Saturated packings of convex anisotropic objects under random sequential adsorption protocol

The paper presents an algorithm to generate two-dimensional, saturated random sequential adsorption packings of identical, unoriented, anisotropic convex shapes. The method consists in tracking regions where the next shape can be added. The algorithm was tested on packings built of spherocylinders and ellipses of different width-to-height ratio. Obtained packing fractions at saturation agree with earlier predictions using an estimation of the saturated packing fraction based on the kinetics of packing growth, but they are much more accurate. Additionally, we discuss how to increase packing generation speed using parallel computations.

Figure 1

Illustration of the generation of saturated packing of disks of a unit surface area using voxels of decreasing size. The gray rectangle represents the packing. Its size is $7\times 7$. Disks of radius $\sqrt{1/\pi }\approx 0.5642$ are red, and their exclusion zones are marked with black circles. Voxels approximate available space outside the exclusion zones. Due to periodic boundary conditions, some regions near boundaries, which seem to be uncovered, are inside exclusion zones of disks from the other side of a packing. The sizes of a single voxel in panels (a) and (b) are 0.25 and 0.125, respectively. Some voxels visible in panels (a) and (b) are filled with a new disk, while others disappear due to splitting and analysis. Panels (a) and (b) show nonsaturated packings containing 21 and 23 disks, respectively. Panel (c) shows saturated packing built of 24 disks where no active voxels are present.

Figure 2

Left panel: exclusion zone (the orange area) for a spherocylinder and a virtual spherocylinder rotated by $\pi /6$. Right panel: several exclusion zones for different orientations of a virtual spherocylinder.

Figure 3

The left panel shows exclusion zones for a spherocylinder for relative orientation of a virtual particle, $\alpha \in [0,\pi /3]$. Red and yellow regions correspond to exclusion zones for ${\alpha }_1=0$ and ${\alpha }_2=\pi /3$, respectively. The exclusion zone for $\alpha \in [0,\pi /3]$ is the intersection of these zones except for the two purple regions. One of them has been enlarged in the right panel.

Figure 4

The intersection of exclusion zones for an ellipse and $\alpha \in [0,\pi /3]$. The red region is the exclusion zone for ${\alpha }_1=0$, and the yellow region corresponds to ${\alpha }_2=\pi /3$. The exclusion zone for the whole range $\alpha \in [0,\pi /3]$ is the intersection of exclusion zones for ${\alpha }_1$ and ${\alpha }_2$, except for the two purple regions. One of them has been enlarged in the right panel.

Figure 5

Illustration of the generation of saturated packing of ellipses of unit surface area and a width-to-height ratio of $2:1$ (the lengths of their semiaxes are $\sqrt{2/\pi }\approx 0.7979$ and $1/\sqrt{2\pi }\approx 0.3989$) using the presented algorithm. Active voxels form dark regions. The gray rectangle represents the packing. Its size is $7\times 7$. Periodic boundary conditions are used. The spatial side lengths of voxels in panels (a), (b), and (c) are 0.25, 0,125, and 0.0625, respectively, Some voxels visible in panels (a), (b), and (c) are covered by a new ellipse, while others disappear due to splitting and analysis. Panels (a), (b), and (c) show nonsaturated packings containing 21, 24, and 24 ellipses, respectively. Panel (d) shows saturated packing built of 28 ellipses where no active voxels are present.

Figure 6

Dependence of the mean saturated packing fraction on shape anisotropy for ellipses and spherocylinders. Symbols represent data from numerical simulations, and solid lines are to guide the eye. The error bars (around 0.000 02) are smaller than the symbol size. The inset shows the same data near the maximum. Here the solid lines are quadratic fits: $0.5039+0.086221x-0.023213x^2$ and $0.5115+0.078371x-0.021517x^2$ for ellipses and spherocylinders, respectively. Parameter $x$ is the width-to-height ratio.

Figure 7

Packing generation speed measured in packings-per-minute dependence on the number of OpenMP threads. Symbols are data, the solid line is to guide the eye, and the dashed line is a linear scaling $0.088009+0.33883x$ fitted to data for the number of used threads less than 15. Here, variable $x$ denotes the number of threads. The data were obtained for packings of a surface ${10}^4$ containing ellipses of a width-to-height ratio of 2.0.

Figure 8

Packing generation speed measured in packings-per-minute dependence on packing size. Symbols are data, and the dashed line corresponds to the least-squares fit $28661/x$, where $x$ is a packing surface. The fit uses data obtained for $x>300$. The data were obtained using eight OpenMP threads for packings containing ellipses of a width-to-height ratio of 2.0.