2 4 Se p 19 98 Sluggish Kinetics in the Parking Lot Model

We investigate, both analytically and by computer simulation, the kinetics of a microscopic model of hard rods adsorbing on a linear substrate. For a small, but finite desorption rate, the system reaches the 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. A mean-field approach fails to predict the relaxation rate associated with the latter. We show that the correct answer can only be provided by using a systematic description based on a gap-distribution approach. In many situations, thermal energies are significantly smaller than the energy needed for hard particles to diffuse, and under applied external forces the system evolves towards non-equilibrium configurations, metastable configurations, or eventually reaches a stable equilibrium state after a very slow process. For example, the adsorption of some proteins and colloidal particles on solid surfaces[1] involves particle-surface energies that are so strong that the process is characterized by extremely small desorption and surface diffusion on the experimental time scale. In granular materials, particles are trapped in a metastable configuration, unless external energy is brought to the system. Recent experiments[2, 3] measured the densification of a vibrated granular material. A column containing monodisperse spherical beads was tapped periodically with a given intensity and the powder evolved slowly, essentially as the inverse of the logarithm of the number of taps, from a loosely packed state to a denser steady state whose density depends on the tapping strength. In all these cases, geometric exclusion effects dictate the kinetics of densification, i.e., addition of new particles is exponentially limited by the inverse of the free volume[3, 4]. These effects are accounted for in a simple adsorption-desorption model[3, 5, 6]. Its one-dimensional version, also known as the parking lot model, was shown to display a 1/ ln(t) approach to the final state for vanishing desorption rate[5, 6]. More recently, Ben Naim et al.[7] proposed an approximate solution of the model for a small but non-zero desorption rate. We discuss here the solution and show that the mean-field treatment (or adiabatic approximation) used in the late stage of the process is not valid and greatly underestimates the characteristic time of relaxation. We develop a more consistent approximation which shows good agreement with the numerical simulations. The general definition of the adsorption-desorption model is as …