Electrical conductivity of a monolayer produced by random sequential adsorption of linear $k$-mers onto a square lattice

The electrical conductivity of a monolayer produced by the random sequential adsorption (RSA) of linear $k$-mers (particles occupying $k$ adjacent adsorption sites) onto a square lattice was studied by means of computer simulation. Overlapping with predeposited $k$-mers and detachment from the surface were forbidden. The RSA process continued until the saturation jamming limit, $p_j$. The isotropic (equiprobable orientations of $k$-mers along $x$ and $y$ axes) and anisotropic (all $k$-mers aligned along the $y$ axis) depositions for two different models—of an insulating substrate and conducting $k$-mers (C model) and of a conducting substrate and insulating $k$-mers (I model)—were examined. The Frank-Lobb algorithm was applied to calculate the electrical conductivity in both the $x$ and $y$ directions for different lengths ($k=1$ – 128) and concentrations ($p=0$ – $p_j$) of the $k$-mers. The “intrinsic electrical conductivity” and concentration dependence of the relative electrical conductivity $\mathrm{\Sigma }\left(p\right)$ ($\mathrm{\Sigma }=\sigma /{\sigma }_m$ for the C model and $\mathrm{\Sigma }={\sigma }_m/\sigma$ for the I model, where ${\sigma }_m$ is the electrical conductivity of substrate) in different directions were analyzed. At large values of $k$ the $\mathrm{\Sigma }\left(p\right)$ curves became very similar and they almost coincided at $k=128$. Moreover, for both models the greater the length of the $k$-mers the smoother the functions ${\mathrm{\Sigma }}_{xy}\left(p\right),{\mathrm{\Sigma }}_x\left(p\right)$ and ${\mathrm{\Sigma }}_y\left(p\right)$. For the more practically important C model, the other interesting findings are (i) for large values of $k$ ($k=64,128$), the values of ${\mathrm{\Sigma }}_{xy}$ and ${\mathrm{\Sigma }}_y$ increase rapidly with the initial increase of $p$ from 0 to 0.1; (ii) for $k\ge 16$, all the ${\mathrm{\Sigma }}_{xy}\left(p\right)$ and ${\mathrm{\Sigma }}_x\left(p\right)$ curves intersect with each other at the same isoconductivity points; (iii) for anisotropic deposition, the percolation concentrations are the same in the $x$ and $y$ directions, whereas, at the percolation point the greater the length of the $k$-mers the larger the anisotropy of the electrical conductivity, i.e., the ratio ${\sigma }_y/{\sigma }_x$ ($>1$).

Figure 1

Fragment of a square lattice with two deposited 3-mers of different orientations. Conductivities of bonds are indicated.

Figure 2

Relative electrical conductivity $\mathrm{\Sigma }$ versus the concentration of $k$-mers, $p$, for $k=2,L=100k$, isotropic deposition and the C model. The enlarged section near the percolation threshold is presented. The diagram shows both the results of 100 runs without averaging (a) and the corresponding averaged effective conductivity with error bars (b).

Figure 3

Scaling analysis for $k=16,L=25k,50k,100k$, the C model and isotropic deposition. The data are obtained by averaging over 100 independent runs. Inset shows ${\mathrm{\Sigma }}_{25}=\sigma /{\sigma }_{L=25k}$ versus $1/L$ dependencies at $p=0.4$ and $p=0.7$. The statistical error is smaller than the marker size.

Figure 4

Effective conductivity, $\mathrm{\Sigma }$, and derivative $dln\mathrm{\Sigma }/dp$ versus the concentration of monomers ($k=1$), $p$, for the C model, $L=256$. The results are averaged over 1000 runs. Results obtained from our computer experiments are shown as solid lines. Results of the GEM approximations are shown as dashed lines.

Figure 5

Calculated dependencies of the value of $|\mathrm{\Sigma }-1|$ versus the concentration of monomers, $p$, for the C model and the I model, $L=256$. The results are averaged over 1000 runs. Linear fittings are shown as solid lines. Results of the GEM approximations are shown as dashed lines. The statistical error is smaller than the marker size.

Figure 6

“Intrinsic conductivity”, $\left|\right[\sigma \left]\right|$, versus $k$-mer length. $p\to 0$. Calculations and fits. (a) “Intrinsic conductivity”, $\left|\right[{\sigma }_{xy}\left]\right|$, versus $k+k^{-1}$, isotropic deposition. The C model ($R^2=0.998$) and the I model ($R^2=0.999$). (b) “Intrinsic conductivity” versus $k$, anisotropic deposition, $\left[{\sigma }_x\right]$ for the C model, $\left|\right[{\sigma }_y\left]\right|$ for the I model. (c) “Intrinsic conductivity” versus $k^{-1}$, anisotropic deposition, $\left[{\sigma }_y\right]$ for the C model, $\left|\right[{\sigma }_x\left]\right|$ for the I model. The statistical error is smaller than the marker size when not shown.

Figure 7

Relative electrical conductivity, $\mathrm{\Sigma }$, versus concentration of $k$-mers, $p$. (a) $k=2$. (b) $k=16$. (c) $k=128$.

Figure 8

Concentrations $p_g$ versus $k$-mer length (${log}_2$ scale) for the C model and the I model evaluated from dependencies $\left[{\sigma }_{xy}\left(p\right)\right]$ (isotropic deposition) and $\left[{\sigma }_x\left(p\right)\right],\left[{\sigma }_y\left(p\right)\right]$ (anisotropic deposition). The lines are provided simply as visual guides. The statistical error is smaller than the marker size.

Figure 9

Effective conductivity, $\mathrm{\Sigma }$ versus concentration of $k$-mers, $p$, for different values of $k$. C model. (a) Isotropic deposition. (b) Anisotropic deposition, transversal conductivity ${\mathrm{\Sigma }}_x$. (c) Anisotropic deposition, longitudinal conductivity ${\mathrm{\Sigma }}_y$.

Figure 10

Derivative $dln\mathrm{\Sigma }/dp$ versus concentration of $k$-mers, $p$. $k=64$, C model. Circles at $p=0$ correspond to the “intrinsic conductivity”, ${\left[\sigma \right]}_0=dln\mathrm{\Sigma }/dp$. The maxima at $p=p_c$ correspond to the position of the percolation threshold. Compare with Fig. 4 for monomers.

Figure 11

Deposition patterns for different concentrations of $k$-mers, $p$ in the vicinity of the percolation threshold. $k=128$, anisotropic deposition, C model. Empty sites are shown in white, occupied sites are shown in gray, percolation clusters are shown in black. (a) $p=p_c=0.5118$. One cluster in the vertical direction. (b) $p=0.525$. Several clusters in the vertical direction. (c) $p=0.53$. More clusters in the vertical direction, some of these clusters are merged in the horizontal direction.

Figure 12

Anisotropy of electrical conductivity $\delta$ [Eq. (11)] versus $k$-mer length (${log}_2$ scale). C model. The line is provided simply as a visual guide. The statistical error is smaller than the marker size.