Equal Spheres Research Articles

A new class of infinite sphere packings

The oscillatory or Apollonian packing in two dimensions is well known and is described for example in [13]. Recently we investigated the three dimensional osculatory packing of a sphere [4] However, the results of that paper indicate that, for N > 3, ΛΓ-dimensional osculatory packings are irregular and not invariant under inversion as is the case for N = 2 and 3. In this paper we introduce a class of packings which we call discrete packings, and produce some examples. This class is analogous to the class of lattice packings which appear in the theory of packings of equal spheres. We shall use the systems of polyspherical coordinates developed in [4]. Section 2 contains a description of these as well as the proofs of some additional results needed here. The idea of the separation Δ{X, Y) between two spheres X and Y will again play an important role. In § 3 we consider inversively generated configurations obviously generalizing the construction used in [4]. That is, we begin with a 'cluster' of (N + 2) disjoint spheres and by successive inversions replace the spheres one at a time with new spheres in such a way that the separations between the spheres in the new cluster are the same as for the initial cluster. In terms of polyspherical coordinates the necessary inversions are represented by matrices which preserve a certain indefinite quadratic form. Repetition of the process leads to a configuration of spheres in EN which may or may not be a packing, depending on the initial cluster. In § 4 we give sufficient conditions under which an inversively generated configuration is a packing. The conditions force the separations between the spheres in the configuration to lie in a discrete subset of the rational numbers, hence the name 'discrete packing'. In addition to the two and three dimensional osculatory packings, we give examples of discrete packings for dimensions 2, 3, 4, 5, and 9. We do not know yet whether such packings exist in all dimensions. The examples we have found are given in § 6. The packings described in § 4 are not in general osculatory; that is, the largest possible sphere is not generated at each step. However,

Exact solution of the mean spherical model for simple polar mixtures

Wertheim's solution of the mean spherical model (MSM) for pure fluids composed of hard spheres with embedded dipoles [J. Chem. Phys. 55, 4291 (1971)] is extended to multicomponent polar fluid mixtures. The components are restricted to have equal hard sphere radii but may have different dipole moments. The anisotropic part of the pair correlation functions for an m-component fluid characterized by hard sphere diameter d, temperature parameter β, dipole moments μ1, μ2, ···,μm, and densities ρ1, ρ2, ···,ρm are shown to be expressable in terms of the corresponding functions for an effective pure MSM polar fluid with the same hard sphere radius and the same temperature parameter but with an effective dipole moment μ̂ = [m−1 (μ12 + μ22 + ⋯ + μm2)]1/2 and an effective density ρ̂ = μ̂−2(μ12ρ1 + μ22ρ2 + ⋯ + μm2ρm). The excess thermodynamic properties of the mixture (relative to a pure hard sphere Percus-Yevick fluid) are shown to be those of the effective pure fluid. The dielectric constant of the mixture is also that of the effective pure fluid. The case of polar-nonpolar mixtures is considered by allowing one or more of the dipole moments to vanish. It is found that the potential of mean force and spatial distribution of the nonpolar molecules is independent of the magnitude of the dipoles while the anisotropic correlations between the polar molecules are independent of the presence of the nonpolar species.

Simulated random packing of equal spheres

Simulated random packing of equal spheres

Effects of Particle‐Size Distribution in Initial‐Stage Sintering

Diffusion models for the initial‐stage sintering of spherical particles were developed for distributions of particle sizes. A linear array of particles with different sizes was treated quantitatively; number, rather than weight, distributions are considered directly to permit summation of particle‐pair interactions. Evaluation of a single particle size which will characterize the equivalent shrinkage of the distributed array of random particles yields a size much smaller than that which would designate the division at 50 wt% finer conventionally used. Also, the effective size is smaller for the lattice‐diffusion model than for the grain‐boundary‐diffusion‐controlled model. The results show that the rates for binary mixtures are intermediate between the behaviors of the end‐member sizes, in accordance with previous experimental findings. A square array of equal spheres with contacting interstitial spheres was also analyzed. Coherent shrinkage requires a retardation of shrinkage along the diagonal contacts (due to tensile loads). Procedures for evaluating the stress transfer were generated for this and similar particle packings. The calculated shrinkage rate is again intermediate between that predicted from previous models and the end‐member sizes.

