A new view of diffusion in nanoporous materials

Transcription

1 Chemie Ingenieur Technik DOI: /cite Diffusion 779 A new view of diffusion in nanoporous materials Christian Chmelik, Lars Heinke, Rustem Valiullin und Jörg Kärger* Diffusion is among the rate-controlling processes in the technological application of nanoporous materials, including separation and conversion processes. Over decades, the different techniques of diffusion measurements yielded controversial results. The benefit of novel measuring techniques which, by immediate visual evidence, exemplify the self-consistency of the resulting diffusivities is shown. Furthermore, by quantifying the permeabilities through the particle surfaces and by correlating the rate of molecular uptake and release with the molecular mobilities, these techniques are able to identify and to explore additional transport resistances which so far, though being rate-limiting in numerous cases, were outside the range of direct experimental observation. Keywords: diffusion, interference microscopy, IR microscopy, nanoporous materials, NMR, transport resistances Received: February 18, 2010; revised: February 26, 2010; accepted: March 02, Introduction Being omnipresent in all states of matter, diffusion is among the most important and, simultaneously, most fascinating phenomena in nature and technology. This is particularly true for diffusion in nanoporous materials. With diameters comparable with the pore sizes of the confining host materials, the guest molecules are subject to both guest-guest and guesthost interaction. Depending on the nature and the relative weight of these interactions, molecular diffusion is able to reflect a multitude of fundamental processes governing the behaviour of molecular ensembles, from single-molecule patterns up to soft-matter behaviour. Simultaneously, guest-diffusion in nanoporous host materials is among the rate limiting processes in their technological application [1] for upgrading by heterogeneous catalysis [2], selective adsorption [3] and membrane separation [4]. The scientific progress in diffusion studies with nanoporous materials was accompanied by a series of surprises. With the introduction of pulsed field gradient (PFG) NMR to diffusion studies in zeolites, the intracrystalline diffusivities were suddenly found to exceed some of the previously quite generally accepted data by several orders of magnitude [5]. However, even after most of these discrepancies could be unambiguously referred to shortcomings in the interpretation of the old transient sorption experiments [6, 7] and, correspondingly, were taken account of in subsequent experiments, a series of well-documented studies remained which revealed substantial differences between the results of the different measuring techniques [8]. Possible explanations of these differences included the existence of transport resistances within the zeolite bulk phase (internal barriers), acting in addition to the drag on molecular propagation by the genuine pore space. The existence of such resistances has recently been confirmed by both diffusion measurements over space scales of the order of the distances between these resistances [9-12] and by direct observation by high-resolution transmission electron microscopy [13]. Over the past few years, the understanding of diffusion in nanoporous materials has been notably enhanced by two innovations in the experimental procedure [14]. They include the microscopic observation of the evolution of the guest profiles in the interior of the individual crystallites and the determination of guest self-diffusivities as a function of the externally applied guest pressure. After a repetition of some basic relations of diffusion theory and experiment in Sect. 2, Sect. 3 focusses on these novel options of diffusion measurement. Examples illustrating the novel quality of information provided by these techniques are presented in the subsequent Sect.s 4 to 6. They refer to phenomena in the intracrystalline space, at the interface between the nanoporous particle and the surroundings and in mesopores. With diameters comparable with the pore sizes of the confining host materials, guest molecules are subject to both guest-guest and guest-host interaction. The understanding of diffusion in nanoporous materials has been notably enhanced by two innovations in the experimental procedure. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

2 780 Übersichtsbeiträge Ch. Chmelik Random molecular movement which, quite generally, is referred to as diffusion, occurs under both non-equilibrium conditions and under equilibrium conditions. Correlating the diffusivities over the whole concentration range requires a suitable model with appropriate assumptions. In pioneering work diffusion in nanopores is treated within the frame of the Maxwell-Stefan model. 2 Diffusion Fundamentals Diffusion is the process of random motion of the elementary constituents of matter, in particular of atoms and molecules, sustained by their thermal energy. It may be quantified by Fick s first law j ˆ D c c (1) x correlating the flux density j of the particles under study (the diffusants ) with the gradient of their concentration c. The gradient direction is assumed to define the direction of the x coordinate. D is referred to as the coefficient of diffusion (the diffusivity ). Fig. 1a provides a scheme illustrating the typical situation of diffusion measurements. The small spheres indicate the individual molecules and the larger ones represent the pores of the host system (in the given case the metal organic framework (MOF) ZIF-8 to which we shall refer in more detail in Sect. 4.2). It is obvious that, on their random movement, more molecules will pass the indicated plane from left to right than from right to left, giving rise to a net flux. In real experiments, the concentration gradients will amount to typically one molecule per cavity divided by the distance of more than thousands of cavities rather than by the distance between adjacent cavities, as assumed in Fig. 1a for simplification. Thus, all effects of non-linearity are definitely excluded and a relation of the type of Eq. (1) may be assumed to hold quite generally: doubling the concentration gradient doubles the flux. Although the diffusivity may Figure 1. Scheme illustrating typical situations of diffusion measurements. The small spheres indicate individual molecules. The cage system symbolizes the structure of the host system MOF (metal organic framework) ZIF-8 (see sect. 4.2). The following molecular fluxes through the virtual plane in the middle are considered: (a) transport diffusion fluxes under the influence of an overall concentration gradient, and (b) self-diffusion the opposing (identical!) fluxes of differently labelled but otherwise identical molecules. be assumed to be independent of the concentration gradient, it will in general depend on the guest concentration, as well as on the temperature, the nature of the diffusants and the host system under study. Random molecular movement which, quite generally, is referred to as diffusion, occurs under both non-equilibrium conditions (as shown in Fig. 1a) and under equilibrium conditions. Also in the latter case, diffusivity may be defined by an equation of the type of Eq. (1). For this purpose, as illustrated by Fig. 1b, molecules which are differently labelled (the open and filled spheres in Fig. 1b), but coincide in their microdynamic properties are considered. Now again fluxes of the differently labelled molecules may be observed. They are opposite to each other and of identical magnitude so that the overall situation (total concentration of labelled and unlabelled molecules) remains unchanged. The diffusivity measured under such equilibrium conditions is referred to as the self- or tracer diffusivity. This diffusivity depends on the overall concentration rather than on only the concentration of the labelled species considered. For distinction, the diffusivity measured under an overall concentration gradient is referred to as the transport diffusivity. In this overview, the symbol D was used for the self-diffusivity and D T for the transport diffusivity. It is important to note that, completely equivalent with Eq. (1), the self-diffusivity may also be defined by the Einstein relation [15] hx 2 t i ˆ 2Dt (2) correlating the molecular mean square displacement in x direction with the diffusion ( observation ) time t. Referring to these different situations it should be noted that there is no universal formal relationship between the coefficients of transport diffusion and self-diffusion except that in the limit of negligibly small concentrations where both coefficients have to coincide. This may be easily understood by considering equal gradients of the overall concentration in Fig. 1a and of the labelled species (filled spheres) in Fig. 1b. In the limit of sufficiently small concentrations, where there is negligible molecular interaction, both diffusion fluxes will be identical so that, given the identical concentration gradients, it is evident from Eq. (1) that the diffusivities must coincide. Correlating the diffusivities over the whole concentration range requires a suitable model with appropriate assumptions. In the pioneering work of Rajamani Krishna, diffusion in nanopores is treated within the frame of the Maxwell-Stefan model [7, 16-18]. It is based Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

