Steady-state properties of single-file systems with conversion

Abstract

We have used Monte Carlo methods and analytical techniques to investigate the influence of the characteristic parameters, such as pipe length, diffusion, adsorption, desorption, and reaction rate constants on the steady-state properties of single-file systems with a reaction. We looked at cases when all the sites are reactive and when only some of them are reactive. Comparisons between mean-field predictions and Monte Carlo simulations for the occupancy profiles and reactivity are made. Substantial differences between mean-field and the simulations are found when rates of diffusion are high. Mean-field results only include single-file behavior by changing the diffusion rate constant, but it effectively allows passing of particles. Reactivity converges to a limit value if more reactive sites are added: sites in the middle of the system have little or no effect on the kinetics. Occupancy profiles show approximately exponential behavior from the ends to the middle of the system.

Conversion Reactions in Surface-Functionalized Mesoporous Materials: Effect of Restricted Transport and Catalytic Site Distribution

“…An exact description of our discrete reaction-diffusion model is provided by the master equations for the evolution of probabilities of various configurations within the pore. Often these are written in hierarchical form [8,11,[12][13][14]. Here, we use <C n > to denote the probability or ensemble averaged concentration for species C = A or B at site n, <C n E n+1 > for the probability that C is at site n and for site n+1 to be empty (E), etc.…”

“…A simple MF factorization approximation, <C n E n+1 >  <C n ><E n+1 >, etc., produces a closed set of discrete reaction-diffusion equations (dRDE) for single-site concentrations. A higher-level pair approximation retains pair quantities, but factorizes triplets, e.g., <C n M n+1 N n+2 >  <C n M n+1 ><M n+1 N n+2 >/<M n+1 >, with C, M, N = A, B, or E. This generates a closed set of equations for single-site and pair quantities [8,11,[12][13][14].…”

“…For AB conversion in the case where all sites are catalytic, one has that [12][13][14] . In (1), we have defined the discrete derivative, K n = K n -K n-1 .…”

“…In the steady state, sites are randomly occupied by particles, X, with probability X eq = <X out > [12].…”

“…Passing of A with other A (or B with B) is not significant. Several previous analyses exist for the case of single-file diffusion [8][9][10][11][12][13][14][15] revealing that reactivity is strongly localized near the pore openings in this case of no-passing [9]. A perception exists that a simple mean-field (MF) type treatment of chemical diffusion (see below) is adequate [11].…”