Sphere Packing Research Articles

Deformation, shape transformations, and stability of elastic rod loops within spherical confinement

Mechanical insight into the packing of slender objects within confinement is essential for understanding how polymers, filaments, or wires organize and rearrange in limited space. Here we combine theoretical modeling, numerical optimization, and experimental studies to reveal spherical packing behavior of thin elastic rod loops of homogeneous or inhomogeneous stiffness. Across varying loop lengths, a rich array of configurations including circle, saddle, figure-eight, and more intricate patterns are identified. A theoretical framework rooted in the local equilibrium of force and moment is proposed for the rod loop deformation, facilitating the determination of internal and contact forces experienced by the rods during deformation. For the confined homogeneous rod loops, their stable and metastable configurations are well described using proposed Euler rotation curves, which offer a concise and effective approach for configuration prediction. Moreover, formulated analysis on the stability and critical force for homogeneous rod loops on great circles of the spherical confinement are performed. For inhomogeneous rod loops with two segments of differing stiffness, the stiffer segment exhibits less deviation from the great circle, while the softer segment undergoes more pronounced deformation. These findings not only enhance our understanding of buckling and post-buckling phenomena but also offer insights into filament patterning within confining environments.

Resolving Flow Structures Inside Three Dimensional Packed Beds With Gas Flow By Ray Tracing Particle Image Velocimetry

Two different configurations are investigated in this study by ray tracing Particle Image Velocimetry to access the interior of different spherical packing structures. Configuration 1 consists of a regular bidisperse packed bed of 10mm and 40mm spheres based on a body-centered cubic arrangement. Ray tracing Particle Image Velocimetry was performed behind two spheres in the 40mm layer for particle Reynolds numbers from 200 to 700. The averaged velocity flow fields show the jet structures according to the pores between the underlying smaller spheres and the influence of the large spheres. Since ray-tracing Particle Image Velocimetry suffers from experimental and numerical limitations, planar 2D Particle Image Velocimetry measurements are performed to obtain spatially highly resolved validation data in the optically accessible void pores. These measurements were performed at different height positions in the bed showcasing that similar flow fields appear in the different pores. Configuration 2 deals with the jet flow (Re=5900) around an individual sphere, part of a body-centred cubic arrangement of d=40mm spheres. The emphasis lays on the temporally resolved snapshot data of the turbulent jet flow obtained by ray tracing Particle Image velocimetry. Data obtained from both configurations is used for a comprehensive discussion of the chances and challenges of ray tracing Particle Image Velocimetry. Limitations appear on the experimental side, like generation of depth of field, illumination and reflections, sufficient measurement signal, and accuracy of the geometry, but also on the numerical side, like the reproduction of the geometry, material properties, and real-world conditions. Since all these phenomena are interlinked, performing the ray tracing method in more complex systems is questionable.

Sintering behavior, microstructure and dielectric properties of a novel low loss SrEr2O4 microwave ceramic

In this paper, a novel SrEr2O4 ceramic with relative densities exceeding 96% was synthesized by a conventional solid-state reaction method. Furthermore, investigations of the correlations between its crystal structure, microstructure, and microwave dielectric properties were carried out. Rietveld refinements show that the SrEr2O4 ceramics crystallize in the orthorhombic with the space group Pnam. The SrEr2O4 ceramic sintered at 1575 °C exhibits the good performance of permittivity (εr) = 14.44, quality factor (Q×f) = 23271 GHz, and resonant temperature frequency coefficient (τf) = –21.5 ppm/°C, which is attributed to its high lattice energy, packing fraction, and relative density. In addition, to satisfy the requirement of temperature stability for communication devices, near-zero τf value is obtained by adding CaTiO3 at the cost of a small quality factor loss. The SrEr2O4 + 1.25 wt% CaTiO3 composite exhibits decent properties with εr = 18.57, Q×f = 19940 GHz, and τf = +3.99 ppm/°C, making it an ideal candidate for microwave band applications.

Numerical mass transfer simulations of Venturi-type solid phase oxygen control with mass exchanger in UPBEAT loop

The Venturi-type solid-phase oxygen control system with a bypass mass exchanger is widely regarded as an effective approach to keep the oxygen concentration in the LBE under reasonable ranges to reduce the corrosion. In this work, numerical simulations of the particle–fluid mass transfer of the oxygen concentration in LBE are performed for the 1:1 scale 3D UPBEAT loop under various conditions. The discrete element method is applied to generate the geometry of the packing of the PbO spheres. The velocity profile and oxygen concentration governing equations are solved by using the SST turbulence model. At the same temperature, the oxygen mass transfer coefficient increases monotonically with the flow rate. At the same mass flow rate, the mass transfer coefficient increases with the temperature, reaching a maximum at about 475 °C. It provides guidance for future research in the design and optimization of oxygen control systems under high temperature in lead–bismuth nuclear reactors.

