Sphere Packing Research Articles

Theoretical Study of Phase Behaviors of Symmetric Linear B1A1B2A2B3 Pentablock Copolymer.

The nanostructures that are self-assembled from block copolymer systems have attracted interest. Generally, it is believed that the dominating stable spherical phase is body-centered cubic (BCC) in linear AB-type block copolymer systems. The question of how to obtain spherical phases with other arrangements, such as the face-centered cubic (FCC) phase, has become a very interesting scientific problem. In this work, the phase behaviors of a symmetric linear B1A1B2A2B3 (fA1 = fA2, fB1 = fB3) pentablock copolymer are studied using the self-consistent field theory (SCFT), from which the influence of the relative length of the bridging B2-block on the formation of ordered nanostructures is revealed. By calculating the free energy of the candidate ordered phases, we determine that the stability regime of the BCC phase can be replaced by the FCC phase completely by tuning the length ratio of the middle bridging B2-block, demonstrating the key role of B2-block in stabilizing the spherical packing phase. More interestingly, the unusual phase transitions between the BCC and FCC spherical phases, i.e., BCC → FCC → BCC → FCC → BCC, are observed as the length of the bridging B2-block increases. Even though the topology of the phase diagrams is less affected, the phase windows of the several ordered nanostructures are dramatically changed. Specifically, the changing of the bridging B2-block can significantly adjust the asymmetrical phase regime of the Fddd network phase.

Pore-scale morphology effects on colloid deposition by trajectory tracking simulations

The migration and deposition of colloids in natural porous media is a phenomenon of high importance in hydrocarbon recovery, wastewater treatment and contaminant hydrogeology. Commonly described by colloid filtration theory (CFT), and using a single representative collector element, the theoretical framework is not accounting for morphological effects and heterogeneity of realistic porous media, including the local variations of balance of forces or presence of hydraulically stagnant regions in the vicinity of grain-to-grain contacts. So far developed numerical models of colloidal transport in porous media accounting for all major interactions and pore structure were applied to 2D systems. Known 3D simulations are limited to particulate flow with a reduced set of interactions.We propose a coupled colloid transport model combining computational fluid dynamics (CFD) and discrete element model (DEM) approaches, where the former provides the velocity field at a given time step, while the second calculates the updated colloid positions accounting for a set of forces acting on each colloid. Relationships between colloid retention and morphological characteristics, such as the pore coordination number and surface to volume ratio of a random sphere pack are investigated by pore partitioning of a digitized representation. The role of stagnation zones is quantified by expressing the capture probability of colloids as a function of distance to the nearest grain-to-grain contact measured along the curved surface.The analysis of pore space morphology of the collector and the simulated filtration process reveals a strong relation between colloid retention and pore connectivity. The simulations successfully mimic the expected effects of ionic strength of the carrier fluid, such as the increasing role of grain-to-grain contacts in colloid retention at low ionic strength.

Pore-scale numerical investigation on the thermal storage properties of packed-bed systems with phase change material microcapsules

Phase change material (PCM) microcapsules are gaining increasing attention because it can overcome the shortage of PCM macrocapsules and increase the heat transfer area, while the pore-scale thermal performance of the PCM microcapsule packed-bed systems, as well as the influence of packing methods and microcapsule sizes on thermal storage, is still poorly understood. This paper introduces various spherical PCM microcapsule pack models, including single-layered, diameter-changed two-layered, and three-layered packing models, to investigate the charging process at pore scale. The indices of thermal storage properties such as charging time, temperature distribution, phase change fraction, pressure drop and thermocline degradation are studied with the finite element method. In the single-layered thermal storage system, the results show that the charging process can be improved by reducing the capsule size and using cross-packing methods. When the capsule size is reduced from 2 to 0.5 mm with cross packing, the charging time can be shortened by 72.97%, but remarkable pressure drop is created. The results indicate that the overall thermal storage performance of multi-layered packed-bed systems is better than that of single-layered packed-bed systems. The charging time of the diameter-changed two-layered and three-layered thermal storage systems can be reduced up to 16.21% and 24.32%, respectively, compared to the single-layered in-line pack with a d = 2 mm. The filling sequence plays a key role in multi-layered thermal systems. A packed-bed system with a diameter decreasing along the heat transfer fluid (HTF) flow direction has a better capacity to reduce thermocline degradation than other packed-bed systems.