Simulated random packing of equal spheres

AbstractPACS, a computer program to simulate very slow settling of rigid equal spheres into a randomly packed bed, is described. Simulated packings of 6,000, 10,000, and 15,000 spheres are anisotropic. Mean void fraction is 0.42 and mean number of nearest neighbours 6.0. There are numerous cases of four spheres arranged in a tetrahedron, but no larger ordered regions. These findings, plus local voidage, distribution of circles in a section, angular distribution, and radial distribution are compared to theoretical and experimental results.

Potential flow past a sphere tangent to a plane

Uniform potential flow past two equal spheres in contact, parallel to the common tangent plane, is analyzed. The velocity is found to diverge approaching the point of contact as r −2+√2. Application of this result to critical velocity phenomena in superfluid helium is discussed.

Some geometrical properties of packings of equal spheres in cylindrical vessels

Experimental and analytical methods are described for establishing some of the mean geometrical properties of unbiased random packings of equal spheres in cylindrical vessels. Experimental determinations have been made of the volumes of the spheres in an outer region of thickness equal to the sphere diameter, and in the remaining central region. Related properties, such as the mean void fractions of the two regions, can be calculated from these. Several preparation methods were used, including some which give rise to packings with axial or radial bias. A model is developed for computing the relevant properties of packings in a semi-infinite vessel with a plane wall. The model is then adapted to unbiased packings in cylindrical vessels, by means of quadratic equations whose coefficients depend on the cylinder-to-sphere diameter ratio. It is shown that commonly used preparation methods, such as placing or pouring spheres on the top surface, can result in packings with radial bias. Two methods of preparing unbiased random packings are described.

The problem of paired spheres and of straight chains of spheres in slices or polished sections

SUMMARYThe starting point is the problem of finding the percentage q2 of paired nuclei in liver cells, when the percentage p2 of twin cuts in slices is known. The general solution for unequal spheres becomes most simple for equal spheres and leads to an extrapolation for endless straight chains of equal spheres, at first for closed and then for open series with constant intervals. Three total sums of endless series of fractions are a welcome mathematical extra.The microscopist will prefer instructions which are generalized from counts in final rows of spheres. With them he can analyse the composition of a mixture of limited chains, with different numbers of spheres.

Desaturation of porous media. I. Unconsolidated materials

A model of the pore space in a random packing of rotund particles is suggested. The assumption that the pores are randomly connected enables the residual number of undrained pores to be calculated, independent of pore size distribution. There are two distinct mechanisms by which liquid can leave a pore; 1. (i) by evaporation 2. (ii) by fluid flow Desaturation by fluid-flow without evaporation is considered in two ways: a) flow through the whole pore space; and b) a modification where flow also occurs through incipient pendular rings. The shape of the capillary pressure curve, normalized to a window size distribution, is given for each of these cases. The relative permeability to both wetting and nonwetting phases is given for monosized windows. The relative permeability for an idealized window size distribution during desorption is derived. An approximation to Archie's Law is obtained. The analysis indicates deficiencies in both mercury porosimetry and desorption as methods for measuring “pore size distributions”. When applied to the random packing of equal spheres, for which an approximate window size distribution is already known, the theory gives a capillary pressure curve which is in reasonable agreement with experiment.

Random Packing of Equal and Unequal Spheres in Two and Three Dimensions

A new computer simulation of random packing of spheres is applied to two- and three-dimensional problems with a uni-directional gravitational force.

