Impact of defects on percolation in random sequential adsorption of linear k-mers on square lattices (original) (raw)
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The jamming and percolation for two generalized models of random sequential adsorption (RSA) of linear kkk-mers (particles occupying kkk 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 kkk-mers and a lattice with defects. Some of the occupying kkk 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 kkk-mers. The second model is the cooperative sequential adsorption (CSA) one, where, for each new kkk-mer, only a restricted number of lateral contacts zzz with previously deposited kkk-mers is allowed. Deposition occurs in the case when zleq(1−d)zmz\leq (1-d)z_mzleq(1−d)zm where zm=2(k+1)z_m=2(k+1)zm=2(k+1) ...
Physical Review E, 2012
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 × 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 RL(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 pc = a/k α + b log 10 k + c, where a, b, c, α are the fitting parameters varying with anisotropy. We predict that for large kmers (k 1.2 × 10 4 ) isotropic placed at the lattice, percolation cannot occur even at jamming concentration.
2012
Numerical simulations by means of Monte Carlo method and finite-size scaling analysis have been performed to study the percolation behavior of linear kkk-mers (also denoted in the literature as rigid rods, needles, sticks) on two-dimensional square lattices LtimesLL \times LLtimesL with periodic boundary conditions. Percolation phenomena are investigated for anisotropic relaxation random sequential adsorption of linear kkk-mers. Especially, effect of anisotropic placement of the objects on the percolation threshold has been investigated. Moreover, the behavior of percolation probability RL(p)R_L(p)RL(p) that a lattice of size LLL percolates at concentration ppp has been studied in details in dependence on kkk, anisotropy and lattice size LLL. A nonmonotonic size dependence for the percolation threshold has been confirmed in isotropic case. We propose a fitting formula for percolation threshold pc=a/kalpha+blog10k+cp_c = a/k^{\alpha}+b\log_{10} k+ cpc=a/kalpha+blog10k+c, where aaa, bbb, ccc, alpha\alphaalpha are the fitting parameters varying with anisotropy. We predict that for large kkk-mers ($k\gtrapprox 1.2\times10^4$) isotropic placed at the lattice, percolation cannot occur even at jamming concentration.
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 pre-deposited k-mers and detachment from the surface were forbidden. The RSA process continued until the saturation jamming limit, pj. 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 – pj) of the k-mers. The 'intrinsic electrical conductivity' and concentration dependence of the relative electrical conductivity Σ(p) (Σ = σ/σm for the C-model and Σ = σm/σ for the I-model, where σm 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 Σxy(p), Σx(p) and Σy(p). For the more practically important C-model, the other interesting findings are (i) for large values of k (k = 64, 128), the values of Σxy and Σy increase rapidly with the initial increase of p from 0 to 0.1; (ii) for k ≥ 16, all the Σxy(p) and Σx(p) curves intersect with each other at the same iso-conductivity 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 σy/σx (> 1).
Random sequential adsorption of partially oriented linear k-mers on a square lattice
Physical Review E, 2011
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 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.
Random sequential adsorption and diffusion of dimers and k-mers on a square lattice
The Journal of Chemical Physics, 2001
We have performed extensive simulations of random sequential adsorption and diffusion of kmers, up to k = 5 in two dimensions with particular attention to the case k = 2. We focus on the behaviour of the coverage and of vacancy dynamics as a function of time. We observe that for k = 2, 3 a complete coverage of the lattice is never reached, because of the existence of frozen configurations that prevent isolated vacancies in the lattice to join. From this result we argue that complete coverage is never attained for any value of k. The long time behaviour of the coverage is not mean field and non analytic, with t −1/2 as leading term. Long time coverage regimes are independent of the initial conditions while strongly depend on the diffusion probability and deposition rate and, in particular, different values of these parameters lead to different final values of the coverage. The geometrical complexity of these systems is also highlighted through an investigation of the vacancy population dynamics.
Jamming and percolation for deposition ofk2-mers on square lattices: A Monte Carlo simulation study
Physical Review E, 2019
Jamming and percolation of square objects of size k × k (k 2-mers) isotropically deposited on simple cubic lattices have been studied by numerical simulations complemented with finite-size scaling theory. The k 2-mers were irreversibly deposited into the lattice. Jamming coverage θ j,k was determined for a wide range of k (2 ≤ k ≤ 200). θ j,k exhibits a decreasing behavior with increasing k, being θ j,k→∞ = 0.4285(6) the limit value for large k 2-mer sizes. On the other hand, the obtained results shows that percolation threshold, θ c,k , has a strong dependence on k. It is a decreasing function in the range 2 ≤ k ≤ 18 with a minimum around k = 18 and, for k ≥ 18, it increases smoothly towards a saturation value. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the 3D random percolation, regardless of the size k considered.