Chem. Mater. 2007, 19, 3917-3923
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Assessing Guest Diffusion in Nanoporous Materials by Boltzmann’s Integration Method Pavel Kortunov,†,‡ Lars Heinke,† and Jo¨rg Ka¨rger*,† Fakulta¨t fu¨r Physik und Geowissenschaften, UniVersita¨t Leipzig, Linne´ strasse 5, 04103 Leipzig, Germany, and Corporate Strategic Research, ExxonMobil Research and Engineering Company, 1545 Route 22 East, Annandale, New Jersey 08801 ReceiVed February 9, 2007. ReVised Manuscript ReceiVed May 15, 2007
With the introduction of interference microscopy to diffusion studies in zeolites, the direct, spaceresolved observation of transport diffusion in nanoporous materials has become possible for the first time. The method is based on recording transient concentration profiles during molecular uptake or release so that the local intracrystalline diffusivities directly result by a microscopic (“differential”) application of Fick’s second law. We show that under certain conditions the thus accessible wealth of information on the transport characteristics of nanoporous materials may even be surpassed by the application of an integration method that Ludwig Boltzmann invented more than a century ago and that thus encounters a large new field of application. The procedure is illustrated by discussing its application to the adsorption process of methanol on FER-type zeolite crystallites. The advantages and limitations of the method in comparison with the differential application of Fick’s second law are discussed.
1. Introduction Diffusion, i.e., the irregular movement of particles induced by their thermal energy, occurs in all states of matter. As one of the most fundamental, omnipresent phenomena in nature and technology, it is a perpetual focus of research. A literature search in Web of Science 2005 yields as much as 19 206 papers with “diffusion” in the topic or title. Under the multitude of subjects of research, nanoporous host-guest systems1 have attained particular interest. This is primarily related to their practical relevance for future technologies2 and their crucial role in catalysis3 and separation,4 because it is the rate of mass transfer that in many cases decides the systems’ technological performance.5,6 Moreover, many exciting phenomena of (anomalous) mass transfer generated by the peculiarities of molecular confinement in such systems,7-12 and last but not least, the discrepancies between * Corresponding author. E-mail:
[email protected]. Tel: 49 341 97 32 502. Fax: 49 341 97 32 549. † Universita ¨ t Leipzig. ‡ ExxonMobil Research and Engineering Company.
(1) Laeri, F.; Schu¨th, F.; Simon, U.; Wark, M. Host-Guest Systems Based on Nanoporous Crystals; Wiley-VCH: Weinheim, Germany, 2003. (2) Davis, M. E. Nature 2002, 417, 813-821. (3) Weitkamp, J.; Puppe, L. Catalysis and Zeolites; Springer: Berlin, 1999; p 564. (4) Schu¨th, F.; Sing, K. S. W.; Weitkamp, J. Handbook of Porous Solids; Wiley-VCH: Weinheim, Germany, 2002. (5) Ka¨rger, J.; Vasenkov, S. Microporous Mesoporous Mater. 2005, 85, (3), 195-206. (6) Ruthven, D. M.; Post, M. F. M. Diffusion in Zeolite Molecular SieVes, 2nd ed.; Elsevier: Amsterdam, 2001; pp 525-578. (7) Beerdsen, E.; Dubbeldam, D.; Smit, B. Phys. ReV. Lett. 2006, 96 (4), Art. No. 044501. (8) Hahn, K.; Ka¨rger, J.; Kukla, V. Phys. ReV. Lett. 1996, 76 (15), 27622765. (9) Kumar, A. V. A.; Bhatia, S. K. J. Phys. Chem. B 2006, 110 (7), 31093113. (10) Saravanan, C.; Jousse, F.; Auerbach, S. M. Phys. ReV. Lett. 1998, 80 (26), 5754-5757.
