Sphere Packing Research Articles

Adhesion behavior and deposition morphology of cement grout flowing through granular materials

The liquid distribution of unsaturated wet granular media composed of granular materials and Newtonian fluid has been in detail studied. However, when a granular material is mixed with a certain amount of Bingham fluid, the effect of rheological properties of Bingham fluid on the deposition morphologies is not clearly explored. In this study, the adhesion rules of fresh cement grout flowing through granular materials without considering the filtration effect is investigated, and the deposition morphologies of cement grout in random packing of spheres is observed with the help of X-ray microtomography device. The results show that the deposition mechanism of cement grout is significantly dependent on its yield stress. The cement grout spontaneously forms open structures with large liquid–air interface. The morphologies of cement grout within cemented granular materials vary with grout content. With the increase of the yield stress of cement grout, neighbouring smaller clusters gradually touch each other and coalesce into more complex structure, and the largest cluster can contain almost all the cement grout.

Universal scaling for the permeability of random packs of overlapping and nonoverlapping particles.

Constraining fluid permeability in porous media is central to a wide range of theoretical, industrial, and natural processes. In this Letter, we validate a scaling for fluid permeability in random and lattice packs of spheres and show that the permeability of packs of both hard and overlapping spheres of any sphere size or size distribution collapse to a universal curve across all porosity ϕ in the range of ϕ_{c}<ϕ<1, where ϕ_{c} is the percolation threshold. We use this universality to demonstrate that permeability can be predicted using percolation theory at ϕ_{c}<ϕ≲0.30, Kozeny-Carman models at 0.30≲ϕ≲0.40, and dilute expansions of Stokes theory for lattice models at ϕ≳0.40. This result leads us to conclude that the inverse specific surface area, rather than an effective sphere size or pore size is a universal controlling length scale for hydraulic properties of packs of spheres. Finally, we extend this result to predict the permeability for some packs of concave nonspherical particles.

Descriptor-free unsupervised learning method for local structure identification in particle packings.

Local structure identification is of great importance in many scientific and engineering fields. However, mathematical and supervised learning methods mostly rely on specific descriptors of local structures and can only be applied to particular packing configurations. In this work, we propose an improved unsupervised learning method, which is descriptor-free, for local structure identification in particle packing. The point cloud is used as the input of the improved method, which directly comes from spatial positions of particles and does not rely on specific descriptors. The improved method constructs an autoencoder based on the point cloud network combined with Gaussian mixture models for dimension reduction and clustering. Numerical examples show that the improved method performs well in local structure identification of quasicrystal disk and sphere packings, achieving comparable accuracy with previous methods. For disordered packings, which have been considered having nearly no local structures, the improved method identifies a nontrivial seven-neighbor motif in the maximally dense random packing of disks and finds acentric structural motifs in the random close packing of spheres, which demonstrate the ability on identification of new and unknown local structures. The improved unsupervised learning method would help obtain information from massive simulation and experimental results as well as devising new order parameters for particle packings.

The modification of the Kozeny-Carman equation through the lattice Boltzmann simulation and experimental verification

In this paper, the numerical simulation based on the lattice Boltzmann method (LBM) and experimental studies are conducted to investigate the permeability of porous media with sphere packing. The accuracy of LBM is verified through the comparison of theoretical solutions and the numerical result of flat channel flows. The resolution and the size of representative element volume used for simulation are evaluated. Then, the effects of porosity and particle size on the permeability of porous media are studied through the combination of LBM and discrete element method, and the factors that affect the fluid flow in porous media are explored. It is found that there are very high linear correlations between n3/(1-n)2 and permeability. Meanwhile, the relationship between the particle size, d, and permeability is also established. The physical constant, Ck, is found closely related to the porosity, n, based on which, the modification of the Kozeny-Carman (K-C) equation is proposed. After that, the constant head test is conducted for the glass beads with different porosity and particle size distribution, and the proposed equation is validated. The experimental results show that the modified K-C equation presents a higher accuracy than the classical K-C equation for the porous media from sphere packing.

On the Mordell–Weil lattice of y^2 = x^3 + b x + t^{3^n + 1} in characteristic 3

We study the elliptic curves given by y^2 = x^3 + b x +t^{3^n+1} over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u^3 + b u = v^{3^n + 1}. In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension 2 cdot 3^n for every n ge 1, which improve on the currently best known sphere packing densities in dimensions 162 (case n=4) and 486 (case n=5). For n=3, the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n=1 it has the same density as the lattice E_6 in dimension 6.

