Disk Packings Research Articles

Nearest-neighbor statistics for packings of hard spheres and disks

The probability of finding a nearest neighbor at some radial distance from a given particle in a system of interacting particles is of fundamental importance in a host of fields in the physical as well as biological sciences. A procedure is developed to obtain analytical expressions for nearest-neighbor probability functions for random isotropic packings of hard D-dimensional spheres that are accurate for all densities, i.e., up to the random close-packing fraction. Using these results, the mean nearest-neighbor distance \ensuremath{\lambda} as a function of the packing fraction is computed for such many-body systems and compared to rigorous bounds on \ensuremath{\lambda} derived here. Our theoretical results are found to be in excellent agreement with available computer-simulation data.

Patterns of broken symmetry in the impurity-perturbed rigid-disk crystal

As a new example of spontaneous pattern formation in many-body systems, we examine the collective means by which a close-packed disk crystal reacts to the presence of a single oversized impurity disk. Computer simulation has been used for this purpose; it creates the jammed impurity-containing packings by a kinetic particle-growth algorithm. Hexagonal primitive cells with periodic boundary conditions were employed, and the “natural” number 3n 2 of disks (including the impurity) ranged upt to 10,800. For impurity diameter 1.2 times that of the other disks, the patterns of observed crystal perturbation displayed several remarkable features. Particle displacements relative to the unperturbed triangular crystal possess local irregularity but long-range coherence. The symmetry of the coherent patterns preserved that of the hexagonal cell for rapid growth, but was lower for slower growth. The final jammed packings contain “rattler” disks of the sort known to apper in random disk packings. Finally, the area increase induced by the presence of a fixed-size impurity appears to grow without bound as the system size (i.e., 3n 2) itself increases.

Comment on “Particle Size Segregation by Shaking in Two-Dimensional Disk Packings” by R. Jullien et al

In relation to size segregation in shaken granular materials we highlight the important role of non-sequential particle dynamics and the complex nature of the coupling between the driving forces and the granular structures. We also discuss recent observations of two-dimensional shaking-induced size segregation.

Dense Packings of Equal Disks in an Equilateral Triangle

Previously published packings of equal disks in an equilateral triangle have dealt with up to 21 disks. We use a new discrete-event simulation algorithm to produce packings for up to 34 disks. For each $n$ in the range $22 \le n \le 34$ we present what we believe to be the densest possible packing of $n$ equal disks in an equilateral triangle. For these $n$ we also list the second, often the third and sometimes the fourth best packings among those that we found. In each case, the structure of the packing implies that the minimum distance $d(n)$ between disk centers is the root of polynomial $P_n$ with integer coefficients. In most cases we do not explicitly compute $P_n$ but in all cases we do compute and report $d(n)$ to 15 significant decimal digits. Disk packings in equilateral triangles differ from those in squares or circles in that for triangles there are an infinite number of values of $n$ for which the exact value of $d(n)$ is known, namely, when $n$ is of the form $\Delta (k) := \frac{k(k+1)}{2}$. It has also been conjectured that $d(n-1) = d(n)$ in this case. Based on our computations, we present conjectured optimal packings for seven other infinite classes of $n$, namely \begin{align*} n & = & \Delta (2k) +1, \Delta (2k+1) +1, \Delta (k+2) -2 , \Delta (2k+3) -3, \\ && \Delta (3k+1)+2 , 4 \Delta (k), \text{ and } 2 \Delta (k+1) + 2 \Delta (k) -1 . \end{align*} We also report the best packings we found for other values of $n$ in these forms which are larger than 34, namely, $n=37$, 40, 42, 43, 46, 49, 56, 57, 60, 63, 67, 71, 79, 84, 92, 93, 106, 112, 121, and 254, and also for $n=58$, 95, 108, 175, 255, 256, 258, and 260. We say that an infinite class of packings of $n$ disks, $n=n(1), n(2),...n(k),...$, is tight , if [$1/d(n(k)+1) - 1/d(n(k))$] is bounded away from zero as $k$ goes to infinity. We conjecture that some of our infinite classes are tight, others are not tight, and that there are infinitely many tight classes.

Atomistic octagonal random-tiling model

A random-tiling model with octagonal symmetry is presented. The tiles are the square and various hexagons. When each vertex of the tiling is decorated by a disc, a valid disc packing is formed, which is an octagonal disc packing of maximum density under certain constraints. A simple inflation rule is given that produces one member of the random-tiling ensemble. The space group of this tiling is non-symmorphic. The projection description of this tiling involves a fractal acceptance domain with fourfold symmetry for the even nodes of a four-dimensional cubic lattice and the same acceptance domain rotated by pi /4 in perpendicular space for the odd nodes. Other tilings can be generated by an inflation rule with constrained randomness. Additional members of the random-tiling ensemble can be created from inflation-generated tilings by a set of update moves that rearrange the positions of the discs along closed loops. An update move does not always conserve the number of each kind of tile.

