Sphere Packing Research Articles

Packing spheres in high dimensions with moderate computational effort.

We generate nonlattice packings of spheres in up to 22 dimensions using the geometrical constraint satisfaction algorithm RRR. Our aggregated data suggest that it is easy to double the density of Ball's lower bound and, more tentatively, that the exponential decay rate of the density can be improved relative to Minkowski's longstanding 1/2.

Color gamut volume and the maximum number of mutually discernible colors based on a Riemannian metric.

For the calculation of the color gamut volume and the maximum number of mutually discernible colors, an algorithm based on a Riemannian metric and the densest packing of spheres is proposed. With this algorithm, the color gamut volume was calculated for the conditions of experiments reported in literature. Good agreement was found with the experimental findings of the color gamut volume as a function of the peak luminance. Using the new algorithm, the color gamut volume and the maximum number of mutually discernible colors was calculated for various sets of primary colors corresponding to display standards and various dynamic ranges. Comparisons were made with state-of-the-art methods which are based on the Euclidean metric in approximately uniform color spaces and a simple cubic lattice. It was found that the state-of-the-art methods underestimate the maximum number of mutually discernible colors. However, the relative differences decrease as the primary colors are more saturated. Based on the new algorithm the maximum number of mutually discernible colors was calculated for a range of peak retinal illuminance levels and various sets of primary colors. We found that, for a given set of primary colors, the maximum number of mutually discernible colors is proportional to the logarithm of the ratio of the peak retinal illuminance level and a fitting parameter.

On the Ding and Helleseth's 7th open problem about optimal ternary cyclic codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. The first two and the 6th problems were completely solved, and the third and last three problems were partially solved. In this paper, we focus on the 7th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer and then present a counterexample. Furthermore, we construct two new classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound by studying some special polynomials over finite fields.

Correlations in randomly stacked solids.

Packing of spheres is a problem with a long history dating back to Kepler's conjecture in 1611. The highest density is realized in face-centered-cubic (FCC) and hexagonal-close-packed (HCP) arrangements. These are only limiting examples of an infinite family of maximal-density structures called Barlow stackings. They are constructed by stacking triangular layers, with each layer shifted with respect to the one below. At the other extreme, Torquato-Stillinger stackings are believed to yield the lowest possible density while preserving mechanical stability. They form an infinite family of structures composed of stacked honeycomb layers. In this article, we characterize layer-correlations in both families when the stacking is random. To do so, we take advantage of the Hägg code-a mapping between a Barlow stacking and a one-dimensional Ising magnet. The layer correlation is related to a moment-generating function of the Ising model. We first determine the layer correlation for random Barlow stacking, finding exponential decay. We next introduce a bias favoring one of two stacking chiralities-equivalent to a magnetic field in the Ising model. Although this bias favors FCC ordering, there is no long-ranged order as correlations still decay exponentially. Finally, we consider Torquato-Stillinger stackings, which map to a combination of an Ising magnet and a three-state Potts model. With random stacking, the correlations decay exponentially with a form that is similar to the Barlow problem. We discuss relevance to ordering in clusters of stacked solids and for layer-deposition-based synthesis methods.

Numerical modeling of Van der Waals interaction between a spherical particle and rough surfaces with different planar asperity distributions

Roughness and microstructures at the micro and the nano scales have significant influence on the van der Waals forces between the adhering particles and the surface. Previous theoretical models may treat the particle-asperity interactions in an unprecise manner, which may result in inaccurate estimations of the van der Waals force especially when the sphere is much larger than the asperities. In this paper, we proposed a numerical model to calculate the van der Waals force between a spherical particle and a rough surface with periodic nanostructures. The 3D sinusoidal rough surfaces are simplified with their asperities modelled as hemispheres, and the contact is within the Derjaguin-Muller-Toporov (DMT) regime. The proposed model calculates the van der Waals forces between the sphere and all the asperities in the contact surface, which will provide more accurate estimations of the van der Waals force. Moreover, the effect of the planar distribution of asperities on the van der Waals forces is considered, which includes the cases when asperities have close-packed hexagonal lattice, square, and hollow-centered hexagonal packing distributions. The theoretical results calculated using the proposed model are verified by comparing the results to previous theoretical and experimental results and the comparison shows that the proposed numerical model predicts the van der Waals force accurately.

