Analytical Estimate of the Entering Probability of Molecules into

J. Phys. Chem. C 2007, 111, 11285-11290

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Analytical Estimate of the Entering Probability of Molecules into Crystalline Nanoporous Materials Andreas Schu1 ring Institute for Theoretical Physics, UniVersity of Leipzig, Vor dem Hospitaltore 1, D-04103 Leipzig, Germany ReceiVed: February 14, 2007; In Final Form: April 19, 2007

This paper considers the probability that a molecule at the surface of a nanoporous material, e.g., a zeolite, will be able to enter the intracrystalline space. A relation is derived between the entering probability and the self-diffusion coefficient for the case of an ideal surface. It is known that the flux through the surface is limited mainly by a certain energy difference, ∆E, (Barrer, R. J. Chem. Soc., Faraday Trans. 1990, 86, 1123) and, for real crystals, the fraction of open pores b e 1. With the help of the relation presented here, it is possible to separate the two different effects responsible for transport resistance caused by the surface. This opens new perspectives for the characterization of real surfaces. Furthermore, it is shown that ∆E must be less than 50% of the heat of adsorption and, surprisingly, can even practically vanish in special cases. The relation is tested using molecular dynamics simulations of ethane adsorbing into silicious zeolite LTA. The results of the simulations show in all cases excellent agreement with the theoretical predictions. A discussion of the discrepancies between previously reported values of the entering probability shows that, indeed, values differing over several orders of magnitude may be observed.

1. Introduction Crystalline nanoporous materials such as zeolites1 have welldefined pore dimensions, which make them selective with regard to the shape and the size of the molecules trying to enter into the crystals. This beneficial property is utilized on the industrial scale for processes of catalysis, gas separation, and absorption. The laws of diffusional transport are widely studied for the intracrystalline diffusion of nanoporous materials.2-7 However, in all such applications, molecules have to pass through the surfaces of the crystals. Consequently, the focus of research using molecular simulations has increasingly switched over to the boundaries of the crystals to study the elementary steps of adsorption and desorption and their influence on the overall transport. Starting from simulations considering the steric hindrance during adsorption,8-10 now computationally more expensive grand-canonical molecular dynamics (GCMD) simulations studying cases of nonequilibrium,11-14 and even simulations of systems of the size of zeolite crystals used in catalysis15,16 have been carried out. Theoretical studies considering the one-dimensional transport through membranes are given in refs 17-20. The different steps during the transport through a membrane are classified as follows:19 (1) Adsorption from the gas phase to the external surface. (2) Mass transfer from the external surface into the zeolite pores. (3) Intracrystalline diffusion. (4) Mass transport out of the zeolite pores to the external surface. (5) Desorption from the external surface to the gas phase. The probability that a molecule that comes from the gas phase will be able to enter the intracrystalline space of a porous material in a subsequent step is therefore a key quantity for the understanding of sorption kinetics and molecular transport through membranes. The first papers focusing on this quantity21,22 reported values differing over several orders of magnitude. * Tel. ++49 341 235 2589, Fax ++49 341 235 2307, E-mail: [email protected].

The entering probability is defined as the ratio between the number Nin of molecules entering the crystal and the total number Nenc of molecules encountering the surface during a certain time interval

Penter )

Nin Nenc

(1)

In this article, it is proposed to use the term entering probability, Penter, for this quantity, since the terms sticking probability and sticking coefficient (used in refs 21-24) are widely used in literature for the fraction of molecules which adsorb on surfaces of impermeable solids (e.g., ref 25) and clusters (e.g., ref 26). The sticking coefficient is related to the first step in the above scheme of membrane transport, while in this paper, the second step is considered. The same differentiation has been used also in ref 18. The phenomenon of flux reduction by the surface is widely called the surface barrier (also surface resistance and surface diffusion are in use). However, behind this term hide a number of different effects. Some part of the molecules encountering the surface will be scattered and immediately return to the gas phase. The fraction of molecules which actually adsorb on the surface for a longer period is given by the sticking coefficient. This part of molecules can be assumed to relax, or thermalize, on the surface so that equilibrium statistical mechanics may be applied to determine the transition rates in the direction of either the crystal or the gas phase. At this point, it must be distinguished whether the surface is ideal, i.e., the crystal is simply “cut” and open bonds are saturated by hydrogen, or nonideal. Cases of nonideality are studied widely in experiments, e.g., the effect of hydrothermal treatment of the crystals leading to structural defects,27,28 the effect of removing defects by etching the crystals,29 or the obstruction of the surface by other molecular species which can also be used to tailor the desired pore diameter.30 Both cases can, in principle, lead to obstruction of a fraction of the pores or to a narrowing of the pores which still allows the molecules to enter. The latter case can lead to a rise or increase of a potential energy barrier for entering the pore. It is the idea of this article to show, by molecular dynamics simulations, that the reduction of the flux caused by an ideal

