Random sequential adsorption on Euclidean, fractal, and random lattices

Abstract

Irreversible adsorption of objects of different shapes and sizes on Euclidean, fractal and random lattices is studied. The adsorption process is modeled by using random sequential adsorption (RSA) algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension d between 1 and 2, and on Erdos-Renyi random graphs. The number of sites is M = L d for Euclidean and fractal lattices, where L is a characteristic length of the system. In the case of random graphs it does not exist such characteristic length, and the substrate can be characterized by a fixed set of M vertices (sites) and an average connectivity (or degree) g. The paper concentrates on measuring (1) the probability W L(M ) (θ) that a lattice composed of L d (M ) elements reaches a coverage θ, and (2) the exponent νj characterizing the so-called "jamming transition". The results obtained for Euclidean, fractal and random lattices indicate that the main quantities derived from the jamming probability W L(M ) (θ) behave asymptotically as M 1/2 . In the case of Euclidean and fractal lattices, where L and d can be defined, the asymptotic behavior can be written as M 1/2 = L d/2 = L 1/ν j , and νj = 2/d. arXiv:1907.02572v2 [cond-mat.stat-mech]

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

“…In both cases (main figure and inset), the critical exponent obtained from the slope of the curves is close to 3/2. The procedure was repeated for different values of k. In all the cases, the values obtained for ν j : (1) remain close to 3/2, and (2) coincide, within the numerical errors, with the values previously reported by us in other 3D systems [22,26].…”

“…Using extensive simulations supplemented by finite-size scaling analysis, jamming coverages and percolation thresholds were determined for a wide range of k values. This study (i) completes previous work on jamming and percolation of extended objects on D-dimensional lattices [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]; and (ii) allow us to extract general conclusions about the behavior of the system and its dependence on the relationship between the dimension of the depositing object and the dimension of the substrate.…”

“…Thus, θ j,k=∞ =0.4045 (19) [22], 0.4285(6) [26] and 0.4204(9), for k-mers, k 2 -mers and k 3mers, respectively. The limiting values of θ j,k were obtained by simulations for relatively small k sizes and then extrapolated to represent very long objects.…”

“…straight rigid k-mers on 2D square lattices [11][12][13][14][15][16][17][18][19][20]; straight rigid k-mers on 3D simple cubic lattices [21] and k × k tiles (k 2 -mers) on 3D simple cubic lattices [26]. In these systems, percolating and non-percolating phases extend to infinity in the parameter space k and, consequently, the model presents percolation transition in all ranges of k-mer size.…”

“…Thus, the jamming coverage has an important role on the percolation threshold and the interplay between jamming and percolation has been discussed in several works [1,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”