3 Chemie Ingenieur Technik Diffusion 781 on equating the generalized forces stimulating the diffusion fluxes with the drag they are subject to by either mutual collisions or by interaction with the pore walls. Self- and transport diffusion may thus be shown to be interrelated by 1=D ˆ 1=D T0 h=d AA. (3) Here, D T0 is the corrected diffusivity, which is related to D T via ln c D T0 ˆ D T (4) lnp with c(p) denoting the adsorption isotherm, i.e. guest concentration c in equilibrium with the surrounding pressure p [7, 17]. The logarithmic derivative lnp= lnc(the reciprocal value of which appears on the right side of Eq. (4)) is known as the thermodynamic factor. Ð AA is referred to as the mutual Maxwell-Stefan diffusivity. Actually, given that the mutual Maxwell-Stefan diffusivity Ð AA is a free parameter, Eq. (3) is not much more than a confirmation of the above statement that there is no universal correlation between the self- and transport diffusivities. It provides, however, a useful distinction between the two different mechanisms which, intuitively, may be understood to control the flow rate during counter diffusion of differently labelled molecules. The first term on the right hand side of Eq. (3) is related to the drag to which the molecules are subject under both the conditions of transport diffusion and tracer exchange, i.e. to the friction with the host lattice. Irreversible thermodynamics identifies the gradients of the chemical potential as the driving forces of diffusion [6, 7]. Any deviation from ideality, i.e. from proportionality between concentration and pressure, may thus be shown to lead to an extra force which does not appear in the selfdiffusivity. This is the reason that the corrected transport diffusivity, rather than the transport diffusivity itself, appears in this term. The second term, i.e. the reciprocal value of the mutual Maxwell-Stefan diffusivity multiplied with the relative pore filling h, characterizes the drag by counter-diffusion, i.e. between the fluxes of the filled and open molecules in Fig. 1b. In the limit of small guest concentrations Henry s law applies (c p) so lnc= lnp= 1. This yields, with equation (3), equality of the self- and transport diffusivities, in conformity with the previous discussion. Molecular transport under single-file conditions [19] is characterized by an infinitely large drag between the counter-diffusing molecules, since any mutual passage between adjacent molecules is prohibited. This leads to the wellknown fact that assuming ideally permeable one-dimensional channels with diameters close to those of the diffusants transport diffusion does not yield any peculiarities while, for infinitely long channels, self-diffusion drops to zero (leading to mean square displacements increasing only with the square root of time rather than with the time itself, as would have been required by Eq. (2) for finite self-diffusivities). For diffusion in two- and three-dimensional pore spaces, however, it is often the opposite limiting case which may be implied for a comparison of self- and transport diffusion. This is in particular true if the diameters of the windows between adjacent cavities are comparable with the molecular sizes so that molecular propagation is controlled by jumps through the windows rather than by redistribution within the cavities. In fact, for the MOFs of type ZIF-8 forming the background of Figs. 1a and b, the propagation of short alkanes may be expected to follow exactly this pattern, as is considered in more detail in Sect Under such conditions, any drag between counter-diffusing molecules becomes negligibly small, i.e. Ð AA becomes infinite and the self-diffusivity coincides with the corrected diffusivity: ln c D ˆ D T0 ˆ D T lnp. (5) By similar reasoning, Eq. (5) may also be shown to follow from irreversible thermodynamics (by assuming negligibly small phenomenological cross coefficients) and from transition state theory (if it is assumed that molecules in the transition state are isolated from the remaining molecules) [6]. In most technical applications, one is concerned with diffusion phenomena within guest mixtures rather than with single-component diffusion. For diffusion within a mixture of n guest components, the diffusivity of Eq. (1) has to be replaced by a diffusion matrix D ij, yielding j i ˆ Xn D ij c 1 ; c 2 ; c 3 :::c n c j (6) x jˆ1 where the matrix elements D ij denote the factors of proportionality between the flux of component i and the concentration gradient of component j. This generalized Fick s 1 st law holds as generally as Eq. (1). However, like Eq. (1), this relation is of immediate practical use only if the individual components of the diffusion matrix are known. The recently introduced technique of IR micro-imaging (see Sect. 3.1) has opened the way to such measurements. However, such measurements are far There is no universal correlation between the selfand transport diffusivities. In most technical applications, one is concerned with diffusion phenomena within guest mixtures rather than with singlecomponent diffusion. Molecular transport under single-file conditions is characterized by an infinitely large drag between the counterdiffusing molecules. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

