Percolation of linear $k$-mers on a square lattice: From isotropic through partially ordered to completely aligned states

Numerical simulations by means of Monte Carlo method and finite-size scaling analysis have been performed to study the percolation behavior of linear $k$-mers (also denoted in publications as rigid rods, needles, sticks) on two-dimensional square lattices $L\times L$ with periodic boundary conditions. Percolation phenomena are investigated for anisotropic relaxation random sequential adsorption of linear $k$-mers. Especially, effect of anisotropic placement of the objects on the percolation threshold has been investigated. A detailed study of the behavior of percolation probability $R_L\left(p\right)$ that a lattice of size $L$ percolates at concentration $p$ in dependence on $k$, anisotropy, and lattice size $L$ has been performed. A nonmonotonic size dependence for the percolation threshold has been confirmed in the isotropic case. We propose a fitting formula for percolation threshold, $p_c=a/k^{\alpha }+b{\log }_{10}k+c$, where $a$, $b$, $c$, and $\alpha$ are the fitting parameters depending on anisotropy. We predict that for large $k$-mers ($k⪆1.2\times {10}^4$) isotropically placed at the lattice, percolation cannot occur, even at jamming concentration.

Figure 1

Percolating clusters of different sorts on a plane and on a torus. (a) Crossing clusters; (b) spiral-like wrapping clusters; (c) ringlike wrapping clusters.

Figure 2

Comparison of $R_L^{\mathrm{or}}$ and $R_L^{\mathrm{and}}$ versus $p$ dependencies for monomer problem ($k=1$) (physical percolation) and crossing clusters (with open boundary conditions) and different size of square lattice, $L$.

Figure 3

Critical exponent $\nu$ extracting for $k=2$, $s=0.5$. ${\nu }^{-1}=0.750\pm 0.001$. Log-log scale.

Figure 4

Percolation concentration $p_c$ versus size of the lattice $L$ for different criteria for physical percolation. (a) Isotropic case, $k=2$, $s=0.0$. (b) Slightly anisotropic case, $k=4$, $s=0.1$. (c) Anisotropic case, $k=16$, $s=0.8$.

Figure 5

Probability curves for isotropic systems, $s=0$, and different values of $k$ and $L$. Arrows indicate the intersection points.

Figure 6

Probability curves for $k=32$, $s=0.0,0.7,1.0$. Arrows indicate the intersection points.

Figure 7

Intersection point of percolation probability $R^{*}$ versus $k$ at $s=0.0,0.7,1.0$.

Figure 8

Examples of wrapping clusters incipient in vertical direction for completely ordered system ($s=1.0$) for different length of $k$-mers. The size of a square lattice is $L=128k$. Periodical boundary conditions. (a) $k=2$; (b) $k=8$; (c) $k=32$.

Figure 9

Mean degree of the system anisotropy $\delta$ versus order parameter $s$: (a) at different concentrations $p$ and the fixed length of a $k$-mer, $k=32$; (b) at different $k$-mer length $k$ and concentrations that corresponds to the percolation transition for given systems. The size of lattice is $L=4096$ and the data averaged over 100 independent runs.

Figure 10

Percolation threshold $p_c$ versus order parameter $s$ for $k$-mers of different length.

Figure 11

Percolation threshold $p_c$ versus $k$-mer length $k$ at different values of order parameter $s$. Here, the different data are presented that have been obtained in: (1) this work, (2) Ref. [5], (3) Ref. [7], (4) Ref. [6], (5) Ref. [8] and (6) Ref. [13]. The dashed lines have been obtained by least-square fitting of the data points using Eqs. [(10), (11)].

Figure 12

Examples of percolation configurations of $k$-mers ($k=128$) on a square lattice of size $L=4096$ at $s=0$. Vertical and horizontal orientations are represented by different gray levels in printed version and in red and blue in online version; empty sites are labeled black and sites of percolation cluster are labeled white. Here, (a) shows the whole lattice; (b) shows the magnification of the pattern (a) in the central square; (c) shows the magnification of the pattern (b) in the central square.

Figure 13

Ratio of percolation and jamming concentration $p_c/p_j$ versus $k$-mer length $k$ for disordered ($s=0$) and completely ordered $s=1$ systems. The dashed lines for $s=0$ has been obtained by least-square fitting of the data points using Eq. (12).