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NMR Diffusometry with Beds of Nanoporous Host Particles: An Assessment of Mass Transfer in Compartmented Two-Phase Systems Margarita Krutyeva and Jo¨rg Ka¨rger* UniVersity of Leipzig, Department of Interface Physics, Linne´strasse 5, 04103, Leipzig, Germany ReceiVed May 8, 2008. ReVised Manuscript ReceiVed June 19, 2008 Molecular diffusion in a bed of zeolite crystallites is mimicked by dynamic Monte Carlo simulation of a particle hopping on a two-dimensional square lattice. The resulting probability distribution of molecular propagation (the “mean propagator”) is used for a rigorous determination of the resulting dependencies of diffusion measurement by pulsed field gradient NMR. In the limiting cases of intracrystalline, restricted, and long-range diffusion, these dependencies are found to coincide with the well-known relations resulting from the application of a simplifying exchange model (the “two-region” approximation). The intensity of transport resistances on the boundary between the intra- and intercrystalline spaces, i.e., on the compartment boundaries, is only accessible in the intermediate case, i.e., for observation times comparable with the mean lifetimes within the different compartments. In this case, significant differences between the results of the rigorous treatment and the “two-region” approximation may occur.
Introduction Diffusion, i.e., the irregular, thermally driven motion of the molecules within molecular ensembles, is among the most important, quite general phenomena in nature.1,2 In numerous cases, diffusion is quintessential for the functionality of living organisms3,4 and the performance of technological processes, including mass separation5,6 and chemical conversion by heterogeneous catalysis.6-8 In all these cases, molecular transport occurs in highly complex systems. In first-order approximation, they may be considered to be composed of regions with differing transport properties. A complete assessment of the internal molecular dynamics of such systems has to include, therefore, the rate of propagation within each individual region and the exchange rate between different regions. NMR diffusometry9 has proven to be a most versatile tool for the exploration of molecular dynamics within such complex systems.10-12 Among the different options of diffusion measurement by NMR, the pulsed field gradient (PFG) NMR method (also referred to as pulsed gradient spin echo (PGSE) technique) with ultrahigh magnetic field gradients13,14 has attained particular relevance. In * Corresponding author. Address: University of Leipzig, Department of Interface Physics, Linne´strasse 5 04103, Leipzig, Germany. Tel: +49 (0) 341 97 32502. Fax: +49 (0) 341 97 32549. E-mail:
[email protected]. (1) Jost, W. Diffusion in Solids, Liquids and Gases; Academic Press: New York, 1960. (2) Heitjans, P.; Ka¨rger, J. Diffusion in Condensed Matter: Methods, Materials, Models; Springer: Berlin, Heidelberg, 2005. (3) Howard, J. Mechanics of Motor Proteins and the Cytoskeleton; Sinauer Associates: Sunderland, U.K., 2001. (4) Scheidt, H. A.; Huster, D.; Gawrisch, K. Biophys. J. 2005, 89, 2504–2512. (5) Ruthven, D. M.; Farooq, S.; Knaebel, K. S. Pressure Swing Adsorption; VCH: New York, 1994. (6) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley & Sons: New York, 1992. (7) Ruthven, D. M. In Diffusion Fundamentals 2; Ka¨rger, J., Grinberg, F., Heitjans, P., Eds.; Leipziger Universita¨tsverlag: Leipzig, Germany, 2005; p 77. (8) Chen, N. Y.; Degnan, T. F.; Smith, C. M. Molecular Transport and Reaction in Zeolites; VCH: New York, 1994. (9) Kimmich, R. NMR Tomography, Diffusometry, Relaxometry; Springer: Berlin, 1997. (10) Ka¨rger, J.; Pfeifer, H.; Heink, W. AdV. Magn. Reson. 1988, 12, 2–89. (11) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1–45. (12) Price, W. S. In Annual Reports on NMR Spectroscopy; Webb, G. A., Ed.; Academic Press: London, 1996; p 51. (13) Callaghan, P. T. Principles of NMR Microscopy; Clarendon Press: Oxford, 1991.
