17694
J. Phys. Chem. B 2006, 110, 17694-17695
On the Sticking Probability of Aromatic Molecules on Zeolites. Reply to “Comment on ‘STICKING PROBABILITY ON ZEOLITES’” J. Ka¨rger*,† and S. Vasenkov‡ Department of Interface Physics, Leipzig UniVersity, Linne´ strasse 5, 04103 Leipzig, Germany, and Chemical Engineering Department, UniVersity of Florida, P.O. Box 116005, GainesVille, Florida 32611-6005. ReceiVed: July 11, 2006 In ref 1 we used the results of molecular dynamics simulations (for n-butane in NaX) and the evidence of long-range diffusion as provided by pulsed field gradient (PFG) NMR (notably for n-alkanes in zeolite NaX) for an estimate of the sticking probability, that is, of the probability that molecules upon encountering the outer surface of a zeolite crystal are able to continue their trajectory into the interior. In both ways of analysis, the sticking probabilities turned out to be close to 1, that is, orders of magnitude larger than the only data for sticking probabilities communicated so far for zeolitic host-guest systems.2,3 In the present comment4 to our paper,1 a careful reconsideration of the arguments given in ref 2 and extended in ref 3 is provided, confirming the estimates of the previous studies yielding sticking probabilities of aromatics in the MFItype zeolites under study on the order of 10-6. We completely agree with the authors of this comment that there is no nature-given lower limit of the sticking probability. Irrespective of its great practical relevance for transportoptimized nanoporous catalysts and adsorbents and its attractiveness from a quite fundamental point of view, our understanding of the phenomena associated with the transition from the gas phase into the intracrystalline space and vice versa is still at the very beginning. There is no question that all new information has to be primarily based on experimental evidence, so that any additional method of experimental observation is more than welcome. In our response, we would like to highlight two important consequences on the molecular trajectories one may immediately deduce from the measured sticking probabilities. Complementing the experiments2-4 and molecular modeling, as presented, for example, in refs 1 and 5, these quite general considerations are inevitable for the complete understanding of matter exchange between the gaseous and adsorbed state within nanoporous materials. (i) Interrelation between the Sticking Probability and the Effectiveness Factor of Ensembles of Nanoporous Particles (Zeolite Crystallites). With R denoting the sticking probability, its reciprocal value, R-1, represents the mean number of collisions between a molecule and the surface of a nanoporous particle necessary to let it enter, that is, to permit the molecule to be subject to the conditions relevant for the desired technological effect of, for example, matter separation or conversion. Thus, with sticking probabilities of about 10-6, as reported in refs 2 and 3, only after one million collisions will molecules from the gas phase be able to enter the intraparticle space. To ensure an adequate technological performance, a large enough * Corresponding author. † Leipzig University. ‡ University of Florida.
number of particles within a bed (and within a compacted multiparticle pellet, respectively) has to be chosen. Under the Knudsen conditions (i.e., for a negligibly small probability of mutual encounters of the molecules in the intercrystalline space as the relevant case for diffusion through the transport pores of compacted zeolite catalysts6), this means in particular that each catalyst particle should contain on the order of at least 106 zeolite crystallites to ensure that a notable percentage of the molecules that have entered the catalyst particle will be captured by one of the zeolite crystallites of this particle. (ii) Dynamic Equilibrium between the Molecules Entering and Leaving the Nanoporous Particles (Zeolite Crystallites). The number of molecules colliding with the particle surface from the gas phase may be estimated via the relation7
1 p jgs ) cgu ) 4 x2πkTm
(1)
with cg, p, u, and m respectively denoting the gas-phase concentration, the pressure, and the mean thermal velocity of the molecules of mass m. This flux, multiplied with the sticking probability, has to coincide with the flux jig of particles leaving the particle under (dynamic) equilibrium. Hence, knowledge of jig would provide an alternative access to the sticking probability. The exploration of procedures to determine this quantity experimentally is among the current challenges of zeolite science and technology. Approximatively, however, one may extend a procedure introduced in ref 8 for intracrystalline diffusion to estimate the rate of molecular escape from the last layer of cavities into the gas phase. In refs 8 and 9, intracrystalline diffusion is treated as if it would take place in a three-dimensional sinusoidal potential landscape, leading to the simple expression
