Scaling behavior of jamming fluctuations upon random sequential adsorption

TL;DR

This paper derives a power-law scaling law for fluctuations in jamming coverage during random sequential adsorption, relating it to substrate dimension and fractal properties, and confirms it with numerical simulations.

Contribution

It introduces a universal scaling relation for jamming fluctuations in RSA processes based on substrate and fractal dimensions, validated by simulations.

Findings

Fluctuations decay as a power-law with lattice size.

The exponent relates to substrate and fractal dimensions.

Numerical results agree with the theoretical prediction.

Abstract

It is shown that the fluctuations of the jamming coverage upon Random Sequential Adsorption ($\sigma_{\theta_J}$), decay with the lattice size according to the power-law $\sigma_{\theta_J} \propto L^{-1 / \nu_J}$, with $\nu_{J} = 2 / (2D - d_f)$, where $D$ is the dimension of the substrate and $d_{\rm f}$ is the fractal dimension of the set of sites belonging to the substrate where the RSA process actually takes place. This result is in excellent agreement with the figure recently reported by Vandewalle {\it et al} ({\it Eur. Phys. J.} B. {\bf 14}, 407 (2000)), namely $\nu_J = 1.0 (1)$ for the RSA of needles with $D = 2$ and $d_f = 2$, that gives $\nu_J = 1$. Furthermore, our prediction is in excellent agreement with different previous numerical results. The derived relationships are also confirmed by means of extensive numerical simulations applied to the RSA of dimers on both stochastic and deterministic fractal substrates.