Jamming and percolation of parallel squares in single-cluster growth model

Abstract

This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size $k\times k$ squares (E-problem) or a mixture of $k\times k$ and $m\times m$ ($m⩽k$) squares (M-problem). The larger $k\times k$ squares were assumed to be active (conductive) and the smaller $m\times m$ squares were assumed to be blocked (non-conductive). For equal size $k\times k$ squares (E-problem) the value of $p_j=0.638\pm 0.001$ was obtained for the jamming concentration in the limit of $k\to \infty$. This value was noticeably larger than that previously reported for a random sequential adsorption model, $p_j=0.564\pm 0.002$. It was observed that the value of percolation threshold $p_{\mathrm{c}}$ (i.e., the ratio of the area of active $k\times k$ squares and the total area of $k\times k$ squares in the percolation point) increased with an increase of $k$. For mixture of $k\times k$ and $m\times m$ squares (M-problem), the value of $p_{\mathrm{c}}$ noticeably increased with an increase of $k$ at a fixed value of $m$ and approached 1 at $k⩾10m$. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.

11 pages, 9 figures

References in corpus (2)

• Percolation of linear $k$-mers on square lattice: from isotropic through partially ordered to completely aligned state
• Random Loose Packing in Granular Matter