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The effect of defects on the behaviour of electrical conductivity, σ , in a monolayer produced by the random sequential adsorption of linear k -mers (particles occupying k adjacent sites) onto a square lattice is studied by means of a Monte Carlo simulation. The k -mers are deposited on the substrate until a jamming state is reached. The presence of defects in the lattice (impurities) and of defects in the k -mers with concentrations of d l and d k , respectively, is assumed. The defects in the lattice are distributed randomly before deposition and these lattice sites are forbidden for the deposition of k -mers. The defects of the k -mers are distributed randomly on the deposited k -mers. The sites filled with k -mers have high electrical conductivity, σ k , whereas the empty sites, and the sites filled by either types of defect have a low electrical conductivity, σ l , i.e., a high-contrast, σ k ∕ σ l ≫ 1 , is assumed. We examined isotropic (both the possible x and y orientations of a particle are equiprobable) and anisotropic (all particles are aligned along one given direction, y ) deposition. To calculate the effective electrical conductivity, the monolayer was presented as a random resistor network and the Frank–Lobb algorithm was used. The effects of the concentrations of defects d l and d k on the electrical conductivity for the values of k = 2 n , where n = 1 , 2 , … , 5 , were studied. Increase of both the d l and d k parameters values resulted in decreases in the value of σ and the suppression of percolation. Moreover, for anisotropic deposition the electrical conductivity along the y direction was noticeably larger than in the perpendicular direction, x . Phase diagrams in the ( d l , d k )-plane for different values of k were obtained.

Diffusion in a two-species 2D system has been simulated using a lattice approach. Rodlike particles were considered as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice. These $k_x$- and $k_y$-mers were treated as species of two kinds. A random sequential adsorption model was used to produce an initial homogeneous distribution of $k$-mers. The concentration of $k$-mers, $p$, was varied in the range from 0.1 to the jamming concentration, $p_j$. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a random walk, while rotational diffusion was ignored. We demonstrated that the diffusion coefficients are strongly anisotropic and nonlinearly concentration-dependent. For sufficiently large concentrations (packing densities) and $k \geq 6$, the system tends toward a well-organized steady state. Boundary conditions (BC) predetermine the final state of the system. When periodic BCs are applied along both directions of the square lattice, the system tends to a steady state in the form of diagonal stripes. The formation of stripe domains takes longer time the larger the lattice size, and is observed only for concentrations above a particular critical value. When insulating (zero flux) BCs are applied along both directions of the square lattice, each kind of $k$-mer tries to completely occupy a half of the lattice divided by a diagonal, e.g., $k_x$-mers locate in the upper left corner, while the $k_y$-mers are situated in the lower right corner ("yin-yang" pattern). From time to time, regions built of $k_x$- and $k_y$-mers exchange their locations through irregular patterns. When mixed BCs are used (periodic BCs are applied along one direction whereas insulating BCs are applied along the other one), the system still tends to form the stripes, but they are unstable and change their spatial orientation., Comment: 18 pages, 9 figures, 24 references

Pattern formation in a two-dimensional system of rod-like particles has been simulated using a lattice approach. Rod-like particles were modelled as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice with periodic boundary conditions (torus). Two kinds of random sequential adsorption model were used to produce the initial homogeneous and isotropic distribution of $k$-mers with different values of packing fraction. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a random walk, while rotational diffusion was ignored, so, $k_x$- and $k_y$-mers were considered as individual species. The system tends toward a well-organized nonequilibrium steady state in the form of diagonal stripes for the relatively long $k$-mer ($k \geq 6$) and moderate packing densities (in the interval $p_{down} < p < p_{up}$, where both the critical packing fractions $p_{down}$ and $p_{up}$ are depended on $k$)., 10 pages, 10 figures, 20 references; XXIX IUPAP Conference in Computational Physics (CCP2017) https://ccp2017.sciencesconf.org/; submitted to J.Phys.Conf.Ser

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 $\Sigma (p)$ ($\Sigma=\sigma/ \sigma_m$ for the C-model and $\Sigma=\sigma_m /\sigma$ for the I-model, where $\sigma_m$ is the electrical conductivity of substrate) in different directions were analyzed. At large values of $k$ the $\Sigma (p)$ 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 $\Sigma_{xy}(p)$, $\Sigma_{x}(p)$ and $\Sigma_{y}(p)$. For the C-model, the other interesting findings are: for large values of $k$ ($k=64, 128$), the values of $\Sigma_{xy}$ and $\Sigma_{y}$ increase rapidly with the initial increase of $p$ from 0 to 0.1; for $k \geq 16$, all the $\Sigma_{xy}(p)$ and $\Sigma_{x}(p)$ 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 $\sigma_y/\sigma_x$ ($>1$)., Comment: 11 pages, 12 figures, 2 tables, 44 references, submitted to Phys.Rev.E

