the critical state reached in simple shearing of such aggregates. At such states the distribution density is shown to be uniform. Furthermore, it is shown that for random close packings the distribution density is skewed towards the denser ‘Voronoi cells’. Exact expressions are given for the expected values of the critical void ratios as well as critical porosities for random loose packing states of both three- and two-dimensional granular aggregates. Similar to three-dimensional packings, it is shown that there exist two critical porosities for the random two-dimensional packing; one corresponding to random loose packing for which n cr ≈ 0.160 and another for random close packing for which n cr ≈ 0.111. Finally, in an Appendix, the connections between the statistical mechanical considerations, the information theoretic considerations, and Jayne's postulate on minimally biased distribution are presented.

Variation in the local packing density near the wall of a randomly packed bed of equal spheres

In randomly packed beds of spherical particles, a damped oscillatory variation in the local packing density appears near the containing walls and dies away in the interior. This paper presents detailed data obtained from a computer-simulated random loose packing. The molecular pair-correlation function from statistical mechanics yields a theoretical variation in local packing density which agrees with values from the simulation. The variation is similar to the short-range order of the arrangement of spheres in the interior of the random assemblage.

Liquid Structure and the Coordination of Equal Spheres in Random Assemblage

THE random assemblage of equal spheres can provide a useful model of liquid structure1–4 and so many investigations have been made into the spatial distribution of the spheres. Smith et al.5 and Bernai and Mason6 examined the number of points of contact between a reference sphere and the adjacent spheres (the coordination number) in which the range of the bulk mean particle volume fraction was ∼0.55 to 0.64. In this communication I shall discuss theoretically the bulk mean of the coordination number for a wide range of the bulk mean particle volume fraction Cv. The result compares well with the known structure of liquids. The discussion continues with a consideration of the distribution of the coordination number, which explains the occurrence of two critical types of random packing6: random loose packing (Cv = 0.60) and random close packing (Cv = 0.64).

Random Loose Packing of Binary Mixtures of Spheres

THE attention which J. D. Bernal1 has directed to the importance of geometrical models for representing the structure of liquids has elicited considerable recent interest in the packing of spheres2–7. While previous investigations were often concerned with the dense random state8,9, some of the more recent work has been devoted to looser states3,4,6. Both Scott3 and MacRae and Gray4 obtained loose deposits of spheres by a rolling procedure, while Rutgers6, in common with Leva et al.10 some years earlier, obtained a loose packing by dumping or pouring. Porosities thus reported, though significantly higher than those found in the dense random state, are nevertheless still lower than those obtained either by rapidly inverting a cylinder11 or by gradually diminishing to zero the gas (or liquid) flow through a fluidized bed12,13. It is the latter porosities which correspond to those observed in both moving packed14 and incipiently fluidized13 beds.