10.1021/jp071276x CCC: $37.00 © 2007 American Chemical Society Published on Web 07/12/2007

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surface is predictable because it is related to the diffusion coefficient and the equilibrium constant. On the other hand, if the entering probability is determined experimentally like in ref 21 and the diffusion coefficient and the equilibrium constant are known, it is possible to determine the parameters characterizing the transport resistance of the surface caused by the energetic effect and by the pore blockage, respectively. The derivation of the relation between the diffusion coefficient and the entering probability follows in section 2. Details of the simulations carried out are given in section 3, and the results are presented in section 4, which, furthermore, contains a discussion of values of Penter reported in the literature. 2. The Entering Probability For the derivation of Penter, the following assumptions are made about the considered system: (1) In the periodic zeolite lattice (latticed constant L), the molecules prefer certain positions (adsorption sites) which are separated by regions of finite but considerably lower residence probability (pore openings). (2) The intracrystalline diffusion is a random walk with a correlation length equal to the lattice constant. (3) Transition-state theory (TST) can be applied to calculate the jump rate of the diffusion process. (4) For an adsorption site at the crystal edge, the pore opening facing the surface has the same geometry as the opposite pore opening facing the crystal. (5) The potential energy in the surface-facing pore opening is (comparing equivalent positions) by a difference ∆E higher than in the zeolite-facing pore opening, which is, furthermore, assumed to behave like any intracrystalline pore opening. (6) The distribution of the molecules in the gas phase is uniform. (7) All molecules which reach the surface relax on the surface so that equilibrium statistical mechanics may be applied to determine the transition rates in direction of either the crystal or the gas phase. In the context, the assumptions are discussed in more detail. The derivation is made considering a zeolite membrane. In the case of relaxation, the numbers of molecules entering the crystal or returning to the gas phase, respectively, are proportional to the number of molecules that arrived at the surface. The proportionality factors are the transition rates, which can be obtained following transition-state theory by the equation

k ) Vj

qA QV

(2)

where Vj ) xkBT/2πm is the mean thermal velocity of the molecules of mass m at temperature T perpendicular to the separation plane31 of area A. qA is the configuration integral in the separation plane between two states. The configuration integral in the initial state, which is delimited within a certain volume V, is QV. kB is the Boltzmann constant. By using eq 2, the definition of the entering probability, eq 1, transforms to a relation containing the configuration integrals in the separation planes (see Figure 1) between zeolite surface and the sites at the crystal edges, qsz, and between the zeolite surface and the gas phase, qsg

Penter )

qsz qsz + qsg

(3)

This equation is of general validity, since it may be extended to the whole crystal surface including effects of nonideality as discussed above.

Figure 1. Left: Definition of the configuration integrals used in the derivation; see text. Right: Position where the crystal is cut. Within the zeolite fragments shown, oxygen is given in dark gray and silicon in light gray.

Figure 2. Schematic to depict the origin of the energy difference ∆E. Close to the surface (left), a molecule is surrounded by fewer zeolite atoms compared to an equivalent position inside the crystal (right).

