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J. Phys. Chem. C 2008, 112, 17030–17031
COMMENTS Comment on “High-Accuracy Estimation of ‘Slow’ Molecular Diffusion Rates in Zeolite Nanopores, Based on Free Energy Calculations at an Ultrahigh Temperature” P. Demontis,† G. B. Suffritti,† and S. Yashonath*,†,‡,§,# Dipartimento di Chimica, UniVersita di Sassari, Via Vienna 2 I-07100 Sassari, Italy, and Solid State and Structural Chemistry Unit, and Center for Condensed Matter Theory, Indian Institute of Science, Bangalore-560012, India ReceiVed: April 7, 2008; ReVised Manuscript ReceiVed: July 25, 2008 Introduction In a recent article, Nagumo et al.1 proposed a method to obtain slow diffusivity, D, at low temperatures of guest molecules in zeolites, where the usual molecular dynamics method is unable to give accurate results. There are several computational methods already in the literature that attempt to estimate slow diffusivities.2-4 Although the method proposed by Nagumo et al. is a timely addition to the available computational techniques, certain aspects discussed in their article need to be carefully examined. They report calculations over a rather wide range of temperatures (800-1800 K). LTA zeolite is stable only up to about 1200 K.5,6 However, for the purposes of demonstration of their methodology, they can, in principle, go beyond this temperature, although it is not of any practical relevance. Rigid Zeolite Cage. Over this wide range of temperatures, the authors assume the zeolite framework to be rigid. This is rather drastic, considering the fact that simulations have been carried out at rather high temperatures. A rigid zeolite framework is a good approximation at low temperatures since the modes associated with the zeolite framework are usually higher in frequency than those associated with the guests. However, their contribution cannot be neglected at higher temperatures. Even at 600 K, many of the modes associated with the zeolite framework are active, leading to significant vibrational amplitude. Thus, their contributions to guest motion cannot be neglected. As Nagumo et al. simulate, at as high a temperature as 1800 K, the rigid zeolite approximation becomes untenable. It is therefore essential that the framework motion be included in the simulation. In fact, as we show below, some of the observed behavior is an artifact of the rigid zeolite approximation. Different force fields are available for modeling the zeolite framework. Demontis et al.7 modeled the framework motion in terms of a harmonic potential for the bond vibration, while Brickmann and co-workers have proposed8 a more sophisticated potential. Deviation from Arrhenius Behavior. The authors find a non-Arrhenius behavior for the self-diffusivity of methane over * To whom correspondence should be addressed. † Universita di Sassari. ‡ Solid State and Structural Chemistry Unit, Indian Institute of Science. § Center for Condensed Matter Theory, Indian Institute of Science. # Also at the Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore.
Figure 1. Variation of window area during the intercage diffusion event. The curve is shown from 4 ps before to 4 ps after the intercage diffusion for levitation parameter γ values of (a) 0.99 and (b) 0.80, corresponding to guest diameters of 3.41 and 2.56 Å (taken from Santikary, P.; Yashonath, S. J. Phys. Chem. 1994, 98, 9252).
the range of 800-1800 K. They argue that this is due to a larger change in the potential energy ∆V along the reaction or diffusion coordinate. Their analysis of various thermodynamic properties suggests that the potential energy change, ∆V, along the reaction coordinate is larger at higher temperatures. They attribute this to an increased collision frequency, particularly when methane is passing through the window, since the free volume is small at this point along the diffusion path. This leads to a higher activation energy barrier at higher temperatures as compared to that at lower tempeatures. This is responsible for the nonArrhenius behavior. The analysis of Nagumo et al. appears to be reasonable, and this interpretation of why the downward convex curve is seen for the Arrhenius plot of self-diffusivity appears reasonable. Previous9 investigations of the effect of diffusion of different sized guest particles on the window area during passage through this eight-ring window in LTA zeolite have been reported. In these simulations, the zeolite framework was modeled as a flexible framework using the potential of Demontis et al.7 Their results suggest that when the size of the guest is not too small relative to that of the eight-ring window in LTA zeolite, there is a noticeable expansion of the window during the passage of the guest through that window. This is shown in Figure 1. The figure demonstrates the expansion in the window for the case of σss ) 3.41 Å. The σss for methane is generally larger than 3.41 Å.10 Quirke and co-workers11 have used 3.817 Å, while others have used 3.46 Å.10 Clearly, the eight-ring window would exhibit considerable expansion during the passage of methane in LTA zeolite. The effect of such an expansion is to reduce the frequency and intensity of the collision at the window. This would lead to lowering of the potential energy V at the window, thus reducing ∆V along the reaction coordinate. Such expansion of the window is greater at higher temperatures. Further, Figure 1 reports results for the harmonic potential model of Demontis et al.7 In the case of the anharmonic potential, the expansion of the window is likely to be larger. A lower ∆V at higher
10.1021/jp802989n CCC: $40.75 2008 American Chemical Society Published on Web 09/25/2008