New framework for upscaling gas-solid heat transfer in dense packing

The objective of this work is to provide a suitable upscaling framework for modeling gas-solid heat transfer in dense packings. In a macroscopic two-temperature model, the pore-scale heat transfer is modeled by determining a Nusselt number. We introduce a new surface statistical filter to extract the Nusselt number from pore-scale simulations. A thermal offset boundary condition is developed to deal with dense packings that exhibit high heat transfer. The new method is applied to sphere packings. Several pore-scale simulations over a range of Reynolds number (10–100) and solid volume fraction (0.1–0.6) are performed. The extracted Nusselt numbers show very good agreement with results available in the literature for lower solid volume fractions (0.1–0.5). We show that the Nusselt number to be used with the two-temperature model is the Nusselt number extracted using macroscopic (filtered) temperature. We propose a valid Nusselt number correlation for a dense packing of spheres. Nevertheless, the proposed scaling framework is not restricted to sphere packings.

Dynamic Measure of Hyperuniformity and Nonhyperuniformity in Heterogeneous Media via the Diffusion Spreadability

The recently developed concept of spreadability, $\mathcal{S}(t)$, provides a direct link between time-dependent diffusive transport and the microstructure of two-phase media across length scales. We explicitly compute $\mathcal{S}(t)$ for well-known two-dimensional and three-dimensional model structures, including nonhyperuniform fully penetrable spheres and equilibrium hard spheres, as well as hyperuniform sphere packings derived from perfect glasses, uniformly randomized lattices (URL), disordered stealthy point processes and Bravais lattices. We further confirm that the small-, intermediate- and long-time behaviors of $\mathcal{S}(t)$ sensitively capture the small-, intermediate- and large-scale characteristics of the models. In instances in which the spectral density $\tilde{\chi}_{_V}(\mathbf{k})$ has a power-law form $B|\mathbf{k}|^\alpha$ in the limit $|\mathbf{k}|\rightarrow 0$, the long-time spreadability provides a simple means to extract the value of the coefficients $\alpha$ and $B$ that is robust against noise in $\tilde{\chi}_{_V}(\mathbf{k})$ at small wavenumbers. Interestingly, the excess spreadability $\mathcal{S}(\infty)-\mathcal{S}(t)$ for URL packings has nearly exponential decay at small to intermediate $t$, but transforms to a power-law decay at large $t$. Our study of the aforementioned models enables us to devise an algorithm that accurately extracts large-scale behaviors from diffusion data alone. Lessons learned from such analyses of our models are used to determine accurately the large-scale structural characteristics of a sample Fontainebleau sandstone, which we show is nonhyperuniform. Our study demonstrates the practical applicability of the diffusion spreadability to extract crucial microstructural information from real data across length scales and provides a basis for the inverse design of materials with desirable time-dependent diffusion properties.

Novel method for the facile control of molecular weight cut-off (MWCO) of ceramic membranes

This study demonstrates a simple and novel preparation method to prepare ceramic nanofiltration membranes with a precise and tunable molecular weight cut-off (MWCO) by packing variously sized nanoparticles into existing membrane pores. As a result, ceramic membranes with a MWCO from 1000 Da to 10,000 Da were successfully prepared with the narrow distribution of the pore size after the filtration-coating process. In addition, the effective porosity of the ceramic membranes was calculated from the results of the membrane properties by the Hagen-Poiseuille equation which fit within the range of the sphere packing theory from 17.3% to 41.8%. Furthermore, the results of nonlinear curve fitting between the MWCO and the nanoparticle size show a high accuracy, which implies that the MWCO of the ceramic membranes can be predicted using the curve fitting model with variously sized nanoparticles in the filtration-coating process. In conclusion, the novel filtration-coating method enables precise pore control and provides a tunable MWCO to ceramic membranes by preparing various sizes of nanoparticles.

Soft Alloys Constructed with Distinct Mesoatoms via Self-Sorting Assembly of Giant Shape Amphiphiles.