4 782 Übersichtsbeiträge Ch. Chmelik The two measuring techniques applied for microscopic imaging were interference microscopy (IFM) and IR microimaging (IRM, IR microscopy). Knowledge on diffusion in nanoporous materials has benefitted from the application of microscopic measuring techniques operating under nonequilibrium conditions. from straightforward so that even in future the coefficients appearing in Eq. (6) will be mostly determined on the basis of theoretical approaches. Within the approximation leading to Eq. (5), the matrix elements D ij may be shown to obey the relation [20] D ij c 1 ; c 2 ; c 3 :::c i c 1 ;c 2 ;c 3 :::c n lnp i c 1 ;c 2 ;c 3 :::c n n ˆciD ; c j lnc j (7) where D i (c 1, c 2, c 3...c n ) denotes the self-diffusivity of component i measured under equilibrium conditions for the given set of concentrations c 1, c 2, c 3...c n. PFG NMR has been established as a versatile technique for this type of measurements [21, 22]. The partial derivatives lnp i = lnc j result from the set of the multi-component adsorption isotherms c i (p 1, p 2, p 3...p n ). It is due to these derivatives that the fluxes of one component are also affected by the concentration gradients of the other components. For sufficiently low concentrations p i c i and the diffusivity matrix has non-vanishing elements only in the diagonal. They coincide with the self-diffusivities of the individual components. This is a simple consequence of the fact that, in the limit of small concentrations, the diffusivity of a given molecule is unaffected by the existence of all other molecules, as explained already in the paragraph following Eq. (2). 3 Key techniques of experimental observation In addition to the impressively wide range of nanoporous materials which have emerged over the past few years [3, 23-25], present knowledge on diffusion in nanoporous materials has greatly benefitted from the application of microscopic measuring techniques operating under non-equilibrium conditions. This concerns both the pulsed field gradient technique of nuclear magnetic resonance (PFG NMR [26-29]) and the novel techniques of microscopic imaging by IR and interference microscopy [30-33]. They are key to the novel insights on mass transfer in nanoporous materials which are referred to in this overview. It is noteworthy that PFG NMR studies of selfdiffusion during molecular uptake by beds of microporous particles (zeolite crystallites of type NaX [34, 35]) have already been performed some time ago. However, they did not yield any remarkable deviation from the expected behaviour: the rate constants governing the approach to equilibrium in response to a change in gas pressure (or gas composition) could be estimated reasonably well on the basis of the diffusivity data resulting from the PFG NMR measurements. It will be shown in Sect. 5 that the situation is dramatically different for mesoporous materials. 3.1 Monitoring intracrystalline concentration profiles Figs. 2 and 3 illustrate the principles of the two measuring techniques applied for microscopic imaging, namely interference microscopy (IFM) and IR micro-imaging (IRM, also referred to as IR microscopy). Both techniques operate under the conditions known from conventional macroscopic uptake and release experiments [36], including sample activation (i.e. sample heating and evacuation). Operating with a single crystal rather than a macroscopic crystal ensemble leads to particular challenges, especially in relation to the disturbing influence of omnipresent impurities (with water as a prominent representative) which may affect the sorption and diffusion properties of a small single crystal much more than macroscopic amounts of host particles. Transient sorption experiments are initiated by changing the gas pressure in the surrounding atmosphere. All IRM and IFM measurements reported in this overview have been performed at room temperature, although there is also the option of temperature variation [37]. In both IRM and IFM concentration integrals in the observation direction are recorded. In IFM this measurement is based on comparing the optical path length of the light beams passing through the crystal with their counterparts (generated by the use of semi-transparent mirrors) in the surrounding atmosphere (Fig. 2d). The optical path length is determined by the refractive index of the crystal which varies with the concentration. Variation in the refractive index during transient sorption phenomena and, hence, in the optical path lengths appear as changes in the interference patterns. These can be analyzed (Fig. 2c) to yield the changes in the concentration integral responsible for these changes [33, 38]. Spatial resolution in the plane perpendicular to observation is limited by the optical resolution of about 0.5 lm. In IRM, information about the concentration integrals in the observation direction is given by the area under a characteristic IR band of the guest molecule under study (top centre of Fig. 3). The IR microscope may operate in two modes. By considering the spectrum of the whole crystal (by a single-element (SE) detector), the device is able to record the total uptake or release. This information is related Wiley-VCH Verlag GmbH & Co. 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5 Chemie Ingenieur Technik Diffusion 783 Figure 2. Experimental set-up and basic principle of interference microscopy (IFM) for diffusion measurements in nanoporous systems. a) Schematic representation of the vacuum system (static) with the optical cell containing the sample. b) Interference microscope with CCD camera on top, directly connected to c) the computer. d) Basic principle: changes in the intracrystalline concentration during diffusion of guest molecules cause changes in the refractive index of the crystal (n 1 ) and, hence, in the phase difference Dj of the two beams. After measuring the difference of the optical path length the difference of the intracrystalline concentration by the given equation can be evaluated. e) Close-up view of the optical cell containing the crystal under study. Figure 3. Experimental set-up and basic principle of IR microscopy (IRM). An optical cell is connected to a vacuum system. After mounting it on a movable platform under the microscope one individual crystal is selected for the measurement. Changes in area under IR bands of the guest species are related to the guest concentration. The spectra can be recorded as signal integrated over the whole crystal (SE single-element detector) or with a spatial resolution of up to 2.7 lm by the imaging detector (FPA focal plane array). Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

6 784 Übersichtsbeiträge Ch. Chmelik Being able to distinguish between different molecular species, IRM is also able to record diffusion phenomena in multi component systems. Both IFM and IRM are able to record the evolution of concentration profiles. Microscopic profile imaging provides no direct information on the propagation rates of the individual guest molecules. to that of conventional transient sorption experiments, with the significant difference that one is now able to monitor uptake and release on a single crystal. Measurements of this type yield information intermediate between microscopic and macroscopic measurements and are, therefore, referred to as mesoscopic [39]. Even without resolving intracrystalline concentration profiles, single-crystal measurements have a great advantage in comparison with bed experiments: Due to the dramatically enlarged surface-to-volume ratio, transient sorption experiments with single crystals are essentially unaffected by the disturbing effects of heat release [40] which, with fast sorption processes in beds of zeolite crystallites [6, 41], is known to become the rate limiting process. In the integral mode of IRM, information about the underlying diffusion phenomena may be deduced by adopting the conventional procedure, i.e. by implying that uptake and release is controlled by intracrystalline diffusion and that the adsorption/desorption steps are small enough so that the diffusivity is uniform over the sample and constant. Assuming that the crystal shape is approximated by a sphere of radius r, relative uptake obeys the relation [6, 42] A norm ˆ At A 0 A A 0 ˆ 1 6 p 2 X nˆ1 1 n 2 exp n2 p 2 D t r 2 : 8 Deviations from spherical shape lead only to minor changes in the shape of the uptake curve [6]. As an alternative to the measurement of integral uptake, by the use of a focal plane array (FPA) detector, it is possible to record separately and simultaneously, the spectra over each individual space element (top right of Fig. 3), with a minimum pixel size of 2.7 lm. Both IFM and IRM are able to record the evolution of concentration profiles as exemplified in the centre right of Fig. 3. These profiles represent the integrals over the concentration in observation direction. For crystals with oneand two-dimensional pore networks extended perpendicular to the direction of observation, these integrals simply result as the product of the crystal thickness and the local concentration and directly reflect therefore, in the usual case of constant thickness, the local concentration. Note that, conventionally, the direction of observation (e.g., Fig. 2d) is assigned to the x coordinate. Conventionally, the diffusion equations are also noted with respect to the x coordinate (see Eq. (1)). If diffusion is in fact restricted to the two directions perpendicular to the observation direction, the diffusion equations must, correspondingly, be expressed in terms of the other coordinates. Being able to distinguish between different molecular species, IRM is also able to record diffusion phenomena in multicomponent systems. Thus, also the elements of the diffusion matrix as introduced by Eq. (6) become accessible by direct measurement. This includes, in particular, the observation of the counter flux of differently labelled, but otherwise identical molecules, yielding, via Eq. (1), the self- or tracer diffusivity. Among several possibilities to deduce the relevant diffusivities from transient concentration profiles [43-45], the most direct one is based on Fick s 1 st law (Eq. 1). The flux at a certain position at time t simply results as the integral between the two concentration profiles at the times adjacent to t from this position to the crystal centre, divided by the time difference. Dividing this flux by the mean value of the concentration gradient at these times and position yields the diffusivity. For asymmetric profiles the determination of fluxes becomes more complicated. As an alternative, the diffusivities may be determined on the basis of Fick s 2 nd law c t ˆ x D c x ˆ D 2 c x 2 D c c 2 (9) x which results from combining Fick s 1 st law with the equation of matter conservation. In addition to the uncertainty in the determination of the second derivative of the concentration, this way of analysis is complicated by the fact that the diffusivities appear also as their derivatives D/ c. One may get rid of this complication by considering the profiles at their minima, (where c/ x = 0 and the disturbing term D/ c disappears). These considerations are only necessary for transient sorption experiments (where D stands for the transport diffusivity and c denotes the total concentration). In self-diffusion experiments, i.e. during tracer exchange, c denotes the concentration of the labelled molecules within a uniform overall concentration. It has already been mentioned that the self-diffusivity is clearly independent of the percentage of labelled molecules so that the troublesome term D/ c becomes zero automatically. 3.2 Recording guest diffusion during transient sorption experiments Microscopic profile imaging as described in the previous section reveals the evolution of guest distribution in response to changes in the surrounding atmosphere. It provides, however, no direct information on the propagation rates of the individual guest molecules, i.e. on their self-diffusion behaviour, during this pro Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