In-situ monitoring of changes in ultrastructure and mechanical properties of flax cell walls during controlled heat treatment

Plant fibres are increasingly used as reinforcements, especially in thermoplastic composites. Understanding the impact of temperature on the properties of these fibres is an important issue for the manufacturing of high-performance materials with minimal defects. In this work, the structural evolution and mechanical behaviour of flax fibre cell walls were dynamically monitored by temperature-controlled X-ray diffraction and nanoindentation from 25 to 230 °C; detailed biochemical analysis was also conducted on fibre samples after each heating step. With increasing temperature up to 230 °C, a decrease in the local mechanical performance of the flax cell walls, of about −72 % for the indentation modulus and −35 % for the hardness, was measured. This was associated with a decrease in the packing of the cellulose crystal lattice (increase in d-spacing d200), as well as significant mass losses measured by thermogravimetric analysis and changes in the biochemical composition, i.e. non-cellulosic polysaccharides attributed to the middle lamellae but also to the cell walls. This work, which proposes for the first time an in-situ investigation of the dynamic temperature evolution of the flax cell wall properties, highlights the reversible behaviour of their crystalline structure (i.e. cellulose) and local mechanical properties after cooling to room temperature, even after exposure to high temperatures.

Kleinian sphere packings, reflection groups, and arithmeticity

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study “properly integral” packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher–dimensional hyperbolic polyhedra.

Regulating the self‐assembly of AB/CD diblock copolymer blends via secondary interactions

AbstractDesigned multiblock copolymers with complex architectures offer unlimited opportunities to obtain novel nanostructured phases, however, their synthesis could be challenging and expensive. An alternative approach to access desired nanostructures is to use blends of block copolymers with simple chain architectures and designed block‐block interactions. We use binary blends composed of AB and CD diblock copolymers as a model system to establish design principles of polymeric blends containing block copolymers. Specifically, we explore the phase behavior of AB/CD blends by using the polymeric self‐consistent field theory to construct phase diagrams of the blends focusing on the sphere‐forming regions in the phase space. We predict the formation of various spherical packing phases composed of either core‐shell‐structured spheres or binary spheres resembling metallic alloys. We demonstrate that the equilibrium morphology can be regulated by adjusting the blend composition and molecular parameters such as block fractions, conformational asymmetry, and segment‐segment interactions. The strategy of using secondary interaction in polymeric blends to control the phase behavior explored in the current study can also be generalized to other soft matter systems.

Evaporation-driven assembly of colloidal nanoparticles into clusters: A dissipative particle dynamics study.

In this work we consider a simulation strategy for assembling Janus nanoparticles in oil-in-water emulsion droplets by evaporation based on the dissipative particle dynamics method. Our simple method reproduces all the observed cluster configurations that have been explored experimentally. In addition, the kinetic process of cluster formation is systematically investigated. We observe a structural transition from spherical packings to minimal second-moment configurations via visual inspection and a simple angle parameter. We reveal that the critical volume at which the transition occurs is a cubic function of the number of particles, N. Our approach also allows us to anticipate higher-order clusters, overcoming the limitations of the standard methods in the literature. Similarly to small N values, we find that for each N in the range of 16-39, all final clusters have a unique configuration.

XProtoSphere: an eXtended multi-sized sphere packing algorithm driven by particle size distribution

XProtoSphere: an eXtended multi-sized sphere packing algorithm driven by particle size distribution

Ray tracing particle image velocimetry – Challenges in the application to a packed bed

Ray tracing Particle Image Velocimetry (RT-PIV) is an optical technique for high resolution velocity measurements in challenging optical systems, such as transparent packed beds, that uses ray tracing to correct for distortions introduced by transparent geometries in the light paths. The ray tracing based correction is a post processing step applied to the raw PIV particle images before classical PIV evaluation. In this study, RT-PIV is performed in the top layer of a body centred cubic (bcc) sphere packing with gaseous flow, where optical access is obtained by the use of transparent N-BK7 glass balls with a diameter of d=40 mm. RT-PIV introduces new experimental and numerical challenges, for example a limited field of view, illumination difficulties, a very large required depth of field and high sensitivity to geometric parameters used in the ray tracing correction. These challenges and their implications are the main scope and discussed in the present work. Further, the validation of the ray tracing reconstruction step is presented and examples for the obtained corrected vector fields in a packed bed are given. The results show the strength of the method in reconstructing velocity fields behind transparent spheres that would not have been accessible by optical measurement techniques without the ray tracing correction.