The direct computation of configuration integrals is difficult and might be impossible in practice. However, for the case of an ideal crystal surface we can make use of the fact that these configuration integrals are related to other quantities which are accessible by MD simulation as well as by experiments: the diffusion coefficient D and the equilibrium constant K. The configuration integral qsz occurs in the transition rate of desorption, i.e., for molecular movement from an adsorption site at the crystal edge to the surface

kdes ) Vj

qsz Qzeol

(4)

Qzeol is the configuration integral in the adsorption site at the crystal edge. For an ideal crystal, it can be assumed that the geometry of the pore openings to the surface is practically the same as that between the sites within the crystal, but the molecule is surrounded by fewer zeolite atoms; see Figure 2. Then, the potential energy values for respective configurations are shifted by a potential energy difference ∆E,11,16,19 so that the transition rate for intracrystalline diffusion, kD ) D/L2 (this relation is valid, as long as correlation effects, which occur at high loadings due to mutual hindrance2,32 but also at zero loading due to geometric effects,33 are negligible), where L is the cageto-cage distance, may be used to obtain kdes

kdes )

D -β∆E ‚e L2

(5)

For a more general description (including effects of nonideality like changes in the pore geometry), the difference in the free energy has to be taken. However, it is necessary to point out that surface effects for ideal crystals are primarily of energetic nature. An estimation of ∆E is difficult for the following reason: the exact configuration of zeolite atoms which build the surface is not known. The values for reasonable configurations can be obtained from the potential energy landscape resulting from force field calculations. Later in this paper, it is shown how ∆E can be obtained from experiments. A compari-

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son to the values obtained from the force field could then allow one to draw conclusions about the surface of the real crystals under study. The range of possible values lies reasonably between 0 < ∆E < (1/2)Hads, where Hads is the heat of adsorption. This is because the neighboring atoms are removed only at one side; hence, maximal 50% of the atoms are removed compared to an equivalent position inside the crystal. The actual value will depend on the density of the zeolite atoms and the distance between the surface forming atoms and the pore opening. Interestingly, the same Boltzmann factor, e-β∆E, slowing down the desorption11,16 (compared to the intracrystalline diffusion) occurs in the description of the adsorption process. To the author’s knowledge, this has been recognized first by Barrer19 (symbol δ∆E in Barrer’s analysis). The configuration integral qsg can be obtained from the equilibrium constant for which

K)

Qzeol Qgas

(6)

holds. If we assume a uniform density in the gas phase (which holds if no liquid ordering effects are given), the configuration integral Qgas is simply

∫x

xsg

Qgas )

q dx ) qsgL -L sg

(7)

sg

where xsg is the position of the separation plane between surface and gas phase. Using the eqs 4-7, eq 3 may be transformed to

(

Penter ) 1 + ≈

LVj DK e-β∆E

)

-1

DK -β∆E ‚e LVj

(8) (9)

The latter approximation holds for entering probabilities of ,1. This value of Penter may be further reduced by pore blockage, which may be captured by multiplying Penter by the fraction of open pores, b. Provided that the pore blockage does not depend on the temperature (this could be the case for structural defects closing some pores), it should be possible to obtain ∆E and b experimentally from the temperature dependence from the slope and intercept of

ln

LVj(T)Penter(T)

)-

D(T)K(T)

∆E + ln b kBT

(10)

On the left-hand side, the values of the quantities at the respective temperature have to be taken. Finally, an upper limit of the entering probability can be calculated by

Penter e

DK LVj

(11)

since all other effects that may influence the entering probability will actually lower it: cases of reflected molecules, values of ∆E > 0, cases of pore blockage on nonideal surfaces, and also the relation eq 8 give a lower value than the approximated one in eq 9. 3. Computational Details Model of the Guest-Host System. The system under consideration is the silicious variant of zeolite LTA34 with ethane

as a guest molecule. Zeolite LTA has a cubic structure of large, almost spherical R-cages in a distance of a ) L ) 11.87 Å. These cages are separated from each other by narrow ring structures (8-rings) formed by 8 silicon and 8 oxygen atoms, which can be seen in Figure 1. The size of a CH3 group of ethane is approximately equal to the size of the 8-ring. Therefore, the ethane molecule can pass the ring only when the C-C vector is almost perpendicular to the 8-ring plane. This system has been thoroughly studied in previous works,33,35 where the computational model is described in detail. For the current simulations of surface processes, a membrane of zeolite LTA is considered. The surface has been modeled by cutting the crystal in the center of the large cage of zeolite LTA; see Figure 1. Computational Model. The zeolite is modeled with a flexible lattice in order to allow energy exchange between the ethane molecules and the lattice atoms. This is of importance for the relaxation of the guest molecules at the surface. The energy exchange with the vibrating lattice is the most realistic method to thermalize the guest molecules. Therefore, an additional thermostat is needed only in the equilibrium phase of the simulation to bring the whole system close to the desired temperature. Lennard-Jones potentials

