Packing Of Uniform Spheres Research Articles

Local rotational symmetry in the packing of uniform spheres.

Local rotational symmetry (LRS) of a particulate system is important for understanding its structure and phase transition. However, how to properly characterize LRS for this system is still a challenge as the system normally includes both ordered and disordered local structures. Herein, based on the so-called common neighbour subcluster (CNS), we proposed a method to characterize the LRS of uniform spheres packings with the packing fraction ρ ranging within 0.20 and 0.74. It was found that different fold LRSs coexist in most packings, and their maximum degree increases at ρ < 0.64, except for the 2-fold LRS held by 6-sphere CNS that continuously increases to form the fcc crystal at ρ = 0.74. The overall LRS involving all the CNSs monotonically increases with two critical changes at ρ = (0.35-0.40) and 0.64; the evolution of individual LRSs held by specific CNS groups critically changes at ρ ≈ (0.35-0.40), 0.50, 0.55-0.60, and 0.64. The physics corresponding to these critical changes has also been discussed. The findings will significantly enrich the understanding of the structural symmetry of materials including atoms and particles.

Quasi-universality in the packing of uniform spheres under gravity

A hypothesis that packing fraction alone can be used to characterize the structure of a sphere packing, known as the quasi-universality in the literature, is tested. The analysis, conducted in terms of coordination number, radial distribution function, and structural properties from the Voronoi/Delaunay tessellation, is based on the packing results generated under different conditions, covering a wide packing fraction range. The results show strong similarities in these properties for a given packing fraction, indicating that although not generally valid, the quasi-universality approximately holds for the packing of spheres formed when the gravity is the driving force. The usefulness of this finding is also demonstrated through representative examples.

Analysis of bender element test interpretation using the discrete element method

While bender element testing is now well-established as a laboratory technique to determine soil stiffness, a robust technique to interpret the data remains elusive. A discrete element method (DEM) model of a face-centred cubic packing of uniform spheres was created to simulate bender element tests to investigate this test from a fundamental perspective. During the DEM simulations transmitter and receiver signals were recorded, analogous to the data available in laboratory tests, and these macro-scale data were supplemented with particle scale measurements (forces, stresses and displacements). A range of approaches previously applied in experimental and numerical studies were used to analyse the resulting data in both the time and frequency domains. The shortcomings in these approaches are clear from the differences in the resultant shear stiffness values and the frequency-dependent nature of the values. The particle-scale data enabled visualization of the passage of the wave through the sample, and it was found not to be possible to precisely link the arrival of the shear wave at the receiver and any of the previously proposed characteristic points along the signal recorded at the receiver. The most reliable determination of the shear wave velocity was obtained by applying a two-dimensional fast Fourier transform (2D FFT) to the data describing the velocity of the particles lying between the transmitter and receiver elements. Use of the DEM model and this 2D FFT approach facilitated the sensitivity of the system response to small variations in the interparticle force–displacement law (the contact model) to be established.

Structural evolution in the packing of uniform spheres.

Structural analysis is very important to understanding the physics of atomic or particle systems of various types. However, properly characterizing the structures at different packing fraction ρ is still a challenge. Here we analyze the local structure, in terms of the so-called common-neighbor-subcluster (CNS), of sphere packings with ρ ∈ (0.2, 0.74). We show that although complicated in structure, there are totally 39 kinds of CNSs of which 12 are dominant. The evolution of these CNSs with the increase of ρ is quantified, and the rules governing the evolution are explored. The results are found to be useful in constructing a comprehensive picture about the critical states and their transition in sphere packing.

Experimental study on the packing of uniform spheres under three-dimensional vibration

Densification of mono-sized sphere packings under three-dimensional (3D) vibration is experimentally studied. The effects of an operational condition, such as vibration amplitude and frequency and feeding method, on packing density are systematically investigated. The results indicate that the dense packings can be achieved by proper control of both vibration amplitude and frequency. The feeding method plays an important role in densification. Higher packing densities can be obtained when the number of particles fed per batch is less than one layer. Packing density decreases with increasing number of particles fed per batch, but keeps constant when the number of particles per batch is larger than three layers. Through the extrapolation on packing density obtained from different sized containers, the maximum packing density is 0.69 for the total feeding method and 0.74 for the batch-wise feeding under the present experimental condition. The formation of ordered structure is discussed based on the particle interlayer diffusion.