the results of different measuring techniques13-16 (including their assessment within quite general concepts of theoretical treatment) substantially contributed to the continuously increasing interest. With respect to the latter item, the introduction of interference microscopy to studying zeolitic diffusion17,18 meant a decisive breakthrough. By measuring transient intracrystalline concentration profiles, one has become able to correlate the change in local concentrations with the adjacent concentration gradients. This, however, is exactly the correlation expressed by Fick’s second law. It has thus become possible to determine the transport diffusivity, i.e., the factor of proportionality within this equation, by analyzing the variation of local concentrations with time and space, which has now become accessible to direct experimental observation. Among the spectrum of techniques applicable to monitoring concentration profiles, including, for example, magnetic resonance imaging,19-21 positron emission profiling,22 and IR imaging,23 interference micros(11) Clark, L. A.; Ye, G. T.; Snurr, R. Q. Phys. ReV. Lett. 2000, 84 (13), 2893-2896. (12) Ka¨rger, J.; Vasenkov, S.; Auerbach, S. M. Diffusion in Zeolites; Marcel Dekker: New York, 2003; pp 341-423. (13) Ramanan, H.; Auerbach, S. M.; Tsapatsis, M. J. Phys. Chem. B 2004, 108, (44), 17171-17178. (14) Chen, N. Y.; Degnan, T. F.; Smith, C. M. Molecular Transport and Reaction in Zeolites; Wiley-VCH: New York, 1994. (15) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley & Sons: New York, 1992. (16) Jobic, H.; Theodorou, D. N. J. Phys. Chem. B 2006, 110 (5), 19641967. (17) Schemmert, U.; Ka¨rger, J.; Krause, C.; Rakoczy, R. A.; Weitkamp, J. Europhys. Lett. 1999, 46, (2), 204-210. (18) Kortunov, P.; Vasenkov, S.; Chmelik, C.; Ka¨rger, J.; Ruthven, D. M.; Wloch, J. Chem. Mater. 2004, 16 (18), 3552-3558. (19) Stapf, S.; Han, S. NMR Imaging in Chemical Engineering; WileyVCH: Weinheim, Germany, 2006; 620. (20) Gladden, L. F. AIChE J. 2003, 49, 2-9. (21) Ba¨r, N. K.; Bauer, F.; Ruthven, D. M.; Balcom, B. J. J. Catal. 2002, 208, 224-228.
10.1021/cm070404r CCC: $37.00 © 2007 American Chemical Society Published on Web 07/04/2007
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Figure 1. Schematics of interference microscopy. (a) Two light beams; one passing through the ferrierite crystal with the two-dimensional pore structure and the other through the surrounding atmosphere. (b) Interference microscope. (c) Interference patterns generated because of different optical properties of the media passed by the two beams. (d) Concentration profiles calculated from the changes in interference patterns with time.
copy is the only technique allowing spatial resolution down into the range of micrometers, which is quintessential for the observation of concentration profiles within crystals of sizes that are scarcely much larger than these dimensions. In comparison with these techniques, however, interference microscopy does not provide the option of selective diffusion measurements in multicomponent systems. In this respect, with the advent of the focal plane array detector24 and their enhanced temporal and spatial resolution, we may expect a breakthrough in the investigation of multicomponent systems in the near future. Because the translational mobility of molecules notably depends on their mutual interaction, for many systems, the diffusivities turned out to depend dramatically on the loading. It is well-known25 that under such conditions the conventional techniques, which are based on an analysis of the time dependence of the overall molecular uptake, fail to provide detailed information on these dependencies if too large concentration ranges are covered. It is one of the big advantages of interference microscopy that, as a literally “microscopic” technique, it is not affected by such type of problems. At the same time, however, these first measurements also revealed that the determination of the diffusivities over large ranges of concentration is extremely time-consuming. Moreover, as a typical feature of a differential measuring technique, the accuracy of the calculated diffusivities was found to be highly impaired by the noise in the primary concentration data. By adopting Ludwig Boltzmann’s inte(22) Schumacher, R. R.; Anderson, B. G.; Noordhoek, N. J.; de Gauw, F. J. M. M.; de Jong, A. M.; de Voigt, M. J. A.; van Santen, R. A. Microporous Mesoporous Mater. 2000, 6, 315-326. (23) Karge, H. G.; Niessen, W.; Bludau, H. Appl. Catal., A 1996, 146, (2), 339-349. (24) Lewis, E. N.; Treado, P. J.; Reeder, R. C.; Story, G. M.; Dowrey, A. E.; Marcott, C.; Levin, I. W. Anal. Chem. 1995, 67 (19), 3377-3381. (25) Crank, J. Mathematics of Diffusion; Oxford University Press: Oxford, UK, 1956; p 347.
