Saturated random packing built of arbitrary polygons under random sequential adsorption protocol

Random packings and their properties are a popular and active field of research. Numerical algorithms that can efficiently generate them are useful tools in their study. This paper focuses on random packings produced according to the random sequential adsorption (RSA) protocol. Developing the idea presented by G. Zhang [Phys. Rev. E 97, 043311 (2018)], where saturated random packings built of regular polygons were studied, we create an algorithm that generates strictly saturated packings built of any polygons. Then, the algorithm was used to determine the packing fractions for arbitrary triangles. The highest mean packing density, $0.552814\pm 0.000063$, was observed for triangles of side lengths $0.63:1:1$. Additionally, microstructural properties of such packings, kinetics of their growth, as well as distributions of saturated packing fractions and the number of RSA iterations needed to reach saturation were analyzed.

Figure 1

Stages involved in generating an RSA packing of 2D disks. Red disks represent nonpenetrable particles. Yellow indicates regions where it is impossible to place the center of another disk due to intersection with the existing ones. Black square voxels approximate an area where placing of a subsequent disk is possible. In each subsequent stage, these voxels are divided into $2^d$ subvoxels to better approximate available space as well as are filled up with new disks. In panel (d), there are no black voxels; thus, the packing is saturated.

Figure 2

Example of a voxel (the black square) that would never be removed by the algorithm described in [13]. The solid rectangle is already part of the packing while the dashed one is the virtual one with the center inside the black square voxel. Rectangles partially overlap but their edges do not, according to the criterion (3).

Figure 3

An unavailable voxel that could not be identified as such using the original algorithm [13]. Blue pentagons are objects already inserted into the packing and the black pentagon represents the unavailable voxel.

Figure 4

Illustration of the two ways to add helper segments to an irregular pentagon to solve the problem detailed in the text.

Figure 5

Model of a triangle of side lengths ratio $a\le 1\le b$. Dashed lines are helper segments from the triangle mass center to its vertices.

Figure 6

Fragments of example saturated random packings of triangles for (a) $a=b=1.0$; (b) $a=0.63$, $b=1.0$; and (c) $a=1.0$, $b=1.31$.  

Figure 7

The mean saturated packing fraction for isosceles triangles. The parameter $x$ denotes the length ratio of the isosceles triangle base to its arm, e.g., $x=1$ corresponds to equilateral triangle. Dots are numerical results, and the solid line is to guide the eye. Statistical errors are smaller than the size of dots.

Figure 8

The mean saturated packing fraction for arbitrary triangles.

Figure 9

Density autocorrelation functions for packings built of three different triangles.

Figure 10

The dependence of saturated mean packing fraction on packing size. For all packing sizes except $S=90000$, packing fraction was calculated by averaging over 100 independent random packings. For $S=90000$, ${10}^4$ independent packings were used. Error bars correspond to the standard deviation of the mean packing fraction. Solid black line corresponds to $\theta (S=9000)=0.5258119$. Dashed blue lines correspond to the range of $\theta$ determined by Zhang [13].

Figure 11

The dependence of increments of packing fraction on dimensionless time for equilateral triangles. Black dots are measured data and red dashed line is a power-law fit $d\theta /dt=0.12728t^{-1.3387}$. Inset shows the value of parameter $d$ estimated from data obtained for times within $[0.01t,t]$. Red dashed line corresponds to $d=3$.

Figure 12

The dependence of the median of the number of iterations (expressed in dimensionless time) needed to generate saturated RSA packing built of equilateral triangles on packing size. Squares represent numerical data obtained from 100 packings. Solid line is a power fit: $M\left(t\right)=0.043177S^{3.0649}$.

Figure 13

Histograms of the number of iterations (expressed in dimensionless time) needed to generate saturated packing and the packing fraction (inset) obtained from studying ${10}^4$ packings built of equilateral triangles.