Serially Deposited Amorphous Aggregates of Hard Spheres

Aggregates of several thousand equal hard spheres were constructed by depositing additional spheres, one at a time, at surface sites on a small seed cluster, placing each new sphere in contact with three already present and not moving it afterward. Always choosing the site closest to the center of the original seed results in an aggregate showing no evidence of crystallinity, with a pair correlation function quite similar to that of the dense random packings which have been prepared from ball bearings, and to the pair correlation functions calculated from x-ray diffraction work on amorphous alloys. The packing density ⅙πNσ3/V and the mean coordination number both decrease with distance from the center; the large-aggregate limits are, respectively, 0.61 (extrapolated, 4% lower than the ball-bearing value), and exactly 6.0 (in agreement with the ball-bearing value). The apparent difficulty of homogeneously nucleating crystallization in hard-sphere systems, even when simultaneous relaxation of many particle positions is allowed, is attributed to the fact that small hcp and fcc fragments are more bulky and have no more hard contacts than amorphous arragements of the same number of atoms.

On the Stokes resistance of two equal spheres in contact in a linear shear field

The Stokes resistance for two spheres in contact in a linear shear field is determined exactly, and using the results, the elements of the grand resistance matrix and the shear resistance matrix for a rigid dumb-bell are calculated. The simplicity of the matrix representation of the Stokes resistance is illustrated in deriving the translational and angular velocities for a neutrally buoyant dumb-bell moving in a linearly sheared fluid.

Equation of State of Simple Liquids from Coordination Number in Random Assemblage

USING the corresponding states principle, Renon et al.1 have derived the following equation of state of simple liquids, which is the same in form as that obtained by Flory2. illustration where T, Ṽ and P are respectively the absolute temperature, volume and pressure reduced by the characteristic parameters1 (denoted by the superscript*). Bhatia and Reed3 recently obtained the same equation from the smoothed potential cell model using an attracting hard-sphere pair potential with attraction interaction varying as r−3. In this communication, I shall also derive equation (1) using the coordination number4 in a random assemblage of equal spheres. The characteristic parameters of Renon et al.1 are related to the Lennard-Jones (6–12) pair potential parameters. Open image in new window

Dielectric measurements using sphere gaps - a simple approximation for the potential distribution

A simple approximate solution is derived for the current and potential distributions around a sphere gap with small separation. The method is based on the fact that the problem has an elementary solution for two equal spheres in contact. Its accuracy is about 1% for ratios of gap/sphere radius not greater than 0·1. Explicit elementary expressions are obtained for the stress distribution, the total conductance for spheres and segments of spheres, the fraction of the total current flowing within a given radius from the axis, and the same fraction within a radius having a specified ratio of stress to maximum stress. The implications of the results are discussed in relation to measurements of the dielectric properties of liquids and solids.

On the Densest Packing of Equal Spheres in a Cube

On the Densest Packing of Equal Spheres in a Cube

On the Densest Packing of Equal Spheres in a Cube

(1971). On the Densest Packing of Equal Spheres in a Cube. Mathematics Magazine: Vol. 44, No. 4, pp. 199-208.

Sphere Packings and Error-Correcting Codes

Error-correcting codes are used in several constructions for packings of equal spheres in n-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m ≧ 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in En for n = 2m satisfiesIn this paper we make systematic use of error-correcting codes to obtain sphere packings in En, including several of the densest packings known and several new packings.

Molecular Configuration of Spherical, Non-Polar, Dense Gases and Liquids

IN a previous communication1, I discussed the coordination of equal spheres in a random assemblage and liquid structure and supported Bernal's idea2,3 that the liquid structure can be characterized as a homogeneous, coherent and essentially irregular assemblage of particles. In view of the bulk-mean particle volume fraction, this communication provides a comparison between the random assemblage of equal spheres and the molecular configuration of spherical, non-polar, dense gases and liquids.

Velocity and turbulence measurements of air flow through a packed bed

AbstractMean velocity and turbulence measurements in the void spaces of a cubic packing of equal spheres have been made at Reynolds numbers of 27,000, 10,000, 5,000 and 2,500, where NRe is based on superficial air velocity and a sphere diameter of 7 cm. Two cubic arrangements were used: in the regular arrangement, the mean flow was parallel to one of the principal axes, while in the skewed arrangement, the mean flow made equal angles with the three principal axes of the packing.Transverse mean velocity and turbulence intensity profiles across the center line of a central pore have been measured behind every bank and behind the bed for the regular arrangement of ten banks of spheres. The power spectrum and probability distribution of the fluctuating velocity have also been determined.

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).