Experimental study of radical tesselations of assemblies of discs with size distribution

We had previously shown that topological and metric properties of 2d mosaics obtained from the Voronoitesselation of monosize packings of discs deviate from those of a totally random mosaic. Here, we describe a mosaic obtained from the radical tesselation of a two-size mixture of dises at different packing fractions. Two kinds of packings are considered, experimental (dises moving on an air table) and numerical. The deviations are even more striking, particularly at large packing fraction where neither Aboav's nor Lewis' law hold ; moreover, some distributions, such as the distribution of the number of sides of the cells, or distribution of the cell area, are split into two parts, each of them related to one species of dises. Finally, we consider polydisperse packings : as to their topological properties, the mosaics obtained from those packings obey approximately the laws of random mosaics but this is not so for their metric properties, which are still largely affected by steric exclusion.

Crystalline?amorphous interface packings for disks and spheres

We have employed a computer simulation method for uniaxial compression to create random, but spatially inhomogeneous, disk and sphere packings in contact with exposed faces of their own close-packed crystals. The disk calculations involved 7920 movable particles, while the sphere cases utilized over 4000 particles. Rates of compression to the jamming limit were varied over two orders of magnitude, and in three dimensions this produced a clear distinction between the cases of jamming against (001) and (111) faces of the sphere crystal. Specifically, epitaxial order next to the (001) face was markedly enhanced by slowing the compression; for the (111) face the epitaxial order was quite insensitive to the compression rate.

INVESTIGATION OF PERFORMANCE IMPROVEMENT OF INTERNAL COMBUSTION ENGINES USING NEW OIL ADDITIVES.

A new oil additive is tested to justify its influences on the performance of the Internal Combustion Engines (ICE). This additive belongs to the fluoroplastics group. It contains polytetrafluoroethylene (PTFE), Some forms of which were developed before in USA under the brand names Teflon, Fluon or Hostaflon. Also a high-polymer form of PTFE was used for years in advanced bearing, clutch disk and valve stem packing design in the automotive industry, however its use was limited to components of racing engines. In this research low polymer PTFE is tried as an additive to ICE oils, to produce a self-lubricating coating on the concerned engine components. Tests are carried out to investigate the effects of PTFE on the behavior of ICE. The results reflected an increase of about 9% in the compression pressure, 6% in the oil pressure, 8% in the engine power. The results also showed a decrease in fuel consumption of about 4%, a noticeable reduction in engine noise, no contamination effects and no change in oil viscosity. It is concluded that the PTFE low polymers are deposited on the metal surface of the internal parts of the engine forming a durable slippery coat which significantly reduces the friction, consequently the wear, and improve the tolerances.

A constraint on the random packing of disks

This paper addresses random packing of equal-sized disks in a manner such that no disk has a gap on its circumference large enough to accommodate an extra touching neighbour. This structure generalises the deterministic packing models discussed in classical geometry (Coxeter (1961), Hilbert and Cohn-Vossen (1952)). Relationships with the dual mosaic formed by joining the centres of touching disks are established. Constraints on the neighbourhood of disks and on the packing density are established.

A constraint on the random packing of disks

This paper addresses random packing of equal-sized disks in a manner such that no disk has a gap on its circumference large enough to accommodate an extra touching neighbour. This structure generalises the deterministic packing models discussed in classical geometry (Coxeter (1961), Hilbert and Cohn-Vossen (1952)). Relationships with the dual mosaic formed by joining the centres of touching disks are established. Constraints on the neighbourhood of disks and on the packing density are established.

Arrangement of cells in Voronoi tesselations of monosize packing of discs

Abstract We describe a two-dimensional mosaic obtained by the Voronoi tesselation of a monosize assembly of discs at different packing fractions. The experimental device (hard discs moving on an air table) produces, for every concentration of the discs, a succession of mosaics in statistical equilibrium, which constitutes a statistical ensemble. This ensemble is large enough for fluctuations from the most probable distributions to be negligible. Both topological and metric properties show deviations from those of a totally random mosaic. These deviations can be ascribed to steric exclusions. In particular, distributions of the numbers of sides, of the perimeters and of the areas of the polygons differ from those observed in biological celi assemblies. The Aboav law holds, but with a slope which can be as Iow as 4–66. The Lewis law is obeyed only for smali packing fractions. The variance of the distribution of the number n of polygon sides is a universal function of p 6, the probability of finding a six-si...

Flight in a packing of disks

For a packing in the plane consisting of open convex domains, none of which contains the originO, letλ(r) be the length of the shortest path which evades the domains and connectsO with a point at a distancer fromO. Upper and lower bounds are given for the supremum ofλ(r) taken over all packings whose members are domains of diameter at least 2.

Space filling tiling by random packing of discs

We study here a space filling tiling generated by random non-overlapping packing of discs. To place the nth disc we randomly choose a point in the empty pore space of the n−1 discs and place a disc with the centre at this point and touching the nearest disc. We measure the fractal dimension of the pore space left in the asymptotic limit for these space filling patterns.