gration method to this new field of application in material sciences, we are going to show how these shortcomings may be remedied if the prerequisites for the application of this method are given. Considering the uptake of methanol by a zeolite crystal of type ferrierite, this method will be shown to provide easy and straightforward access to the whole range of diffusivities within this host-guest system, which is found to increase over as much as 2 orders of magnitude with increasing guest concentration. 2. Materials and Methods 2.1. Sample of Zeolite Ferrierite. We have applied the new method of diffusion analysis to a cation-free silica ferrierite, which has been synthesized at the University of Stuttgart as described in ref 26.26 The zeolite crystals under study were about 50 µm × 200 µm × 10 µm in size and had the shape of a rectangular solid body with a “roof” on both elongated sides, as schematically indicated by Figure 1a. To ensure that there are no organic residues, the crystals were exposed to oxygen at 700 °C for 12 h. Prior to monitoring their adsorption-desorption behavior in an atmosphere of methanol, we activated the crystals under a high vacuum at a temperature of 400 °C for 12 h. 2.2. Adsorption-Desorption Runs. The adsorption-desorption experiments were performed at room temperature with one selected crystal. The adsorbate was pure methanol obtained from SigmaAldrich, with a purity of 99.9%. The complete data set for the transient methanol concentrations recorded in the experiments (pressure steps from 0 to 5, 5 to 10, 0 to 10, 0 to 20, 0 to 40, and 0 to 80 mbar, and reverse pressure steps in corresponding desorption runs) is presented in refs 27 and 28.27,28 The present analysis is based on the largest pressure step from 0 to 80 mbar. 2.3. Basics of Interference Microscopy. Figure 1 illustrates the application of interference microscopy to diffusion studies with nanoporous materials. It is on the basis of an analysis of the (26) Rakoczy, R. A.; Traa, Y.; Kortunov, P.; Vasenkov, S.; Ka¨rger, J.; Weitkamp, J. Microporous Mesoporous Mater. 2007, in press.
Assessing Guest Diffusion in Nanoporous Materials interference pattern generated by superimposing light beams passing the nanoporous crystal (generally zeolites3,4,14,15,29) and the surrounding atmosphere. Because the optical density of the crystal is a function of the concentration of the guest molecules, changes in local concentration directly appear in corresponding changes in the interference patterns (panels b and c in Figure 1). On the other hand, one may deduce from the interference patterns the corresponding concentration profiles (Figure 1d). The quantity directly accessible is thus the integral over the intracrystalline concentration in the observation direction (x-axis in Figure 1a) with a spatial resolution of ∆y × ∆z ≈ 0.5 × 0.5 µm2. If, because of a corresponding blockage of the relevant crystal faces or by the architecture of the pore system, diffusion in the x-direction is excluded, there is also no concentration variation in the x-direction. In this case, interference microscopy directly yields the local concentrations c(y, z). Such a situation is in particular given for materials with pore systems consisting of arrays of parallel channels like the zeolite framework type AFI.29 Most astonishingly, however, the channels of these systems were found to be obstructed by internal resistances.30,31 In fact, the novel option of interference microscopy did identify internal transport resistances as a quite general feature of nanoporous zeolitic materials. This is in particular true for the important framework type MFI,29 with ZSM-5 and silicalite-1 as their most significant representatives.18,29 Besides cracks notably accelerating molecules uptake and release,18 in these materials, interference microscopy revealed notable transport resistances at the internal interfaces separating the intergrowth sections.32,33 2.4. Interference Microscopy Applied to FER-Type Crystals. Crystallites of structure type FER (ferrierite)29 turned out to be the first zeolite material in which, using methanol as a probe molecule, any substantial influence of intracrystalline transport resistances could be excluded.27,28 Figure 1a provides a schematic view of the pore system of zeolite ferrierite. It consists of two arrays of parallel channels perpendicular to each other. The channels are framed by 10-membered rings (formed by 10 silicon and 10 oxygen atoms) and 8-membered rings in the z- and y-directions, respectively. Their peculiar structural features make them excellent candidates for our studies: whereas in the rooflike upper and lower parts of the crystals the 10-membered channels are easily accessible, their entrances to the main rectangular part of the crystal are blocked.27,34 Therefore, as a consequence of the high molecular mobility in the 10membered channels, a pressure change in the surrounding atmosphere will essentially instantaneously be followed by the establishment of the new equilibrium concentration