V(r) ) 4

[(σr ) - (σr ) ] 12

6

(12)

have been assumed for the interaction between the CH3 groups and the lattice oxygen atoms. The energy parameter was chosen to  ) 1.18 kJ mol-1, 35 and the cutoff was 0.8 nm. In order to demonstrate the significant dependence of Penter on the size of the pore opening if the molecule is equally large or larger, the size-determining parameter σ has been varied within the range of values (0.30 nm < σ < 0.34 nm) known from the literature36 for the CH3-O interaction. The lattice vibrations are modeled using the anharmonic model introduced in ref 37 and taking the experimental equilibrium distances given in the supplementary information of ref 34. Simulations: Simulations have been carried out at a temperature of T ) 300 K, 2 different particle numbers, N, in the system (24 and 96 molecules, respectively), and 5 different window diameters (by varying the interaction potential as described above). Simulation times where chosen between 50 and 500 ns, depending on the magnitude of the entering probability, aiming to register at least 100 successful (entering) events (only the simulations with the smallest pore diameter, σ ) 3.4 Å, could not fulfill this condition). After equilibration, intracrystalline concentrations of 1.3 (at N ) 24) and 4.4 (at N ) 96) molecules per R-cage resulted, while the pressures in the gas phase were 1.7 and 18 bar, respectively. Additional simulations of the intracrystalline diffusion were carried out using these concentrations in order to determine the intracrystalline diffusion coefficients. Evaluation of the Quantities. Penter is evaluated directly from the molecular trajectories applying the definition in eq 1. During the simulation, the position of the center of mass of the molecule is calculated. All crossings of the transition states are registered. Due to the flexibility of the lattice, the positions of the transition states vary and the crystal structure can translate in the simulation box. In order to reduce the technical problems which arise from this fact, some of the surface atoms (chosen in such a way that an influence on the dynamics of the atoms forming the pore openings may be neglected) have been fixed by harmonic potentials around their equilibrium positions in the simulation box. This prevents translational movement of the

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Figure 3. The potential energy of an ethane molecule on an axis through the pore openings of the LTA zeolite. The molecule is oriented parallel to this axis. Note that the potential energy is (for small values of σ) minimal in the pore openings. Curve 1 results if the LTA zeolite is cut in the center of the large cage. Curve 2 results if the zeolite is cut directly before the pore opening. In the latter case, ∆E ≈ 2.5 kJ mol-1.

zeolite structure but allows oscillations. The equilibrium xcoordinate of the atoms forming the outer pore openings (these all lie in one plane) on each side of the membrane is considered the position of the respective transition state. The transition state between the surface and the gas phase is assumed at a distance of 12 Å from the transition-state surface-zeolite. A molecule that came from the gas phase to the surface is considered to have entered the crystal successfully, if the center of mass has moved 1 Å beyond the transition-state surface-crystal into the crystal. This assures the entering of both CH3 groups into the R-cage, and furthermore, this takes into account the movement of the lattice. The self-diffusion coefficient is evaluated from additional simulations of a bulk crystal at the same intracrystalline concentration as in the membrane simulations. It is calculated from the mean square displacement, 〈(∆r b)2〉, of the 2 particles using the Einstein relation 〈(∆r b) 〉 ) 6Dt. The equilibrium constant, K, is obtained as the ratio between the average number of molecules in the crystal unit cell and an equivalent volume in the gas phase. The quantity ∆E has been obtained from the potential energy curve of an ethane molecule lying on and oriented parallel to an axis going through the centers of the 8-rings of zeolite LTA. 4. Results and Discussion For the surface shown in Figure 1, the potential energy difference ∆E was found to be practically zero; see curve 1 in Figure 3. This can be explained by the distance of 0.65 nm between the plane containing the pore opening and the plane where the crystal has been cut. In a distance of 2σ, the value of the Lennard-Jones potential, eq 12, is only the 16th part of the minimal value. The second curve in Figure 3 shows the potential energy if the crystal is cut directly before the pore opening. In this case, the difference is significant, ∆E ≈ 2.5 kJ mol-1. However, this value is small compared to the minimal value of the potential. This result is remarkable, since the major part of the energetic step between gas and zeolite occurs already at the surface. In another case, for the zeolite-like material AlPO4-5, the ratio between ∆E and the heat of adsorption was >0.3,11 so that e-β∆E may become considerably small. Figure 4 shows the entering probabilities for the considered guest-host system as a function of the self-diffusion coefficient. Reference to the respectively used values of the σ-parameter

