Dense Sphere Packings Research Articles

Special Hysteresis Effects in N2-Sorption and Mercury-Porosimetry Measurements

AbstractModel pore structures were prepared from dispersions of submicron monodispersed silica particles by a sedimentation process. Ordered dense sphere packing structures were observedwith scanning electron microscopy. Nitrogen sorption- as well as Hg-porosimetry measurements confirmed the calculated values of the pore openings in those structures. In Hgporosimetry measurements a two step extrusion curve was observed, when the pore system was only partially filled during the intrusion process. This two step curve was not observed in case the pore system was filled with mercury to more than 95% during the intrusion run. The mercury porosimetry results can be interpreted in terms of the coexistence of octahedral and tetrahedral voids (pores) in the examined sphere packing structure and their special arrangement within the structure (connectivity). Two models will be described to explain thegeneral occurrence of hysteresis in Hg-porosimetry. The actual pore geometry is shown to have a profound influence on the hysteresis shape as well as a change in the contact angle (constant within each measurement) can result in totally different hysteresis curves. Nitrogen adsorption and desorption measurements on the same powders did not reveal any fine structure within the hysteresis range.

Effective medium treatment of laminar flow in magnetic filtration

A theory for laminar flow through randomly distributed magnetic spheres which uses an effective medium treatment (EMT) is presented. In the EMT, the system of fluid and spheres is replaced by a composite sphere—a representative sphere of radius a enclosed by a fluid shell of radius b—embedded in an effective medium of different viscosity. The velocity fields in the fluid shell and in the effective medium as a function of sphere volume packing fraction (γ3=a3/b3) or density of spheres in liquid are determined by using Green’s theorem and proper boundary conditions. The results are applied to study the capture of paramagnetic and diamagnetic particles by an assemblage of spheres. The capture radius as a function of γ and applied magnetic field (H0) is obtained and compared with results from previous study based on Happel’s theory. The dependence of capture radius on H0 is also given for various fixed values of γ. Finally, filter efficiency is predicted for varying γ and H0.

A model for high-frequency acoustic backscatter from gas bubbles in sandy sediments at shallow grazing angles

A model for acoustic backscatter from trapped bubbles in sandy sediments was developed. The model combines a Biot acoustic penetration model with a resonance scattering mechanism from trapped bubbles. The bubble size distribution is assumed to mirror the size distribution of the fluid pores that exist between sand grains. An estimate of the pore size distribution is constructed from the grain size distribution, based on the known pore structure between dense random packings of hard spheres. The principle of acoustic reciprocity is employed to compute backscattered acoustic pressure in terms of the incident pressure and the scattering cross section of a bubble distribution. The model is applied to data from experiments recently taken at sea. It is concluded that trapped gas is often a likely cause of observed backscatter from sandy sediments. Very small amounts of gas appear to be sufficient to produce significant backscatter.

Nanostructured amorphous SiAu alloys — Structure and atomic correlations

Nanostructured amorphous Si 1− x Au x alloys ( 0.1 ⪕ x Au ⪕ 0.35 ) were produced by an inert gas condensation and in situ consolidation technique. Small-angle neutron scattering (SANS), density measurements and wide-angle neutron scattering (WANS) were employed to characterize the microstructure and local atomic order. The SANS results are interpreted in terms of a microstructure which consists of nanometre-sized amorphous grains embedded in a matrix of lower density, formed by grain boundaries and free volumes. The relative densities depend on the average particle size, but not on the alloy composition. In all samples, the SANS intensity decreases at low Q according to a power law − Q −3.37, which is attributed to a surface fractal structure of dimension D s = 2.63. WANS reveals two contributions: (a) a continuously decreasing part resulting from the presence of about 10 at.% hydrogen, and (b) a modulation of the intensity as a function of the scattering angle, characteristic of amorphous material. The total pair correlation function g( R) presents maxima corresponding to the first and second atomic neighbours. Nanostructured amorphous SiAu alloys exhibit a reduced short-range order compared with systems with dense random packing of hard spheres, e.g. metallic glasses, or continuous random network structures, e.g. covalent systems.

