Inverse percolation by removing straight rigid rods from square lattices in the presence of impurities

Abstract

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing straight rigid rods from square lattices contaminated with non-conducting impurities. The presence of impurities provides a more realistic approach to the deposited monolayer, which usually presents inhomogeneities due to the irregular arrangement of surface and bulk atoms, the presence of various chemical species, etc. The process starts with an initial configuration, where all lattice sites are occupied by an impurity (with a concentration ) or a conducting particle (with a concentration ). Then, the system is diluted by randomly removing linear k-mers (linear clusters of k consecutive conducting particles) from the lattice. The impurities remain fixed in its position and cannot be removed. The central idea of this paper is based on finding the maximum concentration of conducting sites (minimum concentration of empty sites) for which the connectivity disappears. This particular value of the concentration is called inverse percolation threshold, and determines a well-defined geometrical phase transition in the system. On the other hand, the inverse jamming coverage is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbour sites of appropriate size. The dependence of percolation and jamming thresholds on the concentration of defects was investigated for different values of k, ranging from 1 to 120. The obtained results show that the behaviour of the system is significantly affected by the presence of impurities. In addition, the nature of the jamming and percolation transitions was studied. In the first case, the corresponding spatial correlation length critical exponent was measured, being . This value coincides with previous calculations of this exponent for the standard random sequential adsorption of linear k-mers on square lattices. Critical exponents were also calculated for the percolation phase transition, showing that the universality class corresponding to ordinary percolation is preserved regardless the values of k and considered.

Jamming and percolation of k 2-mers on simple cubic lattices

“…To conclude with the analysis of jamming properties, the critical exponent ν j was measured for different values of the size k. In all cases, the values obtained for ν j (1) remain close to 3/2, (2) coincide, within numerical errors, with the same value of the critical exponent obtained by us for other three dimensional systems [22,26], and (3) differs clearly from the value ν j ≈ 1 reported by Vandewalle et al [12] for the case of linear k-mers on square lattices, and from other 2D systems [25,42].…”

Kinetics of random sequential adsorption of two-dimensional shapes on a one-dimensional line

“…The mean packing fraction at the limit of infinite packing can be estimated by finding the crossing of the CDF's for different packing sizes. This method is especially useful for studying RSA on lattices [30][31][32][33][34][35]. However, in our case, the precision given by 2 is enough due to quite large size of packings used in this study.…”

Random sequential adsorption on Euclidean, fractal, and random lattices

“…Following the scheme given in Eqs. (1)(2)(3)(4)(5), different systems were characterized in previous work from our group: (1) linear k-mers on 1D lattices [45]; (2) linear k-mers on 2D square lattices with and without the presence of impurities [48,49]; (3) linear k-mers on 2D triangular lattices [50]; (4) k × k square tiles (k 2 -mers) on 2D square lattices [27]; (5) linear k-mers on 3D simple cubic lattices [45]; (6) k 2 -mers on 3D simple cubic lattices [51] and (7) k × k × k cubic objects k 3 -mers on 3D simple cubic lattices [52]. In all cases, ν j was determined from Eqs.…”

“…More recently, the exponent ν j was measured for different systems in 1D, 2D and 3D Euclidean lattices [27,[49][50][51][52]. The obtained results reveal a simple dependence of ν j with the dimensionality of the lattice.…”