Packing Of Equal Spheres Research Articles

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.

CFD-DEM numerical study on air impacted packing densification of equiaxed cylindrical particles

A CFD-DEM model was developed to reproduce the packing densification process of mono-sized equiaxed cylindrical particles under air impact. The effects of operating parameters on packing density were firstly studied. Then various microscopic properties of packing structures such as coordination number (CN), contact types, particle orientations, pore features were characterized and compared. And corresponding densification mechanisms were analysed based on particle motion behaviour, local structure evolution, and forces. Results indicate that the air impact can realize the packing densification of cylindrical particles under appropriate conditions. The pore size distribution in the packing of cylindrical particles shows a tail at larger pore sizes compared with that in the packing of equal spheres. Both the size and the sphericity of the pores decrease in the final dense packing; also, more surface-surface and less surface-edge contacts between two particles therein can be formed. More cylindrical particles tend to be in parallel or perpendicular contact with each other to form more stable local structures during air impact. Most particles at higher position move down (direction of gravity/air impact) with about one particle length during the densification process and most particles exhibit translational motion to realize the local rearrangement for pore filling through air impact induced inter-particle forces.

Densely packed tetrahedra clusters displaying diamond-like superstructures

Two special tetrahedra packing clusters were analytically calculated and discussed. Diamond and double-diamond, periodic superstructures emerge. Both structures exemplify an efficient, congruent tetrahedra space tiling strategy. The diamond superstructure exhibits a remarkable packing density of φ∼0.76 exceeding maximal density for the packing of equal spheres. The presented architectures display long range order with specific, repetitive structural motifs. Due to their simple tetrahedral symmetry, the investigated clusters have the potential to become interesting, unique sub-units both in construction technology and in self-assembled systems design. The modified structure of presented cluster can entirely tile three-dimensional Euclidean space ℝ3 through periodic tessellations. Moreover, detailed analysis of the cluster enables definition of a new plesiohedron.

Densification mechanism and microstructure characteristics of nano- and micro- crystalline alumina by high-pressure and low temperature sintering

Fully dense ceramics with retarded grain growth can be attained effectively at relatively low temperatures using a high-pressure sintering method. However, there is a paucity of in-depth research on the densification mechanism, grain growth process, grain boundary characterization, and residual stress. Using a strong, reliable die made from a carbon-fiber-reinforced carbon (Cf/C) composite for spark plasma sintering, two kinds of commercially pure α-Al2O3 powders, with average particle sizes of 220 nm and 3 μm, were sintered at relatively low temperatures and under high pressures of up to 200 MPa. The sintering densification temperature and the starting threshold temperature of grain growth (Tsg) were determined by the applied pressure and the surface energy relative to grain size, as they were both observed to increase with grain size and to decrease with applied pressure. Densification with limited grain coarsening occurred under an applied pressure of 200 MPa at 1050 °C for the 220 nm Al2O3 powder and 1400 °C for the 3 μm Al2O3 powder. The grain boundary energy, residual stress, and dislocation density of the ceramics sintered under high pressure and low temperature were higher than those of the samples sintered without additional pressure. Plastic deformation occurring at the contact area of the adjacent particles was proved to be the dominant mechanism for sintering under high pressure, and a mathematical model based on the plasticity mechanics and close packing of equal spheres was established. Based on the mathematical model, the predicted relative density of an Al2O3 compact can reach ∼80 % via the plastic deformation mechanism, which fits well with experimental observations. The densification kinetics were investigated from the sintering parameters, i.e., the holding temperature, dwell time, and applied pressure. Diffusion, grain boundary sliding, and dislocation motion were assistant mechanisms in the final stage of sintering, as indicated by the stress exponent and the microstructural evolution. During the sintering of the 220 nm alumina at 1125 °C and 100 MPa, the deformation tends to increase defects and vacancies generation, both of which accelerate lattice diffusion and thus enhance grain growth.

