Percolation and jamming of linear $k$-mers on square lattice with defects: effect of anisotropy

TL;DR

This study investigates how defects and anisotropy influence the percolation and jamming behavior of rods on a square lattice, revealing critical defect concentrations and cluster properties for different models and rod lengths.

Contribution

It introduces analysis of defect effects on percolation and jamming of rods with anisotropic orientations using two models, highlighting new insights into cluster formation and critical thresholds.

Findings

Critical defect concentration decreases with rod length in RRSA model.

Cluster density at critical defect varies complexly with anisotropy and rod length.

Ordered systems form denser clusters compared to disordered ones.

Abstract

We study the percolation and jamming of rods ($k$-mers) on a square lattice that contains defects. The point defects are placed randomly and uniformly on the substrate before deposition of the rods. The general case of unequal probabilities for orientation of depositing of rods along different directions of the lattice is analyzed. Two different models of deposition are used. In the relaxation random sequential adsorption model (RRSA), the deposition of rods is distributed over different sites on the substrate. In the single cluster relaxation model (RSC), the single cluster grows by the random accumulation of rods on the boundary of the cluster. For both models, a suppression of growth of the infinite cluster at some critical concentration of defects $d_c$ is observed. In the zero defect lattices, the jamming concentration $p_j$ (RRSA) and the density of single clusters $p_s$ (RSC) decrease with increasing length rods and with a decrease in the order parameter. For the RRSA model, the value of $d_c$ decreases for short rods as the value of $s$ increases. For longer rods, the value of $d_c$ is almost independent of $s$. Moreover, for short rods, the percolation threshold is almost insensitive to the defect concentration for all values of $s$. For the RSC model, the growth of clusters with ellipse-like shapes is observed for non-zero 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 $k$. For disordered systems, the value of $p_s$ tends towards zero in the limits of the very long rods and very small critical concentrations $d_c \to 0$. In this case, the introduction of defects results in a suppression of rods 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.