No abstract

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 publications as rigid rods, needles, sticks) on two-dimensional square lattices L × 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. A detailed study of the behavior of percolation probability R(L)(p) that a lattice of size L percolates at concentration p in dependence on k, anisotropy, and lattice size L has been performed. A nonmonotonic size dependence for the percolation threshold has been confirmed in the isotropic case. We propose a fitting formula for percolation threshold, p(c) = a/k(α)+blog(10)k+c, where a, b, c, and α are the fitting parameters depending on anisotropy. We predict that for large k-mers (k >/≈ 1.2 × 10(4)) isotropically placed at the lattice, percolation cannot occur, even at jamming concentration.

The behavior of the percolation threshold and the jamming coverage for isotropic random sequential adsorption samples has been studied by means of numerical simulations. A parallel algorithm that is very efficient in terms of its speed and memory usage has been developed and applied to the model involving large linear k-mers on a square lattice with periodic boundary conditions. We have obtained the percolation thresholds and jamming concentrations for lengths of k-mers up to 2 17 . New large k regime of the percolation threshold behavior has been identified. The structure of the percolating and jamming states has been investigated. The theorem of G. Kondrat, Z. Koza, and P. Brzeski [Phys. Rev. E 96, 022154 (2017)] has been generalized to the case of periodic boundary conditions. We have proved that any cluster at jamming is percolating cluster and that percolation occurs before jamming.

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 256. The 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 one-dimensional systems. In particular, the jamming threshold tends to the Rényi'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⩽4), denser configurations are observed in disordered systems as compared to those of completely ordered systems. However, longer k-mers exhibit the opposite behavior.

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, pc, vs the concentration of defects, d, were analyzed for different values of k. Above some critical concentration of defects, dm, percolation is blocked in both models, even at the jamming concentration of k-mers. For long k-mers, the values of dm are well fitted by the functions dm ∝ k −α m − k −α (α = 1.28 ± 0.01, km = 5900 ± 500) and dm ∝ log(km/k) (km = 4700 ± 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 6 × 10 3 .