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

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 $s=1.0$.

Figure 1

Examples of the jamming states (top row) and the percolation clusters (bottom row) of anisotropically deposited $k$-mers on a square lattice with defects for (a) $s=0.25$, (b) $s=0.5$, and (c) $s=1$. Both pictures (top and bottom) in each pair (a)–(c) comes from the same Monte Carlo run. The lattice size is $1024\times 1024,k=8$, and a fragment of the lattices with $128\times 128$ sites is shown. The concentration of defects on the lattice is 0.09. The horizontal $k$-mers are shown in red, vertical $k$-mers are shown in blue, $k$-mers not belonging to the percolation cluster are shown in the same colors but with a different hue, empty sites are shown in white, and defects on the lattice are shown in black.

Figure 2

Probability curves for $k=4,s=0.75$, and the criterion and: (a) $R\left(p\right)$ and (b) $R\left(d\right)$. The insets show the scaling. The statistical error is smaller than the marker size.

Figure 3

Examples of the clusters in the absence (top row) and in the presence (bottom row) of defects for the RSC model at different values of the order parameter: (a) $s=0.0$, (b) $s=0.5$, and (c) $s=1$. The lattice size is $1024\times 1024,k=8$, and a fragment of the lattices with $128\times 128$ sites is shown. The concentration of defects on the lattice $d$ is near the critical $d_c$: 0.215 ($s=0.0$), 0.186 ($s=0.5$), and 0.134 ($s=1.0$). The horizontal $k$-mers are shown in red, vertical $k$-mers are shown in blue, empty sites are shown in white, and defects on the lattice are shown in black.

Figure 4

Probability curves $R_b\left(d\right)$ for the RSC model with (a) $k=2$ and (b) $k=32$. The data are presented for disordered systems $s=0$. The statistical errors are smaller than the marker size. The lines are provided simply as a guide for the eye.

Figure 5

Concentration of defects $d_c$ versus the scaled size of the system $L^{1/\nu }$ for the RSC model ($k=2$) (a) at different values of probability $R_b$ ($s=0$) and (b) at different values of the order parameter $s$ ($R_b=0.5$). Here $\nu =4/3$ is the critical exponent of the correlation length [12].

Figure 6

Jamming concentration $p_j$ versus the defect concentration $d$ for different values of $k$ and a fixed order parameter $s=0.5$. The solid lines are guides for the eye. The statistical error is smaller than the marker size.

Figure 7

Relative jamming concentration $p_j/p_j(s=0)$ versus the order parameter $s$ at different concentrations of defects $d$. The length of the $k$-mers equals 8. The solid lines are guides for the eye. The statistical error is smaller than the marker size.

Figure 8

Percolation threshold $p_c$ and jamming concentration $p_j$ as a function of the defect concentration $d$ for the anisotropy $s=0.5$ and $k=2$. The hatched region corresponds to the concentrations of the objects and the defects when percolation is possible. Here $d_c$ is a critical concentration of defects that suppresses the formation of an infinite (percolation) cluster. The solid lines are simply guides for the eye. The statistical error is smaller than the marker size when not shown explicitly. The inset shows the percolation threshold $p_c$ as a function of the defect concentration $d$. The scale of the vertical axis is strongly increased to demonstrate changes in the value of the percolation threshold in the range of a few thousandths.

Figure 9

Percolation concentration $p_c$ versus the critical concentration of defects $d_c$ for different values of the order parameter $s$ and lengths of the $k$-mers. The lines are simply guides for the eye. The statistical error is smaller than the marker size.

Figure 10

Examples of the degree of the cluster anisotropy $\delta$ versus the order parameter $s$ for different values of $k$ for zero-defect lattices ($d=0$). The inset shows examples of the clusters for $k=8$.

Figure 11

Density of cluster $p_s$ versus the value of $k$ for the RSC model and the jamming concentration $p_j$ versus the value of $k$ for the RRSA model [24]. The data are presented for disordered $s=0$ and completely ordered $s=1$ zero-defect systems ($d=0$). The inset shows $p_s-p_s^{\infty }$ and $p_j-p_j^{\infty }$ versus $k$. Here $p_s^{\infty }$ and $p_j^{\infty }$ are the limiting values at $k\to \infty$.

Figure 12

Density of cluster $p_s$ versus the concentration of defects $d$ for the RSC model at different values of the linear segment length $k$ for disordered $s=0$ (solid lines) and completely ordered $s=1$ (dashed lines) systems.

Figure 13

Critical concentration of defects $d_c$ versus the value of $k$ for the RSC model at different values of the order parameter $s$. The inset shows $1/d_c$ versus $k-1$ dependences.

Figure 14

Density of clusters $p_s$ versus the critical concentration of defects $d_c$ for the RSC model at different values of the order parameter $s$ and of $k$. Arrows show the values of $p_s$ in the limit of very long $k$-mers $k\to \infty$.