The packing structures of spherical motifs affect the properties of resultant condensed materials such as in metal alloys. Inspired by the classic metallurgy, developing complex alloy-like packing phases in soft matter (also called "soft alloys") is promising for the next-generation superlattice engineering. Nevertheless, the formation of many alloy-like phases in single-component soft matter is usually thermodynamically unfavourable and technically challenging. Here, we utilize a novel self-sorting assembly approach to tackle this challenge in binary blends of soft matter. Two types of giant shape amphiphiles self-sort to form their discrete spherical motifs, which further simultaneously pack into alloy-like phases. Three unconventional spherical packing phases have been observed in these binary systems, including MgZn2 , NaZn13 , and CaCu5 phases. It's the first time that the CaCu5 phase is experimentally observed in soft matter. This work demonstrates a general approach to constructing unconventional spherical packing phases and other complex superlattices in soft matter.

Soft Alloys Constructed with Distinct Mesoatoms via Self‐Sorting Assembly of Giant Shape Amphiphiles

AbstractThe packing structures of spherical motifs affect the properties of resultant condensed materials such as in metal alloys. Inspired by the classic metallurgy, developing complex alloy‐like packing phases in soft matter (also called “soft alloys”) is promising for the next‐generation superlattice engineering. Nevertheless, the formation of many alloy‐like phases in single‐component soft matter is usually thermodynamically unfavourable and technically challenging. Here, we utilize a novel self‐sorting assembly approach to tackle this challenge in binary blends of soft matter. Two types of giant shape amphiphiles self‐sort to form their discrete spherical motifs, which further simultaneously pack into alloy‐like phases. Three unconventional spherical packing phases have been observed in these binary systems, including MgZn2, NaZn13, and CaCu5 phases. It's the first time that the CaCu5 phase is experimentally observed in soft matter. This work demonstrates a general approach to constructing unconventional spherical packing phases and other complex superlattices in soft matter.

A local orientational order parameter for systems of interacting particles.

Many physical systems are well modeled as collections of interacting particles. Nevertheless, a general approach to quantifying the absolute degree of order immediately surrounding a particle has yet to be described. Motivated thus, we introduce a quantity E that captures the amount of pairwise informational redundancy among the bonds formed by a particle. Particles with larger E have less diversity in bond angles and thus simpler neighborhoods. We show that E possesses a number of intuitive mathematical properties, such as increasing monotonicity in the coordination number of Platonic polyhedral geometries. We demonstrate analytically that E is, in principle, able to distinguish a wide range of structures and conjecture that it is maximized by the icosahedral geometry under the constraint of equal sphere packing. An algorithm for computing E is described and is applied to the structural characterization of crystals and glasses. The findings of this study are generally consistent with existing knowledge on the structure of such systems. We compare E to the Steinhardt order parameter Q6 and polyhedral template matching (PTM). We observe that E has resolution comparable to Q6 and robustness similar to PTM despite being much simpler than the former and far more informative than the latter.

Numerical approach to synthesizing realistic asteroid surfaces from morphological parameters

Context. The complex shape of asteroids and comets is a critical parameter in many scientific and operational studies. From the global irregular shape down to the local surface details, these topographies reflect the formation and evolutionary processes that remould the celestial body. Furthermore, these processes control how the surface will continue to evolve: from mass wasting on high slopes to spin-up due to anisotropic re-emission of thermal radiation. In addition, for space missions, the irregular coarse shape and complex landscape are a hazard to navigation, which must be accounted for in the planning phase. Aims. In this paper, we propose a novel method to synthesize physically correct 3D shape models of small celestial bodies, such as asteroids, to support the testing of a wide range of parameters in scientific and operational studies. Methods. We modeled virtual asteroid shapes using non-uniform sphere packings to represent the coarse shape, define an implicit surface, and then synthesize high-resolution topography with user-defined, locally controlled spot noise models. This effectively replaces the random noise model (e.g., Perlin noise) used in traditional approaches and allows us to construct a morphology based on actual physical shapes of the most common features observed on asteroids and comets. As an example of such a feature, we propose several kernel functions to add virtual craters to the coarse shape of the asteroid, of which the spatial distribution is controlled by typical crater production functions (e.g., a power law). Results. We demonstrate how this technique can be used to generate a variety of asteroid shapes and topographies using different cratering parameters and distributions. We apply our technique to artificially increase the resolution of existing models of the Didymos-Dimorphos system, the target of the Double Asteroid Redirection Test, and Hera missions. We show that our approach generates models that are suitable for typical analysis relying on detailed asteroid shapes, as well as operational scenarios for space missions. The meshes created with our algorithm can be directly used with existing visualization software and operations or science pipelines to generate data suitable for mission planning and to validate data analysis techniques.

