Transient behavior in single-file systems

Abstract

We have used Monte Carlo methods and analytical techniques to investigate the influence of the characteristics, such as pipe length, diffusion, adsorption, desorption, and reaction rates on the transient properties of single-file systems. The transient or the relaxation regime is the period in which the system is evolving to equilibrium. We have studied the system when all the sites are reactive and also when only some of them are reactive. Comparisons between mean-field predictions, cluster approximation predictions, and Monte Carlo simulations for the relaxation time of the system are shown. We outline the cases where the mean-field analysis gives good results compared to dynamic Monte Carlo results. For some specific cases we can analytically derive the relaxation time. Occupancy profiles for different distributions of the sites both for the mean field and simulations are compared. Different results for slow and fast reaction systems and different distributions of reactive sites are discussed.

Controlling reactivity of nanoporous catalyst materials by tuning reaction product-pore interior interactions: Statistical mechanical modeling

“…Then, A out will decrease from an initial value of X out , and B out will increase from zero with increasing fraction, F = B out / X out (= 1 − A out / X out ), of the initial reactant converted to product. 14,15 Following most previous stochastic modeling of reactiondiffusion processes in linear nanopores, [8][9][10][11][12][13][14][15] the simplest prescription for diffusion dynamics within the pores is that A and B hop to adjacent empty (E) sites at rate h, corresponding to a diffusion rate of D 0 = a 2 h for isolated particles. This simple prescription would correspond to single-file diffusion with a strict no-passing constraint.…”

“…Often these are written in hierarchical form. 8,[11][12][13][14][15] Here, we use C n to denote the probability or ensemble averaged concentration for species C = A or B at site n (or for this site to be empty when C = E), C n E n+1 for the probability that C is at site n and for site n + 1 to be empty (E), etc. Then, the lowest-order equations in the hierarchy describe the evolution of single-site occupancies.…”

“…Our model for catalytic conversion describes nanoporous materials which consist of a parallel array of linear pores by partitioning the continuous-space pores into adjacent cells labeled n = 1 to L. [8][9][10][11][12][13][14] The cell width "a" is selected to be comparable to the species size ∼1 nm. Species within pores are regarded as localized to specific cells, and diffusive transport is treated as hopping or exchange between adjacent cells.…”

“…Previous analyses for SFD or restricted passing [8][9][10][11][12][13][14][15] reveal that reaction is strongly localized near the pore openings. 9 While simple mean-field (MF) type reactiondiffusion equations 8,[11][12][13] are not adequate, recent studies have shown that behavior in this regime is captured by a "generalized hydrodynamic" (GH) formulation which accounts for both the effect of restricted passing on chemical diffusion as well as fluctuation effects in adsorption-desorption at pore openings. 14 Here, we adopt the latter rather than computationally more expensive kinetic Monte Carlo (KMC) simulation which could also provide a precise characterization of model behavior.…”

Polymer length distributions for catalytic polymerization within mesoporous materials: Non-Markovian behavior associated with partial extrusion

“…This approach has already been successfully applied for reaction-diffusion processes in mesoporous systems with single-file diffusion. 7,14 In our formulation, monomers and polymers reside on individual sites or adjacent strings of sites on this lattice which runs through the pore. The lattice constant is taken as the monomer dimension, and all lengths below are quoted in lattice constants ͑so polymer lengths are measured in monomers͒.…”