Dense Sphere Packings Research Articles

Liquid Organic Frameworks: A Liquid Crystalline 8‐Connected Network with Body‐Centered Cubic Symmetry

AbstractLiquid state self‐assembly is important for the understanding of the complex structures developed in abiogenesis and biogenesis as well as for numerous potential technological applications. Herein we report the first body‐centered cubic liquid crystalline phase with 8‐connected network topology and open octahedral network structure. It is formed by dynamic soft self‐assembly of X‐shaped polyphiles with oligo(para‐phenylene‐ethynylene) cores. The π‐conjugated rods with perfluorinated inner benzene rings form networks conjoined by eight‐way junctions, which are formed by nano‐segregated spheres involving hydrogen‐bonded polar end groups, while the branched aliphatic chains at opposite sides of the cores fill the continuum. This novel cubic phase is based on the I‐WP minimal surface separating the frameworks of polyaromatic cores from the most disordered chain segments. It can also be considered as a dense sphere packing. Such liquid organic frameworks, representing hybrids of sphere packings and networks could be of interest for organic photonics and other technologies.

A pore-scale thermo–hydro-mechanical model for particulate systems

A pore scale numerical method dedicated to the simulation of heat transfer and associated thermo–hydro-mechanical couplings in granular media is described. The proposed thermo–hydro-mechanical approach builds on an existing hydro-mechanical model that employs the discrete element method for simulating the mechanical behavior of dense sphere packings and combines it with the finite volume method for simulating pore space fluid flow and the resulting hydro-mechanical coupling. Within the hydro-mechanical framework, the pore space is discretized as a tetrahedral network whose geometry is defined by the triangulation of discrete element method (DEM) particle centers. It is this discretization of DEM particle contacts and tetrahedral pore spaces that enables the efficient conductive and advective heat transfer models proposed herein. In particular, conductive heat transfer is modeled explicitly between and within solid and fluid phases: across DEM particle contacts, between adjacent tetrahedral pores, and between pores and incident particles. Meanwhile, advective heat transfer is added to the existing implicit fluid flow scheme by estimating mass–energy–flux from pressure induced fluid fluxes. In addition to the heat transfer model, a thermo-mechanical coupling is implemented by considering volume changes based on the thermal expansion of particles and fluid. The conduction and advection models are verified by presenting comparisons to an analytical solution for conduction and a fully resolved numerical solution for conduction and advection. Finally, the relevance of the fully coupled thermo–hydro-mechanical model is illustrated by simulating an experiment where a saturated porous rock sample is subjected to a cyclic temperature loading.

Constructions of Dense Lattices of Full Diversity

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

Constructions of Dense Lattices of Full Diversity

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

A discrete element study of the effect of particle shape on packing density of fine and cohesive powders

Fine and cohesive powders typically exhibit low packing density, with solid volume fraction around 0.3. Discrete element modelling (DEM) of particulate materials and processes typically employs spherical particles which have much larger solid fractions (e.g. 0.64 for dense random packing of frictionless spheres). In this work a range of quasi-spherical particles are designed, represented by a number of small satellites connected rigidly to a larger centre sphere. Using DEM, packing density is found to be controlled by the interplay between particle shape, size and inter-particle cohesion and friction. Low packing density is obtained for an appropriate combination of (1) particle shape that allows the creation of geometrically loose structures via separation of the central particles by the satellites, (2) particle size that should be sufficiently small so that adhesive forces between particles become dominant over gravity, (3) adhesive forces, determined from surface energy, should be sufficiently large, and (4) friction (static friction was found to have a dominant role compared to rolling friction, but negligible compared to adhesive forces for small particle size). By using the proposed quasi-spherical particle designs it becomes possible to calibrate more realistic DEM models for particulate processes that reproduces not only packing, but also other behaviours of bulk powders.

Sphere packing and quantum gravity

We establish a precise relation between the modular bootstrap, used to con- strain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra U(1)c maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in d = 2c dimensions. We also show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context. For c = 4 and c = 12, these functionals exactly repro- duce the “magic functions” used recently by Viazovska [1] and Cohn et al. [2] to solve the sphere packing problem in dimensions 8 and 24. The same functionals are also applied to general 2D CFTs, with only Virasoro symmetry. In the limit of large central charge, we relate sphere packing to bounds on the black hole spectrum in 3D quantum gravity, and prove analytically that any such theory must have a nontrivial primary state of dimension {Delta}_0underset{sim }{<}c/mathrm{8.503.}

Lattice-Packing by Spheres and Eutactic Forms

We consider a semi-random walk on the space X of lattices in Euclidean n-space which attempts to maximize the sphere-packing density function Φ. A lattice (or its corresponding quadratic form) is called “sticky” if the set of directions in X emanating from it along which Φ is infinitesimally increasing has measure 0 in the set of all directions. Thus the random walk will tend to get “stuck” in the vicinity of a sticky lattice. We prove that a lattice is sticky if and only if the corresponding quadratic form is semi-eutactic. We prove our results in the more general setting of self-adjoint homogeneous cones. We also present results from our experiments with semi-random walks on X. These indicate some idea about the landscape of eutactic lattices in the space of all lattices.

ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS

We prove a lower bound on the entropy of sphere packings of $\mathbb{R}^{d}$ of density $\unicode[STIX]{x1D6E9}(d\cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $\unicode[STIX]{x1D6FA}(d\cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_{d}(1))\log (2/\sqrt{3})d\cdot 2^{-d}$ when the ratio of the fugacity parameter to the volume covered by a single sphere is at least $3^{-d/2}$. Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh.

Dense Packing of Poly-Fractional Powder of Ceramic Materials

The article describes the results of theoretical and experimental studies of obtaining powder fillings methods with relative density more than 0.5 by adjusting the fractional composition. To achieve the indicated density the method of selecting fractional composition of corundum spherical shape powder was used based on known theoretical models of regular and random dense packing of polydisperse spheres. Three fractional compositions were chosen, with calculated theoretical max packing density respectively equal to 0.824, 0.837 and 0.846. Powder compacts’ degree of theoretical density reduction due to the influence of the walls of the mold was evaluated, as well as the deviation of the real packing structure of powder particles from dense regular and that due to the form deviation of real powder particles from a perfect sphere and the influence of the surface characteristics of the particles. To obtain maximum packing density in the experiments the method of vibration compaction with a mechanical axial pressure was used. In the present work the assessment was made of the application prospects of the described methods in the technology of ceramics production.

Generating dense packings of hard spheres by soft interaction design

Packing spheres efficiently in large dimension dd is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on the packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be efficiently constructed by this method, up to a packing fraction close to 7 \, d \, 2^{-d}7d2−d. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space of the interaction potentials. Our method crucially exploits a recent exact formulation of the thermodynamics and the dynamics of simple liquids in infinite dimension.

On the crushing of polydisperse foams

Open-cell low-density foams are widely used in energy absorption and impact mitigation applications because of the low stress crushing response that extends to strains of 50–60%. In a continuing effort to connect the microstructure to these unique properties, we use micromechanically accurate foam models of the same relative density to examine the effect of cell-size distribution (polydispersity) on the full crushing response of aluminum open-cell foams under both quasi-static and dynamic loadings. A foam model starts as a Laguerre tessellation originating from a dense packing of polydisperse spheres with prescribed probability distributions of radii. The tessellation then becomes the initial condition in Surface Evolver to generate soap froth that satisfies Plateau's laws. Such soap froths are subsequently “dressed” with the desired solid material and discretized with finite elements to create foam models of different polydispersity levels. Quasi-static simulations show that the limit stress decreases with increasing polydispersity while the plateau stress is less affected and governed by the relative density, which in this case remains constant. Foams of different polydispersity were also crushed by impact under a range of super-critical speeds. The Hugoniot of a highly polydisperse foam is reported and is compared to the corresponding one of a monodisperse foam of the same density. The results indicate only a modest variation of the shock kinetics with polydispersity, but a drop is recorded in both the distal and proximal stresses, as well as the associated energy absorption capacity of the foam.

A FORMAL PROOF OF THE KEPLER CONJECTURE

This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

Coupled flow and deformations in granular systems beyond the pendular regime

A pore-scale numerical model is proposed for simulating the quasi-static primary drainage and the hydro-mechanical couplings in multiphase granular systems. The solid skeleton is idealized to a dense random packing of polydisperse spheres by DEM. The fluids (nonwetting and wetting phases) space is decomposed to a network of tetrahedral pores based on the Regular Triangulation method. The local drainage rules and invasion logic are defined. The fluid forces acting on solid grains are formulated. The model can simulate the hydraulic evolution from a fully saturated state to a low level of saturation but beyond the pendular regime. The features of wetting phase entrapments and capillary fingering can also be reproduced. Finally, a primary drainage test is performed on a 40,000 spheres of sample. The water retention curve is obtained. The solid skeleton first shrinks then swells.

Geometric method for large Morse clusters formation

This paper describes a geometrically founded method for constructing the initial configuration of a complete icosahedral structure of an atomic cluster. The method finds a global minimum of the cluster energy as a result of a single local optimization procedure. The method is based on the proposed algorithm for the stratified accommodation of atom centers, that allows one to form structural configurations commonly found among the clusters with global minimum of interatomic interaction energy. This algorithm provides the ability to build a spatial configuration of the dense packing of spheres for the formation of icosahedral and decahedral structures, as well as building complete icosahedrons with a large number of atoms. With the proposed method, we have reached the global minima for large Morse clusters (N = 817, 923, and 1415), which is a record for Morse clusters with r = 6.

Concentrated dispersions and melts from block-copolymer micelles: Computer simulation

A technique for simulation of concentrated solutions and melts of spherical micelles composed of diblock copolymers is presented; the degrees of polymerization of blocks are 20 and 80, respectively. The concentration of solutions is varied in a wide range up to 67 vol %, which corresponds to the dense packing of spheres in the volume-centered lattice. It is shown that, at all the values of the Flory–Huggins parameter, such inverted phases are instable and fairly rapidly transformed into more equilibrated phases without any long-range order.

