Study of random sequential adsorption by means of the gradient method

Abstract

By using the gradient method (GM) we study random sequential adsorption (RSA) processes in two dimensions under a gradient constraint that is imposed on the adsorption probability along one axis of the sample. The GM has previously been applied successfully to absorbing phase transitions (both first and second order), and also to the percolation transition. Now, we show that by using the GM the two transitions involved in RSA processes, namely percolation and jamming, can be studied simultaneously by means of the same set of simulations and by using the same theoretical background. For this purpose we theoretically derive the relevant scaling relationships for the RSA of monomers and we tested our analytical results by means of numerical simulations performed upon RSA of both monomers and dimers. We also show that two differently defined interfaces, which run in the direction perpendicular to the axis where the adsorption probability gradient is applied and separate the high-density (large-adsorption probability) and the low-density (low-adsorption probability) regimes, capture the main features of the jamming and percolation transitions, respectively. According to the GM, the scaling behaviour of those interfaces is governed by the roughness exponent $alpha=1/(1+u)$, where $u$ is the suitable correlation length exponent. Besides, we present and discuss in a brief overview some achievements of the GM as applied to different physical situations, including a comparison of the critical exponents determined in the present paper with those already published in the literature.