Sphere packings with three contacts per sphere and the problem of the least dense sphere packing

AbstractTwo procedures to derive all types of sphere packings with three contacts per sphere are described. 52 such types have been found. They are characterized by their shortest meshs and related to Wells' three-connected nets, if possible. It is proved that there exists no sphere packing with contact number three and a density lower than 5.5%, i.e. the density of the Heesch-Laves packing.

Surface decontamination treatments of undoped BaTiO 3—part I: Powder and green body properties

Five commercial barium titanate powders were subjected to various surface decontamination treatments: calcination at 500 °C, followed by leaching either in pure water or acetate buffer solution. They were then de-agglomerated by ultrasonic treatment and finally characterised by XRD, TGA, XPS, EGA, pyrolysis/FT-IR, BET specific surface area and particle size analysis. The main contaminating elements in raw powders were carbon and sulphur, present as barium carbonate (between 0.7 and 1.2wt%) and barium sulphate (about 0.5 wt%). Barium sulphate is thermally stable up to 1400 °C and is not affected by the treatments. Barium carbonate decomposes in two steps, between 500 and 900 °C and 1200 and 1400 °C by reaction with TiO 2 and BaTiO 3 to release CO 2. Carbonation is slightly reduced by calcination, but subsequent water cleaning promotes re-carbonation. Acid cleaning substantially decreases the BaCO 3 content, down to 0.07–0.7 wt%. The remaining quantity corresponds to the most stable form which decomposes near 1200–1400 °C and is still partially present at the surface of the particles. Forming of ceramics green bodies by slip casting of aqueous suspensions stabilised by ammonium poly (acrylate) was then investigated. The dispersing solution compositions have been optimised for each powder and treatment, in order to minimise the liquid/solid ratio of the slip. In all cases, the properties of the slips and of the cast specimens have been improved by the cleaning treatments: higher solid volume fractions in the slip up to 60% for the best case and higher green densities. In particular green densities as high as 70% have been obtained which is very close to the theoretical maximum packing density of log-normally polydispersed spheres. This enhancement can be ascribed to the beneficial effect of decarbonation on the deagglomeration of the powders.

On the determination of the structure of metallic glasses by reverse Monte Carlo simulation: comparison with hard sphere Monte Carlo results

Reverse Monte Carlo (RMC) simulation has recently been used for modelling the atomic level structure of several amorphous metallic alloys. Using the particle configuraitons that were provided by the RMC calculations, characteristics of the short-range structure were evaluated. Standard (Metropolis) Monte Carlo simulations on hard sphere systems mimicking three of the metallic glassy materials modelled by RMC have also been carried out. Particle configurations obtained by these two methods were compared. It was shown that dense random packing of hard spheres is most similar to “real” metallic glassy structures when the chemical nature of the components differs the least.

Nonconnected atomic surfaces for quasicrystalline sphere packings.

The problem of maximizing the density of quasiperiodic sphere packings is investigated. The investigation is limited primarily to the case where the centers of the spheres are determined by a cut through a higher-dimensional Bravais lattice of a single atomic surface. For certain cases, the sphere packing obtained by the largest connected atomic surface can be made more dense by replacing it with an atomic surface that is nonconnected. The nonconnected atomic surfaces differ from the connected ones only in specific regions near the atomic surface boundaries. These regions are determined by the quasicrystalline point-group symmetry and by the radius of the spheres. This method of improving sphere packings is applied to find nonconnected atomic surfaces for the dodecagonal, octagonal, and icosahedral cases. Using this method in conjunction with other methods, a packing fraction of approximately 0.5645 for a simple icosahedral b-c network is achieved.