Mechanical and hydraulic properties of carbonate rock: The critical role of porosity

Carbonate rocks are extensively used in civil infrastructure and play a critical role in geoenergy geoengineering, either as hydrocarbon reservoirs or potential repositories for CO 2 geological storage. Carbonate genesis and diagenetic overprint determine the properties of carbonate rocks. This study combines recent data gathered from Madison Limestone and an extensive dataset compiled from published sources to analyze the hydraulic and mechanical properties of limestone carbonate rocks. Physical models and data analyses recognize the inherently granular genesis of carbonate rocks and explain the strong dependency of physical properties on porosity. The asymptotically-correct power model in terms of (1- ϕ / ϕ ∗) α is a good approximation to global trends of unconfined stiffness E and unconfined compressive strength UCS , cohesive intercept in Mohr-Coulomb failure envelopes, and the brittle-to-ductile transition stress. This power model is the analytical solution for the mechanical properties of percolating granular structures. We adopted a limiting granular porosity ϕ ∗ = 0.5 for all models, which was consistent with the loosest packing of monosize spheres. The fitted power model has exponent ( α = 2) in agreement with percolation theory and highlights the sensitivity of mechanical properties to porosity. Data and models confirm a porosity-independent ratio between unconfined stiffness and strength, and the ratio follows a log-normal distribution with mean ( E/UCS ) ≈ 300. The high angle of internal shear strength measured for carbonate rocks reflects delayed contact failure with increased confinement, and it is not sensitive to porosity. Permeability spans more than six orders of magnitude. Grain size controls pore size and determines the reference permeability k ∗ at the limiting porosity ϕ ∗ = 0.5. For a given grain size from fine to coarse-grained dominant carbonates, permeability is very sensitive to changes in porosity, suggesting preferential changes in the internal pore network during compaction.

Perturbation-Based Thresholding Search for Packing Equal Circles and Spheres

This paper presents an effective perturbation-based thresholding search for two popular and challenging packing problems with minimal containers: packing N identical circles in a square and packing N identical spheres in a cube. Following the penalty function approach, we handle these constrained optimization problems by solving a series of unconstrained optimization subproblems with fixed containers. The proposed algorithm relies on a two-phase search strategy that combines a thresholding search method reinforced by two general-purpose perturbation operators and a container adjustment method. The performance of the algorithm is assessed relative to a large number of benchmark instances widely studied in the literature. Computational results show a high performance of the algorithm on both problems compared with the state-of-the-art results. For circle packing, the algorithm improves 156 best-known results (new upper bounds) in the range of [Formula: see text] and matches 242 other best-known results. For sphere packing, the algorithm improves 66 best-known results in the range of [Formula: see text], whereas matching the best-known results for 124 other instances. Experimental analyses are conducted to shed light on the main search ingredients of the proposed algorithm consisting of the two-phase search strategy, the mixed perturbation and the parameters. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by the National Natural Science Foundation of China [Grants 61703213 and 61933005]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1290 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0004 ) at ( http://dx.doi.org/10.5281/zenodo.7579558 ).

On superintegral Kleinian sphere packings, bugs, and arithmetic groups

Abstract We develop the notion of a Kleinian Sphere Packing, a generalization of “crystallographic” (Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA 116 2019, 2, 436–441]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do “superintegral” such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from ℚ {{\mathbb{Q}}} -arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles π m {\frac{\pi}{m}} for finitely many m. We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.

On Packing of Minkowski Balls

We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conjecture about the critical determinant we divide Minkowski balls into 3 classes: Minkowski balls, Davis balls and Chebyshev–Cohn balls. We investigate lattice packings of these balls on planes with varying Minkowski metric and search among these packings the optimal packings. In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev–Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding balls.

