Cooperative sequential adsorption models on a Cayley tree: analytical results and applications

Abstract

We present a class of cooperative sequential adsorption models on a Cayley tree with constant and variable attachment rates and their possible applications for ionic self-assembly of thin films and drug encapsulation of nanoparticles. Using the empty interval method, and generalizing results known from reaction-diffusion processes on Cayley trees, we calculate a variety of quantities such as time-dependent surface coverage and time-dependent probabilities of certain particle configurations.

Layer-by-Layer Cell Encapsulation for Drug Delivery: The History, Technique Basis, and Applications

“…As suggested, the degree of material attaching is proportional to concentration, which was even verified in a math model [127]. This effect is common in most cases, although hydrodynamics affects it sometimes [138]. As concentration of polyelectrolytes increased, the adsorption speed increased accordingly.…”

An introduction to stochastic self-assembly: theory, simulation and experimental applications

“…The dendrimers are not rigid structures and they can change their shape and orientation depending on a variety of factors, such as pH, temperature, and concentration of drug molecule suspension. Over the years, dendritic polymers and their interactions with drug molecules have been modeled using the following general methods: equilibrium and non-equilibrium molecular dynamics [44,45]; equilibrium partition function calculations using the Ising model for localized electrostatic interactions between the drug molecules and the charged nodes of the dendrimer [46]; shell-like dendrimer models with a continuous and uniform charge distribution [47]; cooperative sequential adsorption models solved using the empty interval method [15]. CSAE models are ideal for describing drug encapsulation and release because (i) the deposition process of the drug nanoparticles is stochastic and can be modeled by sequential adsorption models; (ii) the deposited drug nanoparticles are electrically charged, as are the substrate deposition sites, suggesting a cooperative model with deposition rates dependent on nearest-neighbor site occupation; (iii) the drug nanoparticles have a probability of detachment, which is incorporated in the model via an evaporation rate.…”

“…There are many computational adsorption models [11], but few analytical solutions have been developed for the general two-dimensional case. Recently, analytical results have been reported for the random sequential process [8] and reaction-diffusion processes on Cayley trees and Bethe lattices [12,13,14,15,16,17]. Adding the possibility of particle detachment, or evaporation, to such models brings additional complications.…”

A two-state stochastic model for nanoparticle self-assembly: theory, computer simulations and applications

“…The dendrimers are not rigid structures and they can change their shape and orientation depending on a variety of factors, such as pH, temperature, and concentration of drug molecule suspension. Over the years, dendritic polymers and their interactions with drug molecules have been modelled using the following general methods: equilibrium and nonequilibrium molecular dynamics [19,20]; equilibrium partition function calculations using the Ising model for localized electrostatic interactions between the drug molecules and the charged nodes of the dendrimer [21]; shell-like dendrimer models with a continuous and uniform charge distribution [22]; CSA models solved using the empty interval method [23]. We model the dendrimer as a Cayley tree and address the drug encapsulation process using the CSAE model presented in equation (1).…”