New approach to the dense random packing of soft spheres. Dependence of model characteristics on potential shape

A new sequential method of amorphous structure simulation which does not use the hard sphere approximation is proposed. Every atom added is presented with a prescribed potential from the beginning. A method of direct minimization of cluster energy was used for generation and relaxation of the model structure. Two different empirical potentials of Morse and Bullough were used for a search for the effect of potential shape on the features of simulated structures. Two 800-atomic clusters of amorphous copper were constructed using these potentials. For both models, all structure-dependent characteristics were calculated. The long-range attraction of a Morse potential resulted in a more dense, compressed structure, while the more complicated shape of a Bullough potential resulted in another form of the pair correlation function. Also it is shown that the structure of short-range order in both cases is nearly the same and distributions of microstrains differ markedly.

ON THE SPHERE PACKING PROBLEM AND THE PROOF OF KEPLER'S CONJECTURE

International Journal of MathematicsVol. 04, No. 05, pp. 739-831 (1993) No AccessON THE SPHERE PACKING PROBLEM AND THE PROOF OF KEPLER'S CONJECTUREWU-YI HSIANGWU-YI HSIANGDepartment of Mathematics, University of California, Berkeley, CA 94720, USAhttps://doi.org/10.1142/S0129167X93000364Cited by:35 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Research partially supported by NSF Grant.Some improvements of the proofs were obtained during my visits at IHES in October of 1991 and at the Mathematics Institute of Geneva University in June of 1992. 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Guerci1 Jan 2014Mathematical modeling of high-rate Anammox UASB reactor based on granular packing patternsChong-Jian Tang, Rui He, Ping Zheng, Li-Yuan Chai and Xiao-Bo Min1 Apr 2013 | Journal of Hazardous Materials, Vol. 250-251Unit Sphere PackingsKároly Bezdek25 June 2013Bibliography28 October 2011Surface reconstruction using Delaunay triangulation for applications in life sciencesGraziela Grise and Michael Meyer-Hermann1 Apr 2011 | Computer Physics Communications, Vol. 182, No. 4The Kepler Conjecture and Its ProofJeffrey C. Lagarias20 October 2011Sphere Packings, IIT. C. Hales20 October 2011Bounds for Local Density of Sphere Packings and the Kepler ConjectureJ. C. Lagarias20 October 2011Historical Overview of the Kepler ConjectureThomas C. Hales20 October 2011Towards Sub-cellular Modeling with Delaunay TriangulationG. Grise and M. Meyer-Hermann3 February 2010 | Mathematical Modelling of Natural Phenomena, Vol. 5, No. 1Lattices and packings in higher dimensionsMarcel Berger5 February 2010Sphere packings revisitedKároly Bezdek1 Aug 2006 | European Journal of Combinatorics, Vol. 27, No. 6Geometrical learning, descriptive geometry, and biomimetic pattern recognitionWang Shoujue and Lai Jiangliang1 Aug 2005 | Neurocomputing, Vol. 67Orbits of Orbs: Sphere Packing Meets Penrose TilingsCharles Radin1 February 2018 | The American Mathematical Monthly, Vol. 111, No. 2Does the proof stack up?George Szpiro1 Jul 2003 | Nature, Vol. 424, No. 6944Kugeln im Computer — Die Kepler-VermutungMartin Henk and Günter M. Ziegler1 Jan 2002Continuous Covering Location ProblemsFrank Plastria1 Jan 2002Remarks on sphere packings, clusters and Hales Ferguson theoremJean-Louis Verger-Gaugry1 January 2001 | Séminaire de théorie spectrale et géométrie, Vol. 19Kugeln im Computer — Die Kepler-VermutungMartin Henk and Günter M. Ziegler1 Jan 2000Low–dimensional lattices. VII. Coordination sequencesJ. H. Conway and N. J. A. Sloane8 November 1997 | Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 453, No. 1966Asymptotically dense spherical codes. I. Wrapped spherical codesJ. Hamkins and K. Zeger1 Nov 1997 | IEEE Transactions on Information Theory, Vol. 43, No. 6Finite-Size Effects on the Closest Packing of Hard SpheresS. Neser, C. Bechinger, P. Leiderer and T. Palberg22 September 1997 | Physical Review Letters, Vol. 79, No. 12Kugelpackungen und Wurstkatastrophen oder Zur Theorie der finiten und infiniten PackungenMax Leppmeier1 Jan 1997A Rejoinder to Hales’s ArticleWu-Yi Hsiang10 January 2009 | The Mathematical Intelligencer, Vol. 17, No. 1Minimal-energy clusters of hard spheresN. J. A. Sloane, R. H. Hardin, T. D. S. Duff and J. H. Conway1 October 1995 | Discrete & Computational Geometry, Vol. 14, No. 3What are all the best sphere packings in low dimensions?J. H. Conway and N. J. A. Sloane1 June 1995 | Discrete & Computational Geometry, Vol. 13, No. 3-4Sphere packings with three contacts per sphere and the problem of the least dense sphere packingE. Koch and W. Fischer28 July 2010 | Zeitschrift für Kristallographie - Crystalline Materials, Vol. 210, No. 6The status of the kepler conjectureThomas C. Hales9 January 2009 | The Mathematical Intelligencer, Vol. 16, No. 3 Recommended Vol. 04, No. 05 Metrics History Received 17 November 1992 Revised 9 March 1993 PDF download

