Paris car parking problem for partially oriented discorectangles on a line

The random sequential adsorption (RSA) of identical elongated particles (discorectangles) on a line (“Paris car parking problem”) was studied numerically. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The possible orientations of the discorectanles were restricted between $\theta \in [-{\theta }_{\text{m}};{\theta }_{\text{m}}]$ while the aspect ratio (length-to-width ratio) for the discorectangles was varied within the range $\varepsilon \in [1;100]$. Additionally, the limiting case $\varepsilon =\infty$ (i.e., widthless sticks) was considered. We observed that the RSA deposition for the problem under consideration was governed by the formation of rarefied holes (containing particles oriented along a line) surrounded by comparatively dense stacks (filled with almost parallel particles oriented in the vertical direction). The kinetics of the changes of the order parameter and the packing density are discussed. Partial ordering of the discorectangles significantly affected the packing density at the jamming state, ${\phi }_{\text{j}}$, and shifted the cusps in the ${\phi }_{\text{j}}\left(\varepsilon \right)$ dependencies. This can be explained by the effects on the competition between the particles' orientational degrees of freedom and the excluded volume effects.

Figure 1

A description of the RSA packing of discorectangles on a line. Intersections of the particles are forbidden. Each deposited particle covers a distance $a$ on the line. Here $L$ is the total length of the line, and $r$ is the distance between the particle centers.

Figure 2

Examples of packing patterns of infinitely thin particles (sticks) with $\varepsilon =\infty$ at different values of the preassigned order parameter $S_0$. Length of the line is $L=2^{15}$ and time is $t={10}^{10}$.

Figure 3

The order parameter, $S$, and mean number density, ${\rho }_0$, versus the deposition time, $t$, at a zero value of the preassigned order parameter, $S_0=0$. The data are presented for deposition of sticks ($\varepsilon =\infty$) on the line with length $L$. Inset presents the limiting order parameter $S_{\text{f}}$ (at $t\to \infty$) versus the inverse length of the line $1/L$.

Figure 4

Order parameter, $S$ (a), and mean number density, ${\rho }_0$ (b), versus the deposition time, $t$, at different preassigned order parameters, $S_0$. The data are presented for the deposition of sticks ($\varepsilon =\infty$) on a line with length $L=2^{15}$. Inset in (b) presents the enlarged portion of the ${\rho }_0\left(t\right)$ dependencies.

Figure 5

The density $g_2\left(r\right)$ (a) and orientation $s_2\left(r\right)$ (b) pair correlation functions for different values of the preassigned order parameter, $S_0$. The data are presented for the deposition of sticks ($\varepsilon =\infty$) on a line with length $L=2^{15}$, and a deposition time $t={10}^6$.

Figure 6

Differential distribution functions of number density, $f$, versus the reduced number density in cells, ${\rho }^{*}=\rho /{\rho }_0$ at different preassigned order parameters, $S_0$. The data are presented for the deposition of sticks ($\varepsilon =\infty$) on a line with length $L=2^{15}$.

Figure 7

Order parameter, $S$, versus the deposition time, $t$, for disordered RSA packings ($S_0=0$). The data are presented for the deposition of discorectangles with aspect ratio $\varepsilon$ on a line with length $L=2^{15}$.

Figure 8

Final order parameter in the jamming state, $S_{\text{f}}$, versus the aspect ratio, $\varepsilon$, at different values of the preassigned order parameter, $S_0$ (a), and $S_{\text{f}}$ versus $S_0$ at different values of $\varepsilon$. The data are presented for the deposition of discorectangles on a line with length $L=2^{15}$.

Figure 9

Packing density, $\phi$, versus the deposition time, $t$, for disordered RSA packings ($S_0=0$). The time derivatives $d\phi /d{log}_{10}t$ were calculated to estimate the characteristic deposition times $\tau$ and ${\tau }_s$. The data are presented for the deposition of discorectangles with aspect ratio $\varepsilon$ on a line with length $L=2^{15}$.

Figure 10

Characteristic RSA deposition times $\tau$ and ${\tau }_s$ versus the aspect ratio of discorectangles, $\varepsilon$, for disordered RSA packings ($S_0=0$) (a), and $\tau$ versus $S_0$ at fixed value of $\varepsilon =10$ (b). The data are presented for the deposition of discorectangles with aspect ratios $\varepsilon$ on a line with length $L=2^{15}$.

Figure 11

Packing density at jamming state, ${\phi }_{\text{j}}$, versus the discorectangle aspect ratio, $\varepsilon$, at different values of the preassigned order parameter, $S_0$, (a) and enlarged portion of the ${\phi }_{\text{j}}\left(\varepsilon \right)$ dependencies for $1⩽\varepsilon ⩽5$ (b). For $S_0=0$, the maximum ${\phi }_{\text{j}}=0.7822\pm 0.004$ was observed at $\varepsilon \approx 1.46$. Dashed horizontal lines correspond to the jamming limit of disks (${\phi }_{\text{j}}=C_{\text{R}}=0.7476\cdots$ at $\varepsilon \approx 1.0$). The open squares in (b) correspond to the data obtained in [27]. The data are presented for deposition on a line with length $L=2^{15}$.

Figure 12

Examples of the density $g_2\left(r\right)$ (a) and orientation $s_2\left(r\right)$ (b) pair correlation functions with fragments of illustrative RSA disordered packings of discorectangles ($S_0=0.0$) at the jamming concentrations for different aspect ratios $\varepsilon =2$ (a), $\varepsilon =5$ (b), $\varepsilon =20$ (c), and $\varepsilon =100$ (d). The data are presented for deposition on a line with length $L=2^{15}$.