in this part of the crystal. Such enhanced guest mobility in channel-like systems has been repeatedly reported in the literature.8,35 Molecular uptake by (27) Ka¨rger, J.; Kortunov, P.; Vasenkov, S.; Heinke, L.; Shah, D. B.; Rakoczy, R. A.; Traa, Y.; Weitkamp, J. Angew. Chem., Int. Ed. 2006, 45, 7846-7849. (28) Kortunov, P.; Heinke, L.; Vasenkov, S.; Chmelik, C.; Shah, D. B.; Ka¨rger, J.; Rakoczy, R. A.; Traa, Y.; Weitkamp, J. J. Phys. Chem. B 2006, 110, 23821-23828. (29) Baerlocher, C.; Meier, W. M.; Olson, D. H. Atlas of Zeolite Framework Types, 5th ed.; Elsevier: Amsterdam, 2001; p 302. (30) Ka¨rger, J.; Valiullin, R.; Vasenkov, S. New J. Phys. 2005, 7, no. 15. (31) Lehmann, E.; Vasenkov, S.; Ka¨rger, J.; Zadrozna, G.; Kornatowski, J. J. Chem. Phys. 2003, 118 (14), 6129-6132. (32) Lehmann, E.; Chmelik, C.; Scheidt, H.; Vasenkov, S.; Staudte, B.; Ka¨rger, J.; Kremer, F.; Zadrozna, G.; Kornatowski, J. J. Am. Chem. Soc. 2002, 124, 8690-8692. (33) Geier, O.; Vasenkov, S.; Lehmann, E.; Ka¨rger, J.; Schemmert, U.; Rakoczy, R. A.; Weitkamp, J. J. Phys. Chem. B 2001, 105, 1021710222. (34) Kortunov, P. Rate Controlling Processes of Diffusion in Nanoporous Materials. Ph.D. Thesis, Leipzig University, Leipzig, Germany, 2005. (35) Jakobtorweihen, S.; Verbeek, M. G.; Lowe, C. P.; Keil, F. J.; Smit, B. Phys. ReV. Lett. 2005, 95 (4), Art. No. 044501.
Chem. Mater., Vol. 19, No. 16, 2007 3919 and release from these rooflike parts of the crystal may therefore be quite easily subtracted from the general response, yielding the concentration evolution of exclusively the central rectangular part of the crystal. Because the entrances to the 10-membered channels are blocked in this part of the crystals, regardless of the high diffusivity in these channels, there is no net mass transfer in the z-direction; molecular uptake and release predominantly occur along the 8-membered rings in the y-direction. Thus, with respect to the application of interference microscopy, ferrierite turns out to be an ideal host system for the observation of diffusion along one direction. This is the prerequisite for the application of Boltzmann’s integration method. At the end of our discussion in section 3.3 we have to return to this point and take into account that, actually, transport in z-direction is not totally excluded. It is this deficiency in the prerequisites that eventually leads to a systematic deviation by a factor of about 1.6, which the diffusivities derived by Boltzmann’s integration method shifts toward lower values. This aberration is clearly much smaller than the 2 orders of magnitude over which, essentially, the diffusivity is found to vary over the considered concentration range. 2.5. Boltzmann’s Integration Method. Boltzmann’s integration method36-38 represents a particularly efficient and elegant means to deduce diffusivities from transient concentration profiles. It may be applied if these concentration profiles evolve by one-dimensional diffusion (which in our case shall be considered to be directed along the crystallographic y-direction) under the condition that (i) the concentration is initially constant throughout the whole system (c(y,t)0) ) c0), (ii) the evolution of transient concentration profiles is initiated by a step change in the boundary concentration on one face to a new value c∞, which remains constant during the whole experiment (c(y)0, t) ) c∞), and (iii) the time interval considered is small enough so that the diffusion front evolving from one side has not yet reached the opposite one (nor did it interfere with any other diffusion fronts) so that the system may be considered to be semi-infinitely extended (c(y)∞, t) ) c0). Mathematically, the situation is described by Fick’s second law ∂ ∂c ∂c ∂D ∂c 2 ∂2c ) D(c(y)) ) + D(c(y)) 2 ∂t ∂y ∂y ∂c ∂y ∂y
(
)
( )
(1)
with the boundary and initial conditions c(0,t) ) c∞ and c(y,0) ) c(∞,t) ) c0. The transport diffusivity D(c) is introduced by Fick’s first law as the factor of proportionality between the particle flux density and the concentration gradient. Though this definition implies that (as also fulfilled in reality) the transport diffusivity must not depend on the concentration gradient, it clearly may depend on the concentration. This leads to the first term on the utmost right of eq 1, which prohibits a general solution of this differential equation. Following Ludwig Boltzmann,36,37 by introducing a new variable η ) y/xt, eq 1 may be transferred into d2c η dc dD/dc dc 2 + )0 + 2 2D dη D dη dη
( )
(2)
where the concentration now appears as a function of the sole variable η ) y/xt. Thus, the above stated complication resulting from the concentration-dependent prefactor of the second derivative in (36) Boltzmann, L. Wied. Ann. 1894, 53, 959-964. (37) Jost, W. Diffusion in Solids, Liquids and Gases; Academic Press: New York, 1960; p 558. (38) Bird, R.; Stewart, W.; Lightfoot, E. Role of Transport Phenomena in Chemical-Engineering Teaching and ResearchsPast, Present and Future. American Chemical Society: Washington, DC, 1979.
