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

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 $m \times m$ ($m \leqslant k$) 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 $k\rightarrow\infty$. 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 $k\geqslant 10m$. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.Comment: 11 pages, 9 figure

Percolation of the aligned dimers on a square lattice

Percolation and jamming phenomena are investigated for anisotropic sequential deposition of dimers (particles occupying two adjacent adsorption sites) on a square lattice. The influence of dimer alignment on the electrical conductivity was examined. The percolation threshold for deposition of dimers was lower than for deposition of monomers. Nevertheless, the problem belongs to the universality class of random percolation. The lowest percolation threshold (pc = 0.562) was observed for isotropic orientation of dimers. It was higher (pc = 0.586) in the case of dimers aligned strictly along one direction. The state of dimer orientation influenced the concentration dependence of electrical conductivity. The proposed model seems to be useful for description of the percolating properties of anisotropic conductors.Comment: 6 pages, 9 figures, submitted to EPJ