The jamming and percolation for two generalized models of random sequential adsorption (RSA) of linear $k$-mers (particles occupying $k$ adjacent sites) on a square lattice are studied by means of Monte Carlo simulation. The classical random sequential adsorption (RSA) model assumes the absence of overlapping of the new incoming particle with the previously deposited ones. The first model LK$_d$ is a generalized variant of the RSA model for both $k$-mers and a lattice with defects. Some of the occupying $k$ adjacent sites are considered as insulating and some of the lattice sites are occupied by defects (impurities). For this model even a small concentration of defects can inhibit percolation for relatively long $k$-mers. The second model is the cooperative sequential adsorption (CSA) one, where, for each new $k$-mer, only a restricted number of lateral contacts $z$ with previously deposited $k$-mers is allowed. Deposition occurs in the case when $z\leq (1-d)z_m$ where $z_m=2(k+1)$ is the maximum numbers of the contacts of $k$-mer, and $d$ is the fraction of forbidden NN contacts. Percolation is observed only at some interval $k_{min}\leq k\leq k_{max}$ where the values $k_{min}$ and $k_{max}$ depend upon the fraction of forbidden contacts $d$. The value $k_{max}$ decreases as $d$ increases. A logarithmic dependence of the type $\log(k_{max})=a+bd$, where $a=-4.03 \pm 0.22$, $b=4.93 \pm 0.57 $, is obtained., Comment: 7 pages, 8 figures, 32 references

The effect of defects on the percolation of linear $k$-mers (particles occupying $k$ adjacent sites) on a square lattice is studied by means of Monte Carlo simulation. The $k$-mers are deposited using a random sequential adsorption mechanism. Two models, $L_d$ and $K_d$, are analyzed. In the $L_d$ model, it is assumed that the initial square lattice is non-ideal and some fraction of sites, $d$, is occupied by non-conducting point defects (impurities). In the $K_d$ model, the initial square lattice is perfect. However, it is assumed that some fraction of the sites in the $k$-mers, $d$, consists of defects, i.e., are non-conducting. The length of the $k$-mers, $k$, varies from 2 to 256. Periodic boundary conditions are applied to the square lattice. The dependencies of the percolation threshold concentration of the conducting sites, $p_c$, vs the concentration of defects, $d$, were analyzed for different values of $k$. Above some critical concentration of defects, $d_m$, percolation is blocked in both models, even at the jamming concentration of $k$-mers. For long $k$-mers, the values of $d_m$ are well fitted by the functions $d_m \propto k_m^{-\alpha}-k^{-\alpha}$ ($\alpha = 1.28 \pm 0.01$, $k_m = 5900 \pm 500$) and $d_m \propto \log (k_m/k)$ ($k_m = 4700 \pm 1000$ ), for the $L_d$ and $K_d$ models, respectively. Thus, our estimation indicates that the percolation of $k$-mers on a square lattice is impossible even for a lattice without any defects if $k\gtrapprox 6 \times 10^3$., Comment: Submitted to Physical Review E

Numerical simulations by means of Monte Carlo method and finite-size scaling analysis have been performed to study the percolation behavior of linear $k$-mers (also denoted in the literature as rigid rods, needles, sticks) on two-dimensional square lattices $L \times L$ with periodic boundary conditions. Percolation phenomena are investigated for anisotropic relaxation random sequential adsorption of linear $k$-mers. Especially, effect of anisotropic placement of the objects on the percolation threshold has been investigated. Moreover, the behavior of percolation probability $R_L(p)$ that a lattice of size $L$ percolates at concentration $p$ has been studied in details in dependence on $k$, anisotropy and lattice size $L$. A nonmonotonic size dependence for the percolation threshold has been confirmed in isotropic case. We propose a fitting formula for percolation threshold $p_c = a/k^{\alpha}+b\log_{10} k+ c$, where $a$, $b$, $c$, $\alpha$ are the fitting parameters varying with anisotropy. We predict that for large $k$-mers ($k\gtrapprox 1.2\times10^4$) isotropic placed at the lattice, percolation cannot occur even at jamming concentration., Comment: 11 pages, 12 figures

Jamming phenomena on a square lattice are investigated for two different models of anisotropic random sequential adsorption (RSA) of linear $k$-mers (particles occupying $k$ adjacent adsorption sites along a line). The length of a $k$-mer varies from 2 to 128. Effect of $k$-mer alignment on the jamming threshold is examined. For completely ordered systems where all the $k$-mers are aligned along one direction (e.g., vertical), the obtained simulation data are very close to the known analytical results for 1d systems. In particular, the jamming threshold tends to the R{\'e}nyi's Parking Constant for large $k$. In the other extreme case, when $k$-mers are fully disordered, our results correspond to the published results for short $k$-mers. It was observed that for partially oriented systems the jamming configurations consist of the blocks of vertically and horizontally oriented $k$-mers ($v$- and $h$-blocks, respectively) and large voids between them. The relative areas of different blocks and voids depend on the order parameter $s$, $k$-mer length and type of the model. For small $k$-mers ($k\leqslant 4$), denser configurations are observed in disordered systems as compared to those of completely ordered systems. However, longer $k$-mers exhibit the opposite behavior., Comment: 12 pages, 12 figures