D)a
( )
x
ED ED exp 2m kT
(2)
with a and ED denoting the separation of adjacent potential minima and the difference in the heights of the potential maxima and minima (“activation energy” of self-diffusion), respectively. Estimates based on eq 2 have been shown to correctly yield the order of magnitude of the experimental data.9 Considering molecular diffusion as a simple random walk by jumps between the minima of the potential energy, the jump rate is simply
1/τ ) 6D/a2 )
( )
x
6 a
ED ED exp 2m kT
(3)
The jump from the last potential minimum close to the crystal boundary into the surrounding atmosphere is now modeled in the most simple way: we assume that it only differs from a jump between adjacent minima in the intracrystalline space by an enhancement of the activation energy from ED to EA, that is, to the heat of adsorption. Using Fick’s first law, this leads to the following expression for the flux of molecules leaving the crystal:
jig )
( )
D0 EA n exp 4 kT a
(4)
where D0 denotes the preexponential term of the intracrystalline 10.1021/jp068072l CCC: $33.50 © 2006 American Chemical Society Published on Web 08/09/2006
Comments diffusivity and n stands for the number of guest molecules per volume element a3. Let us have a look now at the sticking coefficients R ≡ jig/jgs which would result on the basis of this rough estimate from the literature data for the two main systems discussed in these comments, viz. for benzene in zeolite MFI and for ethane in zeolite NaX. In both cases, as a typical length scale of the intracrystalline zeolitic potential, we set a ) 1 nm, following ref 9. For benzene in silicalite-1, at T ) 395 K and p ) 1.53 mbar, one may find in ref 10 that D0 ) 10-8 m2 s-1, n ) 0.59 molecules per unit cell, and EA ) 50 kJ/mol. Inserting these values into eqs 1 and 4 leads to jig/jgs ≡ R ) 1.2 × 10-4. In ref 9, one may find D0 ) 4 × 10-7 m2 s-1, T ) 300 K, and a value of (n/p) exp(- (EA/kT)) ) 4 × 10-8 molecules per unit cell and Pascal () preexponential factor of the Henry coefficient) for ethane in NaX. Following the same procedure, the sticking coefficient for this system is now found to be R ) 0.04. As evidenced already by this simple estimate, the sticking coefficients of different systems may differ dramatically. Moreover, the present data nicely reflect the experimentally observed tendency that the sticking probability of benzene on silicalite-1 is expected to be much smaller than that of ethane on zeolite NaX. The roughness of the model applied will clearly lead to some arbitrariness in the attained values, which may have contributed to the differences between the estimates by these model considerations and the outcome of the experimental studies, where a difference of up to 5 orders of magnitude between the considered systems was stated. It has to be emphasized that the present estimate is unable to reflect the effect of surface collapse or pore blockage. Any of such effects would simultaneously lead to a decrease of the
J. Phys. Chem. B, Vol. 110, No. 35, 2006 17695 fluxes entering and leaving the crystal so that dynamic equilibrium is maintained without any change in the equilibrium properties. Such a situation would lead to a dramatic reduction of the sticking probability in comparison with the present estimate and provides the most likely explanation for the dramatic reduction of the sticking probability of benzene by the MFI-type zeolites observed experimentally2,3 in comparison with the present estimate. This situation has to be clearly differentiated from an alternative case of reduced sticking probability, viz. if (too bulky) molecules are only able to enter the pore system by assuming very special molecular conformation. In this case, dynamic equilibrium is established by a simple reduction in the equilibrium concentration n, which reduces jig to exactly the same extent to which the sticking probability decreases. References and Notes (1) Simon, J.-M.; Bellat, J.-B.; Vasenkov, S.; Ka¨rger, J. J. Phys. Chem. B 2005, 109, 13523. (2) Tanaka, H.; Zheng, S.; Jentys, A.; Lercher, J. A. Stud. Surf. Sci. Catal. 2002, 142, 1619. (3) Jentys, A.; Tanaka, H.; Lercher, J. A. J. Phys. Chem. B 2005, 109, 2254. (4) Jentys, A.; Mukti, R. R.; Lercher, J. A. J. Phys. Chem. B 2006, 110, 17691. (5) Skoulidas, A. I.; Sholl, D. S. J. Chem. Phys. 2000, 113, 4379. (6) Ka¨rger, J.; Vasenkov, S. Microporous Mesoporous Mater. 2005, 85, 195. (7) Evans, J. W.; Abbasi, M. H.; Sarin, A. J. Chem. Phys. 1980, 72, 2967. (8) Ruthven, D. M.; Doetsch, I. H. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1043. (9) Ka¨rger, J.; Pfeifer, H.; Rauscher, M.; Walter, A. J. Chem. Soc., Faraday Trans. 1 1980, 76, 717. (10) Song, L. J.; Rees, L. V. C. Microporous Mesoporous Mater. 2000, 6, 301.