Applications of Centroid Fractional Coordinates in Locating Interstices in Close Packings of Equal Spheres

Applications of Centroid Fractional Coordinates in Locating Interstices in Close Packings of Equal Spheres

On Limits of Dense Packing of Equal Spheres in a Cube

We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous best-known arrangements were usually derived from a ccp by omission of a certain number of spheres without changing the initial structure. In this paper, we show that better arrangements exist for all $n\leq\lceil p^{3}/2\rceil-2$. We introduce an optimization method to reveal improvements of these packings, and present many new improvements for $n\leq4629$.

Experiments on densifying packing of equal spheres by two-dimensional vibration

Densification of mono-sized sphere packings using two-dimensional (2D) vibration was experimentally studied. The effects of vibration mode, amplitude and frequency, feeding method, and container size on packing density were systematically analyzed. Useful results were obtained.

Partition of Euclidean space into polyhedra induced by a small deformation of the densest lattice sphere packing

The number of combinatorially nonequivalent Dirichlet-Voronoi diagrams constructed for the centers of balls in the packing obtained from the densest lattice packing of equal spheres by a small displacement of the spheres is estimated.

Efficiency of the Conversion of Work of Drainage to Surface Energy for Sandstone and Carbonate

SummaryThe increase in surface energy resulting from drainage of a wetting phase from a porous medium is often equated to the work of displacement determined from the area under its capillary pressure curve. However, capillary pressure vs. saturation relationships are not reversible and do not represent quasistatic displacement. The increase in surface energy is less than the work done because of inherent capillary instabilities that are the basic cause of capillary pressure hysteresis. Nevertheless, relating the area under a capillary pressure curve to the thermodynamic work of displacement can be justified by interpreting the curve as a series of alternating isons (reversible displacements) and rheons (spontaneous redistribution at constant saturation). The efficiency of conversion of work to surface energy, Ed, depends on the increase in surface area that accompanies drainage. Surface areas of nonwetting phase/solid and nonwetting phase/wetting phase have been determined through displacement of a colored low-viscosity liquid resin that can be solidified so that thin sections reveal the distribution of phases and surfaces within the pore space of the rock. Two-dimensional images obtained from thin sections were analyzed using stereology to obtain estimates of saturations and interfacial areas in three dimensions. For drainage of Berea sandstone to 20% wetting-phase saturation, Ed was 36%, which was less than one-half of the efficiency of 85% for the same range of change in saturation determined previously for random packings of equal spheres. Values of Ed for the tested carbonate were approximately one-half of those for sandstone. The wide variation is explained in terms of a simple pore model that relates Ed to aspect ratio.

Influence of wettability on petrophysical properties during imbibition in a random dense packing of equal spheres

Wettability, defined in terms of the value of contact angle of wetting–nonwetting–solid phase interface, is an important feature governing flow and transport processes in porous media. In this work we investigate the influence of wettability on the capillary pressure vs. saturation curve and on relative permeability during imbibition. To describe imbibition, we propose a new way of modeling it, based upon a physically consistent dynamic criterion for the imbibition of an individual pore, originally suggested by Melrose. We illustrate this approach in a simple but physically representative porous medium, the “ideal soil” (a dense random packing of equal spheres). Complete knowledge of the pore space geometry allows making a priori predictions of pore-level processes, without invoking adjustable parameters. This approach allows modeling imbibition for different values of contact angle and different initial conditions (as different drainage endpoints). The simulation yields the pore-level configuration of both fluid phases at each value of saturation. Relative permeabilities are calculated by simulating steady flow through the pore space occupied by each phase. The geometric and topological features of the flow network are directly connected to the geometry of the model rock and distribution of liquid phases in it. The simulated imbibition capillary pressure curves and relative permeability to the wetting phase are very sensitive to the value of contact angle in the whole range of wetting phase saturation. At the same time, relative permeability to the non-wetting phase is not much influenced by the wettability. Predicted capillary pressures and relative permeabilities compare favorably with experimental data for a range of wettabilities, providing strong support for the dynamic imbibition criterion.

