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Four balls represent a tetrahedron that is the simplest body in 3D space (simplex). In 2D three disks play the same role forming a triangle, in 4D five hyperspheres represent a four-dimensional simplex. First I would like to discuss that the features of the perfect simplices are the determinant factor in formation of dense packings of spheres. Second I will demonstrate that the structure of packings can be investigated using decomposition of a system of particles into simplices.
1. Perfect simplices. A simplex of perfect shape corresponds to the densest local configuration of balls (disks). A feature of the perfect 3D simplex (tetrahedron) is the irrationality of the dihedral angle. It is a fundamental difference between 3D and 2D where the similar angle is an integer part of the full angle. Just this feature of 3D is the origin of many well-known problems discussed in science, e.g. the Kepler conjecture, the existence of simple glasses and supercooled liquids, the phenomenon of the maximal (Bernal) density for non-crystalline packing. Actually, it follows from the contradiction between the densest local order and the global structure. What is the densest packing of the perfect tetrahedral configurations of balls? Starting from one tetrahedral configuration one can add a new ball to the hole between three balls. As a result a new tetrahedron configuration is settled to the initial one. In this way a new tetrahedron is added to an "open" face of the initial tetrahedron [1]. This procedure can be continued up to exhaustion of all open faces on the produced cluster. Recently such "a saturated polytetrahedron" was generated [2]. It this case the balls are distributed in space homogeneously with packing fraction 0.435. Its radial distribution function demonstrates the features of "icosahedral local order" [3] in spite of the icosahedron not being a principal element of such packing. There is an interesting question, maybe the saturated tetrahedron can play a role of an idealized model of noncrystalline packing (or simple liquids), like the ideal lattice is a mathematical model for crystalline solids? 2. Distorted simplices. The perfect simplicial configurations are absent in real packings. However dense packings contain a sufficient fraction of the configurations with shape close to the perfect tetrahedron. One can think about a class of "good tetrahedra" (simply - "class of tetrahedra"). Two questions spring up on this way: (a) how to divide a system of spheres into the simplicial fours of spheres? (b) what is the criterion for assignment of the simplices to the class of tetrahedra? The answer to the first question is known. Delaunay tessellation presents a decomposition into the simplices (Delaunay simplices) [4]. The second question is not so principal. Selection of a discrete class from a continuum is always conventional. However the experience says that there are reasonable ways to estimate the proximity of simplex shape to a given perfect shape [5]. Dense disordered packings of 3D balls have a polytetrahedral structure, i.e., contain clusters of Delaunay simplices of the class of tetrahedra, adjacent by faces [6]. Polytetrahedral clusters grow with increasing packing density and join in the percolation cluster at the Bernal density (0.64) [7]. Note that they contain only 1/3 of the total amount of simplices of the model. What is the nature of the other simplices which are not included in the class of tetrahedra? 3. Dimensionality does matter. Perfect triangles cover plane. It means that the "genuine" disordered phase cannot exist in 2D. In particular, a phenomenon like the maximal Bernal density in 3D is impossible in 2D. What about 4D ? References 1. A. H. Boerdijk, Some remarks concerning close packing of equal spheres, Philips Res. Rep. 7 (1953), 303-313. 2. Nikolai Medvedev, Ekaterina Pilyugina "Three-dimensional packing of perfect tetrahedra". in book "Voronon's Impact on Modern Science", Book 4, Vol. 2: Proceedings of the 5th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD-2008), the Institute of Mathematics, Nat. Acad. Sci. Ukraine, Ed.Kokichi Sugihara and Deok-Soo Kim. pp 144-156, 2008. 3. F. C. Frank, Supercooling of liquids, Proc. R. Soc. Lond. Ser. A 215 (1952) 43-46. P. Ganesh and M. Widom, Signature of nearly icosahedral structures in liquid and supercooled liquid copper, Phys. Rev. B 74 (2006), 134205. 4. Okabe, A., Boots, B., Sugihara, K., Chiu, S.: Spatial tessellations - concepts and applications of Voronoi diagrams. Wiley, New-York (2000). Medvedev, N.N.: Voronoi-Delaunay method for non-crystalline structures. SB Russian Academy of Science, Novosibirsk (2000) (in Russian). 5. Alexey V. Anikeenko, Marina L. Gavrilova, Nikolai N. Medvedev. "Shapes of Delaunay Simplexes and Structural Analysis of Hard Sphere Packings". in book GeneralizedVoronoi Diagram: A Geometry-Based Approach to Computational Intelligence. pp 13-45. In serial Studies in Computational Intelligence,Volume 158. Editor-in-Chief Prof. Janusz Kacprzyk, 2008 Springer-Verlag Berlin Heidelberg. 6. A.V. Anikeenko and N. N. Medvedev. Polytetrahedral Nature of the Dense Disordered Packings of Hard Spheres. PRL 98, 235504 (2007). 7. A.V. Anikeenko, N.N. Medvedev, "Structure of hard sphere packings near Bernal density", Journal of Structural Chemistry, Volume 50, Number 4, pp. 761-768, 2009. |
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