Ефективна пластична поведінка пористих матеріалів зі структурою інверсного опалу

Based on the theoretical principles of the mechanics of composites, the effective plastic behavior of a porous material with a periodic inverse opal structure under uniaxial loading was studied in detail by means of finite element modeling. The creation of such materials is based on the inversion of pores and skeleton of partially sintered dense packing of polystyrene spheres. Electrodeposited nickel was used as the skeleton of the porous material. According to the macroscopic uniaxial loading or unloading, was finding a stress-strain state at the meso-level. For this, equilibrium equations were solved at the meso-level using special boundary conditions for a periodic unit cell. Such boundary conditions relate the problem of equilibrium at the meso-level with the "effective" deformations of the composite. This made it possible to calculate macroscopic residual strains after a cycle of uniaxial loading and unloading and iteratively find the value of effective stress corresponding to residual strains of 0.2%. In this way, the yield strength of inverse opal for uniaxial loading is calculated. At the same time, as a result of finite-element calculations, the transverse deformations coefficient (plastic Poisson ratio) is determined. This coefficient, in turn, makes it possible to approximate the general plastic behavior of the metamaterial by an elliptic yield curve in the plane of invariants of the stress tensor. Invariants mean average pressure and von Mises stress. These calculations were performing for several cases of the inverse opal structure, both with and without an additional coating. Yield stresses under any type of material loading are very sensitive to porosity. In particular, the application of an additional coating, even with a thickness less than 0,05 of the diameter of the spherical pores (initial polymer particles), causes an increase in the yield strength several times. Keywords: metamaterials, inverse opal, porous plasticity model, micromechanics, theory of plasticity.

Stability maps for columnar structures

ABSTRACT We have previously explored the hysteresis and reversibility of transitions between ordered packings of soft spheres of diameter d in cylindrical channels of diameter D [A. Mughal, J. Winkelmann, D. Weaire, S. Hutzler, Phys. Rev. E 98, 043303 (2018)]. Here we extend these initial results to include transitions between all columnar structures without inner spheres (i.e. packings in which all of the spheres are in contact with the cylindrical boundary). These results can be represented by a directed network showing permissible transitions between structures. From the hard sphere limit we deduce that there are two different types of transitions, reversible and irreversible. We explore the nature of these transitions for soft spheres as a function of pressure and due to changes in the ratio D/d. These results are illustrated by the use of schematic diagrams, indicating the topological features of each transition. Specific cases are tabulated and can be understood by reference to the appropriate schematic diagram.

Tuning the physicochemical properties of axitinib by crystallization: Preparation, calculation and Structure-property relationship

Crystal engineering is a radical method to modulate the physicochemical properties of the drugs to meet the needs of the pharmaceutical industry. Examined herein are six axitinib solid-state forms, including anhydrous forms (Ⅳ and XLI), solvates (acetic acid, nitromethane and n-propanol), and cocrystal (nicotinamide). Comprehensive characterization containing morphology, hygroscopicity, solubility, fluorescence and powder tabletability were performed. Calculations of lattice energy, packing coefficient and energy framework rationalized the structure–property relationship of these six forms from the packing landscape. This system illustrates that form IV and cocrystal improve the solubility of axitinib, but the humidity stability is also decreased. Six crystal forms show polychromatic solid-state luminescence ranging from violet to indigo to green. Form IV and cocrystal with slip plane structures exhibited superior tensile strength and compactibility than isotropic form XLI. Elusive form VI can be obtained by controlling the humidity levels during desolvation of the solvate. This work not only realizes the regulation of better physicochemical properties of axtinnib for industrial manufacturing, but also provides a detailed understanding of the structure–property relationship of drugs in complex polymorphic systems.

Computational Modelling of Nanoporous Materials

AbstractRecently proposed constitutive models that describe the mechanical behavior of open‐porous cellular materials take into account the random pore sizes of the materials but not the randomness in the pore shapes [1]. In the present contribution, a computational model to reconstruct the highly irregular structure of porous materials is developed using sphere packing and Laguerre‐Voronoi tessellation [2]. The mechanical structure‐property relationship is then investigated by means of finite element methods. Under compression, porous materials demonstrate linear‐elastic behaviour followed by an irreversible deformation resulting in a plateau region followed by densification [3]. In this contribution, such mechanical responses of porous materials under cyclic loading are investigated using elasto‐plastic material models. Furthermore, effects of tensile loading on the model are also discussed.

Multiple mixed interior and boundary peaks synchronized solutions for nonlinear coupled elliptic systems

This paper is devoted to a class of singularly perturbed nonlinear Schrödinger systems defined on a smooth bounded domain in RN(N=2,3). We use the Lyapunov–Schmidt reduction method to construct synchronized vector solutions with multiple spikes both on the boundary and in the interior of the domain. For each vector solution that has been constructed, we point out that the interior spikes locate near sphere packing points in the domain, and the boundary spikes locate near the critical points of the mean curvature function related to the boundary of the domain.