A simulation study of the effects of dynamic variables on the packing of spheres

The packing of uniform spheres has been studied by means of Discrete Element Method with special reference to variables affecting the packing dynamics, with the results analysed in terms of packing density, radial distribution function (RDF) and coordination number. It is shown that packing density increases with dropping height and restitution coefficient, and decreases with deposition intensity and friction coefficient, which is consistent with previous experimental findings. Both RDF and coordination number distribution vary with these variables, in line with packing density. For a packing of high density, it has a clear split second peak in its RDF, like that observed for the dense random packing. However, as packing density decreases, the first component of the split second peak will gradually vanish, giving an RDF more comparable to those observed in a sequential addition simulation. Mean coordination number can be correlated with packing density; it increases with dropping height and restitution coefficient, and decreases with deposition intensity and friction coefficient.

From Cannon Balls to Yeast Cells

In connection with the News Focus article “Random packing puts mathematics in a box” (Charles Seife, 17 Mar., p. [1910][1]) about “random” versus “close” packing of uniform spheres, it is worth mentioning that English mathematician Thomas Harriot (1560-1621) performed some of the first packing calculations. His correspondence with Johannes Kepler ([1][2]) subsequently led to Kepler's famous “conjecture” on the most efficient density, which is close to 74%. Harriot was intellectual advisor to Sir Walter Raleigh, who was at the time most interested in stacking cannonballs on his ships. The same principles hold for all sizes of spheres. For example, mature yeast cells have a packing density of about 78% ([2][3]); the small difference from theory is primarily attributed to a slight departure from spherical shape. 1. [↵][4]T. C. Hales, [http://www.math.lsa.umich.edu/∼hales/countdown/][5], and references therein. 2. [↵][6]1. W. N. Arnold, 2. J. S. Lacey , J. Bacteriol. 131, 564 (1977). [OpenUrl][7][Abstract/FREE Full Text][8] [1]: /lookup/doi/10.1126/science.287.5460.1910c [2]: #ref-1 [3]: #ref-2 [4]: #xref-ref-1-1 View reference 1 in text [5]: http://www.math.lsa.umich.edu/~hales/countdown/ [6]: #xref-ref-2-1 View reference 2 in text [7]: {openurl}?query=rft.jtitle%253DJournal%2Bof%2BBacteriology%26rft.stitle%253DJ.%2BBacteriol.%26rft.aulast%253DArnold%26rft.auinit1%253DW.%2BN.%26rft.volume%253D131%26rft.issue%253D2%26rft.spage%253D564%26rft.epage%253D571%26rft.atitle%253DPermeability%2Bof%2Bthe%2BCell%2BEnvelope%2Band%2BOsmotic%2BBehavior%2Bin%2BSaccharomyces%2Bcerevisiae%26rft_id%253Dinfo%253Apmid%252F407215%26rft.genre%253Darticle%26rft_val_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Ajournal%26ctx_ver%253DZ39.88-2004%26url_ver%253DZ39.88-2004%26url_ctx_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Actx [8]: /lookup/ijlink/YTozOntzOjQ6InBhdGgiO3M6MTQ6Ii9sb29rdXAvaWpsaW5rIjtzOjU6InF1ZXJ5IjthOjQ6e3M6ODoibGlua1R5cGUiO3M6NDoiQUJTVCI7czoxMToiam91cm5hbENvZGUiO3M6MjoiamIiO3M6NToicmVzaWQiO3M6OToiMTMxLzIvNTY0IjtzOjQ6ImF0b20iO3M6MjM6Ii9zY2kvMjg4LzU0NjMvNTUuMi5hdG9tIjt9czo4OiJmcmFnbWVudCI7czowOiIiO30=