Random ensembles of particles with pentagonal symmetry: densification and properties

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Abstract

The paper studies the densities and statistical-geometric characteristics of random packings of regular pentagons on a plane. The initial ensemble was generated by random sequential adsorption (RSA). A packing densification algorithm is proposed, which is a modification of the Lubachevsky-Stillinger (LS) method. The final ensemble was obtained by stepwise increasing the linear dimensions of 2-D particles at a fixed density of the square “box”. It is shown that the packing density of the final ensemble for this algorithm is practically independent of the density of the initial ensemble (with a total number of particles of about 104 or more). The maximum packing density of the starting ensemble of regular pentagons obtained by the RSA method was 0.54306 ± 0.00220, which is in good agreement with the literature value of 0.54132. The highest (final) density achieved after compaction of the starting ensemble was 0.8381 ± 0.0020 for pentagons. This value is close to the value found by a similar algorithm for packing hard disks (0.84-0.86). The correlation functions of hard disks and pentagons demonstrate a number of common patterns. At the same time, the “crystallization” of the ensemble of hard disks at relatively high densities close to the maximum achieved is expressed more sharply. At the same time, the “peaks” of the correlation function for pentagons (compared to disks) are expected to have a smaller height and a larger width, a more complex structure. Ensembles of non-convex particles with pentagonal symmetry (such as five-pointed stars) demonstrate significantly lower packing densities and do not compact to partial “crystallization”. A relatively simple algorithm for compacting “starting” random packings of polygons, applied in the work, allows “compacting” two-dimensional ensembles of any polygons (without self-intersections). However, partial ordering and sufficiently high densities (corresponding to the beginning of “crystallization” of the ensemble) are achieved when using it only for convex polygonal particles.

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About the authors

A. B. Shubin

Institute of Metallurgy of the Ural Branch of the Russian Academy of Sciences

Author for correspondence.
Email: fortran@list.ru
Russian Federation, Ekaterinburg

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Supplementary files

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2. Fig. 1. Dependence of the packing density η on the maximum given number of iterations (K) on the last placed particle.

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3. Fig. 2. Initial (a) and final (b) ensembles of identical regular pentagons in a square box at initial ratio D / L ≈ 0.1.

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4. Fig. 3. Dependence of the number of displacement attempts M on the achieved packing density η.

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5. Fig. 4. Dependence of the maximum achieved packing density ηmax in the starting and final (jammed) packing on the given largest number of iterations (KRSA) per placed particle in the starting RSA ensemble.

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6. Fig. 5. Pairwise correlation functions of the pentagon ensemble for packing densities η = 0.445 (a), 0.650 (b), 0.751 (c), and 0.839 (d).

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7. Fig. 6. Radial distribution functions for the ensemble of hard discs at packing densities η = 0.518 (a), 0.650 (b), 0.750 (c) and 0.836 (d).

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8. Fig. 7. Correlation function G (x / d) for a random ensemble of regular pentagons at different packing densities η.

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9. Fig. 8. Dependence of the maximum achieved packing density on the ratio of the radii of the incircle and circumcircle for star-shaped figures with pentagonal symmetry.

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