Links in orthoplicial Apollonian packings

In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation x4+y4+z4=2t2.

Sign uncertainty principles and low-degree polynomials

We prove an asymptotically sharp version of the Bourgain–Clozel–Kahane and Cohn–Gonçalves sign uncertainty principles for polynomials of sublinear degree times a Gaussian, as the dimension tends to infinity. In particular, we show that polynomials whose degree is sublinear in the dimension cannot improve asymptotically on those of degree at most three. This question arises naturally in the study of both linear programming bounds for sphere packing and the spinless modular bootstrap bound for free bosons.

Packing spheres with quasi-containment conditions

A novel sphere packing problem is introduced. A maximum number of spheres of different radii should be placed such that the spheres do not overlap and their centers fulfill a quasi-containment condition. The latter allows the spheres to lie partially outside the given cuboidal container. Moreover, specified ratios between the placed spheres of different radii must be satisfied. A corresponding mixed-integer nonlinear programming model is formulated. It enables the exact solution of small instances. For larger instances, a heuristic strategy is proposed, which relies on techniques for the generation of feasible points and the decomposition of open dimension problems. Numerical results are presented to demonstrate the viability of the approach.

Local and global expansivity in water.

The supra-molecular structure of a liquid is strongly connected to its dynamics, which in turn control macroscopic properties such as viscosity. Consequently, detailed knowledge about how this structure changes with temperature is essential to understand the thermal evolution of the dynamics ranging from the liquid to the glass. Here, we combine infrared spectroscopy (IR) measurements of the hydrogen (H) bond stretching vibration of water with molecular dynamics simulations and employ a quantitative analysis to extract the inter-molecular H-bond length in a wide temperature range of the liquid. The extracted expansivity of this H-bond differs strongly from that of the average nearest neighbor distance of oxygen atoms obtained through a common conversion of mass density. However, both properties can be connected through a simple model based on a random loose packing of spheres with a variable coordination number, which demonstrates the relevance of supra-molecular arrangement. Furthermore, the exclusion of the expansivity of the inter-molecular H-bonds reveals that the most compact molecular arrangement is formed in the range of ∼316-331K (i.e., above the density maximum) close to the temperature of several pressure-related anomalies, which indicates a characteristic point in the supra-molecular arrangement. These results confirm our earlier approach to deduce inter-molecular H-bond lengths via IR in polyalcohols [Gabriel et al. J. Chem. Phys. 154, 024503 (2021)] quantitatively and open a new alley to investigate the role of inter-molecular expansion as a precursor of molecular fluctuations on a bond-specific level.

Spheroid models to elaborate the broken symmetry and equivalent volume of molecules in crystalline phase.

Dense packing of particles has provided powerful models to elaborate the important structural features of matter in various systems such as liquid, glassy, and crystalline phases. The simplest sphere packing models can represent and capture salient properties of the building blocks for covalent, metallic, and ionic crystals; it, however, becomes insufficient to reflect the broken symmetry of the commonly anisotropic molecules in molecular crystals. Here, we develop spheroid models with a minimal degree of anisotropy, which serve as a simple geometrical representation for a rich spectrum of molecules-including both isotropic and anisotropic, convex and concave ones-in crystalline phases. Our models are determined via an inverse packing approach: Given a molecular crystal, an optimal spheroid model is constructed using a contact diagram, which depicts the packing relationship between neighboring molecules within the crystal. The spheroid models are capable of accurately capturing the broken symmetry and characterizing the equivalent volume of molecules in the crystalline phases. Moreover, our model retrieves such molecular information from low-quality x-ray diffraction data with poorly resolved structures, and by using soft spheroids, it can also describe the packing behavior in cocrystals.