On the road toward highly efficient and large volume three-dimensional-printed liquid chromatography columns?

The current performance of commercially packed liquid chromatography columns is limited by the random structure of the packed bed and by the wall-to-center heterogeneity of its structure. The minimum reduced plate heights observed are not smaller than 1.4, whereas they could theoretically be as low as 0.1 for dense and perfectly ordered packings of spheres. To bridge this gap, a wide inner diameter column with an ordered macroporous structure is printed in three dimensions by stereolithography of poly(ethylene glycol diacrylate) resin. Feature sizes below 100μm are achieved by combining conventional polymer stereolithography with photolithography using photomasks. A layer-by-layer polymerization is performed by alternating two distinct photomasks having horizontally and vertically oriented patterns. Despite the inevitable printing imperfections, minimum reduced plate heights around unity are measured for nonretained analytes. The next challenges for the successful printing of highly efficient and large volume liquid chromatography columns are threefold: reducing the feature size down to below 10μm, keeping minimum the unevenness of the flow channel dimensions, and tackling additive manufacturing of silica aerogels at such small feature sizes for higher mechanical stability and broader range of retention/selectivity than those delivered by polymer materials.

Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures

Ultrasonic testing of polycrystalline media relies heavily on simulation of the expected signals in order to detect and correctly interpret deviations due to defects. Many effects disturb ultrasonic waves propagating in polycrystalline media. One of them is scattering due to the granular microstructure of the polycrystal. The thus arising so-called microstructural noise changes with grain size distribution and testing frequency. Here, a method for simulating this noise is introduced. We geometrically model the granular microstructure to determine its influence on the backscattered ultrasonic signal. To this end, we utilize Laguerre tessellations generated by random sphere packings dividing space into convex polytopes—the cells. The cells represent grains in a real polycrystal. Cells are characterized by their volume and act as single scatterers. We compute scattering coefficients cellwise by the Born approximation. We then combine the Generalized Point Source Superposition technique with the backscattered contributions resulting from the cell structure to compute the backscattered ultrasonic signal. Applying this new methodology, we compute the backscattered signals in a pulse-echo experiment for a coarse grain cubic crystallized Inconel-617 and a fine grain hexagonal crystallized titanium. Fitting random Laguerre tessellations to the observed grain structure allows for simulating within multiple realizations of the proposed model and thus to study the variation of the backscattered signal due to microstructural variation.

Savonius Wind Turbine Numerical Parametric Analysis Using Space-Filling Design and Gaussian Stochastic Process

Wind energy is an alternative source of clean energy to address the growing energy demand and provide pollution-free electricity. With the rapid development of urban areas, high wind energy resources such as high-rise building rooftops are excellent locations for urban wind turbine installation. One of the practical and simple urban wind turbines is the Savonius design. It has a simple design, easy to maintain, and is very affordable. This work focuses on the design evaluation of a Savonius wind turbine (SWT) by varying the rotor diameter, rotor height, and twist angle for urban applications. A transient computational fluid dynamics (CFD) approach is applied to assess the various design treatments using a space-filling design of experiments. To address the spaces in the hypercube statistical design, a sphere packing design method was adopted which suited the evaluation of computational simulations results such as that of the CFD. The Gaussian stochastic process model was applied to establish the trend of the parametric performance of the optimized SWT design through the model fitting. The results have shown that optimized SWT performs well with its self-starting capability compared to the traditional Savonius design. In addition, the optimized SWT has shown a better peak power coefficient compared with the results of previous works on the design of SWT.

Experimental investigation of the pre\u2013Darcy regime

The validity of Darcy’s law at very low Reynolds numbers is discussed controversially in literature, as some authors propose a pre–Darcy flow regime below some critical Reynolds number. The scope of this work is to investigate this problem experimentally. Therefore, a packing of glass spheres is perfused by different glycerin–water solutions. A linear behaviour between the flow velocity and the pressure drop through the packed spheres is found in the complete investigated range of Reynolds number {Re}_{d'}, based on the mean-pore diameter d' and mean-pore velocity v' with 10^{-9} le {Re}_{d'} le 10^{-1}. This contradicts the results of different authors like Fand et al. (1987) or Kececioglu and Jiang (1994), postulating a pre–Darcy regime for {Re}_{d'} le 2.8 cdot 10^{-6} or {Re}_{d'} le 0.13, respectively.Graphical abstract

Magnetic correlations of iron oxide nanoparticles as probed by polarized SANS in stretched magnetic nanoparticle-elastomer composites.