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Figure 2. (a) Experimental concentration profiles in the y-direction of ferrierite type crystals for a pressure step from 0 to 80 mbar of methanol in the surrounding atmosphere at selected times. (b) Normalized concentration profiles, resulting after subtraction of molecular uptake through the 10-membered ring channels in the z-direction.
eq 1 is eliminated, and the partial differential equation with two variables (y and t) has now become an ordinary differential equation in the sole variable η ) y/xt. The initial and boundary conditions of eq 1 are transferred into c(η ) 0) ) c∞ and c(η ) ∞) ) c0. Plotting the concentration profiles c(y,t) as a function of this new parameter, i.e., as c(η ) y/xt), for different times t should therefore yield coinciding representations. Equation 2 cannot be integrated either. However, being interested only in the concentration dependence D(c) of the diffusivity as the key quantity of our study, following Ludwig Boltzmann, we may rewrite eq 2 as d dc η dc D )dη dη 2 dη
( )
(3)
Integration of eq 3 over η from ∞ to η(c) with (dc)/(dη)|η)∞ ) 0 yields D(c) ) -
1 dη 2 dc
∫
c
c)c0
ηdc
(4)
In this way, the diffusivity at any concentration c0 < c < c∞ covered during the whole process of uptake or release may simply be c determined from the respective areas ∫c)c ηdc under the concen0 tration profile (plotted as a function of the unifying variable η ) y/xt) and its slope dη/dc at a given concentration. In the following, we are going to explain in more detail the first application of this method to diffusion studies and, hence, to the structural characterization of nanoporous materials.
3. Results and Discussion 3.1. Transient Concentration Profiles during Methanol Uptake by Ferrierite. It has been illustrated in the section on Materials and Methods that the concentration profiles as recorded by interference microscopy (Figure 1d) may be easily separated into the constituents stemming from molecular uptake by the rooflike parts of the crystallites and by the central main part of the crystals (Figure 1a). This is due to the fact that, in the rooflike parts (and in contrast to the main body), the apertures of the “large” channels with 10-membered rings are unobstructed, thus allowing an essentially instantaneous equilibration of the guest molecules within this part of the crystal. Therefore, the final guest distribution within these rooflike parts becomes easily accessible immediately after the onset of the sorption
experiments, and the evolution of the concentration profiles within the main crystal body result by simply subtracting this contribution from the total profile. Figure 2a represents the thus-determined transient concentration profiles in the main body of a ferrierite-type crystal during the uptake of methanol initiated by a pressure step from 0 up to 80 mbar in the surrounding atmosphere. Such records of transient concentration profiles with spatial resolution in the range of micrometers and time resolution down to seconds were inconceivable before the introduction of interference microscopy.27,28 Closer inspection of Figure 2a reveals that purely onedimensional diffusion fails to correctly describe the observed dependencies. Considering the central parts of the profiles to about t ) 130 s, a notable increase in the concentrations with increasing time is observed, although there is no perceptible curvature in the representations, i.e., both ∂2c/ ∂y2 and ∂c/∂y are equal to zero. Therefore, on the basis of eq 1 for diffusion in a purely one-dimensional system, there should also be no change in the concentrations. One has to conclude, therefore, that regardless of their partial blockage, there is also a finite rate of molecular access from outside into the (large) 10-membered-ring channels in the main body of the ferrierite crystal. It is exactly this process of fast diffusion along the large channels in the z-direction, controlled