Spherical Packing Superlattices in Self-Assembly of Homogenous Soft Matter: Progresses and Potentials

Spherical Packing Superlattices in Self-Assembly of Homogenous Soft Matter: Progresses and Potentials

Decomposition kinetics of sodium amalgam in fixed bed of WC‐coated tubular SS‐304 packing: Modelling and validation

AbstractErosion of graphite packing in continuously operating fixed bed sodium amalgam decomposers leads to performance degradation accompanied by clogging in the downstream process lines which not only increases the maintenance and catalyst refilling costs but also aggravates the risk of mercury vapour exposure. To overcome this drawback, the present work utilizes tubular SS‐304 packing coated with tungsten carbide (WC) on the outer surface. The coating is obtained using an indigenously developed laser‐directed energy deposition facility. Dedicated experimental setup is developed for each set of process parameters to comparatively study the fixed bed decomposition kinetics at steady state for three types of WC‐coated SS‐304 packing and a conventional spherical graphite packing. In parallel, a numerical 3D–1D model is implemented using open source computational solvers and is experimentally validated. The algorithm of this model is based on proposed drop–fill approach for WC‐coated SS‐304 packing and modified drop approach for spherical graphite packing, respectively. In terms of decomposition rates, the performance of WC‐coated SS‐304 packing surpasses that of spherical graphite packing at high amalgam flowrates; meanwhile, the average alteration in experimental steady‐state temperature profile after continuous operation of 2 years is around 2% for WC‐coated SS‐304 packing as compared to nearly 17% for spherical graphite packing.

Does Time Smoothen Space? Implications for Space-Time Representation

The continuous nature of space and time is a fundamental tenet of many scientific endeavors. That digital representation imposes granularity is well recognized, but whether it is possible to address space completely remains unanswered. This paper argues Hales’ proof of Kepler’s conjecture on the packing of hard spheres suggests the answer to be “no”, providing examples of why this matters in GIS generally and considering implications for spatio-temporal GIS in particular. It seeks to resolve the dichotomy between continuous and granular space by showing how a continuous space may be emergent over a random graph. However, the projection of this latent space into 3D/4D imposes granularity. Perhaps surprisingly, representing space and time as locally conjugate may be key to addressing a “smooth” spatial continuum. This insight leads to the suggestion of Face Centered Cubic Packing as a space-time topology but also raises further questions for spatio-temporal representation.

Exact solutions for Janssen effect of lattice disk packings

The rhombic lattice packing model proposed here clearly shows the micro causes of the Janssen effect and is solved analytically. Based on the analysis of force equilibrium equations with an exterior product method, a recurrence relation with a force transmission matrix Q is obtained which shows that a contact force will be transmitted and accumulated along the lattice vector direction and then be reflected by lateral walls, and eventually, reach a saturation value if the packing is frictional. A method based on tensor algebra and difference equation is proposed for the diagonalization of Q. Using roundup and remainder functions, we give the exact and approximate expressions which can describe the piecewise smoothness of the pressure function data given by numerical calculations. Numerical results also show the non-uniformity of pressure distribution. These quantitative results have certain deviations with Janssen model except for very small top load or large force reflectivity.

Stress-field driven conformal lattice design using circle packing algorithm

Reliable extreme lightweight is the pursuit in many high-end manufacturing areas. Aided by additive manufacturing (AM), lattice material has become a promising candidate for lightweight optimization. Configuration of lattice units at the material level and the distribution of lattice units at the structure level are the two main research directions recently. This paper proposes a generative strategy for lattice infilling optimization using organic strut-based lattices. A sphere packing algorithm driven by von Mises stress fields determines the lattice distribution density. Two typical configurations, Voronoi polygons and Delaunay triangles, are adopted to constitute the frames, respectively. Based on finite element analysis, a simplified truss model is utilized to evaluate the lattice distribution in terms of mechanical properties. Optimization parameters, including node number, mapping gradient, and the range of varying circle size, are investigated through the genetic algorithm (GA). Multiple feasible solutions are obtained for further solidification modelling. To avoid the stress concentration, the organic strut-based lattice units are created by the iso-surface modelling method. The effectiveness of the proposed generative approach is illustrated through a classical 3-point bending beam. The stiffness of the optimized structure, verified through experimental testing, has increased 80% over the one using the traditional uniform body center cubic (BCC) lattice distribution.