A new bound on the local density of sphere packings

It is shown that a packing of unit spheres in three-dimensional Euclidean space can have density at most 0.773055..., and that a Voronoi polyhedron defined by such a packing must have volume at least 5.41848... These bounds are superior to the best bounds previously published [5] (0.77836 and 5.382, respectively), but are inferior to the tight bounds of 0.7404... and 5.550... claimed by Hsiang [2]. Our bounds are proved by cutting a Voronoi polyhedron into cones, one for each of its faces. A lower bound is established on the volume of each cone as a function of its solid angle. Convexity arguments then show that the sum of all the cone volume bounds is minimized when there are 13 faces each of solid angle 4?/13.

Remarks on the density of sphere packings in three dimensions

This paper shows how the density of sphere packings of spheres of equal radius may be studied using the Delaunay decomposition. Using this decomposition, a local notion of density for sphere packings in ℝ3 is defined. Conjecturally this approach should yield a bound of 0.740873... on sphere packings in ℝ3, and a small perturbation of this approach should yield the bound of\({\pi \mathord{\left/ {\vphantom {\pi {\sqrt {18} }}} \right. \kern-\nulldelimiterspace} {\sqrt {18} }}\). The face-centered-cubic and hexagonal-close-packings provide local maxima (in a strong sense defined below) to the function which associates to every saturated sphere packing in ℝ3 its density. The local measure of density coincides with the actual density for the face-centered cubic and hexagonal-close-packings.

On the sausage catastrophe in 4‐space

An upper bound for the “sausage catastrophe” of dense sphere packings in 4-space is given.A basic problem in the theory of finite packing is to determine, for a given positive integer k, the minimal volume of all convex bodies into which k translates of the unit ball Bd of the Euclidean d-dimensional space Ed can be packed ([5]). For d = 2 this problem was solved by Groemer ([6]).

Prediction of relative permeability in simple porous media.

We present a predictive calculation of two-phase relative permeabilities in granular porous media formed from a dense random packing of equal spheres. The spatial coordinates of every sphere in the pack have been measured, enabling the microstructure of the medium to be completely determined. From these data we extract a network model that replicates the pore space. By compacting the packing or swelling individual spheres, we may generate model porous media of different porosities whose microstructure is also completely determined. We simulate both viscous- and capillary-dominated invasion of a nonwetting fluid into a wetting fluid contained in these media. During invasion we calculate the average hydraulic conductance of each phase to obtain the relative permeabilities as a function of fluid saturation. Because the microstructure is known, the calculations do not involve any adjustable parameters or supplementary measurements of pore structure. The computed relative permeabilities are successfully compared with experimental values previously measured on sand packs, bead packs, and a simple sandstone.