numerous applications,15-18 PFG NMR data for diffusion in such systems have been analyzed by implying an exchange model. This approach has been introduced in ref 19 for systems where, depending on their state and/or location, the molecules may diffuse at two different rates of propagation. The generalization to several diffusion regimes may be found in refs 10 and 20. Analysis within this exchange model is based on the assumptions that (i) diffusivity variation from one regime to another occurs independently of the given location of the diffusant; (ii) any molecule within a given state of mobility will, after time t, still be in this state with a probability ∝ exp(-t/τi), with τi denoting the molecular mean lifetime within this state; and (iii) within the given state of mobility, the diffusing molecule does not experience any spatial restriction. Under exactly such conditions molecular transport will proceed in solutions if the molecular species under study diffuses either isolated21 or attached to some larger “carrier”.22 As a further prominent example, one may consider molecular transport in (sufficiently large particles of) mesoporous materials that are only partially filled with guest molecules. Consequently, they may propagate in both the fluid and gaseous phases formed within the phase space. Under such conditions, the exchange time between two “states” of mobility is usually much smaller than the observation time of the PFG NMR experiment.23-25 Guest (14) Stallmach, F.; Galvosas, P. Annu. Rep. NMR Spectrosc. 2007, 61, 51– 131. (15) Price, W. S. In Diffusion Fundamentals 2; Ka¨rger, J., Grinberg, F., Heitjans, P., Eds.; Leipziger Universita¨tsverlag: Leipzig, Germany, 2005; p 112. (16) Meier, C.; Dreher, W.; Leibfritz, D. Magn. Reson. Med. 2003, 50, 510– 514. (17) Adalsteinsson, T.; Dong, W. F.; Scho¨nhoff, M. J. Phys. Chem. B 2004, 108, 20056–20063. (18) Hindmarsh, J. P.; Su, J. H.; Flanagan, J.; Singh, H. Langmuir 2005, 21, 9076–9084. (19) Ka¨rger, J. Ann. Phys. 1971, 27, 107–109. (20) Ka¨rger, J. AdV. Colloid Interface Sci. 1985, 23, 129–148. (21) Burstein, D.; Gray, M. L.; Hartman, A. L.; Gipe, R.; Foy, B. D. J. Orthop. Res. 1993, 11, 465–478. (22) Evilevitch, A.; Jo¨nsson, B.; Olsson, U.; Wennerstro¨m, H. Langmuir 2001, 17, 6893–6904. (23) Valiullin, R.; Naumov, S.; Galvosas, P.; Ka¨rger, J.; Woo, H. -J.; Porcheron, F.; Monson, P. A. Nature 2006, 430, 965–968. (24) Dvoyashkin, M.; Valiullin, R.; Ka¨rger, J. Phys. ReV. E 2007, 75, 041202. (25) Ardelean, I.; Farrer, G.; Mattea, C.; Kimmich, R. Magn. Reson. Imaging 2005, 23, 285–289.
