Percolation of defective dimers irreversibly deposited on honeycomb and triangular lattices

The percolation problem of irreversibly deposited dimers on honeycomb and triangular lattices is studied. A dimer is composed of two segments, and occupies two adjacent adsorption sites. Each segment can be either a conductive segment (segment type $A$) or a nonconductive segment (segment type $B$). Three types of dimers are considered: $AA$, $BB$, and $AB$. The connectivity analysis is carried out by accounting only for the conductive segments (segments type $A$), whereas the $B$ segments occupy a site in the lattice but are not taken into account in the percolation study. Different cases were investigated, according to the types of dimers involved in the process. By means of numerical simulations and finite-size scaling techniques, the complete phase diagram separating the percolating from the nonpercolating regions was determined for each considered lattice. The present results are compared with the previous study of deposition of defective dimers on square lattices.

Figure 1

Heteronuclear dimers deposited on (a) a honeycomb lattice with $L=10$, and (b) a triangular lattice with $L=6$. Black circles, gray circles, and open circles represent $A$-dimer units, $B$-dimer units, and empty sites, respectively. The definitions of $r$ and $l$ are given in the text.

Figure 2

Probability of percolation $R_L^X$ ($X=I,U,$ and $A$, as defined in Sec. 2) as a function of ${\theta }_2$ for two lattice sizes: $L=128$ (squares) and $L=384$ (triangles). Two cases are shown: (a) model I, honeycomb lattice, ${\theta }_1=0.20$; and (b) model II, triangular lattice, ${\theta }_1=0.32$. The vertical dashed line denotes the percolation threshold in the thermodynamic limit.

Figure 3

Extrapolation of ${\theta }_{2,c}^X\left(L\right)$ towards the thermodynamic limit according to the theoretical prediction given by Eq. (4). Triangles, circles, and squares denote the values of ${\theta }_{2,c}^X\left(L\right)$ obtained by using the criteria $I$, $A$, and $U$ (as defined in Sec. 2), respectively. Two cases have been considered: (a) model I, honeycomb lattice, ${\theta }_1=0.20$; and (b) model II, triangular lattice, ${\theta }_1=0.32$.

Figure 4

(a) Percolation phase diagram corresponding to heteronuclear dimers on honeycomb lattices. Four regions can be distinguished. Region 1: forbidden region for models I and II; region 2: nonpercolating region for both models I and II; region 3: percolating region for model I and nonpercolating region for model II; and region 4: percolating region for models I and II. (b) Same as in part (a) for triangular lattices. As explained in the text, ${\theta }_1={\theta }_{AB}$ and ${\theta }_2={\theta }_{AA}$ for model I, and ${\theta }_1={\theta }_{BB}$ and ${\theta }_2={\theta }_{AA}$ for model II.

Figure 5

(a) Maximum of the derivative of the $A$ percolation probability ${\left(d{R}_L^A/d{\theta }_2\right)}_{\max }$ as a function of $L$ (in a log-log scale) for two different cases: model I in honeycomb lattice (solid squares) and model II in triangular lattice (solid circles). Inset: standard deviation in Eq. (3) ${\mathrm{\Delta }}_L^A$ as a function of $L$ (in a log-log scale) for the same cases shown in part (a). According to Eq. (6), the slope of each line corresponds to $-1/\nu =-\frac{3}{4}$. (b) Maximum of the susceptibility ${\chi }_{\max }$ as a function of $L$ (in a log-log scale) for the cases in part (a). The error bar in each measurement is smaller than the size of the symbols. The slope of each line corresponds to $\gamma /\nu =\frac{43}{24}$. (c) Maximum of the derivative of the percolation order parameter ${\left(dP/d{\theta }_2\right)}_{\max }$ as a function of $L$ (in a log-log scale) for the same cases reported in part (a). According to Eq. (7), the slope of each line corresponds to $(1-\beta )/\nu =\frac{31}{48}$.

Figure 6

Data collapsing of the percolation probability $R_L^A\left(\theta \right)$ vs $({\theta }_2-{\theta }_{2,c})L^{1/\nu }$. Upper left inset: data collapsing of the percolation order parameter $P{L}^{\beta /\nu }$ vs $|{\theta }_2-{\theta }_{2,c}|L^{1/\nu }$. Bottom right inset: data collapsing of the susceptibility $\chi L^{-\gamma /\nu }$ vs $({\theta }_2-{\theta }_{2,c})L^{1/\nu }$. Two cases have been considered: (a) model I, honeycomb lattice, ${\theta }_1=0.20$; and (b) model II, triangular lattice, ${\theta }_1=0.32$.