The effect of defects on the percolation of linear $k$-mers (particles occupying $k$ adjacent sites) on a square lattice is studied by means of Monte Carlo simulation. The $k$-mers are deposited using a random sequential adsorption mechanism. Two models $L_d$ and $K_d$ are analyzed. In the $L_d$ model it is assumed that the initial square lattice is nonideal and some fraction of sites $d$ is occupied by nonconducting point defects (impurities). In the $K_d$ model the initial square lattice is perfect. However, it is assumed that some fraction of the sites in the $k$-mers $d$ consists of defects, i.e., is nonconducting. The length of the $k$-mers $k$ varies from 2 to 256. Periodic boundary conditions are applied to the square lattice. The dependences of the percolation threshold concentration of the conducting sites $p_c$ vs the concentration of defects $d$ are analyzed for different values of $k$. Above some critical concentration of defects $d_m$, percolation is blocked in both models, even at the jamming concentration of $k$-mers. For long $k$-mers, the values of $d_m$ are well fitted by the functions $d_m\propto k_m^{-\alpha }-k^{-\alpha }$ ($\alpha =1.28\pm 0.01$ and $k_m=5900\pm 500$) and $d_m\propto {log}_{10}(k_m/k)$ ($k_m=4700\pm 1000$) for the $L_d$ and $K_d$ models, respectively. Thus, our estimation indicates that the percolation of $k$-mers on a square lattice is impossible even for a lattice without any defects if $k⪆6\times {10}^3$.

• Received 29 October 2014

DOI:https://doi.org/10.1103/PhysRevE.91.012109