10.1021/la801426f CCC: $40.75 2008 American Chemical Society Published on Web 08/19/2008
Mass Transfer in Compartmented Two-Phase Systems
diffusion in compartmented samples such as polymer multiphase systems,26 emulsions,18 or nanoporous host materials,27 however, deviates from such simplifying assumptions: then and only then may guest molecules change their diffusivity when they pass from one compartment to an adjacent one. Diffusivity variation is therefore correlated with the guest location, and the probability of this variation is no more a sole, simple function of rate constants. Finally, molecular displacements within one compartment (i.e., with one diffusivity) are subject to the confinement by their spatial extension. In refs 18, 28, and 29, this latter effect of confinement by the individual compartments was taken into account by assigning to them time-dependent diffusivities. In its very first application, the exchange model of NMR diffusometry has been used in a comprehensive analysis of the PFG NMR data on molecular diffusion in beds of zeolite crystallites,6 i.e., in a two-region system consisting of the zeolite crystallites and the free (“intercrystalline”) space in between. For exactly such type of systems, however, the prerequisites of application are not rigorously fulfilled: variation in mobility between the highly mobile state in intercrystalline space and the low intracrystalline diffusivity occurs at the interface between these two regions, namely on the external surface of the zeolite crystallites. As a consequence, only in the very special case of “barrier-limited” exchange (i.e., for dominating transport resistance on the surface of the individual crystallites) is the exchange probability expressed by a simple exponential as implied by the above assumption (ii). In general, however, the exchange probability (as appearing in the tracer exchange curve) exhibits a much more complicated dependence.6,30 Nevertheless, the “two-region” approach based on this exchange model proved to be able to provide a comprehensive description of all main features of the measurements. These features include the limiting case of restricted diffusion (by introducing time-dependent diffusivities31) and the regimes of “intracrystalline” diffusion (for observation times short enough so that the mean molecular displacements are much less than the crystallite radii) and of “long-range” diffusion (for molecular displacements notably exceeding the crystallite radii), with the option of the transition between both of them for intermediate observation times. Since, however, the prerequisites of the exchange model are not strictly compatible with reality, it is difficult to decide whether the two-region approach may provide anything more than a qualitative overview of the diffusion patterns provided by PFG NMR. In this paper, we are going to compare the simplified representation of PFG NMR diffusion within the exchange model of the two-region approximation with their rigorous analysis. It will turn out that already a two-dimensional model representation of a bed of zeolite crystallites allows a clear distinction between regimes of essential agreement and of notable differences. Most importantly, rigorous data analysis is found to be quintessential for the correct determination of surface permeabilities. In fact, over the past few decades, the exploration of “surface barriers”, (26) Fu, Q.; Rao, G. V. R.; Ward, T. L.; Lu, Y.; Lopez, G. P. Langmuir 2007, 23, 170–174. (27) Van Bavel, E.; Meynen, V.; Cool, P.; Lebeau, K.; Vansant, E. F. Langmuir 2005, 21, 2447. (28) Price, W. S.; Barzykin, A. W.; Hayamizu, K.; Tachiya, M. Biophys. J. 1998, 74, 2259–2271. (29) Pfeuffer, J.; Flogel, U.; Dreher, W.; Leibfritz, D. NMR Biomed. 1998, 11, 19–31. (30) Barrer, R. M. Langmuir 1987, 3, 309–315. (31) Frey, S.; Ka¨rger, J.; Pfeifer, H.; Walther, P. J. Magn. Reson. 1988, 79, 336–342.
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i.e., of zeolite crystals with notably reduced surface permeabilities, has remained a challenging task for fundamental research and technological application.30,32-35 We start our presentation with a short review of the exchange model and its application in the two-region approximation of diffusion in beds of nanoporous materials. Subsequently, a squarelattice model is presented, which is used to mimic molecular diffusion in beds of zeolite crystallites by dynamic Monte Carlo simulations. The obtained relations are used for the rigorous calculation of the corresponding patterns of PFG NMR diffusometry. We conclude by comparing the thus derived dependencies with those following from the two-region exchange model and by assessing their similarities and differences.