Estimating Porosity of Agglomerated Products Using Optimized Sphere Packing

The article presents the results of studying the dynamics of changes in the porosity of two-component briquettes and pellets depending on the size of particles and the proportions of components in the mixture using an optimized packing of spheres. Knowing the patterns of change in the porosity allows to optimize the strength of the briquette and pellets as well as to improve their behavior in the reduction processes in blast furnaces, steel making furnaces and in direct reduction reactors. A computation experiment based on heuristic simulation model was designed to study the change of the estimated porosity under increasing/substituting the number of larger spherical particles in the mixture of spheres. The results obtained made it possible for the first time to reveal the extreme nature of the change in the porosity of the briquette/pellet with the addition of larger particles, depending on the fractional composition of the briquette. The results obtained open up new opportunities for optimizing the placement of fine-grained materials in the charge of metallurgical furnaces.

Uniqueness of Iris Pattern Based on the Auto-Regressive Model.

In this paper, we evaluate the uniqueness of a hypothetical iris recognition system that relies upon a nonlinear mapping of iris data into a space of Gaussian codewords with independent components. Given the new data representation, we develop and apply a sphere packing bound for Gaussian codewords and a bound similar to Daugman's to characterize the maximum iris population as a function of the relative entropy between Gaussian codewords of distinct iris classes. As a potential theoretical approach leading toward the realization of the hypothetical mapping, we work with the auto-regressive model fitted into iris data, after some data manipulation and preprocessing. The distance between a pair of codewords is measured in terms of the relative entropy (log-likelihood ratio statistic is an alternative) between distributions of codewords, which is also interpreted as a measure of iris quality. The new approach to iris uniqueness is illustrated using two toy examples involving two small datasets of iris images. For both datasets, the maximum sustainable population is presented as a function of image quality expressed in terms of relative entropy. Although the auto-regressive model may not be the best model for iris data, it lays the theoretical framework for the development of a high-performance iris recognition system utilizing a nonlinear mapping from the space of iris data to the space of Gaussian codewords with independent components.

Structural properties of different sphere packings with arbitrary porosities for planetary-science applications

Granular solids in planetary science are found in the regolith that covers planetary surfaces as well as in the bulk of rubble-pile asteroids, comets and planetesimals. To help understand the physics of these planetary bodies, we aim at deriving the structural properties of granular packings over a large range of porosities. Relevant to fluid flow and gas diffusion are the void spaces inside the granular packings so that we analyze the mean free path of point-like particles, their diffusion constant and their total traveled path lengths. For mechanical and heat-transport properties, the coordination number and the absolute chain length of the inter-connected particles are important. Generally, we also derive the homogeneity and isotropy of the granular solids. We compare granular packings generated by four algorithms for porosities in the range between 85 and 42%, which are the upper and lower limits for natural packings of equal-sized spheres. All produced sphere packings arrive at very similar quantities for the mean free path, the free path probability distribution function, the diffusion constant and the total traveled path length for the entire range of porosities. Hence, transport processes governed by the void-space properties are independent of the specific generation algorithm for the granular packing. In contrast, heat conduction or mechanical stresses almost exclusively depend on the existence and properties of particle contacts and particle chains in the network of spheres. In this case, the four algorithms deliver very different results.Graphic abstract

Estimating angoricity and granular equationsof state for monodisperse packings of frictional hard spheres in a wide range of densities.

In this paper, we study the granular equationof state (EOS) for computer-generated three-dimensional mechanically stable packings of frictional monodisperse particles over a wide range of densities (packing fractions), φ=0.56-0.72. As a statistical physics framework, we utilize the statistical ensemble for granular matter, specifically the "angoricity" ensemble, where the compressional component Σ_{p} of the force-moment tensor serves as granular energy and angoricity A_{p} is the corresponding granular "temperature." We demonstrate that the systems under study conform well to this statistical description, and the simple equationof state Σ_{p}=2.8NA_{p} holds very well, where N is the number of particles. We show that granular temperature exhibits a rapid drop around the random-close packing (RCP) limit φ≈0.64-0.65, and, hence, one can say that granular packings "freeze" at the RCP limit. We repeat these calculation for shear angoricity A_{sh} and shear component Σ_{sh} of the force-moment tensor and obtain a similar EOS, Σ_{sh}=0.85NA_{sh}. Additionally, we measure the so-called keramicity, an inverse temperature variable corresponding to the determinant of the force-moment tensor, while pressure angoricity corresponds to its trace. We show that inverse keramicity κ^{-1} and angoricity A_{p} conform to an EOS 1/A_{p}Σ_{p}/N+0.11κ(Σ_{p}/N)^{3}=1.2, whose form is predicted by mean-field theory. Finally, we demonstrate that the alternative statistical ensemble where Voronoi volumes serve as granular energy (and so-called compactivity serves as temperature) does not describe the systems under study well.

