Adsorption, desorption and diffusion of extended objects on a square lattice

Abstract

Kinetics of the adsorption–desorption process with diffusional relaxation is studied by Monte-Carlo simulations on a square lattice. At each Monte-Carlo step three independent processes proceeding in parallel are attempted: adsorption, desorption and diffusion. The time is measured by the number of adsorption attempts per lattice site and the kinetics of the process is governed by the ratios of desorption to adsorption rate and diffusion to adsorption rate. The process is reversible and after long enough time the system reaches an equilibrium state. For a given desorption/adsorption rate ratio equilibrium coverage is not affected by the diffusion rate, while the diffusion fastens the approach to the equilibrium state. These simulations could also be useful for explanation of the densification process in granular materials.

Introduction

A number of processes in physics, chemistry and biology, where events occur essentially irreversibly on the time scales of interest, can be studied as random sequential adsorption (RSA) [1], [2], [3], [4], [5], [6], [7]. For example, the adsorption of large particles such as colloids, proteins or latexes on substrates is often a highly irreversible process. Random sequential adsorption, or irreversible deposition, is a process in which the objects of a specified shape are randomly and sequentially deposited onto a substrate. Once an object is placed it affects the geometry of all later placements, so the dominant effect in RSA is the blocking of the available substrate area and the limiting (“jamming”) coverage ρ(∞) is less than in close packing.

However, in real physical situations one often needs to consider the possibility of desorption or diffusion of deposited particles [8], [9], [10], [11]. Allowing desorption makes the process reversible and the system finally reaches an equilibrium state. The approach to the equilibrium state is exponential [8], [10]. In [12] adsorption–desorption processes of extended objects on a square lattice were studied. In the late times of the process the plots of ln(ρ_eq−ρ(t)) were found to be straight lines for all desorption probabilities which confirms that the approach to the equilibrium state is of the form

$$\text{ρ}{}_{\mathrm{eq}}\text{−ρ(t)}\text{∝}\text{e}{}^{\mathrm{-t/\sigma }}\text{,}$$

where the equilibrium coverage ρ_eq and the relaxation time σ depend only on the ratio of desorption to adsorption probability. When the desorption rate decreases, the equilibrium coverage increases, as well as the time necessary to bring the system to the steady state. When P_des/P_a→0 the system tends to the close packed state, with the corresponding relaxation time σ→∞.

Kinetics of the adsorption process with diffusional relaxation governed by geometric exclusion effects was studied in Ref. [13]. The adsorbing objects were squares of three different sizes, covering four, nine and 16 lattice sites. The diffusion leads to the formation of large clusters of covered sites. After long enough time, that depends on the diffusion/adsorption probability ratio, a configuration is formed in which almost all sites are occupied. Only few frozen defects, whose dimensions are less than the dimensions of the depositing objects, remain unoccupied. The number of blocked sites in the frozen defects is negligible comparing to the total number of lattice sites, so the final coverage is ρ≃1. The plots of ln(1−ρ(t)) are straight lines in the late stages of deposition for all diffusion probabilities and all investigated object sizes, suggesting the kinetics of the exponential form

$$\text{ρ(t)=1−A}\text{e}{}^{\mathrm{-t/\sigma }}\text{.}$$

Generally, all the three processes: adsorption, desorption and diffusion can proceed in parallel. Here we present the results of the Monte-Carlo simulations of adsorption–desorption processes with diffusional relaxation in order to investigate the way in which the diffusion of the adsorbed particles affects the kinetics of the process.