Model Considerations Quantities of Interest. The key function of PFG NMR is the mean propagator P(x,t). It represents the probability (density) that, within the observation time of the PFG NMR experiment, an arbitrarily selected molecule is shifted over a distance x in the direction of the applied field gradients. In the narrow-gradient approximation, i.e., assuming that molecular displacements during the application of the gradient pulses are negligibly small in comparison with the displacements between the pulses, the attenuation Ψ of the signal intensity as accessible in PFG NMR experiments is the Fourier transform of this propagator9,13,36
Ψ(q, t) )
∫ P(x, t) cos(qx)dx
(1)
where q (≡ γδg) stands for the intensity parameter of the field gradient pulses, with γ denoting the gyromagnetic ratio and δ, g, and t denoting the width, amplitude, and separation of the field gradient pulses, respectively. The observation times and mean displacements as accessible by PFG NMR are in the range of milliseconds to seconds and hundreds of nanometers to hundreds of micrometers, respectively. In a homogeneous system, the propagator is a Gaussian
( )
P(x, t) ) (4πDt)-1⁄2 exp -
x2 4Dt
(2)
D denotes the diffusivity which, completely equivalently, may be introduced by either Fick’s first law as the factor of proportionality between the flux and the concentration gradient of labeled molecules or via the Einstein relation
〈x(t) 〉 ) 2Dt
(3)
as a factor of proportionality between the mean square displacement and the observation time. Inserting eq 2 into eq 1 yields
Ψ(γδg, t) ) exp(-q2Dt)
(4)
)exp[-q 〈x (t) 〉 /2]
(5)
2
2
where, with eq 5, we have made use of eq 3. To first order, i.e., for sufficiently small field gradient intensities, eq 5 may be shown to remain a useful approximation for any type of propagator.10 Likewise, eq 4 may also be applied as a good approximation, though, clearly, it is only strictly correct in the limiting case of Gaussian propagators. In this case, D has to simply be understood as an effective diffusivity, defined on the basis of eq 3. In the (32) Ka¨rger, J. Langmuir 1988, 4, 1289–1292. (33) Bu¨low, M.; Micke, A. Adsorption 1995, 1, 29–48. (34) Heinke, L.; Kortunov, P.; Tzoulaki, D.; Ka¨rger, J. Phys. ReV. Lett. 2007, 99, 228301. (35) Vasenkov, S.; Schu¨ring, A.; Fritzsche, S. Langmuir 2006, 22, 5728– 5733. (36) Ka¨rger, J.; Heink, W. J. Magn. Reson. 1983, 51, 1–7.
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case of normal diffusion, i.e., for 〈x2(t)〉 ∝ t, the thus defined effective diffusivity coincides with the genuine diffusivity. In compartmented systems, the propagator deviates from the simple form of eq 4. Within the two-region approach, however, it is still possible to derive an exact analytic expression of the PFG NMR signal attenuation. It assumes a particularly simple form for diffusion in beds of zeolite crystallites. We are going to review it in the subsequent section. Modeling Diffusion in Zeolite Assemblages by the TwoRegion Approach. Following refs 6 and 10, PFG NMR signal attenuation due to molecular diffusion in beds of zeolite crystallites may be represented in terms of the diffusivities D1 and D2 in the inter- and intracrystalline spaces and of their relative populations p1 and p2 (≡ 1 - p1). The molecular mean lifetimes in the interand intracrystalline spaces are denoted by τ1 and τ2, respectively. Detailed balance requires the equality p1/τ1 ) p2/τ2. We consider gas-phase adsorption, so that the diffusant molecules within the intercrystalline space are in the gaseous state. Thus one has D1 . D2 and p1 , p2 (≈ 1), and the somewhat bulky expression for the PFG NMR spin-echo attenuation as resulting from the general theory assumes the simple form
( (
Ψ(q, t) ) exp -q2 D2(t) +
p1D1 q τ2p1D1 + 1 2
)) t
(6)
Following the approaches of refs 18, 28, 29, and 31, the inclusion of a time-dependent coefficient of intracrystalline diffusion D2(t) takes into account the finite space available for molecular displacements in the intracrystalline space. For this purpose, eq 3 is considered to serve quite generally as the definition of an effective diffusivity D2(t). For sufficiently short observation times, i.e., in the absence of notable restrictions of molecular propagation by the boundary, D(t) assumes the value of the genuine, microscopic diffusivity within the compartment. In the opposite limiting case of sufficiently large observation times, D(t) has to scale with 1/t since the quantity 〈x2(t)〉 on the left-hand side of eq 3 has to remain bounded as a result of the finite size of the compartments. The complete time dependence D(t) is a sensitive function of the compartment geometry and the boundary conditions to which the diffusants are subjected.31,37-39 The intracrystalline mean lifetime is introduced as the first statistical moment ∞
τ2 )
∫ (1 - γ(t))dt
(7)
t)0
of the tracer exchange curve γ(t).40 Square-Lattice Model for Simulating Diffusion in Zeolite Assemblages. Figure 1 displays our simulation model. We strived to find a structure as simple as possible, which is still able to reflect all features relevant for molecular transportation in such systems, i.e., intracrystalline diffusion, diffusion in the intercrystalline space, particle accumulation within the crystallites due to adsorption, and additional transport resistances at the boundary. It is worthwhile to mention that the chosen model ensures molecular exchange within a continuous two-dimensional “gas phase”, into which the individual crystals are “embedded”. Thus, in contrast to modeling in one dimension and as a requirement for the validity of our simulations, the molecules (37) Krutyeva, M.; Yang, X.; Vasenkov, S.; Ka¨rger, J. J. Magn. Reson. 2007, 185, 300. (38) Mitra, P. P.; Sen, P. N.; Schwartz, L. M. Phys. ReV. B 1993, 47, 8565– 8574. (39) Sen, P. N. Concepts Magn. Reson. A 2004, 23, 1–21. (40) Barrer, R. M. Zeolites and Clay Minerals as Sorbents and Molecular SieVes; Academic Press: London, 1978.
