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

Abstract

Statistical mechanical modeling is performed of a catalytic conversion reaction within a functionalized nanoporous material to assess the effect of varying the reaction product-pore interior interaction from attractive to repulsive. A strong enhancement in reactivity is observed not just due to the shift in reaction equilibrium towards completion but also due to enhanced transport within the pore resulting from reduced loading. The latter effect is strongest for highly restricted transport (single-file diffusion), and applies even for irreversible reactions. The analysis is performed utilizing a generalized hydrodynamic formulation of the reaction-diffusion equations which can reliably capture the complex interplay between reaction and restricted transport. Statistical mechanical modeling is performed of a catalytic conversion reaction within a functionalized nanoporous material to assess the effect of varying the reaction product-pore interior interaction from attractive to repulsive. A strong enhancement in reactivity is observed not just due to the shift in reaction equilibrium towards completion but also due to enhanced transport within the pore resulting from reduced loading. The latter effect is strongest for highly restricted transport (single-file diffusion), and applies even for irreversible reactions. The analysis is performed utilizing a generalized hydrodynamic formulation of the reaction-diffusion equations which can reliably capture the complex interplay between reaction and restricted transport.

Multi-functionalization of nanoporous catalytic materials to enhance reaction yield: Statistical mechanical modeling for conversion reactions with restricted diffusive transport

“…Here, <C n > denotes the probability for species C = A or B at site n, <C n E n+1 > for the pair probability that C is at site n and for site n+1 to be empty (E), etc. One has for 1<n<L that [6,7] d/dt <A n > = -k<A n > +k′<B n > -∇J A n→n+1 , d/dt <B n > = +k<A n > -k′<B n > -∇J B n→n+1 , (1) with separate equations for terminal sites reflecting adsorption-desorption boundary conditions.…”

“…An exact description of our model is provided by the hierarchical master equations for the evolution of probabilities of various configurations within the pore [6,7]. Here, <C n > denotes the probability for species C = A or B at site n, <C n E n+1 > for the pair probability that C is at site n and for site n+1 to be empty (E), etc.…”

“…SFD within the pores corresponds to A and B hopping only to adjacent empty (E) sites at rate h. SFD applies in the reasonable scenario where, e.g., the effective pore diameter, d, is ~1.5 nm, and the effective linear diameters of reactants and products are around ~0.75 nm or above. Other mechanistic steps are [6]: (i) Adsorption of external species at terminal sites of the pore at rate h<A o > (αh<B o >) for A (B), provided that these end sites are unoccupied or empty (E); α is the key parameter capturing product-pore interaction with α<1 (α>1) corresponding to unfavorable (favorable) interaction. (ii) Desorption of A or B from terminal sites at rate h when the fluid cell just outside the pore is unoccupied (which has probability <E o > = 1 -<X o >); (iii) Conversion A → B at catalytic (c) sites within the pore at rate k, and B → A at rate k′; in our modeling, c-sites occupy the entire pore.…”

“…(ii) Desorption of A or B from terminal sites at rate h when the fluid cell just outside the pore is unoccupied (which has probability <E o > = 1 -<X o >); (iii) Conversion A → B at catalytic (c) sites within the pore at rate k, and B → A at rate k′; in our modeling, c-sites occupy the entire pore. The equilibrium fractional conversion is given by F eq = K eq /(K eq + α) [6]; K eq = k/k′ is the equilibrium constant for the reaction inside the pore. Thus, blocking of product reentry (α=0) ensures that the reaction proceeds to complete conversion.…”

“…The tunable interaction of the pore interior with reaction product(s) is captured by a single parameter, α, which determines the ease of product reentry to the pore [6]. We utilize spatially-discrete coarsegrained (CG) stochastic models wherein continuous space inside the pore is divided into cells of width ~1 nm comparable to reactant/product size, and diffusion is described by hopping or exchange between adjacent cells [6,7].…”