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 .

Percolation and jamming phenomena are investigated for anisotropic sequential deposition of dimers (particles occupying two adjacent adsorption sites) on a square lattice. The influence of dimer alignment on the electrical conductivity was examined. The percolation threshold for deposition of dimers was lower than for deposition of monomers. Nevertheless, the problem belongs to the universality class of random percolation. The lowest percolation threshold (pc = 0.562) was observed for isotropic orientation of dimers. It was higher (pc = 0.586) in the case of dimers aligned strictly along one direction. The state of dimer orientation influenced the concentration dependence of electrical conductivity. The proposed model seems to be useful for description of the percolating properties of anisotropic conductors.PACS. 64.60.Ak Renormalization-group, fractal, and percolation studies of phase transitions -64.60.Cn Order-disorder transformations; statistical mechanics of model systems

The diffusion-driven self-assembly of rodlike particles was studied by means of Monte Carlo simulation. The rods were represented as linear k-mers (i.e., particles occupying k adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size L×L up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, L, was varied from 128 to 2048, and periodic boundary conditions were applied along both x and y axes, while the length of the k-mers (determining the aspect ratio) was varied from 2 to 12. The k-mers oriented along the x and y directions (k_{x}-mers and k_{y}-mers, respectively) were deposited equiprobably. In the course of the simulation, the numbers of intraspecific and interspecific contacts between the same sort and between different sorts of k-mers, respectively, were calculated. Both the shift ratio of the actual number of shifts along the longitudinal or transverse axes of the k-mers and the electrical conductivity of the system were also examined. For the initial random configuration, quite different self-organization behavior was observed for short and long k-mers. For long k-mers (k≥6), three main stages of diffusion-driven spatial segregation (self-assembly) were identified: the initial stage, reflecting destruction of the jamming state; the intermediate stage, reflecting continuous cluster coarsening and labyrinth pattern formation; and the final stage, reflecting the formation of diagonal stripe domains. Additional examination of two artificially constructed initial configurations showed that this pattern of diagonal stripe domains is an attractor, i.e., any spatial distribution of k-mers tends to transform into diagonal stripes. Nevertheless, the time for relaxation to the steady state essentially increases as the lattice size growth.

The percolation behaviour of conductive composites containing particles of different sizes was analysed. A composite was simulated as the media containing small conductive particles distributed in the channels between large insulative particles, where each large particle is covered by n monolayers of the filler particles. The simulations were done for the cases of two-dimensional (2D) and three-dimensional (3D) lattices. It was shown that the percolation filler concentration x* versus the particle size ratio λ = R/r and the number of monolayers n may be approximated as , where d is the space dimensionality; is the site random percolation threshold; neff is the effective number of monolayers, which decreases with increase in n and neff → n in the limit of n → ∞. The scaling behaviour of the percolation threshold inside the layers confined by the large particles was analysed. The data obtained at different values of λ and n gave the same correlation length exponent values as for the classical random percolation both for 2D and 3D cases. Analysis of the electrical conductivity behaviour near the percolation threshold in 2D systems showed the existence of the obvious differences at different values of λ and n, though the conductivity exponents s and t retained their universal values typical for the random percolation. The accuracy of the developed theoretical approach was experimentally tested for the polyvinyl chloride–copper (PVC–Cu) and polycarbonate–copper (PC–Cu) composites.