by the limited uptake through the obstructed channel mouths, which gives rise to the increasing concentrations in this central part of the profiles. On the basis of the complete (two-dimensional) set of Fick’s equations, this process could be shown in the literature to satisfactorily predict the observed profile evolution.28 This result also means that not all molecules contributing to the profiles shown in Figure 2a did in fact get there by diffusion in the y-direction. In fact, this number of molecules, which represent the central part of the profiles and augment without the presence of any perceptible concentration curvature, have entered in the z-direction. The representations of Figure 2b show the concentration profiles after this constant value is subtracted. Thus, the profiles shown in Figure 2b are those of the molecules that have entered through the channels in the y-direction. To take into account
Assessing Guest Diffusion in Nanoporous Materials
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Figure 3. Evolution of the concentration profiles in y-direction in the left (a) and right (b) side of the ferrierite crystal as a function of the parameter η ) c y/xt. Figure 3a also illustrates the way of determining ∫c)c ηdc and dc/dη which, via eq 4, yields the diffusivity. 0
that the local diffusivities of theses molecules clearly depend on the total concentration at the given position (i.e., on the number of all interacting molecules), the ordinate of Figure 2b has been chosen to refer to relative concentrations. The respective absolute concentrations easily follow by referring the relative concentration 1 to the boundary concentration as given by Figure 2a for the corresponding times. 3.2. Application of Boltzmann’s Integration Method. The considerations of the last paragraph have led us to Figure 2b, in which only those molecules are recorded that have entered the ferrierite crystals along the channels in the y-direction. Moreover, it turns out that, until about 90 s after the onset of the adsorption process, the thus-determined concentration profiles are in reasonable accordance with the initial and boundary conditions required for the application of Boltzmann’s integration method (with the initial loading c(y,t)0) ) c0 being equal to zero). The mutual encounter of the diffusion fronts in the crystal center after about 100 s terminates the time range of its application. Figure 3 displays the plots of these concentration profiles as a function of the unifying variable η ) y/xt, considering the diffusion fronts entering in the (y-direction on either face of the crystal. Obviously, in contrast to the behavior expected on the basis of eq 2, there is no complete coincidence between the different plots. We have to accept these differences as an indication that the prerequisites of Boltzmann’s integration method are not ideally fulfilled. It shall be the task of the last part of the Results and Discussion section to identify the origin of these shortcomings and estimate their influence on the obtained data. The insets to Figure 3a illustrates the straightforward way how the key quantities of the conditional equation (eq 4) for the concentration dependence of the intracrystalline diffusivities may be determined from the unified concentrac ηdc tion profiles, namely from the respective integral ∫c)c 0 and (reciprocal) slope dη/dc. They, in addition, provide some first estimate how the differences in the different curves lead to different diffusivities. Figure 4 provides an overview of the diffusivities determined from the uptake profiles after 70 and 90 s on both sides of the crystal. For comparison, we have also indicated the data points that have been determined previously, by conventional data analysis on the basis of the differential
Figure 4. Molecular diffusivity as a function of concentration evaluated via Fick’s second law in one dimension27 and by Boltzmann’s integration method.
