The adsorption-desorption model and its application to vibrated granular materials

TL;DR

This paper studies the kinetics of a microscopic adsorption-desorption model for granular materials, revealing slow relaxation regimes, equilibrium fluctuations, and methods to achieve denser packings by adjusting desorption rates.

Contribution

It introduces an analytical and numerical analysis of a hard-rod adsorption-desorption model, highlighting long-time kinetics and density fluctuation behaviors in granular compaction.

Findings

Three successive kinetic regimes: algebraic, logarithmic, exponential.

Equilibrium density fluctuations follow a power law and exponential decay.

Adjusting desorption rate influences final packing density.

Abstract

We investigate both analytically and by numerical simulation the kinetics of a microscopic model of hard rods adsorbing on a linear substrate, a model which is relevant for compaction of granular materials. The computer simulations use an event-driven algorithm which is particularly efficient at very long times. For a small, but finite desorption rate, the system reaches an equilibrium state very slowly, and the long-time kinetics display three successive regimes: an algebraic one where the density varies as $1/t$, a logarithmic one where the density varies as $1/\ln(t)$, followed by a terminal exponential approach. The characteristic relaxation time of the final regime, though incorrectly predicted by a mean field arguments, can be obtained with a systematic gap-distribution approach. The density fluctuations at equilibrium are also investigated, and the associated time-dependent correlation function exhibits a power law regime followed by a final exponential decay. Finally, we show that denser particle packings can be obtained by varying the desorption rate during the process.