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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 onto a square lattice was studied by means of computer simulation. Overlapping with pre-deposited $k$-mers and detachment from the surface were forbidden. The RSA continued until the saturation jamming limit, $p_j$. The isotropic and anisotropic 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 $Σ(p)$ ($Σ=σ/ σ_m$ for the C-model and $Σ=σ_m /σ$ for the I-model, where $σ_m$ is the electrical conductivity of substrate) in different directions were analyzed. At large values of $k$ the $Σ(p)$ 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 $Σ_{xy}(p)$, $Σ_{x}(p)$ and $Σ_{y}(p)$. For the C-model, the other interesting findings are: for large values of $k$ ($k=64, 128$), the values of $Σ_{xy}$ and $Σ_{y}$ increase rapidly with the initial increase of $p$ from 0 to 0.1; for $k \geq 16$, all the $Σ_{xy}(p)$ and $Σ_{x}(p)$ curves intersect with each other at the same iso-conductivity points; 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 $σ_y/σ_x$ ($>1$).