Schu¨ring

Figure 4. The entering probability Penter. Squares represent the directly computed values from eq 1; triangles hold for the values obtained using eq 8. The filled symbols represent the results obtained from the simulations including N ) 24 molecules, and the open symbols those for N ) 96. The σ labels indicate the different values of the σ parameter of the Lennard-Jones interaction between CH3 groups of ethane and the oxygen atoms of the zeolite. These are σ1 ) 0.30 nm, σ2 ) 0.31 nm, σ3 ) 0.32 nm, σ4 ) 0.33 nm, and σ5 ) 0.34 nm. An increasing value of σ corresponds to a decreasing free diameter of the window and, therefore, to a decreasing self-diffusion coefficient.

Figure 5. The residence time distribution at the surface of the zeolite for the two concentrations considered. The lines are exponential fits to the long-term behavior of the distribution.

of the Lennard-Jones potential are given in the figure. An increasing value of σ corresponds to a decrease of the free diameter of the window and, hence, a smaller self-diffusion coefficient. Compared are the values obtained directly from the trajectories using eq 1 and the values obtained using the formula derived in this article (eq 8). For both molecular concentrations (referenced by the total number N of molecules in the MD box), the agreement is excellent. The predicted values are slightly larger, which is in accordance with the fact that a certain fraction of molecules might be back-scattered, i.e., the sticking coefficient is lower than 1. The factor between the directly computed Penter (eq 1) and the value obtained by eq 8 is in the case of the lower concentration (N ) 24) approximately 0.7 on average, and at the higher concentration (N ) 96) ∼0.9. These values are comparable to those observed for other systems.25,26 In the following, it has been attempted to determine the sticking coefficient from the residence time distribution at the surface, which is shown in Figure 5. The long-term behavior of the distribution may be described by an exponential function, f(t) ) a ‚ exp{-t/τ}, but the number of short visits of the surface is significantly higher than predicted by the exponential. The

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Figure 6. Surface atoms (left: Si, yellow; O, red) assumed in the simulations and the area accessible for the ethane molecules (right). The latter shows the 3D shape of the accessible area defined by the closest visits registered during the simulation. The color dimension represents the number of visits in the respective region.

number of events under the exponential divided by the total number of events is 0.4 for N ) 24 and 0.9 for N ) 96. These fractions correspond for N ) 96 well and for N ) 24 roughly to the sticking coefficients determined above. The time constant, τ, is smaller in the case of the higher concentration. This can be explained with an increased probability for multiple occupations of the surface sites. The distributions in Figure 5 need some further consideration. For N ) 96, a peak at very short times is visible. This is probably related to back-scattering events. However, the significant peak around some picoseconds could rather be interpreted as visits in regions of the surface other than the direct neighborhood of pore openings. Figure 6 gives a view onto such a surface. On the left side, the structure is shown (hydrogen is omitted), and on the right side, the space accessible by the ethane molecules is visualized by the coordinates of the closest visits to the surface. The color dimension gives information about preferred positions on the surface. Clearly, the molecules mostly reside close to the pore opening where the potential energy is lowest. Furthermore, there is a considerable surface area next to the cut R-cages which is less-preferred. The short-term peak in Figure 5 could result from molecules which arrive in these areas and do not migrate to the pore openings. Previous works on the entering probability reported values differing over several orders of magnitude. It must be pointed out that the same zeolite structure (MFI) but two different molecules were considered. In the experimental work of Jentys et al.,21 the entering probabilitiy for benzene was reported as 1.7 × 10-6. On the other hand, Simon et al.22 reported entering probabilities for butane on zeolite silicalite which were close to unity. The diffusion of butane in silicalite is by orders of magnitude faster than the diffusion of benzene, indicating that benzene has to pass higher energy barriers in the channels of silicalite. Above this, the energy ∆E can be large so that the Boltzmann factor e-β∆E bridges several orders of magnitude, as has been shown in ref 11. Diffusivities of benzene in MFI zeolites at T ) 400 K and low loading are reported in ref 38 to range between D ) 5 × 10-15 and D ) 7 × 10-13. The equilibrium constant in the experiments in ref 21 is K ) 1.3 × 105 at the given conditions (external pressure p ) 6 Pa, Si/Al ratio of 45, and a coverage of the SiOHAl groups inside the zeolite by benzene of 0.38). According to these values for D and K, an energy difference of