7 Chemie Ingenieur Technik Diffusion 785 cess of establishing the new equilibrium. This type of information may be provided by the pulsed field gradient technique of NMR (PFG NMR). Fig. 4 (top left) illustrates schematically one possible experimental arrangement which allows PFG NMR measurements to be carried out during a transient sorption rate measurement in order to determine both self- and transport diffusivities under identical conditions. The sample with the material under study is placed within the probe of the spectrometer. It is in contact with the atmosphere of the desired guest species, the pressure of which may be regulated from zero to the saturation pressure of the liquid. Information about the guest molecules is provided by the proton magnetic resonance signal of the sample generated by appropriately chosen sequences of rf pulses [26, 27, 46]. This includes in particular the determination of the total amount of molecules within the sample (resulting from the intensity of the signal following a so-called 90 pulse, the free induction, top centre of Fig. 4) and their self-diffusivity. PFG NMR self-diffusion measurements are based on the principle that the resonance frequency is a function of the intensity of the magnetic field and hence, in an inhomogeneous field (i.e., under the application of field gradients ), a function of the space coordinate in gradient direction. Exactly this dependence is the basis of magnetic resonance tomography (MRT) which, in the last few years, has become the most important imaging technique in medical diagnosis [26, 46]. In diffusion measurements, by the application of gradient pulses, the magnetic field is made inhomogeneous over two short time intervals. By the application of appropriately constructed sequences of rf pulses and gradient pulses (see bottom left of Fig. 4 and further examples in [47]) an NMR signal (the spin echo ) may be generated which is sensitive to the differences in the locations which each individual molecule has assumed during the two gradient pulses. Hence, with the Einstein relation, Eq. (2), direct access to the self-diffusivities is provided. For normal diffusion, i.e., for systems obeying Fick s laws [6, 15], the self-diffusivity results as the slope of the straight line which one obtains by plotting the logarithm of the spin echo signal intensity vs. the observation time (namely the time between the two gradient pulses) multiplied with a parameter characteristic of the intensity of the field gradient pulses (bottom right of Fig. 4). Molecular uptake or release is recorded by plotting the intensity of the free induction si- 90 free induction (FID) signal P Long-time scale uptake Short-time scale Self-diffusion δ g diffusion time t δ g spin-echo signal intensity S Figure 4. Schematic illustration of pulsed field gradient (PFG) NMR experiments that combine the information provided by conventional sorption experiments with the self-diffusivities of the guest molecules in the pore space. The sample is placed within the probe of the spectrometer (top left) and is exposed to the gas phase of the guest species at a certain pressure. Information about the mobility of the guest molecules is provided by the proton magnetic resonance signal of the sample generated by appropriately chosen sequences of rf pulses (bottom left). The total amount of molecules within the sample (via the free induction signal; top right) and their self-diffusivity (bottom right) can be measured. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

8 786 Übersichtsbeiträge Ch. Chmelik IRM and IFM follow the conventional procedure of solidstate diffusion measurements which are based on a layer-by-layer analysis of the concentration of the diffusants in the penetration direction. gnal vs. time (top centre and right of Fig. 4). By implying that the overall process is limited by diffusion, the time dependence for spherical samples is again found to obey Eq. (8), similarly to operating in the integral mode of IRM (see Sect. 3.1). 4 Intracrystalline Diffusion 4.1 Evidence from intracrystalline concentration profiles The IRM and IFM techniques [30, 31] are unique in that they provide a clear and direct visualization of intracrystalline transport in nanoporous materials. In a general way, IRM and IFM follow the conventional procedure of solid-state diffusion measurements which are based on a layer-by-layer analysis of the concentration of the diffusants in the penetration direction [48-50]. In comparison to these techniques, IRM and IFM greatly benefit from the high diffusion rates of the guest molecules in nanoporous materials and from the continuous, non-invasive measurement of the profiles over the whole crystal. These virtues provide the access to data sets with extensions that notably exceed those known from solid-state diffusion studies. These novel options could be demonstrated very nicely in extensive studies of molecular diffusion of methanol in a large all-silica crystal of ferrierite [32, 51-53] since the ferrierite crystals [54] proved to be extremely stable under the conditions of repeated ad- and desorption, allowing dozens of adsorption-desorption cycles to be repeated with the same crystal, without any perceptible change in the sorption and diffusion properties. Fig. 5 illustrates the host system under study and the evolution in the concentration integrals during uptake. The ferrierite pore system allows diffusion in two directions (defining the y and z coordinates, see Fig. 5a) which are chosen to be perpendicular to the direction of observation. From structure analysis, the critical pore diameters in either direction are known to be determined by 8- and 10-membered silica rings. Fig. 5b shows the measured concentration integrals at three instants of time after the start of an adsorption step. Selected concentration profiles in the y and z direction, taken from these profiles, are shown in Figs. 5c and d. It is clear from the presentation in Fig. 5c that there is a substantial fraction of guest molecules which, essentially immediately after the onset of adsorption, has penetrated into the crystal. This fraction of molecules is assumed to be accommodated in the roof-like parts Figure 5. Transient profiles of methanol concentration for uptake in ferrierite (pressure step 0 to 80 mbar in the gas phase, 298 K). The 1d representations on the right side are extracted from the 2d profiles (lower left corner) for a clearer view on the uptake process. The uptake process is dominated by transport along the smaller 8-ring channels (y direction). Except for the formation of a triangular roof at early times no indications for uptake along z direction are found. Obviously, the larger 10-ring channels are blocked in the crystal body and the molecules have to enter through the smaller 8-ring windows in y direction (see sketch in the top left corner) Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

