Standard and inverse site percolation of straight rigid rods on triangular lattices: Isotropic and perfectly oriented deposition and removal

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear $k$-mers (particles occupying $k$ consecutive sites along one of the lattice directions) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of $k$ consecutive monomers (linear $k$-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with size $k$: $\left(\text{i}\right)$ $\left[\right(\text{ii}\left)\right]$ standard isotropic[oriented] percolation threshold ${\theta }_{c,k}\left[{\vartheta }_{c,k}\right]$, which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. ${\theta }_{c,k}\left[{\vartheta }_{c,k}\right]$ is reached by isotropic[oriented] deposition of straight rigid $k$-mers on an initially empty lattice; and $\left(\text{iii}\right)$ $\left[\right(\text{iv}\left)\right]$ inverse isotropic[oriented] percolation threshold ${\theta }_{c,k}^i\left[{\vartheta }_{c,k}^i\right]$, which corresponds to the maximum concentration of occupied sites for which connectivity disappears. ${\theta }_{c,k}^i\left[{\vartheta }_{c,k}^i\right]$ is reached after removing isotropic [completely aligned] straight rigid $k$-mers from an initially fully occupied lattice. ${\theta }_{c,k}$, ${\vartheta }_{c,k}$, ${\theta }_{c,k}^i$, and ${\vartheta }_{c,k}^i$ are determined for a wide range of $k$ ($2\le k\le 512$). The obtained results indicate that $\left(1\right){\theta }_{c,k}\left[{\theta }_{c,k}^i\right]$ exhibits a nonmonotonous dependence on the size $k$. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around $k=11$, and finally increases and asymptotically converges towards a definite value for large segments ${\theta }_{c,k\to \infty }=0.500\left(2\right)$ $[{\theta }_{c,k\to \infty }^i=0.500\left(1\right)]$; $\left(2\right){\vartheta }_{c,k}\left[{\vartheta }_{c,k}^i\right]$ depicts a monotonous behavior in terms of $k$. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long $k$-mers ${\vartheta }_{c,k\to \infty }=0.5334\left(6\right)$ $[{\vartheta }_{c,k\to \infty }^i=0.4666\left(6\right)]$; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line $\theta \left(\vartheta \right)=0.5$. Thus a complementary property is found ${\theta }_{c,k}+{\theta }_{c,k}^i=1$ (and ${\vartheta }_{c,k}+{\vartheta }_{c,k}^i=1$) which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems, and (4) in all cases, the critical concentration curves divide the $\theta$ space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter $k$ so that the model presents percolation transition for the whole range of $k$.

Figure 1

(a) Schematic representation of a typical configuration obtained by isotropically depositing 3-mers ($k=3$) on a $L\times L$ lattice with $L=12$. Solid spheres joined by lines represent the deposited $k$-mers, and gray circles correspond to empty sites. (b) Schematic representation of a typical configuration obtained by removing 3-mers from an initially fully occupied $L\times L$ lattice with $L=12$. In the full occupation state, all lattice sites are occupied by single monomers (each monomer occupies one lattice site). Solid spheres represent occupied sites (monomers), and gray circles indicate the empty sites resulting from the removal of the $k$-mers. Periodic boundary conditions are considered in parts (a) and (b).

Figure 2

(a) Standard (solid circles and solid diamonds) and inverse (solid squares and solid stars) percolation thresholds for straight rigid $k$-mers with $k$ ranging between 1 and 512 on triangular lattices ($k$ axis is presented in log scale). The figure also includes the jamming data corresponding to standard (open circles) and inverse (open squares) problems. Red (dark gray in grayscale) symbols and black symbols represent standard and inverse data. The curves were obtained by following the isotropic deposition-removal scheme. Solid diamonds and solid stars denote results from Papers I and II, respectively. Solid circles and solid squares indicate values obtained in the present work. (b) Standard and inverse percolation curves with $k$ between 1 and 24 and with the vertical axis scaled from 0.4 to 0.6. (c) Standard and inverse percolation thresholds for values of $k$ varying between 16 and 512. The dashed lines corresponds to the fitting curve for each system, from which we get ${\theta }_{c,k\to \infty }=0.500\left(2\right)$ and ${\theta }_{c,k\to \infty }^i=0.500\left(1\right)$.