We have investigated the magnetic correlations among 7 nm iron oxide nanoparticles embedded in stretched silicone elastomers using polarized Small Angle Neutron Scattering (SANS). The magnetic nanoparticle (MNP)-elastomer composite can be stretched during experiments, and macroscopic deformations cause rearrangement of the iron oxide particles on the nanoscale. Polarized neutrons can be used to nondestructively probe the arrangement of magnetic nanoparticles before and after stretching, so that the relationship between applied stress and nanoscale magnetization can be interrogated. We find that stretching the MNP-elastomer composite past a certain threshold dramatically changes the structural and magnetic morphology of the system. The unstretched sample is modeled well by ~40 nm clusters of ~7 nm particles arranged in a hard sphere packing with a "volume fraction" parameter of 0.3. After the sample is stretched 3× its original size, however, the scattering data can be modeled by smaller, 16 nm clusters with a higher volume fraction of 0.4. We suggest that the effect can be understood by considering a stretching transformation on FCC-like crystallites of iron oxide nanoparticles embedded in an elastomeric medium.

Geometric and topological entropies of sphere packing.

We present a statistical mechanical description of randomly packed spherical particles, where the average coordination number is treated as a macroscopic thermodynamic variable. The overall packing entropy is shown to have two contributions: geometric, reflecting statistical weights of individual configurations, and topological, which corresponds to the number of topologically distinct states. Both of them are computed in the thermodynamic limit for isostatic and weakly underconstrained packings in 2D and 3D. The theory generalizes concepts of granular and glassy configurational entropies for the case of nonjammed systems. It is directly applicable to sticky colloids and predicts an asymptotic phase behavior of sticky spheres in the limit of strong binding.

Performance analysis and system parameters optimization of a packed bed solar thermal energy storage having spherical packing elements with pores

World energy demand is rising sharply from last few decades. Solar energy is capable to achieve this increasing energy demand as it is plenty in nature and pollution free by behavior. However, solar energy is associated with unsteady energy supply due to its intermittent nature. Therefore, solar energy systems require a storage unit for continuous supply of energy. The packed bed storage system (PBSS) is a feasible heat storage technique for solar thermal energy systems. Moreover, it is found most suitable technology for the applications of low temperature. The shape of the packing elements influences the performance of the PBSS and packed bed of spheres is found to have highest thermo-hydraulic efficiency among other shapes of packing elements investigated so far. Therefore, the influence of pores over the spherical surface on the performance of PBSS is analyzed and attempted to optimize the packing element parameters under this study. Pore depth to sphere diameter ratio, perforation index and pore to sphere diameter ratio are design parameters, whereas, temperature rise parameter and Reynolds number are considered as operating parameters under this study. The analysis is carried out in terms of thermo-hydraulic efficiency and the optimized value of investigated parameters under different operating conditions are presented. The present study may be useful for the designing of PBSS integrated with low temperature solar thermal applications.

Dionysian Hard Sphere Packings Are Mechanically Stable at Vanishingly Low Densities.

High strength-to-weight ratio materials can be constructed by either maximizing strength or minimizing weight. Tensegrity structures and aerogels take very different paths to achieving high strength-to-weight ratios but both rely on internal tensile forces. In the absence of tensile forces, removing material eventually destabilizes a structure. Attempts to maximize the strength-to-weight ratio with purely repulsive spheres have proceeded by removing spheres from already stable crystalline structures. This results in a modestly low density and a strength-to-weight ratio much worse than can be achieved with tensile materials. Here, we demonstrate the existence of a packing of hard spheres that has asymptotically zero density and yet maintains finite strength, thus achieving an unbounded strength-to-weight ratio. This construction, which we term Dionysian, is the diametric opposite to the Apollonian sphere packing which completely and stably fills space. We create tools to evaluate the stability and strength of compressive sphere packings. Using these we find that our structures have asymptotically finite bulk and shear moduli and are linearly resistant to every applied deformation, both internal and external. By demonstrating that there is no lower bound on the density of stable structures, this work allows for the construction of arbitrarily lightweight high-strength materials.