9 Chemie Ingenieur Technik Diffusion 787 of the crystals which are found on top (and on the bottom) of the slab-like main body of the ferrierite crystals (Fig. 5a). As also indicated in Fig. 5a, this fast uptake implies free access to the 10-ring channels in z direction, while, in the main body of the crystal, access through the 10-ring channels is severely restricted. Such a restriction is necessary to explain why molecular uptake by the main crystal body occurs predominantly along the channels in y direction, irrespective of their smaller diameter. The ferrierite crystals proved to be an excellent model system for considering transient sorption phenomena by diffusion along the 8- ring channels in y direction: it was only necessary to subtract the contribution of uptake by the roof-like components (as immediately visible from Fig. 5c) from the profiles of overall uptake shown in Fig. 5d (which results in a simple shift of the base-line). As an example, Fig. 6b shows the transient concentration profiles during desorption, generated by a sudden decrease of the pressure in the surrounding methanol atmosphere from 10 mbar to zero. It is noteworthy that the boundary concentration (at y= 0 and 50 lm) does not assume the novel equilibrium value c= 0 immediately after the onset of desorption. This indicates the existence of an additional transport resistance at the crystal surface which reduces the rate of molecular exchange between the surrounding atmosphere and the intracrystalline pore space [55]. This resistance may be accounted for quantitatively by introducing a (finite) surface permeability a, defined by the relation [42] j x ˆ 0 ˆa c eq c x ˆ 0 Š (10) where j(x=0) and c(x=0) denote the flux through and the (intracrystalline) concentration at the crystal boundary and c eq indicates the concentration in equilibrium with the surrounding atmosphere which, for constant external pressure, coincides with the finally attained concentration. Again the conventional notation of a diffusion coordinate was used to be in accordance with the notations of Eqs. (1), (6) and (10). Using the notation of Fig. 6b, obviously, x must be replaced by y (where, in addition to y=0, also a second boundary condition y=50lm might be considered). The diffusivities and surface permeabilities displayed in Fig. 6a yield a best fit between the experimentally determined concentration profiles (spheres in Fig. 6b) and the solutions of the diffusion equation (Fick s 2 nd law, Eq. (9)), with Eq. (10) as the boundary condition [51]. The concentrations are normalized with respect to the equilibrium value attained with the largest pressure step. The permeabilities are plotted as a function of the given boundary concentrations. We shall return to this point in Sect. 5.1 which refers in greater detail to the phenomenon of surface resistance. It is noteworthy that data analysis based on desorption with a smaller pressure step yields essentially the same values, confirming the self-consistency of our analysis. Fig. 7 displays the concentration profiles during desorption and adsorption in a unified representation by plotting the concentrations from bottom to top for desorption (left scales in Figs. 7a and b) and from top to bottom for adsorption (right scale) [32]. For simplicity, only one half of the profiles (starting with x=0 in the crystal centre) is shown. During tracer exchange, to maintain overall equilibrium, at any Ferrierite crystals proved to be an excellent model system for considering transient sorption phenomena by diffusion along the 8-ring channels in y direction. (a) diffusivity, D T / m 2 s methanol in FER, 298 K Desorption 5 0 mbar Desorption 10 0 mbar norm. concentration, c norm surf. permeability, α / m s -1 (b) norm. concentration, c norm methanol in FER: desorption 10 0 mbar concentration profiles (IFM) calculation using D(c) and α(c) 0 s 30 s 90 s 210 s 400 s 600 s 900 s 1350 s 1950 s 3450 s 7350 s y /µm Figure 6. a) Loading dependence of transport diffusivity and surface permeability as determined by local analysis of the concentration profiles for the desorption steps from 5 mbar and 10 mbar to vacuum using Fick s 2 nd law. b) Comparison of simulated and experimental profiles for the desorption step from 10 to 0 mbar. The dots are the experimental concentration profiles. The lines are calculated profiles using the concentration dependence of D T (c) anda(c) as shown in a). Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

10 788 Übersichtsbeiträge Ch. Chmelik Figure 7. Methanol in ferrierite: Dramatic impact of the concentration dependence of the transport parameters on molecular uptake (open spheres) and release (full spheres). a) For small loading steps uptake and release proceed essentially complementary (as indicated by the arrows of similar length at t = 200s). b) For large loading steps a dramatic acceleration of the adsorption occurs. The desorption profiles essentially coincide with those from small loading steps. The adsorption profiles are flipped with respect to the desorption profiles to allow their direct comparison. c) Transport diffusivity and surface permeability depend strongly on the loading. For sufficiently small pressure steps and, correspondingly, concentrations steps, the diffusivities and surface permeabilities may be considered to be constant and the diffusion equation becomes linear. instant of time, desorption of one component (i.e. its decrease in concentration) is exactly counterbalanced by adsorption of the other. Implying linearity in both the diffusion equation (Eq. (9)) and the boundary conditions (Eq. (10)), the same has to hold true for comparing subsequent adsorption and desorption over identical pressure steps. For sufficiently small pressure steps and, correspondingly, concentrations steps, the diffusivities and surface permeabilities may be considered to be constant and the diffusion equation becomes linear. This is close to the situation shown in Fig. 7a. The amounts adsorbed and desorbed are similar to each other at any time and place. Fig. 7c illustrates that, over the concentration range covered in the experiments, both the diffusivity and surface permeability do not vary substantially. This is totally different for the large pressure step which yields substantial differences in the transient profiles during adsorption and desorption (Fig. 7b). Most remarkably, the rate of desorption is found to be essentially independent of the pressure step, while the rate of adsorption increases dramatically with the pressure step. This finding may be rationalized by implying that the overall process is dominated by particle exchange between the host particle and the surroundings and, hence, by the magnitude of the diffusivities and permeabilities close to the surface. In the considered cases of desorption these values are essentially identical, corresponding with the similarity in the observed desorption patterns. During adsorption, however, an enhancement in the pressure step leads to a dramatic enhancement in both permeability and diffusivity close to the surface which may easily be understood as the origin of the dramatic acceleration of the uptake process for larger pressure steps. It is well known from classical sorption rate measurements that the measurement of adsorption and desorption over the same pressure step can yield useful insights, especially concerning the concentration dependence of the diffusivity [6, 36]. The insights provided by such measurements are even greater for IFM and IRM measurements since any differences in the form or symmetry of the transient concentration profiles can be seen directly. 4.2 The end of a discrepancy Being able to determine diffusivities under equilibrium and non-equilibrium conditions within the same device, namely by transient Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

11 Chemie Ingenieur Technik Diffusion 789 sorption experiments and by tracer exchange, respectively, IRM is ideally suited for an unbiased comparison of self- and transport diffusivities. With the advent of MOF ZIF-8 [24, 56-58] we also have the advantage of a highly stable nanoporous material with crystals of the necessary size and free from defects, so that the evolution of guest concentration profiles could be easily related directly to intracrystalline transport. A survey of the coefficients of self-diffusion and transport diffusion in ZIF-8 resulting from such experiments is provided by Fig. 8 [59]. Emphasis is on the important finding that, for all guest species under study, self- and transport diffusion are found to merge for sufficiently small concentrations. This behaviour is to be epected quite generally as explained in Sect. 2., However, this has never before been confirmed directly by experiment. Most remarkably, with increasing loading the self-diffusivities of some guest molecules (namely for methanol up to high loadings, see Fig. 8 a, and for ethanol up to medium loadings, see Fig. 8c) are found to actually exceed the transport diffusivities. Such behaviour has never been previously observed. As shown in Fig. 1, the overall gradient for the measurement of transport diffusion (Fig. 1a) coincides with the gradient of the labelled molecules (i.e. full spheres) in the self- (or tracer) diffusion measurements (Fig. 1b). Enhancement of selfdiffusion in comparison with transport diffusion means an enhancement of the flux of the molecules in Fig. 1a due to the presence of the additional (differently labelled) molecules in Fig. 1b. Molecular propagation is evidently promoted rather than hindered under the influence of a counter flux of differently labelled but otherwise identical molecules. At first sight such behavior seems surprising although it is in fact fully understandable. In order to explain this behavior we refer to Eqs. (3) to (5) of Sect. 2 and Figs. 1 and 9. As already indicated in the scheme of the pore space of ZIF-8 in Fig. 1, molecular propagation in ZIF-8 is controlled by the passage through windows between adjacent cavities rather than by the friction between adjacent molecules. Therefore, the second term in Eq. (3) may be neglected. With Eq. (5), the self-diffusivity is thus expected to approach the corrected diffusivity (the product of the transport diffusivity and the inverse lnc= lnp c= p of the c=p thermodynamic factor). Fig. 8 also includes the logarithmic derivatives lnc= lnp determined Most remarkably, with increasing loading the self-diffusivities of some guest molecules are found to actually exceed the transport diffusivities. (a) D T, D T0 or D / m 2 s methanol self-diffusivity, D corrected diffusivity, D T0 transport diffusivity, D T d lnc / d lnp c norm d lnc / d lnp (b) D T, D T0 or D / m 2 s ethane d lnc / d lnp transport diffusivity, D T corrected diffusivity, D T0 self-diffusivity, D c norm 1 d lnc / d lnp (c) D T, D T0 or D / m 2 s self-diffusivity, D ethanol corrected diffusivity, D T0 transport diffusivity, D T d lnc / d lnp d lnc / d lnp c norm Figure 8. Methanol, Ethane and Ethanol in ZIF-8 at 298 K. Loading dependence of transport (D T ), corrected (D T0 ) and selfdiffusivity (D) ofa) methanol, b) ethane and c) ethanol. The inverse thermodynamic factor (obtained from Fig. 9) is also shown. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