Figure 1. Two-dimensional model bed. Periodic structure is organized by repetition of the simulation unit consisting of the square crystal with the edge length La and the outer space (indicated by bold lines). In this picture, the diffusion step length in the outer space exceeds the intracrystalline diffusion step length by a factor 4.
may cover essentially infinitely long distances in the free space between the crystals. Molecular exchange between differently shaped particles, including their representation in two and three dimensions, has been shown to follow transient uptake and release curves that follow essentially coinciding trends (see, e.g., Figure 9.2.b of ref 6). Therefore, for the sake of simplicity and of computing efficiency, we have based our simulations on the presented two-dimensional model, with one of its dimensions being the direction of the field gradients applied, i.e., the direction of diffusion measurement by PFG NMR. Such a model allows the rigorous calculation of the PFG NMR signal attenuation patterns following from eq 1 with the mean propagator determined in the simulations. As the main issue of this paper, we are going to compare the values of p1D1 () pinterDinter ≈ long-range diffusivity) and of τ2 () τintra ) intracrystalline mean lifetime) as resulting from the best fit of the two-region approach (eq 6) to the rigorously calculated spin-echo attenuation, with these very quantities as directly resulting from the simulations. In the simulations, we operate with a unit step length a and a time unit τ characterizing the time interval between two subsequent attempts to perform a diffusion step of length a on a square lattice of extension La × La representing the space of one crystal. Each step direction (of the four possible ones) is selected with equal probability. The probability that the step is actually performed is assumed to be given by the Boltzmann factor exp(-εdiff/RT), where εdiff is the activation energy of
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Figure 2. Energies of activation in the simulation model.
intracrystalline diffusion as illustrated in Figure 2. In the outer (“intercrystalline”) space, each step attempt is successful. Figure 2 illustrates this situation by potential wells of essentially negligible depth. The step length a0 exceeds the diffusion step length inside the crystal by a factor λ . 1, i.e., a0 ) λa. Thus, with varying temperature, the diffusion step length in outer space remains constant. This corresponds to the situation of Knudsen diffusion41,42 which, for not too high temperatures, is a situation typical of PFG NMR diffusion studies with beds of zeolite crystallites.43,44 In our simulations we have chosen the values L ) 100 and λ ) 4. The jump probabilities at the interface between the intraand intercrystalline spaces are determined by activation energies, as illustrated by Figure 2. In the absence of surface barriers (εbarr ) 0), each jump attempt toward the crystal surface from the adjacent gas phase positions will be “successful”. This means that the probe molecule will assume a boundary position within the crystal. Vice versa, a jump attempt from these positions toward intercrystalline space is only successful with the probability exp(-εdes/RT). εdes is the heat of adsorption. Jointly with the respective site numbers, it determines the relative populations p1 and p2 of the diffusants in the intercrystalline space and in the crystallites. The possible presence of surface barriers is taken into account by introducing a further activation energy εbarr, representing the “height” of the surface barrier. It reduces the jump rates both into and out of the crystals by an (additional) factor exp(-εbarr/RT). The mean propagator is calculated using the relation10,36 ∞
P(x, t) )
∫
p(x0)P(x0, x0 + x, t)dx0
(8)
-∞