Comparing methods for permeability computation of porous materials and their limitations

AbstractEfficient numerical simulations of fluid flow on the pore scale allow for the numerical estimation of effective material properties of porous media, e.g. intrinsic permeability or tortuosity. These parameters are essential for various applications where hydro‐mechanical properties on larger scales have to be known. Numerical tools based intrinsically on pore scale simulations are known e.g. as Digital Rock Physics in geosciences and have even more and more replaced physical experiments. For these reasons, the validation of numerical methods as well as the establishment of clear limits regarding the application areas play an important role. Here, we compute single‐phase flow through a porous matrix, e.g. irregular sphere packings, sandstones, artificially created thin porous media, on the pore scale. Therefore we implement on the one hand a Smoothed Particle Hydrodynamics algorithm for solving the Navier‐Stokes equations and on the other hand a Finite Difference solver for the Stokes equations. Both methods work directly and seamlessly on voxel data of porous materials which are generated by µXRCT‐scans or by microfluidic experiments that have undergone segmentation and binarization. We compare both solvers from a parallel performance point of view as well as their results for flows in the Darcy regime. In addition, we investigate the limitations of the solvers using the example of a porous material whose pore geometry changes over time and precipitation affects the flow conditions.

Automated isocenter optimization approach for treatment planning for gyroscopic radiosurgery.

Radiosurgery is a well-established treatment for various intracranial tumors. In contrast to other established radiosurgery platforms, the new ZAP-X® allows for self-shielding gyroscopic radiosurgery. Here, treatment beams with variable beam-on times are targeted towards a small number of isocenters. The existing planning framework relies on a heuristic based on random selection or manual selection of isocenters, which often leads to a higher plan quality in clinical practice. The purpose of this work is to study an improved approach for radiosurgery treatment planning, which automatically selects the isocenter locations for the treatment of brain tumors and diseases in the head and neck area using the new system ZAP-X® . We propose a new method to automatically obtain the locations of the isocenters, which are essential in gyroscopic radiosurgery treatment planning. First, an optimal treatment plan is created based on a randomly selected nonisocentric candidate beam set. The intersections of the resulting subset of weighted beams are then clustered to find isocenters. This approach is compared to sphere-packing, random selection, and selection by an expert planner for generating isocenters. We retrospectively evaluate plan quality on 10 acoustic neuroma cases. Isocenters acquired by the method of clustering result in clinically viable plans for all 10 test cases. When using the same number of isocenters, the clustering approach improves coverage on average by 31 percentage points compared to random selection, 15 percentage points compared to sphere packing and 2 percentage points compared to the coverage achieved with the expert selected isocenters. The automatic determination of location and number of isocenters leads, on average, to a coverage of 97 ± 3% with a conformity index of 1.22 ± 0.22, while using 2.46 ± 3.60 fewer isocenters than manually selected. In terms of algorithm performance, all plans were calculated in less than 2 min with an average runtime of 75 ± 25s. This study demonstrates the feasibility of an automatic isocenter selection by clustering in the treatment planning process with the ZAP-X® system. Even in complex cases where the existing approaches fail to produce feasible plans, the clustering method generates plans that are comparable to those produced by expert selected isocenters. Therefore, our approach can help reduce the effort and time required for treatment planning in gyroscopicradiosurgery.

Quasi-Packing Different Spheres with Ratio Conditions in a Spherical Container

This paper considers the optimized packing of different spheres into a given spherical container under non-standard placement conditions. A sphere is considered placed in the container if at least a certain part of the sphere is in the container. Spheres are allowed to overlap with each other according to predefined parameters. Ratio conditions are introduced to establish correspondence between the number of packed spheres of different radii. The packing aims to maximize the total number of packed spheres subject to ratio, partial overlapping and quasi-containment conditions. A nonlinear mixed-integer optimization model is proposed for this ratio quasi-packing problem. A heuristic algorithm is developed that reduces the original problem to a sequence of continuous open dimension problems for quasi-packing scaled spheres. Computational results for finding global solutions for small instances and good feasible solutions for large instances are provided.

Correction of Stokes drag for non-Newtonian flow through pack of spheres

Non-Newtonian fluids have great relevance in a wide range of porous media applications, such as enhanced oil recovery and filtration processes. The estimation of the drag coefficient of power-law fluids flowing through simple porous media structures in the Stokes regime is typically modeled by adding a direct correction term to the Newtonian model. In this study, to calculate the non-dimensional drag for various fluid indices and volume fractions of packed spheres, extensive numerical simulations of the power-law fluid flow through single cubic, body-centered cubic, and face-centered cubic porous structures are conducted. The numerical results are used to develop a model function of the non-dimensional drag depending on the power-law fluid index and volume fraction of the sphere pack. Consequently, the drag force and coefficient of shear-thinning or shear-thickening fluid flowing through the investigated structures can be predicted. A correction is introduced for the Stokes drag correlation based on the generalized Reynolds number for non-Newtonian fluids. The results of the proposed model and numerical simulations are in excellent agreement with deviations not more than 1.2%. This implies that the modeled predictions are valid for all investigated structures.