(microscopic) use of Fick’s second law.28 This method of data analysis is based on the fact that plots of the spatialtemporal concentration dependence as provided by Figure 2 easily allow the determination of the ratios between the change in concentration and the corresponding changes in time and space, respectively. In view of the relatively small steps in time and space, these difference ratios may be considered to be excellent estimates of the ratios of “differentials”, i.e. of the derivatives needed in eq 1 for the determination of the diffusivities. A detailed presentation and explanation of this procedure may be found elsewhere.28 These data represent a typical extract of several separate adsorption-desorption cycles with varying initial and final pressures. Though the agreement between the results of different runs confirms the reliability, the procedure proves to be rather time-consuming. By contrast, the primary data used for the application of Boltzmann’s integration have been determined during a single run within not more than 100 s. Most remarkably, both data sets indicate a dramatic concentration dependence of the diffusivities that increase by close to 2 orders of magnitude with increasing concentration. This is by far larger than one would expect by the simplifying assumption that the concentration dependence of the transport diffusivities is mainly determined by the thermodynamic factor d(ln p)/d(ln c), with p(c) denoting the adsorbate pressure necessary to maintain the sorbate at concentration c.39 In fact, with the relevant data for equilibrium adsorption,40 for the considered pressure step from
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0 to 80 mbar d(ln p)/d(ln c) would increase from 1 to 3.3. Thus, in turn, the so-called corrected (or Maxwell-Stefan) diffusivity41 D0 ) D(d(ln c))/(d(ln p)) would still result in an increase by a factor of about 30. This indicates increasing mobility with increasing loading following the type-V pattern of the concentration dependence of intracrystalline zeolitic self-diffusion.42 The difference in the absolute values of the diffusivities determined differentially from Fick’s second law and by Boltzmann’s integration method amounts to not more than a factor of 2. In the literature,18,43,44 differences greater than several orders of magnitude are not unusual for intracrystalline diffusivities measured by different techniques. The origin of these differences, however, is generally referred to the different space scales over which the diffusion phenomena are recorded by these different techniques. By contrast, interference microscopy directly yields space-resolved intracrystalline transport diffusivities and the two data sets shown in Figure 4 result from identical primary data, namely the interference patterns and corresponding concentrations (i.e., their integrals in observation direction). Therefore, the differences between these data sets have to be referred to the details of transformation of the primary data to the corresponding diffusivities, which we are going to illuminate in the subsequent section. Clearly, this conclusion is allowed only in the absence of additional internal barriers, which would exclude the application of Boltzmann’s integration method (as an integrating technique) but would not interfere with a microscopic analysis of the primary data as long as the separation between these resistances are larger than the space scale necessary for a reliable determination of the first and second space derivatives of the concentration as required in the analysis. Thus, Boltzmann’s integration method represents a valuable test to check whether the behavior of the total concentration front on passing the crystal turns out to be compatible with the microscopic predictions on the basis of a local application of Fick’s second law. The coinciding trends of the representations of Figure 4 indicate that, over the total range of guest concentrations, the transport properties of the nanoporous crystal under study adequately reveal its microscopic behavior, i.e., any notable influence of additional transport resistances (like internal barriers as observed in the literature45-48) may be excluded. By discussing and evaluating the limitations of the applica(39) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley & Sons: New York, 1992; p 245. (40) Heinke, L.; Chmelik, C.; Kortunov, P.; Ruthven, D. M.; Shah, D. B.; Vasenkov, S.; Ka¨rger, J. Chem. Eng. Technol. 2007, in press. (41) Krishna, R.; Paschek, D. 2002, 85, 7-15. (42) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley & Sons: New York, 1992; p 187. (43) Jobic, H. J. Mol. Catal. A: Chem. 2000, 158, 135-142. (44) Jobic, H.; Ka¨rger, J.; Krause, C.; Brandani, S.; Gunadi, A.; Methivier, A.; Ehlers, G.; Farago, B.; Haeussler, W.; Ruthven, D. M. Adsorption 2005, 11, 403-407. (45) Takaba, H.; Yamamoto, A.; Hayamizu, K.; Nakao, S. J. Phys. Chem. B 2005, 109 (29), 13871-13876. (46) Takaba, H.; Yamamoto, A.; Hayamizu, K.; Oumi, Y.; Sano, T.; Akiba, E.; Nakao, S. Chem. Phys. Lett. 2004, 393 (1-3), 87-91. (47) Adem, Z.; Guenneau, F.; Springuel-Huet, M. A.; Gedeon, A. EVidence of Subdomains in Large Crystals of NaX Zeolite; Leipziger Universita¨tsverlag: Leipzig, Germany, 2005; pp 424-425. (48) Vasenkov, S.; Bo¨hlmann, W.; Galvosas, P.; Geier, O.; Liu, H.; Ka¨rger, J. J. Phys. Chem. B 2001, 105 (25), 5922-5927.