Uniform boundary conditions on models of spherical particles through alpha shape surface tracking and Laguerre–Voronoi diagrams

This study describes the adaptation of the alpha-shape three-dimensional surface reconstruction technique to boundaries of spherical particle packings, such as those modeled using Discrete Elements. The alpha shape technique is employed to reconstruct a boundary surface for a packing as well as to assemble a Laguerre–Voronoi Diagram of the packing's boundary. The reconstructed surface can be used to track the evolution of the packing surface, while the surface diagram can be used to directly obtain the discrete force vectors that result from applying a uniform stress tensor (Static Uniform Boundary Condition), as well as the discrete velocity vectors that result from applying a uniform velocity gradient tensor (Kinematic Uniform Boundary Condition) to a model of spherical discrete particles. Doing so continuously throughout a simulation allows for the application of uniform boundary conditions on spherical discrete particles, whilst enabling precise tracking of stress and deformation tensors of the packing. An implementation of this novel approach in the Yade open-source DEM code is detailed here. The power and versatility of the real-time boundary tracking approach is demonstrated using some boundary value problem simulations on polydisperse spherical particle models.

Patterns of Nanoporous Spherical Packing Emerging under Influence of Curvature and Confinement

Nanoporous membranes are popular in nanotechnology due to biomedical and industrial applications. During the past decade, experimental, theoretical and computational research into porous membranes and soft materials has opened up new mathematical dimensions. In bulk, diblock copolymers exhibit ordered morphologies such as parallel matrices of lamellae, bicontinuous matrices of gyroids, hexagonal matrices of cylinders and body-centred cubic matrices of spheres. In melt, confinement plays an essential role in tuning the frustration of the diblock copolymer system to predict more nanostructures. These nanostructures depend on the composition of the copolymers, their confining geometries and the degree of structural frustration. An isotropic 9-point stencil for Laplacian is constructed. The discrete finite-difference technique is used in polar grids to discretize the macromolecule of the diblock copolymer system to study spherical patterns to study the effect of curvature and confinement with a well-known and efficient cell dynamic simulation model. Intel FORTRAN (IFORT) codes are generated to run the CDS model and visualisation of simulation results is observed with the help of OPENDX. A comparison of the proposed study with existing experimental and computational studies is also presented.

Numerical modelling of contact adhesion in a random assembly of elastic–plastic particles

The prediction of tensile strength properties of powder compacts remains an important industrial issue. In particular, one of the main problems of the powder compaction process is the failure of compacts. Indeed, some powder compacts exhibit cracks which appear during compaction. Such defects occur due to localised tensile or shear stresses, for example close to geometrical singularities. They are also related to the ability of a powder to create enough adhesion at contacts between particles to withstand tensile stresses. Therefore, cracks are a consequence of phenomena occurring at the particle scale and below, down to the molecular scale. To help understanding this mechanism, a particle-scale, numerical method called the multi-particle finite-element method was developed using the finite-element software suite Abaqus (Abaqus 6.14 Documentation, 2016). Such a method allowed to explicitly model the microstructure of a granular media idealised as an assembly of elastic–plastic spheres. The particles were meshed such that their deformations were fully taken into account, using a continuum-mechanics-based material model. The interactions between particles were managed using finite-element contact formulations. A multi-scale, adhesive contact model was developed based on the literature and implemented into the multi-particle finite-element code. The contact model was based on a surface energy formulation weighted by the roughness model developed by Pullen and Williamson (1972). It introduced a novel aspect, the development of adhesion under the effect of external mechanical loads, which is consistent with the cold compaction process. This model was then applied to predict the mesoscopic properties of a packing of spheres, i.e. its response to mechanical stresses of any type, in particular strongly deviatoric stress paths. Such a method intends to be a help toward the development of an efficient continuum model for the modelling of the powder compaction process.