Crashworthiness Study of 3D Printed Lattice Reinforced Thin-Walled Tube Hybrid Structures

Based on the advantages of thin-walled tubes and lattice structures in energy absorption and improved crashworthiness, a hybrid structure of lattice-reinforced thin-walled tubes with different cross-sectional cell numbers and gradient densities was constructed, and a high crashworthiness absorber with adjustable energy absorption was proposed. The experimental and finite element characterization of the impact resistance of uniform density and gradient density hybrid tubes with different lattice arrangements to withstand axial compression was carried out to investigate the interaction mechanism between the lattice packing and the metal shell, and the energy absorption of the hybrid structure was increased by 43.40% relative to the sum of its individual components. The effect of transverse cell number configuration and gradient configuration on the impact resistance of the hybrid structure was investigated, and the results showed that the hybrid structure showed higher energy absorption than the empty tube, and the best specific energy absorption was increased by 83.02%; the transverse cell number configuration had a greater effect on the specific energy absorption of the hybrid structure with uniform density, and the maximum specific energy absorption of the hybrid structure with different configurations was increased by 48.21%. The gradient density configuration had a significant effect on the peak crushing force of the gradient structure. In addition, the effects of wall thickness, density and gradient configuration on energy absorption were quantitatively analyzed. This study provides a new idea to optimize the impact resistance of lattice-structure-filled thin-walled square tube hybrid structures under compressive loading through a combination of experiments and numerical simulations.

Design of Fe36.29Cr28.9Ni26.15Cu4.17Ti1.67V2.48C0.46 HEA using a new criterion based on VEC: Microstructural study and multiscale mechanical response

The present work updated empirical parameter ranges with a high adjustment to the experimental results as obtained by the means of the exploratory data analysis (EDA) method. A new high entropy alloy with FCC structure reinforced by precipitation of intermetallic phases fabricated by vacuum-argon induction melting was designed and modeled. The main results showed that the valence electron concentration (VEC) is the predominant empirical phase prediction parameter in high entropy alloys and that the lattice packing factor is necessary to determine the presence of intermetallic phases effectively. Phase stability ranges based on VEC associated with the base crystal structures and the presence of intermetallics corroborated computationally by CALPHAD and experimentally were reported. The interdendritic zone proved to be a preferential precipitation zone with precipitates of σ, TiC, and γ' phases. The γ' phase showed a size difference between the dendritic and interdendritic zone associated with diffusive Ti mechanisms. The nanoscale mechanical response determined that dislocation creep and reinforcement in the interdendritic zone are predominant creep mechanisms that obtained an effective entanglement of the dislocations increasing the strain hardening coefficient. The mechanical response of the alloy obtained is superior to the average of the alloys with FCC structure. It maintains a high ductility that allows reaching an energy absorption and damage tolerance of 43.56 GPa% showing severe plastic slip lines being in the range of use for aerospace applications and hydrogen tanks.

A note on Fourier eigenfunctions in four dimensions

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem.

Bootstrapping boundaries and branes

The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition function with identical boundaries. We find that this deceivingly simple bootstrap equation, when combined with unitarity, leads to surprisingly strong constraints on admissible boundary states. In particular, we derive universal bounds on the tension (boundary entropy) of stable boundary conditions, which provide a rigorous diagnostic for potential D-brane decays. We also find unique solutions to the bootstrap problem of stable branes in a number of rational CFTs. Along the way, we observe a curious connection between the annulus bootstrap and the sphere packing problem, which is a natural extension of previous work on the modular bootstrap. We also derive bounds on the boundary entropy at large central charge. These potentially have implications for end-of-the-world branes in pure gravity on AdS3.

Pore-scale modelling of CO2 sequestration

Geological sequestration of CO2 in porous sedimentary medium filled with saline water requires a clear view of the gas behavior as a function of topology of the hosting pores, linked to capillary phenomena. The sedimentary medium is modeled as a random packing of equal spheres giving rise to 3D network of pore bodies and throats. The injected CO2 pushes the water from pores of the reservoir. The scenario is simulated by the increase of the curvature of the liquid/CO2 interface. Eventually a critical state is reached where the liquid is drained from pores. When the liquid counteracts this first drainage step (imbibition), there is a progressive decrease of the interfaces curvature until they merge together, eliminating the CO2 from pores. This pore-scale investigation of the mechanisms possibly occurring after a gas injection is studied.