12 790 Übersichtsbeiträge Ch. Chmelik (a) c norm ethanol methanol ethane p / mbar (b) c norm dc/dp c /p dc/dp c /p Thermodynamic factor p / mbar isotherm Figure 9. a). Adsorption isotherms: The isotherms of methanol and ethanol exhibit a pronounced S-shape (type III or V isotherm), whereas the isotherm of ethane shows a usual ( type I ) behaviour. For all molecules the loading at saturated vapour pressure was normalized to reach c norm =1.b) Visualisation of the thermodynamic factor appearing in Eq. 5. For S-shaped isotherms in certain regions the slope ( c= p) exceeds the slope of the straight line connecting the relevant point of the adsorption isotherm with the origin (c/p). With increasing loading, molecular interaction has to be taken into account and two opposing effects become relevant. from the adsorption isotherms shown in Fig. 9a, as well as the resulting corrected diffusivities. Most impressively, for all molecules considered, the self-diffusivities agree well with the corrected diffusivities over the entire ranges of concentration. Finally, on the basis of Eq. (5), also the surprising finding may be explained that, for certain guests and loadings, the self-diffusivities may exceed the transport diffusivities. With Eq. (5), the self-diffusivity is seen to exceed the transport diffusivities under the condition lnc= lnp c= p > 1, i.e. for those parts of c=p the adsorption isotherm where the slope ( c= p) exceeds the slope of the straight line connecting the relevant point of the adsorption isotherm with the origin (c/p). Fig. 9b illustrates that this situation exists in the initial part of an S-shaped isotherm (the so-called type-iii or type-v isotherms) which are typical for the adsorption of polar molecules on hydrophobic surfaces [3]. For the highly polar methanol molecules indeed exactly this behaviour may be expected. When, with increasing loading, molecular interaction has to be taken into account, two opposing effects become relevant; (i) competition for the limited space and (ii) a tendency towards molecular aggregation, driven by the (dipolar) interaction energy of the guest molecules. Item (i) is known to give rise to the typical Langmuir ( type-i ) isotherm, i.e. to c= p < c/p (upper part of Fig. 9b), or, in other words, to the necessity of a less than linear increase in loading with pressure. Item (ii), by contrast, leads to a greater than linear increase in loading with increasing pressure, ensuring that c= p > c/p (lower part of Fig. 9b). Thus, by taking account of the prevailing mechanism, Eq. (5) correctly predicts the correlation between self- and transport diffusion as observed in the experiments. It is noteworthy that, on the basis of these two mechanisms, the experimentally observed behaviour may be explained already intuitively by comparing the situations reflected by Figs. 1a and b: As soon as molecular interactions become relevant, with mechanism (ii) dominating, the intermolecular attractive forces will counteract the driving force due to the concentration gradient and reduce the flux during transport diffusion (to regions of lower overall guest content, Fig. 1a) in comparison with selfor tracer diffusion (in which the overall concentration is uniform, as in Fig. 1b). However, when mechanism (i) is dominant it means that the molecules will even more willingly propagate to regions of lower concentrations, thus enhancing the guest fluxes of transport diffusion (Fig. 1a) in comparison with self- (or tracer) diffusion (Fig. 1b). The differences observed between the different guest molecules nicely illustrate the conditions under which each of these mechanisms will prevail. While, over the whole loading range considered, the diffusion properties of the highly polar methanol molecules are dominated by their mutual interaction (mechanism (ii), transport diffusion exceeded by self-diffusion for all loadings), the reverse is true for nonpolar ethane (mechanism (i), self-diffusion exceeded by transport diffusion). Ethanol assumes an intermediate position, with D > D T for small loadings (prevailing mechanism (ii)) and D < D T for higher loadings (prevailing mechanism (i)). The finding that the self-diffusivities may exceed the transport diffusivities has been illustrated here by referring to the remarkable Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

13 Chemie Ingenieur Technik Diffusion 791 situation in which molecular propagation is promoted rather than slowed down under the influence of a counter flux of labelled molecules. Having rationalized the origin of this finding, the situation may now more correctly be described by referring to the mere presence of the differently labelled but otherwise identical molecules rather than to their flux. By implying an infinitely large mutual Maxwell-Stefan diffusivity Ð AA, i.e. by excluding any intermolecular friction, the flux itself is anyway postulated to be of no influence. It is (only) the very existence of the additional molecules which enhances the propagation rate from cage to cage in comparison with the situation of transport diffusion! 5 Surface Barriers The existence of surface barriers, i.e. of a dramatic reduction of the permeability through a layer close to the surface, has been suggested as one of the possibilities to explain why molecular uptake and release were often found to be much more slowly than expected from the intracrystalline diffusivities provided by PFG NMR [5, 8, 60]. By means of the NMR tracerdesorption technique [61, 62], these surface resistances have even been shown to be accessible to a quantitative estimate. However, it was only with the introduction of the IFM and IRM monitoring techniques that a sufficiently accurate determination has become possible. We report here the first systematic investigations of these resistances and their possible origin. With these novel options, the surface permeabilities themselves emerge as an important parameter for future experimental study. 5.1 In-depth study of a novel quantity Following Crank s textbook on the mathematics of diffusion [42], a surface permeability (see Eq. (10)) is defined by the relation a =j surf / (c eq c surf ) as the ratio between the flux through the particle surface and the difference between the actual surface concentration and the concentration which would be established in equilibrium with the surrounding atmosphere. Complementing Fick s diffusion equation (Fick s 2 nd law, Eq. (9)) by the boundary conditions provided by the surface permeability (Eq. (10)) has proved a reasonable approach to the (in general unknown) real situation. As one option to visualize the situation Fig. 10a represents the surface barrier as an extended layer of dramatically reduced diffusivity (D barr ). (a) (b) c eq c surf (t) c eq c surf (t i ) surface barrier t3 t 1 t 2 bulk crystal t 4 The layer thickness (l) is assumed to be negligibly small in comparison to the crystal extension and is below the limits of spatial resolution by our imaging techniques. With Eq. (10), the permeability through this layer is easily found to be a = D barr /l. Before presenting an example where the micro-structural origin of the surface barrier is studied in detail (and is in fact found to deviate significantly from the picture of Fig. 10a), a more general comment on the nature of surface barrier is needed. Surface barriers notably reduce the rate of molecular exchange between the bulk phase of the nanoporous material and the surroundings, under both equilibrium (for tracer exchange) and non-equilibrium conditions (for transient sorption experiments). In complete analogy with the diffusivities, which are referred to as self- (or tracer) diffusivities and transport diffusivities, one should therefore distinguish between these two types of permeabilities. In general, we are going to investigate surface resistances in transient sorption experiments, so, if not stated otherwise, all permeabilitis referred to in this overview are measured under non-equilibrium conditions. (We leave the option of referring to transport and self - (or tracer ) permeabilities to a future decision.) The importance of such a distinction becomes obvious already when considering the concentration dependence of the surface permeabilities. In complete analogy to the self-diffusivities, permeabilities under equilibrium conditions depend only on the overall concent t 12 t 11 t 10 t 9 t 8 t 7 t 6 t 5 Figure 10. a). Close-up sketch of a crystal surface with a barrier layer on top. The surface permeability depends on the boundary (c surf (t)) and the equilibrium concentration (c eq ). The boundary concentration is the concentration at the inner side of the barrier layer. b) The boundary concentrations (c(x= 0)=c surf )are measured over the indicated sequence of times t i. By means of the NMR tracer-desorption technique surface resistances have even been shown to be accessible to a quantitative estimate. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

