Random sequential adsorption on Euclidean, fractal, and random lattices

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 algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension d between 1 and 2, and on Erdős-Rényi 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, such a characteristic length does not exist, and the substrate can be characterized by a fixed set of M vertices (sites) and an average connectivity (or degree) g. This paper concentrates on measuring (i) the probability W_{L(M)}(θ) that a lattice composed of L^{d}(M) elements reaches a coverage θ and (ii) the exponent ν_{j} characterizing the so-called jamming transition. The results obtained for Euclidean, fractal, and random lattices indicate that the quantities derived from the jamming probability W_{L(M)}(θ), such as (dW_{L}/dθ)_{max} and the inverse of the standard deviation Δ_{L}, 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}}, with ν_{j}=2/d.