Random sequential adsorption of starlike particles

Random packing of surfaceless starlike particles built of 3 to 50 line segments was studied using random sequential adsorption algorithm. Numerical simulations allow us to determine saturated packing densities as well as the first two virial expansion coefficients for such objects. Measured kinetics of the packing growth supports the power law known to be valid for particles with a finite surface; however, the dependence of the exponent in this law on the number of star arms is unexpected. The density autocorrelation function shows fast superexponential decay as for disks, but the typical distance between closest stars is much smaller than between disks of the similar size, especially for a small number of arms.

Figure 1

The starlike particle built of 3, 8, 20, and 50 equal length arms.

Figure 2

Example packings of starlike objects built of 3, 8, 20, and 50 line segments.

Figure 3

Time derivative of the number of starlike particles in packing for stars composed of $3,6,20$, and 50 line segments. Inset shows the dependence of exponent in Eq. (2) on the number of arms.

Figure 4

The example of a single target, a place where the center of a subsequent disk can be put. In case of disk RSA, it is the small, black triangle-like shape between disks.

Figure 5

The dependence of exponent $\alpha$ from relation $w\left(h\right)\sim h^{\alpha }$ on $N$. Dots represent data obtained numerically. Dashed line is drawn to guide the eye. Insets show relation between target sizes and weights for two example cases of starlike particles: $N=3$ and $N=50$. Dots represent numerical data (each dot corresponds to one target) to which power law was fitted (solid lines).

Figure 6

Example targets for $N=3$ and $N=50$. Darker shade of gray corresponds to higher probability (black = 1, white = 0) of choosing particle orientation following its successful addition to the packing.

Figure 7

Graphical interpretation of $C_1$ and $C_2$ parameters for disks. Parameter $C_1$ is equal to the area blocked by a single particle (left). Parameter $C_2$ corresponds to the intersection of these areas for neighboring but not overlapping particles.

Figure 8

Parameters $C_1$ and $C_2$ of ASF Eq. (7). Dots correspond to fitted data, solid and dashed lines are to guide the eye, and dotted lines correspond to the value of $C_1^{\text{disk}}=4\pi \approx 12.57$ and $C_2^{\text{disk}}=6\pi \sqrt{3}\approx 32.65$ taken for disks of unit radius.

Figure 9

Surface density of particles in saturated packings for different number of arms. Dots represent the data from simulations. Solid line is drawn to guide the eye. Dotted line corresponds to the density of saturated packing containing disks ${\rho }_{\text{max}}^{\text{disk}}=0.547/\pi \approx 0.174$. Inset shows example dependence of number of adsorbed particles $n_{\text{max}}$ on surface side length $x$ for $N=6$. Dots represent the data, while the solid line is a fit: $n_{\text{max}}=0.5107x^2+0.158x-9.5$.

Figure 10

The density autocorrelation functions for starlike particles and disks. Solid lines correspond to density autocorrelation function for starlike particles having different number of arms. The dashed line corresponds to density autocorrelation of disks and is drawn as a reference level.

Figure 11

Example of two $N=3$ stars. The distance between them is $r$ and, in general, it can be smaller than the length of an arm.