Comments temperatures will mean a lower Ea. It therefore appears to us that the non-Arrhenius curve for the plot of the logarithm of self-diffusivity against reciprocal temperature, observed by Nagumo et al., is a consequence of the use of a fixed zeolite framework. A flexible framework simulation is likely to change this apparent non-Arrhenius behavior to Arrhenius behavior. It is therefore necessary for care to be taken to see if the observed non-Arrhenius behavior is indeed real. Such a calculation will also be a better test of the method proposed by Nagumo et al., for reasons discussed below. When Available Modes Are Different at Different Temperatures. Although eq 14 in Nagumo’s paper expresses the free energy at temperature T in terms of another temperature Tr, some of the modes accessible at, say, higher temperature Tr may be absent at a lower temperature T. It is not clear how this would be accounted for by the last term on the right-hand side in eq 14. It appears to us that this information about the changes in the modes with temperature is completely left out of eq 14. Nagumo et al. have chosen a problem where the modes available are the same at all temperatures, namely, the translational modes of the spherical monatomic methane. It is not evident how their method will perform when the modes available at temperatures Tr and T are different. This is particularly true for polyatomic guest molecules and flexible zeolite cages. It is therefore, necessary to carry out more detailed studies to see the validity of eq 14 in such situation. The Problem of Phase Space Sampling. Even when the same modes are accessible in the system at all temperatures, like the methane rigid LTA systems studied by Nagumo et al., the sampling of the phase space at lower temperatures is limited. As the temperature increases, other regions of the phase space become accessible. However, the system has no knowledge of these additional regions of the phase space until it actually samples them. Equation 14 does not seem to ensure that information about different regions of the phase space is input. In other words, predicting the diffusivity (or any other property) at higher temperatures from a knowledge of the data at a lower temperature is not expected to be very accurate simply because of the limited phase space accessed at lower temperatures. Consider the converse situation: can a knowledge of the data at higher temperatures enable prediction of the low-temperature properties? The answer to this is somewhat more positive. At higher temperatures, the system samples a large region of the phase space. At lower temperature, the system samples a subset of the phase space sampled by the system at higher temperatures. Therefore, the free energy data at higher temperature is more likely to predict correctly the properties at lower temperatures. Only in this limited sense is the method still a good approach to employ for prediction of ‘slow’ diffusivities, but it needs to be used with care.
J. Phys. Chem. C, Vol. 112, No. 43, 2008 17031 In fact, Figure 5 given by Nagumo et al. corroborates these arguments. We see from Figure 5a that the agreement between the free energy profile along the reaction or diffusion coordinate at 1800 K obtained from data at 800 K shows noticeable deviation from the free energy profile obtained from MD simulations performed at 1800 K. On the contrary, the free energy profile at 800 K predicted from eq 14 from MD simulations at 1800 K shows better agreement with the free energy profile computed from MD simulation at 800 K. Their own results thus suggest that prediction of properties at high temperature from low temperature data is not very good. Finally, the authors state in section 4.5 that “in Figure 3, it is shown that the Arrhenius dependence cannot be always assumed over a wide range of temperatures. Therefore, our methodology will be applied to various diffusion phenomena, regardless of the Arrhenius dependence.” Our arguments above show that deviation from Arrhenius dependence must be carefully examined, and it is not good to disregard deviations from Arrhenius dependence and treat them as inherent. They further state that the temperature dependence is comprehensively predicted. Our analysis suggests that the temperature dependence, even in the simple case employed in their study, is not always good. Therefore, care should be excercised, and there is no room for overconfidence. We suggest that predictions of eq 14 is better when used over a narrower range of temperatures. Acknowledgment. Award of a Visiting Professorship to S.Y. from the University of Sassari is gratefully acknowledged. We also gratefully acknowledge financial support, in part, from the Ramanna Fellowship of the Department of Science and Technology, New Delhi, to S.Y. Support by the Italian Ministry of Research under the CYBERSAR project for a visit of P.D. to Bangalore is gratefully acknowledged. References and Notes (1) Nagumo, R.; Takaba, H.; Nakao, S. J. Phys. Chem. C 2008, 112, 2805. (2) Chandler, D. J. Phys. Chem. 1978, 68, 2959. (3) Voter, A. F.; Doll, J. D. J. Phys. Chem. 1995, 82, 80. (4) Ghorai, P. K. R.; Yashonath, S.; Lynden-Bell, R. M. Mol. Phys. 2002, 100, 641. (5) Weidenthaler, C.; Schmidt, W. Chem. Mater. 2000, 12, 3811. (6) Lutz, W.; Bulow, M.; Feoktistova, N. N.; Vtjurina, L. B.; Shdanov, S. P.; Siegel, H. Cryst. Res. Technol. 2006, 24, 161. (7) Demontis, P.; Suffritti, G. B.; Quartieri, S.; Fois, E. S.; Gamba, A. J. Phys. Chem. 1988, 92, 867. (8) Schrimpf, G.; Schlenkrich, M.; Brickmann, J.; Bopp, P. J. Phys. Chem. 1992, 96, 7404. (9) Santikary, P.; Yashonath, S. J. Phys. Chem. 1994, 98, 9252. (10) Demontis, P.; Fenu, L. A.; Suffritti, G. B. J. Phys. Chem. B 2005, 109, 18081. (11) Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walto, J. P. R. B.; Quirke, N. J. Chem. Soc., Faraday Trans. 1 1991, 87, 1951.
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