Figure 3

The figure shows the sum of the standard and inverse percolation thresholds for the complete range of considered $k$ sizes (note the logarithmic scale on the $k$ axis). The values correspond to the isotropic deposition or removal problem. As it can be observed, ${\theta }_{c,k}+{\theta }_{c,k}^i=1$.

Figure 4

Connectivity of the deposited phase at percolation threshold as a function of $k$. The curves were obtained by following the isotropic deposition and removal scheme. Solid circles represent results obtained for standard percolation $\left[{\xi }_k^s\left({\theta }_{c,k}\right)\right]$. Solid squares correspond to inverse percolation $\left[{\xi }_k^i\left({\theta }_{c,k}^i\right)\right]$. The dashed lines are simply a guide for the eye. In all cases, $L/k=340$.

Figure 5

Typical configurations obtained for $n=2$ and $l=4$. Deposited dimers and empty sites are represented by spheres joined by thick lines and open circles, respectively. $i=1(i=4$) denotes the first(last) column in the cell. (a) Nonpercolating configuration. (b) Percolating configuration. The sites belonging to the percolating cluster are highlighted by large open circles.

Figure 6

(a) Schematic representation of a typical configuration obtained by perfectly oriented deposition of 3-mers ($k=3$) on a $L\times L$ lattice with $L=12$. Solid spheres joined by lines represent the deposited $k$-mers and gray circles correspond to empty sites. (b) Schematic representation of a typical configuration obtained by removing aligned 3-mers from an initially fully occupied $L\times L$ lattice with $L=12$. In the full occupation state, all lattice sites are occupied by single monomers (each monomer occupies one lattice site). Solid spheres represent occupied sites (monomers), and gray circles indicate the empty sites resulting from the removal of the $k$-mers. Periodic boundary conditions are considered in parts (a) and (b).

Figure 7

(a) Standard (solid circles and solid diamonds) and inverse (solid squares) percolation thresholds for straight rigid $k$-mers, with $k$ ranging between 1 and 512 on triangular lattices ($k$ axis is presented in log scale). The figure also includes the jamming data corresponding to standard (open circles) and inverse (open squares) problems. Red (dark gray in grayscale) symbols and black symbols represent standard and inverse data. The curves were obtained by following the perfectly oriented deposition and removal scheme. Solid diamonds denote results from Paper III, respectively. Solid circles and solid squares indicate values obtained in the present work. (b) Standard and inverse percolation thresholds for values of $k$ varying between 1 and 64 and with the vertical axis scaled from 0.4 to 0.6. (c) Standard and inverse percolation thresholds for values of $k$ varying between 64 and 512. The dashed lines correspond to the fitting curve for each system, from which we get that ${\theta }_{c,k\to \infty }=0.5334\left(6\right)$ and ${\theta }_{c,k\to \infty }^i=0.4666\left(6\right)$.

Figure 8

Sum of the standard and inverse percolation thresholds for all the range of considered $k$ sizes (note the logarithmic scale on the $k$ axis). The values correspond to the perfectly oriented deposition problem. As it can be observed, ${\theta }_{c,k}+{\theta }_{c,k}^i=1$.

Figure 9

Connectivity of the deposited phase at percolation threshold as a function of $k$. The curves were obtained by following the perfectly oriented deposition and removal scheme. Solid circles represent results obtained for standard percolation $\left[{\xi }_k^s\left({\vartheta }_{c,k}\right)\right]$. Solid squares correspond to inverse percolation $\left[{\xi }_k^i\left({\vartheta }_{c,k}^i\right)\right]$. The dashed lines are simply a guide for the eye. In all cases, $L/k=340$.

Figure 10

(a) Comparison between isotropic (circles) and oriented (squares) percolation thresholds for the inverse percolation problem of straight rigid $k$-mers on triangular lattices. (b) Same as part (a) but for the standard percolation problem.