The effects of Plateau borders in the two-dimensional soap froth. II. General simulation and analysis of rigidity loss transition

Abstract We have developed a computational simulation of the two-dimensional soap froth, including finite Plateau borders. As the gas fraction фis decreased, there is a loss of rigidity at approximately ф=0·84, at which point the structure is that of a dense random packing of hard discs with contact number equal to four. This result and the general properties of the system are further illuminated by various elementary models and analogies.

Effect of the longitudinal thermal conductivity of a disk packing on the efficiency of regenerators in air separators

Regenerators with disk packing are the most efficient heat-exchange assemblies for air separators. However, it should be noted that manufacuring disk packing is labor intensive. The simplest way to decrease the manufacturing labor in making the disks is to increase their height. To do this, a detailed analysis must be made on how various geometric parameters in the crimped aluminum strip and the plates themselves affect disk packing characteristics: the heat and mass transfer coefficients, the hydraulic resistance, and the longitudinal thermal conductivity. Here the least known characteristic of the packing is its longitudinal thermal conductivity. Detailed investigations [1-4] of the other parameters have shown that when the disk height is increased, the heat and mass evolution is not changed (if the geometric characteristics of the strip are retained), and the hydraulic reistance of the packing is reduced. However, increasing the height of the disks can lead to a large nonuniformity in the temperature difference between the gas and the packing over the height of the disk, as the packing temperature equilibrates due to the thermal conductivity of the disk. This increases underrecuperation and worsens the conditions for cleaning carbon dioxide from the nozzle in the lower regenerator zone. Therefore the data in [i-4] are insufficient for making a decision on increasing the height of the disks. Consequently, we should examine the effect of the disk height on the longitudinal thermal conductivity of the packing, and on the operational efficiency of the regenerators of air separators.

Intrachain dynamics and interchain structures of polymers: A comparison of polyacetylene, polyethylene, polyaniline, and poly(paraphenylene vinylene).

X-ray data for trans-polyacetylene (CH${)}_{\mathit{x}}$ in the range 129 KT513 K show that the a parameter expands twice as fast as b, accompanied by a 6\ifmmode^\circ\else\textdegree\fi{} increase in the setting angle. Polyethylene (${\mathrm{CH}}_{2}$${)}_{\mathit{x}}$ exhibits a similar behavior (but in the opposite sense), such that for both materials the interchain packing becomes closer to that of a two-dimensional triangular lattice at high temperature, as would be expected for close-packed disks. In contrast, poly(paraphenylene vinylene) and the emeraldine-base form of polyaniline both show small, nearly isotropic interchain thermal expansions, and the packing does not approach close packing of disks at high T. We suggest that for polymers consisting of relatively flat and rigid chains, the average interchain structure reflects the T-dependent interchain dynamics, specifically rigid-chain rotational modes about the chain axes. This effect is absent in materials for which ring torsion modes are the lowest-energy deformations. The coupling between rotational and translational degrees of freedom, and between intrachain dynamics and the average interchain structures, is discussed in light of the planar-rotor model of Choi et al. and the rotator phases observed in finite-length alkanes.

Stability of two-dimensional packings of disks built with ballistic deposition

Stability of two-dimensional packings of disks built with ballistic deposition

Disks vs. spheres: Contrasting properties of random packings

Collections of random packings of rigid disks and spheres have been generated by computer using a previously described concurrent algorithm. Particles begin as infinitesimal moving points, grow in size at a uniform rate, undergo energy-onconserving collisions, and eventually jam up. Periodic boundary conditions apply, and various numbers of particles have been considered (N⩽2000 for disks,N⩽8000 for spheres). The irregular disk packings thus formed are clearly polycrystalline with mean grain size dependent upon particle growth rate. By contrast, the sphere packings show a homogeneously amorphous texture substantially devoid of crystalline grains. This distinction strongly influences the respective results for packing pair correlation functions and for the distributions of particles by contact number. Rapidly grown disk packings display occasional vacancies within the crystalline grains; no comparable voids of such distinctive size have been found in the random sphere packings. “Rattler” particles free to move locally but imprisoned by jammed neighbors occur in both the disk and sphere packings.

Performance of Staggered Disc Packing

A new kind of packing is proposed for use in vacuum distillation. The packing consists of circular disks with holes at the center, which are placed at equal distances on vertical rods. The liquid descending the packing is slowed down, mixed and spread on each layer of discs, while the gas flows up in a zig-zag way. Preliminary experiemental results have been obtained in a pilot size distillation column though not yet under vacuum. These show that, at atmospheric pressure, pressure drop and mass transfer efficiency of the packing are better than for conventional packings and comparable to those for other modern packing types at low vapor rates.

Rigidity loss transition in a disordered 2D froth.

Disordered two-dimensional soap froth has been simulated, with a gas fraction \ensuremath{\varphi} less than unity. Upon decreasing \ensuremath{\varphi}, it is found that the system loses its rigidity at ${\mathrm{\ensuremath{\varphi}}}_{\mathit{c}}$=0.84. This value is identified with the dense random packing of hard disks. Results are presented for the variation of yield stress and shear modulus, the latter being tentatively related to elastic-network theory.