14 792 Übersichtsbeiträge Ch. Chmelik The surface permeabilities are considered as a function of the boundary and equilibrium concentrations for which they have been determined. Surface permeation is affected by a whole spectrum of concentrations, from c surf to c eq, rather than by a single one. tration, (i.e. on the sum of the concentration of the labelled and unlabelled molecules). Note that in this case the concentration profile in Fig. 10a would refer to labelled molecules. By adding the concentration of the unlabelled molecules, the overall concentration will be constant through the barrier. With this in mind, we become aware of a rather fundamental problem with the concentration dependence of the surface permeability measured under non-equilibrium conditions. Surface permeation, obviously, is affected by a whole spectrum of concentrations, from c surf to c eq, rather than by a single one. This problem is discussed again when dealing with real experimental permeability data later in this section, and we start with the simplifying assumption that it is sufficient to consider the permeability as a function of only the two extreme concentrations, namely the equilibrium concentration c eq and the actual boundary concentration (i.e. the concentration c surf (t), attributed in Fig. 10a to the interface between the crystal bulk phase with the boundary layer, as accessible in the IRM and IFM experiments). It is important to note that the surface permeability, i.e. the transport resistance on the crystal surface, is not correlated with the step in free energy (i.e. the differences in energy and entropy) between the intra- and intercrystalline spaces. It is true that the increase in potential energy from inside to outside reduces the escape rate of the molecules out of crystals, just as a decrease in entropy from outside to inside (caused by a decrease in the number of degrees of freedom due to pore confinement) decreases the entrance rate. However, these mechanisms appear already in the change of the equilibrium concentrations so that any decrease in the escape or entrance rates is exactly compensated by a corresponding increase in the respective populations and, hence, in the attempt rates to escape out of or to enter the crystals. A first in-depth study of surface barriers has been performed with MOFs of type Zn(tbip) [63]. The Zn(tbip) crystals under study proved to be stable enough to allow a completely reproducible study of adsorption and desorption measurements over many cycles with one and the same crystal. This reproducibility is indispensible since the possibility that the surface permeability may vary from crystal to crystal must be considered. With the equilibrium concentration c eq as determined by the external pressure, and with the fluxes through the surface (j(x= 0)=j surf ) and the boundary concentrations (c(x= 0)=c surf ) as measured over the indicated sequence of times t i, Eq. (10) yields the respective surface permeabilities. These surface permeabilities are considered as a function of the boundary and equilibrium concentrations for which they have been determined. The set of permeabilities resulting from an individual adsorption run (as leading, e.g., to the sequence of profiles shown in Fig. 10b) would be considered, therefore, as a function of a constant equilibrium concentration corresponding to the externally applied pressure and of a set of boundary concentrations that increase from the initial concentration to the equilibrium concentration. In a c eq -vs.- c surf plane, these concentrations would form a straight line, parallel to the c surf axis and ending on the diagonal c surf = c eq, as shown in Fig. 11a. Desorption experiments would lead to a sequence of concentrations forming analogous parallel lines, now, however, leading to the diagonal c surf = c eq from higher surface concentrations. Fig. 11b shows the values of a(c surf, c eq )by little bullets over all c surf - c eq pairs considered in the different sorption experiments. As one may see from the traces in the c eq -vs.-c surf plane, in one case even the external pressure (and hence the relevant equilibrium concentration) was varied during a desorption and an adsorption experiment, leading to deviations from lines strictly parallel to the c surf axis. Within the uncertainty of the measurements, the permeability data lead to the following conclusions: (i) For a given pair of boundary and equilibrium concentrations, i.e. for given values of the two key parameters of Fig. 10a, the resulting permeability is found to be independent of both the direction and the magnitude of the pressure step by which the sorption process is generated. (ii) The measured permeabilities are satisfactorily approached by a function of a sole parameter, namely the mean value (c surf + c eq )/2 of the two limiting concentrations c surf and c eq. The best fit of the surface permeabilities to a function of this sole parameter is indicated by the shaded area in Fig. 11b. Fig. 11c shows the permeability data as a function of this mean concentration (c surf + c eq )/2. Since a projection of the data points (c surf, c eq ) in Fig. 11a onto the diagonal yields exactly this mean value (c surf + c eq )/2, by using identical data points, one is able to indicate (in Fig. 11a) the course of the experimental runs by which the surface permeabilities (shown in Fig. 11c) have been determined. Within a, clearly quite substantial, scattering the permeability data of all experimental runs fit reasonably well to each other, confirming our statement under item (i) Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

15 Chemie Ingenieur Technik Diffusion 793 (c) (b) (a) Figure 11. a). The actual boundary concentration and the equilibrium concentration during the adsorption/desorption runs of propane on nanoporous crystals of type MOF Zn(tbip). b) Surface permeability as a function of the equilibrium and the boundary concentration of the crystal in 3d representation. Large bullets stand for surface permeabilities. Their projection on the bottom plane gives the corresponding pair of equilibrium and boundary concentration. c) The resulting surface permeabilities, plotted (by the same symbols as in a) as a function of the arithmetic mean of these concentrations. The finding that, under the given conditions, the surface permeability may be treated as a function of a single parameter, namely the mean concentration in the perspective of Fig. 10a, is of great practical importance: It notably facilitates the inclusion of such resistances for modelling their influence in technological application. Simultaneously, it is expected to contain implicitly valuable information about the nature of the elementary mechanisms and processes leading to the observed phenomenon of a surface resistance. Fig. 12b shows the best data fit (shaded area in Fig. 11b) of the surface permeabilities of propane in the Zn(tbip) crystal under study as a function of the mean concentration, jointly with the intracrystalline diffusivities [63, 64]. It is worthwhile to mention that in similar studies with a different crystal of type Zn(tbip) of the same batch, the surface permeabilities were found to be enhanced by a factor of about 2, while the intracrystalline diffusivities remained, essentially, unaffected. This finding is in complete agreement with our understanding that the intracrystalline space is much more stable and does not lead so easily to varying diffusivities as structural variations on the surface may lead to variations in the permeability. As a most remarkable finding, the concentration dependencies of the diffusivities and surface permeabilities exhibit identical trends. Even more remarkably, this similarity in the trends is also observed with ethane (Fig. 12 a) and n-butane (Fig. 12 c), when applied to the same crystal as guest molecules. Finally, this similarity in the trends is even observed in tracer exchange experiments with propane on comparing the diffusivities and permeabilities under equilibrium conditions (Fig. 12b). It is interesting to note that this similarity is found for diffusivities and surface permeabilities both increasing (namely under non-equilibrium conditions) and decreasing (during tracer exchange) with increasing concentrations. Considering the actual diffusivity and permeability data, this similarity in the trends may be referred to a surprisingly simple common feature: with all loadings, molecules and modes of measurement considered, the ratio a/d between permeability and diffusivity of the cry- The concentration dependencies of the diffusivities and surface permeabilities exhibit identical trends. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