where p(x0) denotes the (a priori) probability density that a molecule is situated at a position with x ) x0. P(x0, x0 + x, t) indicates the (conditional) probability density that, after time t, a molecule, initially at position x0, has to go to x0 + x. Within the given model, the a priori probability may be assumed to be uniform over all positions in the intracrystalline and intercrystalline spaces, being related to each other by the Boltzmann factor exp(-εdes/RT). As one has to imply because of the system’s ergodicity, this exact dependence followed from following the trajectories of the probe molecules within the system over sufficiently long intervals of time. Figure 3 provides two examples of the thus calculated propagators for identical observation times but for notably different surface permeabilities. For crystallites with surface (41) Levitz, P. J. Phys. Chem. 1993, 97, 3813–318. (42) Papadopoulos, G. K.; Theodorou, D. N.; Vasenkov, S.; Ka¨rger, J. J. Chem. Phys. 2007, 126, 094702. (43) Geier, O.; Vasenkov, S.; Ka¨rger, J. J. Chem. Phys. 2002, 117, 1935–1938. (44) Ka¨rger, J.; Vasenkov, S. Microporous Mesoporuos Mater. 2005, 85, 195– 206.
Figure 3. Probability distribution of the molecular displacements (the “propagator”10,36) for the molecules moving through the two-dimensional model system for εbarr ) 0 (dashed line) and εbarr ) εdes (solid line) and for εdiff/RT ) 1.5, εdes ) 3εdiff, and t ) 107τ. The extension La ) 100a of a crystallite (edge length) is indicated by the two vertical lines.
barriers, the probability distribution of molecular displacements is essentially confined to values smaller than the crystal edge length, nicely reflecting the triangular shape of basis length La, which the mean propagator is known to approach for sufficiently large observation times under ideal one-dimensional restriction over this very distance La.13 There is only a minor fraction of molecules that, during the considered observation (i.e., simulation) time, was able to leave their crystallites as reflected by the broadened “foot” of the distribution. Displacements exceeding the crystal extensions are, obviously, only possible if the molecules have the chance to traverse the free space between the crystallites. The propagator calculated without this surface barrier results in a much smoother curve, without any notable unsteadiness for displacements of the order of the crystal dimensions. Molecular diffusion within a bed of zeolite crystallites is thus found to essentially follow the pattern of ordinary diffusion as if proceeding within a homogeneous system.
Comparing the Simulation Results with the Two-Region Approximation Representation by the Effective-Diffusivity Approach. Figure 4 provides an overview of the effective diffusivities in beds of zeolite crystallites as calculated within the frame of the diffusion model as represented by Figures 1 and 2. The chosen Arrhenius plot reveals all relevant features known from PFG NMR diffusion measurements with beds of nanoporous host particles as exemplified by the experimental data given in Figure 5: (i) For sufficiently low temperatures and short observation times, the effective diffusivity coincides with the genuine intracrystalline diffusivity, with an activation energy εdiff given by the energy barrier between adjacent sites within the zeolite bulk phase. (ii) Intracrystalline diffusion path lengths are confined by the crystal size. As a consequence, the mean square displacements and hence the effective diffusivities of intracrystalline mass transfer remain restricted, approaching a limiting value that appears in the “plateaus” of the Arrhenius plots. For onedimensional diffusion under confinement by a crystal of extension La, the limiting value turns out to be Drestr ) (La)2/12t.10,13,31 Such a situation of “restricted diffusion” is represented in Figure 3 with the propagator under the conditions of restricting boundaries. Following the time dependence of Drestr, the heights
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Figure 4. Dependence of the apparent diffusivity, calculated from the mean squared displacement, as a function of the temperature. Open, full and half-full points correspond to different observation times t1 < t2 < t3 (t1 ) 105τ, t2 ) 106τ, t3 ) 107τ Monte Carlo steps). The simulations have been performed with εdes ) 3εdiff and for εbarr ) 0 (squares) and εbarr ) εdes (circles). The values Drestr ) (La)2/(12t) are shown by horizontal lines. The large circle highlights the data point at εdiff/kBT ) 1.5, which refers to the presentation in Figure 3. The arrows on the bottom point to the representation of the PFG NMR attenuation curves in Figure 6.