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tion of Boltzmann’s integration method to the present system in the following, we shall illustrate that the differences in the two sets of data evaluation may be attributed to these limitations. 3.3. Deficiencies in the Prerequisites for Applying Boltzmann’s Integration Method and Estimate of their Influence. Implying crystal homogeneity and perfect fulfilment of the prerequisites of application, a plot of the concentration files as a function of the sole variable η ) y/ xt of Boltzmann’s integration method should give rise to coinciding representations. By contrast, the representations in Figure 3 show that there are still residual differences in the resulting plots of the profiles at different time spans after the onset of uptake. This deviation from the ideal situation, however, is not unexpected, because the primary concentration profiles as shown in Figure 2a deviate in two items from the behavior to be required for genuine application of Boltzmann’s integration method: (i) the boundary concentrations c(y)0) and c(y)50 µm) at the two opposite sides of the crystal from where the diffusion front starts to propagate into the crystal interior do not instantaneously attain the equilibrium values. Uptake is additionally controlled, therefore, by a surface resistance at (the finite permeability of) the crystal boundary. (ii) Already after 30 s, the concentration in front of the diffusion front does not drop anymore to zero. In the following, we are going to estimate the deviation of the values determined by Boltzmann’s integration method from the real data, brought about by these deficiencies in the prerequisites of its application. For this purpose, we c η dc under a consider eq 4 and the shaded area ∫c)c 0 concentration profile exemplified in Figure 3a. Equation 4 indicates that the diffusivities increase with an increase of this shaded area. Obviously, such an increase would be observed if, on the way from Figure 2a to Figure 2b, we would not have subtracted those molecules that have entered the crystal along the (large) channels in the z-direction. Similarly, the influence of the finite surface permeability may be taken into account by an increase of the shaded area. This may be realized by rationalizing that the transport resistance on the crystal surface could be substituted by a corresponding extension of the crystal in the y-direction over a certain distance l. As a first estimate, one may assume that ∆c/l is equal to the slope ∂c/∂y of the concentration profiles close to the boundary (where ∆c stands for the difference between the actual boundary concentration and the equilibrium value c∞). Including this correction term into eq 4, the equation leading to the diffusivity simply becomes D(c) ) -
c ∫c)c
1 dη 2 dc
(
0
η+
)
l dc xt
(5)
Quantitative analysis of this estimate49 shows that eq 5 leads to an enhancement of the resulting diffusivity in comparison with eq 4. Thus, the shortcomings in the prerequisites of the application of Boltzmann’s integration method are found to reduce the resulting diffusivities by a factor of about 1.6.49 This value is in satisfactory agreement with the difference (49) Heinke, L. Diffus. Fundamentals 2007, 4, 9.1-9.16.
Assessing Guest Diffusion in Nanoporous Materials
between the microscopically determined diffusivities and the results of Boltzmann’s integration method as revealed by Figure 4. 4. Conclusions The technical performance of nanoporous materials for catalysis, molecular sieving, and as host system for optoelectronic devices is often determined by their transport properties. The diffusivity of guest molecules in nanoporous host systems is therefore among the key quantities for characterizing their practical performance. Recently, interference microscopy has been shown to allow straightforward recording of transient, intracrystalline concentration profiles during molecular uptake and release. Thus, for the first time, the space-resolved determination of transport diffusivities in nanoporous materials on the basis of a microscopic application of Fick’s second law has become possible. In the present communication, we have shown that this analysis may be dramatically facilitated by means of Boltzmann’s integration method. Thus, in fact in immediate temporal vicinity to the celebrations honoring Ludwig Boltzmann on the occasion of the 100th anniversary of his death in 1906, in addition to “macroscopic” media like liquids and beds of adsorbent particles in which data analysis based on merging space and time dependencies to a sole variable η ) y/xt is a common technique in chemical engineering, a new field and range of materials for the application of his formula has been found.
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Boltzmann’s integration method provides a straightforward means for correlating the concentration dependence of diffusivities with transient concentration profiles evolving during molecular uptake or release in these materials. Without any need for complicated maths, diffusivities simply result from the slope of and the area under the concentration profiles if plotted as a function of a sole (space and time unifying) variable η ) y/xt. Moreover, because the statistical accuracy of Boltzmann’s integration method notably exceeds the microscopic procedure by a differential application of Fick’s second law, its application as a novel “old” tool for analyzing the evolution of guest profiles in (quasi)one-dimensional nanoporous host-guest systems is strongly recommended. Though, in the system presently studied, minor systematic deviations from the correct values have to be taken into account, in view of the present rapid development in the field of fabrication and characterization of novel nanoporous materials, systems will soon follow in which, because of either larger crystal extensions or smaller surface resistances, the condition for application should be fulfilled completely rigorously. Acknowledgment. Financial support by the Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie is gratefully acknowledged. CM070404R