Impact of fuel and reducing agent on photoluminescence and supercapacitor applications of rhombohedral silver gallium oxide (AgGaO2) nanoparticles

To investigate the benefits of a green-mediated route compared to the chemical route in synthesising AgGaO2 NPs, both methods were employed, and a comprehensive analysis of their characterization, photoluminescence, and electrochemical properties was conducted. In this study, silver gallium oxide (AgGaO2) nanoparticles (NPs) were synthesised using two distinct methods: a chemical route employing urea as a fuel (AgGaO2U) and a green-mediated route using Aloevera gel extract as a reducing agent (AgGaO2A), followed by calcination at 600oC for 3 h. The XRD analysis confirmed the formation of a trigonal structure in both AgGaO2U and AgGaO2A NPs. Rietveld analysis provided insight into lattice parameters, space groups, and packing diagrams. AgGaO2A exhibited smaller NPs compared to AgGaO2U. The direct energy band gap, determined using Wood and Tauc’s relation, was 3.2 eV for AgGaO2U and 3.04 eV for AgGaO2A. AgGaO2A displayed a smaller crystallite size and a higher energy band gap. Photoluminescence spectra, excited at 224 nm, revealed emissions in the violet, blue, green, and red regions, with the highest intensity observed in the blue region. This was further confirmed by CIE coordinates, which fell within the blue region, with AgGaO2A exhibiting higher intensity than AgGaO2U. Electrochemical studies demonstrated supercapacitance values ranging from 24.8 to 69.8 F/g at different scan rates. While the choice of fuel or reducing agent influenced photoluminescence intensity, electrochemical studies showed consistent behaviour in both synthesised samples, suggesting their potential applications in displays and supercapacitors.

Multiple Cubes Growth Algorithms for Simple Representative Elementary Volume Determination on 3D Binary Images

Porosity analysis is fundamental for understanding various material properties and transport phenomena in scientific and engineering disciplines. This study delves into the challenging task of determining the representative elementary volume (REV) in porous media, crucial for accurate analyses. Two novel algorithms, Center-Corner Cubes Growing (3CG) and Random Cubes Growing (RCG), were proposed and tested on synthetic body-centered cubic (BCC) sphere packing and natural porous structures of Berea sandstone and Indiana limestone, obtained using µCT. First algorithm (3CG) operates by analyzing porosity within cubes growing from each of the eight corners and a central region of a 3D binarized stack. In contrast, the Random Cube Growing (RCG) algorithm randomly selects seed points within the 3D stack and grows cubic regions around them. Both algorithms systematically compute porosity for various cube sizes, determining the average porosity and standard deviation for each extent. These visual analytics tools contribute to identifying the specific size ranges where porosity curves converge and stabilize, indicating potential REV within the material. While 3CG simplifies the approach by focusing on a limited number of curves, RCG provides a broader view, capturing diverse porosity patterns. The absence of consistent local minima in certain cases indicates high porosity heterogeneity and the impossibility of achieving REV in certain sample sizes.

Dynamic imaging of force chains in 3D granular media

Granular media constitute the most abundant form of solid matter on Earth and beyond. When external forces are applied to a granular medium, the forces are transmitted through it via chains of contacts among grains-force chains. Understanding the spatial structure and temporal evolution of force chains constitutes a fundamental goal of granular mechanics. Here, we introduce an experimental technique, interference optical projection tomography, to study force chains in three-dimensional (3D) granular packs under triaxial shear loads and illustrate the technique with random assemblies of spheres and icosahedra. We find that, in response to an increasing vertical load, the pack of spheres forms intensifying vertical force chains, while the pack of icosahedra forms more interconnected force-chain networks. This provides microscopic insights into why particles with more angularity are more resistant to shear failure-the interconnected force-chain network is stronger (that is, more resilient to topological collapse) than the isolated force chains in round particles. The longer force chains with less branching in the pack of round particles are more likely to buckle, which leads to the macroscopic failure of the pack. This work paves the way for understanding the grain-scale underpinning of localized failure of 3D granular media, such as shear localization in landslides and stick-slip frictional motion in tectonic and induced earthquakes.

Effective Counting in Sphere Packings

Given a Zariski-dense, discrete group, Γ, of isometries acting on (n + 1)- dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain Γ-orbits. In particular, this allows us to obtain a best-known effective error rate for the Apollonian and (more generally) Kleinian sphere packing counting problems, that is, counting the number of spheres in such with radius bounded by a growing parameter. Our method extends the method of Kontorovich [Kon09], which was itself an extension of the orbit counting method of Lax-Phillips [LP82], in two ways. First, we remove a compactness condition on the discrete subgroups considered via a technical cut- off and smoothing operation. Second, we develop a coordinate system which naturally corresponds to the inversive geometry underlying the sphere counting problem, and give structure theorems on the arising Casimir operator and Haar measure in these coordinates.