Figure 5. Temperature dependence of the effective self-diffusivity for n-hexane in a sample of NaX zeolite crystals of mean diameter 4 µm.6
of the plateaus in Figure 4 scale with the inverse values of the observation times. (iii) With increasing temperature and, hence, with increasing thermal energy, more and more molecules are able to pass the crystallite boundaries so that, eventually, the mean square displacements may increase again. The effective diffusivity is expected to result in the product of the diffusivity of the molecules in the intercrystalline space, D1, and their relative amount, p1. Since D1 has been chosen to be invariant with temperature, the activation energy of the effective diffusivity is now determined by p1, leading to the energy of desorption, εdes. In fact, in our simulations, the long-range diffusivities Dlong-range ) 〈x2(t)〉/2t as resulting for root-mean-square displacements 〈x2(t)〉1/2 much larger than the crystal dimensions La are found to agree with the product of the quantities p1 and D1 within a mean error of less than 15%. The two-region approach as given by eq 6 may be easily shown to yield exactly these special cases, namely, case (i) of intracrystalline diffusion for τ2 . t and D2t , (La)2, case (ii) of restricted diffusion for τ2 . t and D2t . (La)2, and case (iii) of “long-range” diffusion for τ2 , t. Exactly in all these cases,
Figure 6. Comparison of the PFG NMR attenuation curves as resulting by rigorous application of eq 1 with the simulated mean propagator (full lines) with the best fits by the two-region approximation (dashed lines). The attenuation curves refer to three different temperatures represented by εdiff/kBT ) 0.5 (a); 1 (b), and 1.5 (c). They are between the transition range between the region of intracrystalline/restricted diffusion and longrange diffusion as indicated in Figure 4. The mean exchange times and long-range diffusivities, as resulting from the best fitting parameters, are compared with the directly simulated quantities in Table 1.
however, the resulting diffusivities are found to be independent of the intensity of the surface barrier! In fact, the representations of Figure 4 include effective diffusivities that have been determined both with and without surface barriers. In complete agreement with the expected behavior, in each of these special cases, the calculated effective diffusivities turn out to be identical. Only in the transition regime from restricted (intracrystalline) to long-range diffusion does the effective-diffusivity approach fail to satisfactorily reflect the primary experimental data of NMR diffusometry. Reliable information about the existence and intensity of surface barriers has to be based, therefore, on a more rigorous treatment. This includes the analysis of the total PFG NMR attenuation plot. Comparison of the PFG NMR Attenuation Plots. The primary information on molecular diffusion as accessible by
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Table 1. Mean Intracrystalline Lifetimes τintra and Long-Range Diffusivities pinterDinter As Obtained from the Computer Simulations and As Fitting Parameters in the Two-Region Approximation (Eq 6) of the Rigorously Calculated PFG NMR Attenuation Curves Shown in Figure 6a-c sim τintra /τ
(pinterDinter)sim/a2 τ-1
two-reg.mod τintra /τ
(pinterDinter)two-reg.mod/a2 τ-1
t1 ) 105 τ t1 ) 105 τ
εbarr ) 0 εbarr/εdiff ) 3
1 × 104 2.8 × 104
εdiff/RT ) 0.5 (Figure 6a) 6 × 104 6.9 × 104
0.21 0.22
0.5 0.3
t2 ) 106 τ t2 ) 106 τ
εbarr ) 0 εbarr/εdiff ) 3
5 × 104 8.5 × 105
εdiff/RT ) 1 (Figure 6b) 48 × 104 8 × 105
0.03 0.03
0.044 0.043
t3 ) 107 τ t3 ) 107 τ
εbarr ) 0 εbarr/εdiff ) 3
3.5 × 105 300 × 105
εdiff/RT ) 1.5 (Figure 6c) 4 × 106 350 × 105
0.004 0.0041
0.006 0.0036
