The eigenpairs of a Sylvester-Kac type matrix associated with a simple model for one-dimensional deposition and evaporation

This random sequential model is quite general and versatile, and can be customized to describe a variety of two-state physical systems that exhibit adsorption and evaporation processes. One-dimensional sequential adsorption models have been studied thoroughly in different physics contexts [17, 34]. Adsorption in two dimensions is not as well understood, however. There are quite a few computational papers [17] on the topic, but few analytical solutions exist for the general two-dimensional case. The adsorption of particles is exactly solvable in higher dimensions only for tree-like lattices. Recently, analytical results have been reported for the random sequential process [8] and reaction-diffusion processes on Cayley trees and Bethe lattices [2, 19, 29]. The standard method used to study these systems is the empty-interval method [27]. This mathematical method fails when evaporation is considered. We here demonstrate that a matrix theory approach can lead to exact results for a variety of physical systems. Two specific experimental topics motivate our paper. One is the self-assembly mechanism of charged nanoparticles on a glass substrate [24]. Known in literature as ionic self-assembled monolayers (ISAM), this technique has been used successfully in making antireflective coatings [14, 40]. Physical properties of these coatings depend upon the surface coverage of the substrate. The deposition process is stochastic, with particles attaching to and detaching from the substrate, so a random sequential adsorption model with evaporation is appropriate. The second motivating experimental setting involves properties of synthetic polymers called dendrimers, which have potential use as a drug delivery mechanism via drug encapsulation [26]. Dendrimers are physical analogs of Cayley tree structures. They are highly