Hierarchical Assembly of Tungsten Spheres and Epoxy Composites in Three-Dimensional Graphene Foam and Its Enhanced Acoustic Performance as a Backing Material.

Backing materials play important role in enhancing the acoustic performance of an ultrasonic transducer. Most backing materials prepared by conventional methods failed to show both high acoustic impedance and attenuation, which however determine the bandwidth and axial resolution of acoustic transducer, respectively. In the present work, taking advantage of the structural feature of 3D graphene foam as a confined space for dense packing of tungsten spheres with the assistance of centrifugal force, the desired structural requirement for high impedance is obtained. Meanwhile, superior thermal conductivity of graphene contributes to the acoustic attenuation via the conversion of acoustic waves to thermal energy. The tight contact between tungstate spheres, epoxy matrix, or graphene makes the acoustic wave depleted easily for the absence of air barrier. The as-prepared 3DG/W80wt%/epoxy film in 1 mm, prepared using ∼41 μm W spheres in diameter, not only displays acoustic impedance of 13.05 ± 0.11 MRayl but also illustrates acoustic attenuation of 110.15 ± 1.23 dB/cm MHz. Additionally, the composite film exhibits a high acoustic absorption coefficient, which is 94.4% at 1 MHz and 100% at 3 MHz, respectively. Present composite film outperforms most of the reported backing materials consisting of metal fillers/polymer blending in terms of the acoustic impedance and attenuation.

An Energy Efficient QAM Modulation with Multidimensional Signal Constellation

Abstract Packing constellations points in higher dimensions, the concept of multidimensional modulation exploits the idea drawn from geometry for searching dense sphere packings in a given dimension, utilising it to minimise the average energy of the underlying constellations. The following work analyses the impact of spherical shaping of the constellations bound instead of the traditional, hyper-cubical bound. Balanced constellation schemes are obtained with the N-dimensional simplex merging algorithm. The performance of constellations of dimensions 2, 4 and 6 is compared to the performance of QAM modulations of equivalent throughputs in the sense of bits transmitted per complex (two-dimensional) symbols. The considered constellations give an approximately 0:7 dB to 1 dB gain in terms of BER over a standard QAM modulation.

Application of a force field algorithm for creating strongly correlated multiscale sphere packings

This work presents a protocol driven force field algorithm, used to create multiscale correlated dense sphere packings. It was developed as part of a tool chain for the reconstruction of realistic multiscale porous rock samples. It overcomes limitations of Monte-Carlo or deposition based approaches, that are quite common in this field and were used previously. The new algorithm can create large, low porosity sphere packings with radius distributions covering two decades. Highly correlated structures that model pore clogging and sedimentation can be generated. To achieve this, an adequate force field and proper termination strategies are necessary. By changing the algorithm parameters in a controlled way during the simulation, a complex protocol driven process can be established. The implementation of the algorithm targets large parallel computer platforms to perform simulations with more than 10 million spheres. This article includes an application of the algorithm used to generate a highly polydisperse sphere packing with roughly 106 spheres and radii from 1 źtoź 100 µm . The continuum description of this packing is discretized at resolutions from 0.25 źtoź 1 µm and investigated using geometric and statistical characterizations and results from Lattice-Boltzmann flow simulations. These resolution dependent results affirm that reliable, predictive calculations for multiscale porous microstructures depend on the availability of large realistic continuum models. To obtain such models the algorithm presented herein can be used as a starting point.

Применение математических расчетов плотности упаковки сфер к составу легких бетонов

Применение математических расчетов плотности упаковки сфер к составу легких бетонов

Pore-scale simulations of drainage in granular materials: Finite size effects and the representative elementary volume

A pore-scale model is introduced for two-phase flow in dense packings of polydisperse spheres. The model is developed as a component of a more general hydromechanical coupling framework based on the discrete element method, which will be elaborated in future papers and will apply to various processes of interest in soil science, in geomechanics and in oil and gas production. Here the emphasis is on the generation of a network of pores mapping the void space between spherical grains, and the definition of local criteria governing the primary drainage process. The pore space is decomposed by Regular Triangulation, from which a set of pores connected by throats are identified. A local entry capillary pressure is evaluated for each throat, based on the balance of capillary pressure and surface tension at equilibrium. The model reflects the possible entrapment of disconnected patches of the receding wetting phase. It is validated by a comparison with drainage experiments. In the last part of the paper, a series of simulations are reported to illustrate size and boundary effects, key questions when studying small samples made of spherical particles be it in simulations or experiments. Repeated tests on samples of different sizes give evolution of water content which are not only scattered but also strongly biased for small sample sizes. More than 20,000 spheres are needed to reduce the bias on saturation below 0.02. Additional statistics are generated by subsampling a large sample of 64,000 spheres. They suggest that the minimal sampling volume for evaluating saturation is one hundred times greater that the sampling volume needed for measuring porosity with the same accuracy. This requirement in terms of sample size induces a need for efficient computer codes. The method described herein has a low algorithmic complexity in order to satisfy this requirement. It will be well suited to further developments toward coupled flow-deformation problems in which evolution of the microstructure require frequent updates of the pore network.