∆E ) 28 kJ/mol follows, which is approximately half of the heat of adsorption reported in the literature.39 Therefore, it is possible to explain the low entering probability reported in ref 21 with the energy difference ∆E. However, it can be assumed that a considerable part of the pores are blocked, e.g., by structural defects. Further experiments at different temperatures could be used to determine both ∆E and the fraction of open pores, b, via eq 10. 5. Conclusions This article presented a relation between the entering probability and the intracrystalline self-diffusion coefficient. Results of molecular dynamics simulations have shown that for the case of an ideal surface the entering probabilities predicted by the relation are in excellent agreement with the actual values. If the entering probability, the equilibrium constant, and the selfdiffusion coefficient are known, e.g., from experiments, the relation allows to determine separately the two quantities mainly responsible for a surface resistance increased in comparison to diffusion resistance: the energy difference between a pore opening at the zeolite surface and inside the crystal, ∆E, and the fraction of open pores, b. This introduces a new way to characterize surfaces of real zeolites and serves to understand the origin of the surface resistance in the respective case. Acknowledgment. The author is deeply grateful to Dr. habil. Siegfried Fritzsche (Leipzig) and Prof. Dr. Sergey Vasenkov (Gainesville) for the chance to continue the work within their common project. Numerous stimulating discussions with both project leaders and Prof. Jo¨rg Ka¨rger (Leipzig) are gratefully acknowledged. Some gnuplot hints by Cand. Phys. Matthias Mu¨ller (Leipzig) are kindly acknowledged. The author thanks the Deutsche Forschungsgemeinschaft for funding given in the frame of the SPP 1155. References and Notes (1) Baerlocher, C.; Meier, W. M.; Olson, D. H. Atlas of Zeolite Framework Types, 5th ed.; Elsevier: Amsterdam, 2001. (2) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley: New York, 1992. (3) Demontis, P.; Suffritti, G. B. Chem. ReV. 1997, 97, 2845-2878. (4) Bates, S. P.; van Santen, R. A. AdV. Catal. 1998, 42, 1-114. (5) Auerbach, S. M. Int. ReV. Phys. Chem. 2000, 19, 155-198.