Coalescence mechanisms of polymer colloids: I. Coalescence under the influence of particle-water interfacial tension

Coalescence of dense packings of spheres of model latex particles was studied in water. The model particles had a core-shell structure. The core comprised 90% of the mass of the particle, thus largely determining the mechanical properties. The shell contained 10, 15, or 25 wt% methacrylic acid, resulting in three kinds of particles with different particle-water interfacial tensions. It was shown that coalescence in water under the influence of particle-water interfacial tension was possible even at temperatures below the glass transition temperature. The kinetics of coalescence was closely related to interfacial tension. It was shown that the rate of coalescence decreased when interfacial tension decreased, i.e., when the amount of methacrylic acid increased in the shell or when sodium dodecyl sulfate was adsorbed onto the particles, or when the methacrylic acid was neutralized by sodium hydroxide. It was also established that the activation energy of coalescence was equal to the activation energy of the motions of the macromolecules at the glass transition.

Crystals made of close-packed polymeric spheres: A neutron scattering study on latex films.

The dense packing of polymeric spheres in latex films has been investigated by means of small-angle neutron scattering. The films were prepared by evaporation of the water out of a dispersion consisting of monodisperse charged particles (diameter 130 nm). Analysis of the Bragg peaks found in the scattering reveals that the particles are arranged in a face-centered-cubic lattice structure with well defined crystalline domains. Furthermore, it is found that the system still owns a memory of its initial structure upon prolonged annealing at temperatures far above its glass temperature. An interesting suppression of some Bragg peaks due to the influence of the particle scattering form factor is observed.

Some ternary and quaternary codes and associated sphere packings

Tables of good ternary and quaternary codes are presented, and they are used in the construction of dense sphere packings. Results include (1) tables of the best ternary and quaternary constacyclic codes (including cyclic codes) up to block length 50; (2) a class of optimal (n, 2) codes over GF(q); (3) the (u+v+w mod 2u+v mod u) construction, a new ternary code construction technique that can be used to construct the ternary Reed-Muller codes (and others); and (4) tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature. Packings are generated in even dimensions up to 100 using these codes and a sphere-packing construction. In dimensions 36 and 60, new record densities appear to have been achieved. >

Vector quantization for entropy coding of image subbands

Vector quantization for entropy coding of image subbands is investigated. Rate distortion curves are computed with mean square error as a distortion criterion. The authors show that full-search entropy-constrained vector quantization of image subbands results in the best performance, but is computationally expensive. Lattice quantizers yield a coding efficiency almost indistinguishable from optimum full-search entropy-constrained vector quantization. Orthogonal lattice quantizers were found to perform almost as well as lattice quantizers derived from dense sphere packings. An optimum bit allocation rule based on a Lagrange multiplier formulation is applied to subband coding. Coding results are shown for a still image.

Amorphous structure of the immiscible Fe1-y(Cu1-xAgx)y alloy system

Fe1-y(Cu1-xAgx)y alloys sputter deposited on liquid-nitrogen-cooled substrates have been investigated by X-ray diffraction and differential scanning calorimetry measurements. Although this ternary alloy system is of a typically immiscible type in the equilibrium state, a single BCC phase appears at the Fe-rich corner and a single FCC phase at the Cu- and Ag-rich corners, while an amorphous phase appears in the central concentration region. In the interference function of the ternary amorphous alloys, a broad first peak and splitting of the second and third peaks are detected, where the peak intensity is much weaker than those in the binary FCC alloys. The radial distribution function indicates that a tetrahedron is the dominant structure unit of the amorphous alloys, as expected from the dense random packing of hard or soil spheres. The stacking of tetrahedra is not sequential but the main coordination polyhedron is an icosahedron, because the average coordination number is 11-12.

Locally dense finite lattice packings of spheres

We consider finite packings of unit-balls in Euclidean 3-spaceE3 where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL3⊃E3. In particular we show that the facets ofP induced by densest sublattices ofL3 are not too close to the next parallel layers of centres of balls. We further show that the Dirichlet-Voronoi-cells are comparatively small in this direction. The paper was stimulated by the fact that real crystals in general grow slowly in the directions normal to these dense facets.