Catalytic conversion reactions in nanoporous systems with concentration-dependent selectivity: Statistical mechanical modeling

Statistical mechanical modeling is developed to describe a catalytic conversion reaction $A\to B^c$ or $B^t$ with concentration-dependent selectivity of the products, $B^c$ or $B^t$, where reaction occurs inside catalytic particles traversed by narrow linear nanopores. The associated restricted diffusive transport, which in the extreme case is described by single-file diffusion, naturally induces strong concentration gradients. Furthermore, by comparing kinetic Monte Carlo simulation results with analytic treatments, selectivity is shown to be impacted by strong spatial correlations induced by restricted diffusivity in the presence of reaction and also by a subtle clustering of reactants, $A$.

Figure 1

Schematic of spatially discrete stochastic reaction model for concentration-dependent conversion reaction $A\to B^c$ or $B^t$ in catalytically functionalized linear nanopores described by a 1D array of cells.

Figure 2

KMC results for behavior of $\langle A_n{E}_{n+1}\rangle$ and $\langle E_n{A}_{n+1}\rangle$ (solid curves) relative to their MF approximations (dashed curves) versus $n$ near the left end of a pore with $L=100,\langle X_0\rangle =0.8\left(\mathrm{so}\langle E_0\rangle =0.2\right),k/h=0.001$, and $P_{\mathrm{ex}}=0$ (SFD). Since $\langle E_n\rangle =\langle E_0\rangle$, both MF approximations are determined solely by the variation of $\langle A_n\rangle$ with $n$ (as reflected in the “staircase” construction connecting dashed curves).

Figure 3

KMC results for $f_n=\langle A_n{A}_{n+1}\rangle /\left(\langle A_n\rangle \langle A_{n+1}\rangle \right),g_n=\langle A_n{B}_{n+1}\rangle /\left(\langle A_n\rangle \langle B_{n+1}\rangle \right)$, versus $n$ near the left end of a pore with $L=100,\langle X_0\rangle =0.8,k/h=0.001$, and $P_{\mathrm{ex}}=0$ (SFD). Deviations from unity reflect the strength of the associated NN spatial correlation.

Figure 4

Steady-state concentration profiles for SFD with $L=100$ and $k/h=0.001$: (a) $\langle X_0\rangle =0.2$; (b) $\langle X_0\rangle =0.8$. Comparison of precise behavior obtained from KMC simulation (solid curves) with poor MF predictions (dotted curves) and two successful eGH formulations (dashed curves).

Figure 5

Steady-state concentration profiles with exchange, $P_{\mathrm{ex}}=0.25$, for $L=100$ and $k/h=0.001$: (a) $\langle X_0\rangle =0.2$; (b) $\langle X_0\rangle =0.8$. Comparison of precise behavior obtained from KMC simulation (solid curves) with poor MF predictions (dotted curves). eGH formulations are effectively indistinguishable from precise behavior.

Figure 6

Rescaled local production rates: (a) $R_n\left(B^c\right)/k$ and (b) $R_n\left(B^t\right)/k$ versus $n$ near the left end of a pore with $L=100,k=0.001\mathrm{and}h=1,\langle X_0\rangle =0.8$ (high concentration), and $P_{\mathrm{ex}}=0$ (SFD). Precise results from KMC (solid curves) are well described by our eGH formulations (dashed curves), but not by the MF approximation (dotted curves).

Figure 7

Comparison of $\langle A_n{A}_{n+1}{E}_{n+2}\rangle ,\langle A_n{E}_{n+1}{A}_{n+2}\rangle$, and $\langle E_n{A}_{n+1}{A}_{n+2}\rangle$ versus $n$ determined precisely from KMC simulation and MF approximations near the left end of a pore for $L=100,\langle X_0\rangle =0.8,k/h=0.001$, and $P_{\mathrm{ex}}=0$ (SFD).

Figure 8

Comparison of various conditional reactant concentrations versus $n$ determined precisely from KMC simulation for $L=100,\langle X_0\rangle =0.8,k/h=0.001$, and $P_{\mathrm{ex}}=0$ (SFD).

Figure 9

Comparison of various hybrid approximations, including MF(H) and a crude factorization of pair diffusion terms (c) and a pair factorization (pa) with precise behavior (KMC) and our $\text{eGH}\left(f\right)$ treatment for $L=100,\langle X_0\rangle =0.8,k/h=0.001$, and $P_{\mathrm{ex}}=0$ (SFD).

Figure 10

KMC simulation results for the evolution of concentration profiles starting from an empty pore for $L=100,\langle X_0\rangle =0.8$, and $k/h=0.001$. Times are indicated (in units where $h=1)$ and increase from panels (a) to (e).