11290 J. Phys. Chem. C, Vol. 111, No. 30, 2007 (6) Keil, F. J.; Krishna, R.; Coppens, M.-O. Chem. Eng. J. 2000, 16, 71-197. (7) Sholl, D. S. Acc. Chem. Res. 2006, 39, 403-411. (8) Vigne´-Maeder, F.; El Amrani, S.; Ge´lin, P. J. Catal. 1992, 134, 536-541. (9) Ford, D. M.; Glandt, E. D. J. Membr. Sci. 1995, 107, 47-57. (10) Takaba, H.; Koshita, R.; Mizukami, K.; Oumi, Y.; Ito, N.; Kubo, M.; Fahmi, A.; Miyamoto, A. J. Membr. Sci. 1997, 134, 127-139. (11) Arya, G.; Maginn, E. J.; Chang, H. C. J. Phys. Chem. B 2001, 105, 2725-2735. (12) Martin, M. G.; Thompson, A. P.; Nenoff, T. M. J. Chem. Phys. 2001, 114, 7174-7181. (13) Ahunbay, M. G.; Elliott, J. R.; Talu, O. J. Phys. Chem. B 2005, 109, 923-929. (14) Newsome, D. A.; Sholl, D. S. J. Phys. Chem. B 2005, 114, 72377244. (15) Schu¨ring, A.; Vasenkov, S.; Fritzsche, S. J. Phys. Chem. B 2005, 109, 16711-16717. (16) Gulı´n-Gonza´lez, J.; Schu¨ring, A.; Fritzsche, S.; Ka¨rger, J.; Vasenkov, S. Chem. Phys. Lett. 2006, 430, 60-66. (17) Ka¨rger, J. Langmuir 1988, 4, 1289-1292. (18) Kocˇirˇ´ık, M.; Struve, P.; Fiedler, K.; Bu¨low, M. J. Chem. Soc., Faraday Trans. 1 1988, 84, 3001-3013. (19) Barrer, R. M. J. Chem. Soc., Faraday Trans. 1990, 86, 11231130. (20) Kehr, K.; Mussawisade, K.; Wichmann, T. Diffusion of Particles on Lattices. In Diffusion in Condensed Matter; Ka¨rger, J., Heitjans, P., Haberlandt, R., Eds.; Vieweg: Braunschweig, 1998; pp 265-305. (21) Jentys, A.; Tanaka, H.; Lercher, J. A. J. Phys. Chem. B 2005, 109, 2254-2261. (22) Simon, J.-M.; Bellat, J.-P.; Vasenkov, S.; Ka¨rger, J. J. Phys. Chem. B 2005, 109, 13523-13528. (23) Jentys, A.; Mukti, R.; Lercher, J. J. Phys. Chem. B 2006, 110, 17691-17693.

Schu¨ring (24) Ka¨rger, J.; Vasenkov, S. J. Phys. Chem. B 2006, 110, 1769417695. (25) Groβ, A.; Eichler, A.; Hafner, J.; Mehl, M. J.; Papaconstantopoulos, D. A. J. Chem. Phys. 2006, 124, 174713. (26) Batista, E. R.; Ayotte, P.; Bilic´, A.; Kay, B. D.; Jo´nsson, H. Phys. ReV. Lett. 2005, 95, 223201. (27) Ka¨rger, J.; Bu¨low, M.; Millward, G.; Thomas, J. Zeolites 1986, 6, 146-150. (28) Ka¨rger, J.; Pfeifer, H.; Stallmach, F.; Bu¨low, M.; Struve, P.; Entner, R.; Spindler, H.; Seidel, R. AIChE J. 1990, 36, 1500-1504. (29) Kortunov, P.; Vasenkov, S.; Chmelik, C.; Karger, J.; Ruthven, D. M.; Wloch, J. Chem. Mater. 2004, 16, 3552-3558. (30) Zheng, S. R.; Tanaka, H.; Jentys, A.; Lercher, J. A. J. Phys. Chem. B 2004, 108, 1337-1343. (31) In general, the term diViding surface is used in TST. It is not intended to introduce a new term by using separation plane. The purpose is simply to avoid confusion since here surface refers to the zeolite surface. (32) Beerdsen, E.; Dubbeldam, D.; Smit, B. J. Phys. Chem. B 2006, 110, 22754-22772. (33) Schu¨ring, A.; Auerbach, S. M.; Fritzsche, S.; Haberlandt, R. Stud. Surf. Sci. Catal. 2004, 154, 2110-2117. (34) Corma, A.; Rey, F.; Rius, J.; Sabater, M. J.; Valencia, S. Nature (London) 2004, 431, 287-290. (35) Schu¨ring, A.; Auerbach, S. M.; Fritzsche, S.; Haberlandt, R. J. Chem. Phys. 2002, 116, 10890-10894. (36) Fritzsche, S.; Haberlandt, R.; Hofmann, G.; Ka¨rger, J.; Heinzinger, K.; Wolfsberg, M. Chem. Phys. Lett. 1997, 265, 253-258. (37) Demontis, P.; Suffritti, G. B.; Quartieri, S.; Fois, E. S.; Gamba, A. J. Phys. Chem. 1988, 92, 867-871. (38) Jobic, H.; Bee, M.; Pouget, S. J. Phys. Chem. B 2000, 104, 71307133. (39) Zika´nova`, A.; Bu¨low, M.; Schlodder, H. Zeolites 1987, 7, 115118.