PFG NMR is contained in the attenuation of the signal (the “spin-echo”) intensity as a function of the PFG intensity. As indicated by eq 1, it is directly related to the mean propagator, i.e., to the evolution of the probability distribution of molecular displacements within the sample. In the previous section it was shown that the two-region approximation totally complies with the rigorous treatment in the limiting cases of intracrystalline, restricted, and long-range diffusion. In fact, in all these cases, the spin-echo attenuation degenerates to a simple exponential, which is described by a single, effective diffusivity. Hence, deviations between the model representation and rigorous treatment may therefore only be expected in the transition range between intracrystalline and/or restricted and long-range diffusion. In this transition range, i.e., for 0.5 e εdiff/RT e 2 (see Figure 4), we have compared the PFG NMR signal attenuation plots as rigorously resulting via eq 1 from the propagators simulated for our model with the relations of the two-region approach (eq 6). For illustration, Figure 6 shows selected examples of these curves. Table 1 compares the fitting parameters of the two-region two-reg.mod approach, namely, τintra and (pinterDinter)two-reg.mod, with the sim corresponding values τintra and (pinterDinter)sim as resulting directly from the simulations. In the fitting we have included Dintra,eff ) 〈x2(t)〉intra/2t as a known parameter. In all cases, the two-region approach is found to nicely reproduce the real attenuation curves as resulting from their rigorous calculation via eq 1. However, comparison of the “real” long-range diffusivities and intracrystalline mean lifetimes with the fitting parameters τ2(intra) and p1(inter)D1(inter) of the two-region approach yields notable deviations. While the expressions for long-range diffusion differ by less than a factor 2.5, the intracrystalline mean lifetimes resulting as fitting parameters may exceed the value determined by direct simulation by 1 order of magnitude. Note that this great difference is only observed in the absence of surface barriers (εbarr ) 0), while, with barriers, i.e., with additional transport resistances between the two regions of different diffusivities, the intracrystalline mean lifetimes as resulting from the two-region approach are found to nicely reproduce the “true” values as resulting from the model simulations. This finding may be correlated with the fact that barrier-limited exchange complies with the second of the prerequisites of the two-region approximation as given in the Introduction.
Conclusions We have determined the mean propagator, i.e., the evolution of the probability distribution of particle displacements in a twodimensional model system, consisting of an array of (square) crystals (region 2, intracrystalline space) embedded in a medium of high diffusivity and low particle concentration (region 1, intercrystalline space) by dynamic Monte Carlo simulations. By Fourier transform (see eq 1), these propagators have been transferred into the attenuation curves of the NMR signal as the primary quantity accessible in diffusion measurements by PFG NMR. Conventionally, such types of compartmented systems are considered within the frame of the so-called two-region approximation, in which the description of the intrinsic system dynamics is reduced to four parameters, namely, the relative populations in the two spaces, the exchange time, and the respective diffusivities. The present, rigorous treatment of the model system nicely reproduces the three special cases of the two-region approximation that may be described by a single (effective) diffusivity, namely, the coefficients of intracrystalline, restricted, and long-range diffusion, with the last case referring to a fast exchange, and the former two cases referring to essentially no exchange of molecules between the intra- and intercrystalline spaces during the observation time. In the transition range, i.e. for exchange times of the order of the observation time, the dynamic quantities of the two-region model (which result from the best fit to the “experimentally” determined curves, i.e., our simulation curves) are found to notably deviate from the corresponding “real” (i.e., directly simulated) values. This is particularly true for the intracrystalline mean lifetime where the values predicted within the two-region model may exceed the real ones by 1 order of magnitude. The exploration of the conditions under which the benefit of the two-region approach (and its generalization to even more “phases”), namely, the options of its analytical consideration, are not impeded by the observed deviations, is a worthwhile task of further studies. Acknowledgment. Financial support by the Deutsche Forschungsgemeinschaft (Ka 953/19) is gratefully acknowledged. LA801426F
