Percolation and jamming of lineark-mers on a square lattice with defects: Effect of anisotropy

Abstract

Using the Monte Carlo simulation, we study the percolation and jamming of oriented linear $k$-mers on a square lattice that contains defects. The point defects with a concentration $d$ are placed randomly and uniformly on the substrate before deposition of the $k$-mers. The general case of unequal probabilities for orientation of depositing of $k$-mers along different directions of the lattice is analyzed. Two different relaxation models of deposition that preserve the predetermined order parameter $s$ are used. In the relaxation random sequential adsorption (RRSA) model, the deposition of $k$-mers is distributed over different sites on the substrate. In the single-cluster relaxation (RSC) model, the single cluster grows by the random accumulation of $k$-mers on the boundary of the cluster (Eden-like model). For both models, a suppression of growth of the infinite (percolation) cluster at some critical concentration of defects $d_c$ is observed. In the zero-defect lattices, the jamming concentration $p_j$ (RRSA model) and the density of single clusters $p_s$ (RSC model) decrease with increasing length $k$-mers and with a decrease in the order parameter. For the RRSA model, the value of $d_c$ decreases for short $k$-mers $\left(k<16\right)$ as the value of $s$ increases. For $k=16$ and 32, the value of $d_c$ is almost independent of $s$. Moreover, for short $k$-mers, the percolation threshold is almost insensitive to the defect concentration for all values of $s$. For the RSC model, the growth of clusters with ellipselike shapes is observed for nonzero values of $s$. The density of the clusters $p_s$ at the critical concentration of defects $d_c$ depends in a complex manner on the values of $s$ and $k$. An interesting finding for disordered systems $(s=0)$ is that the value of $p_s$ tends towards zero in the limits of the very long $k$-mers, $k\to \infty$, and very small critical concentrations $d_c\to 0$. In this case, the introduction of defects results in a suppression of $k$-mer stacking and in the formation of empty or loose clusters with very low density. On the other hand, denser clusters are formed for ordered systems with $p_s\approx 0.065$ at $s=0.5$ and $p_s\approx 0.38$ at $s=1.0$.