Charge density redistribution with pressure in a zeolite framework

As a result of external compression applied to crystals, ions relax, in addition to shortening the bond lengths, by changing their shape and volume. Modern mineralogy is founded on spherical atoms, i.e., the close packing of spheres, ionic or atomic radii, and Pauling and Goldschmidt rules. More advanced, quantum crystallography has led to detailed quantitative studies of electron density in minerals. Here we innovatively apply it to high-pressure studies up to 4.2 GPa of the mineral hsianghualite. With external pressure, electron density redistributes inside ions and among them. For most ions, their volume decreases; however, for silicon volume increases. With growing pressure, we observed the higher contraction of cations in bonding directions, but a slighter expansion towards nonbonding directions. It is possible to trace the spatial redistribution of the electron density in ions even at the level of hundredths parts of an electron per cubic angstrom. This opens a new perspective to experimentally characterise mineral processes in the Earth’s mantle. The use of diamond anvil cells with quantum crystallography offers more than interatomic distances and elastic properties of minerals. Interactions, energetic features, a branch so far reserved only to the first principle DFT calculations at ultra-high-pressures, become available experimentally.

Design of Fmoc-Phenylalanine Nanofibrillar Hydrogel and Mechanistic Studies of Its Antimicrobial Action against Both Gram-Positive and Gram-Negative Bacteria.

In pursuit of efficient antimicrobial agents, biomaterials such as hydrogels have drawn a considerable amount of attention due to their numerous advantages such as a high degree of hydration, biocompatibility, stability, and direct application at an infectious site. Particularly, biomaterials such as hydrogels based on Fmoc-protected peptides and amino acids have proven to be immensely advantageous. Such biomaterials can undergo gelation by simple pH modulation and can be used for various biological applications. Keeping this in mind, in this work, we reported the synthesis of Fmoc-phenylalanine (Fmoc-F)-based hydrogels using trisodium citrate as a pH modulator and compared them with the previously reported pH modulator glucono-δ-lactone. The gels were compared using various characterization techniques such as rheometry, field emission scanning electron microscopy (FE-SEM), atomic force microscopy (AFM), small angle X-ray scattering (SAXS), FT-IR, thioflavin T (ThT) binding assay, and zeta potential studies. These studies highlighted the role of pH modulators in affecting various parameters such as the ability to alter the zeta potential of the nanofibrils, improve their bactericidal action, reduce the amyloidic characters, shift the lattice packing from amorphous to crystalline, and introduce fluorescence and thermoreversibility. Interestingly, this is the first report where the Fmoc-F-based hydrogel has been shown to be effective against Gram-negative bacteria along with Gram-positive bacteria as well. Additionally, the mechanism of antimicrobial action was investigated using docking and antioxidant studies.

Effects of size ratio on particle packing in binary glasses

Mixtures of binary spheres represent prototypes of amorphous solids and can model metallic, granular, and colloidal glasses. However, the effects of the size ratio λ on amorphous structures are not well understood. Here, we revisit the controversial noncubic scaling law and the local sphere packing by systematically changing λ. Our simulations clarify the existence and mechanism of the noncubic scaling law for mean atomic volume, va∝q1−d, where q1 is the position of the first diffraction peak and d is a constant less than the space dimension D. We find that the scaling law holds at each λ in binary hard-sphere glasses and metallic glasses, but the exponent satisfies a universal power law, d∼(λ−λc)−γ, instead of being a constant. The decreasing trend of d(λ), the abnormal d>D and the divergence of d when λ approaches 1 are theoretically explained. Moreover, d begins to fluctuate at λ<1.2, indicating less stable glasses. Large and small spheres are better dispersed with more disordered structures at λ>1.2. At λ=1.2, various structural parameters change, and the number of icosahedral packing reaches the maximum. The results cast light and pose new challenges on amorphous structures of binary glasses.