Random sequential adsorption of aligned rectangles with two discrete orientations: Finite-size scaling effects

Abstract

We study saturated packings produced according to random sequential adsorption (RSA) protocol built of identical rectangles deposited on a flat, continuous plane. An aspect ratio of rectangles is defined as the length-to-width ratio, $f=l/w$. The rectangles have a fixed unit area (i.e., $l \times w=1$), and therefore, their shape is defined by the value of $f$ ($l=\sqrt{f}$ and $w=1/\sqrt{f}$). The rectangles are allowed to align either vertically or horizontally with equal probability. The particles are deposited on a flat square substrate of side length $L$ (measured in units of particle length, $L \in [20, 1000]$) and periodic boundary conditions are applied along both directions. The finite-size scaling effects are characterized by a scaled anisotropy defined as $\alpha = l/L = \sqrt{f}/L$. We showed that the properties of such packings strongly depend on the value of aspect ratio $f$ and the most significant scaling effects are observed for relatively long rectangles when $l\ge L/2$ (i.e. $\alpha \ge 0.5$). It is especially visible for the mean packing fraction as a function of the scaled anisotropy $\alpha$. The kinetics of packing growth for low to moderate rectangle anisotropy is to be governed by $\ln t/t$ law, where $t$ is proportional to the number of RSA iterations, which is the same as in the case of RSA of parallel squares. We also analyzed global orientational ordering in such packings and properties of domains consisting of a set of neighboring rectangles of the same orientation, and the probability that such domain forms a percolation.