Prediction of imbibition from grain-scale interface movement

Processes in porous media governed by capillary forces, such as drainage or imbibition of wetting phase, are of great importance in different branches of soil science, petroleum engineering and hydrology. In this work we describe a new way of modeling imbibition by considering the movement and merger of interfaces at the pore level. The model is based upon a physically consistent dynamic criterion for the imbibition of a single pore originally proposed by Melrose. The approach can be applied to any porous medium whose grain-scale geometry is well characterized, such as stochastic reconstructions and 3D microtomographic images. Here we illustrate it in a simple but physically representative porous medium, the “ideal soil” (a dense random packing of equal spheres). The spatial location of each sphere is accurately known. Delaunay tessellation of the sphere centers defines the location and geometry of each pore throat in the packing. We simulate imbibition as a quasi-static displacement, so that at any stage of the simulation the geometry of menisci in pore throats and of pendular rings at grain contacts can be determined from the Young–Laplace equation. This approach provides a natural and mechanistic way to account for different values of contact angle and different initial conditions (for example, different drainage endpoints). Predicted capillary pressure curves are consistent with experimental data presented in the literature. This provides strong (though indirect) support for the physically consistent dynamic criterion for the imbibition of a single pore. We illustrate two important and non-trivial consequences of the criterion that also agree with experiments: its implications when wettability (value of contact angle) of the medium varies, and its consistency with the Mayer–Stowe–Princen criterion for drainage at the level of individual pores. We then explore novel features of the criterion. For example, the percolation threshold for imbibition is a manifestation of two kinds of accessibility of the wetting phase, rather than one. Another feature is the strong coupling between the criterion and the spatial correlation of pore-level geometric features, which significantly affects the topology and connectivity of the wetting phase.

Unusually dense crystal packings of ellipsoids.

In this Letter, we report on the densest-known packings of congruent ellipsoids. The family of new packings consists of crystal arrangements of spheroids with a wide range of aspect ratios, and with density phi always surpassing that of the densest Bravais lattice packing phi approximately equal to 0.7405. A remarkable maximum density of phi approximately equal to 0.7707 is achieved for maximal aspect ratios larger than sqrt[3], when each ellipsoid has 14 touching neighbors. Our results are directly relevant to understanding the equilibrium behavior of systems of hard ellipsoids, and, in particular, the solid and glassy phases.

Dense Packings of Equal Spheres in a Cube

We describe an adaptation of the billiard algorithm for finding dense packings of equal spheres inside a domain of the euclidean space. In order to improve the convergence of this stochastic algorithm, we introduce systematic perturbations in it. We apply this perturbed billiard algorithm in the case of $n$ spheres in a cube and display all the optimal and best known packings up to $n=32$. We improve the previous record packings for all $11\leq n\leq 26$ except $n=13,14,18$. We prove the existence of the displayed packings for $n=11,12,15,17,20,21,22,26,32$, by constructing them explicitly. For example, the graph of the conjectured optimal packing of twenty-two spheres is composed of five octahedrons and four isolated points. We also conjecture that the minimum distance $d_n$ between spheres centers of the optimal packings is constant in the range $29\leq n \leq 32$.

Wetting phase connectivity and irreducible saturation in simple granular media

We present drainage simulations that allow trapping of wetting phase in a simple but nontrivial granular medium, a dense random packing of equal spheres. The basis for the simulations is a network model derived directly from the known locations and dimensions of pore space features. This provides a means of evaluating the morphology of trapped wetting phase. The possible morphologies depend on the assumed connectivity of the wetting phase. At one extreme, we assume that the entire wetting phase except for pendular rings is connected. At the opposite extreme, we illustrate a low level of connectivity by assuming that pendular rings are trapped as soon as the pores surrounding them are drained; any wetting phase not yet drained from pore throats connecting these pores is also assumed to be trapped. Finally we consider a set of criteria involving larger neighborhoods within the network, which allows trapping in individual pores. Irreducible wetting phase saturations obtained in the latter case agree with experimental data. The numbers of pendular rings and liquid bridges are also consistent with observations. Because the agreement does not involve adjustable parameters, we conclude that a relatively simple, local evaluation of trapping criteria can yield physically representative wetting phase configurations.

