Percolation of heteronuclear dimers irreversibly deposited on square lattices

The percolation problem of irreversibly deposited heteronuclear dimers on square lattices is studied. A dimer is composed of two segments, and it 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 $A$). The model offers a simplified representation of the problem of percolation of defective (nonideal) particles, where the presence of defects in the system is simulated by introducing a mixture of conductive and nonconductive segments. Different cases were investigated, according to the sequence of deposition of the particles, the types of dimers involved in the process, and the degree of alignment of the deposited objects. By means of numerical simulations and finite-size scaling analysis, the complete phase diagram separating a percolating from a nonpercolating region was determined for each case. Finally, the consistency of our results was examined by comparing with previous data in the literature for linear $k$-mers (particles occupying $k$ adjacent sites) with defects.

Figure 1

Snapshot of a typical configuration of dimers deposited on a homogeneous square surface for model II and the isotropic case. The figure shows only a fraction of the whole lattice. Solid circles correspond to $A$ segments (conductive segments), while open circles represent $B$ segments (nonconductive segments). In the figure, ${\theta }_1={\theta }_{AB}=0.2$ and ${\theta }_2={\theta }_{AA}=0.3$.

Figure 2

Same as in Fig. 1 for model II, the nematic case, ${\theta }_1={\theta }_{AB}=0.2$ and ${\theta }_2={\theta }_{AA}=0.4$.

Figure 3

Fraction of percolating lattices $R_L^U({\theta }_1,{\theta }_2)$ as a function of ${\theta }_2$ for model I, the isotropic case, and ${\theta }_1={\theta }_{AA}=0.4$. Different values of $L$ are considered, as indicated.

Figure 4

Extrapolation of ${\theta }_2^c\left(L\right)$ toward the thermodynamic limit according to the theoretical prediction given by Eq. (1), and the same parameters as in Fig. 3. Symbols represent simulation results, and the solid line is the theoretical fit from Eq. (1).

Figure 5

Percolation phase diagram of isotropic dimers for model I (solid squares), model II (open squares), and model III (solid circles). The symbols divide percolating from nonpercolating regions for each model. Region 1: forbidden region for models I, II, and III; region 2: nonpercolating region for models I, II, and III; region 3, percolating region for models I and II, and nonpercolating region for model III; and region 4, percolating region for models I, II, and III. In all cases, the error bar is smaller than the size of the symbols.

Figure 6

Same as Fig. 5 for perfectly aligned dimers (isotropic case).

Figure 7

Fraction of conductive segments corresponding to percolation (${\theta }_A^c$) as a function of the fraction of defects ($d$) for dimers deposited on square lattices according to model I (solid squares), model II (open squares), model III (solid circles), and the model proposed in Ref. [17] (open circles). The solid lines are simply a guide for the eye. In all cases, the error bar is smaller than the size of the symbols.