Contact thermal conductivity of a powder bed in selective laser sintering

Estimation of the temperature field in the powder bed in selective laser sintering process is a key issue for understanding the sintering/binding mechanisms and for optimising the technique. Heat transfer may be strongly affected by formation and growth of necks between particles due to sintering when the contact conductivity becomes predominant in the powder bed effective thermal conductivity. The necks often remain small as compared to the particle size. To calculate the effective contact conductivity of such structures a model of independent small thermal contacts is proposed. The conductivity of the considered cubic-symmetry lattices and the random packing of equal spheres depends on the three structural parameters: the relative density, the coordination number, and the contact size. The present model agrees with the known numerical calculations in the range of contact radius to particle radius ratio below 0.3. The strong dependence on the contact size is qualitatively confirmed by experimental data.

Densest Packing of Equal Spheres in Hyperbolic Space

Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m≥ 2 , and prove that for all but countably many radii, optimally dense packings must have low symmetry.

Polymorphism in Multisymmetric Close Packings of Equal Spheres on a Spherical Surface

Locally optimum, high-symmetry solutions are sought to the Tammes problem: to determine the maximum angular diameter of n equal circles, which can be packed on a sphere without overlapping. By using triangular surface lattices, in the range 1 ≤ n ≤ 500, those values of n are determined, for which the circles can be close packed in arrangements different from each other with tetrahedral, octahedral, and icosahedral symmetry. Tables are given of the results.

Transport Properties of Reconstructed Alumina and Vycor Membranes

ABSTRACTAim of the present work is the investigation of the transport properties (Knudsen and molecular diffusivity, permeability) of 3-dimensional binary domains that represent two model mesoporous materials, namely Vycor glass and alumina membrane. A process-based technique producing a random packing of equal spheres is used for the reconstruction of the alumina membrane, while a stochastic reconstruction procedure is employed in the case of Vycor. The comparison between computed and reported transport coefficients confirms that the random sphere pack represents quite well the porous structure of the alumina membrane, while the stochastic reconstruction technique is sufficient for an accurate representation of the porous matrix of Vycor glass.

Realistic random sphere pack model for the prediction of relative permeability curves

A three-dimensional network, composed of convergent–divergent flow channels has been employed for the simulation of the relative permeability behavior of a random packing of equal spheres. The relative permeability is calculated as a function of the amount of the sorbed vapor volume, which is the summation of the volume due to multilayer adsorption and capillary condensation. The sizes of the flow channels are determined on the basis of the structural characteristics of the sphere pack. The effect of adsorption–desorption hysteresis on the relative permeability is examined. Results are given for the conditions of vapor sorption of carbon tetrachloride and pentane in different porosities.

Packing fraction and measures of disorder of ultradense irregular packings of equal spheres. II. Transition from dense random packing

Close packings of equal spheres with densities, ϕ, from 0.65 to 0.7405 were obtained from perturbed hexagonal close packing (h.c.p.) using the force-biased algorithm (FBA). Similarly, densities from 0.65 to 0.72 were obtained from face-centered cubic (f.c.c.) packing, completing our earlier study (0.70-0.7405). The disorder of these packings was measured using several transportation metrics, μ and the average local angular disorder, θ For slight or moderate perturbations of either lattice, both μ and θ were approximately constant for almost the whole range of ϕ, but order was re-established much more easily from slightly perturbed h.c.p. Densities from 0.62 to 0.71 were obtained from sets of random points. The disorder, ϕ for these packings decreased steadily with ϕ. The transition from random to ordered packing is frustrated by the occurrence of both f.c.c. and (preferentially) h.c.p. fragments and by the varied orientation of these fragments. Transportation metrics have limited utility as measures of disorder, but θ is versatile and useful.