16 794 Übersichtsbeiträge Ch. Chmelik (a) ethane (b) (c) n-butane propane Figure 12. Diffusivities and surface permeabilities for transient sorption experiments at room temperature with a) ethane, b) propane and c) n-butane. b) as well compares the data of transient sorption with the corresponding equilibrium data (self-diffusivities and tracer exchange surface permeabilities). stal under study is found to be on the order of m 1. In the following Sect. 5.2, this remarkable finding will be correlated with a structure model of the considered host crystal. This section will be concluded by rationalizing the differences in the concentration dependencies observed under equilibrium and nonequilibrium conditions [64]. Following the reasoning of Sects. 2 and 4.2, with an adsorption isotherm of the Langmuir type like that shown in Fig. 9a for ethane in ZIF-8, the increase in diffusivity and permeability with increasing loading under non-equilibrium may easily be referred to an increase in the driving force by the increasing thermodynamic factor lnp= lnc. For the measurement under equilibrium, i.e. for tracer exchange, one has to take into account that the pore system of MOFs of type Zn(tbip) is formed by one-dimensional arrangements of small cages. The cage volume is so small that, under common pressure conditions, it is rather unlikely that they are simultaneously occupied by two molecules of the type considered in this study. This means, in turn, that the molecules residing in a chain of cages are essentially confined to their order since a mutual exchange of the position of adjacent molecules would imply, at least over a short interval of time, the occupation of one cage by two molecules. It is easily understandable, therefore, that an increase in the loading leads to a decrease in the mobility as observed in our self-diffusion experiments. In fact, under ideal conditions of single-file diffusion the diffusivity must even be expected to drop to zero [19, 65-68]. The finite self-diffusivities as observed in our experiments, however, indicate notable deviations from this ideal structure in the system under study, ensuring the possibility that guest molecules may escape from the order as given by their initial position within a given file. In the next section this feature is included in the model considerations to rationalize the close relationship between the diffusivities and surface permeabilities in the Zn(tbip) crystals under study. 5.2 Modelling of a surface barrier Following [69, 70], mass transfer in Zn(tbip) may be simulated in a structure model as provided by Fig. 13. The host system is supposed to consist of channels (1d pores) in which mass transfer occurs by random molecular jumps between adjacent (cubic) segments in x direction. Only a small fraction p open of the channel mouths to the surroundings are open. Filling of the blocked channels is ensured by defects in the channel walls. They occur with the probabilities p y = p z. Jumps are attempted with equal probability in all 6 directions from each channel segment. They are only successful if the jump attempt leads into a direction where the segment is open and if the segment to which they are directed is empty. The equilibrium concentration is adjusted by appropriately choosing the encounter rate of molecules from the reservoir onto the crystal surface, in comparison with the jump rate in the channels Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

17 Chemie Ingenieur Technik Diffusion 795 Fig. 14 shows the result of such a simulation: the resulting concentration profiles perpendicular to the surface (spheres in Fig. 14a) are found to be reproduced by the solution of the diffusion equation, based on the intracrystalline diffusivity and a surface permeability (full lines). These curves correspond with the experimentally obtained ones, as displayed in Fig. 10b. Fig. 14b illustrates that, in the initial state of uptake or release, surface heterogeneity, i.e. the variation between open and blocked entrances, gives rise to heterogeneities in guest concentrations. However, these differences soon disappear with increasing distances from the surface, as a pre-requisite for replacing the effect of permeation through a very small number of open channels by a quasi-homogeneous layer of reduced permeability. In [71, 72], Weiss and co-workers exploited this mean field approach for correlating the bulk diffusivity with the effective permeability through the surface of an isotropic host system where molecular exchange with the surroundings is only possible through circular holes of diameter r. Here, surface permeability a and diffusivity D are found to be correlated by the equation a =2Dr/L 2, where L denotes the distance between adjacent holes. In [69, 70] this concept was successfully transferred to the conditions of anisotropic diffusion. Again, surface permeability and bulk diffusivity are found to be proportional to each other, since both are directly proportional to the jump rate. Hence, any variation in the jump probabilities e.g. by varying the guest molecules and the loading or even by changing from transient uptake to tracer exchange may be expected to affect the diffusivities and permeabilities in the same way, leaving their ratio invariant. Exactly this was the remarkable finding presented in Sect For the investigated crystals of Zn(tbip) it has thus been found that there is no structural correlation to Figure 13. Structure model used to simulate mass transfer in Zn(tbip). The host system is supposed to consist of channels (1d pores) in which mass transfer occurs by random molecular jumps between adjacent (cubic) segments in x direction. Only a small fraction p open of the channel mouths to the surroundings are open. Filling of the blocked channels is ensured by defects in the channel walls. the shaded area in Fig. 10a, which symbolizes the surface barrier as a layer of reduced mobility. The model considered in the present section operates with the blockage of the channel entrances (with the fraction p open unblocked) and the option of guest exchange between adjacent channels (with the probability p y = p z ). With the simulation results shown in Fig. 14a it becomes obvious that the mean concentration close to the surface, i.e. in the first channel segments (n x = 1), coincides with the boundary concentration resulting from the solution of the diffusion equation with the boundary conditions provided by a surface barrier. In contrast to the sketch in Fig. 10a, also the real boundary concentration is far below the equilibrium concentration attained after sufficiently long waiting times. In the initial state of uptake or release, surface heterogeneity gives rise to heterogeneities in guest concentrations. However, these differences soon disappear with increasing distances from the surface. (a) (b) Figure 14. a). Simulated concentration profiles along x and the corresponding solutions of the diffusion equation with a finite surface permeability. b) Dependence of the simulated profiles on the distance x from the crystal surface. With increasing distance differences in the concentration are found to vanish. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

18 796 Übersichtsbeiträge Ch. Chmelik The concentration dependence of the surface permeabilities may be based on the mean value of the concentrations, corresponding to a mean diffusivity in the layer of varying concentrations below the crystal surface. It has become possible to correlate the relative boundary concentration with the total amount of uptake or release at each individual instant of time. It is important, however, to reconsider in this context the meaning of the definition of the concentrations under consideration. In the IRM/IFM measurements as well as in the representations of Fig. 14a, the resulting concentrations are mean values: in our measurements the average in the observation direction, in Fig. 14a the mean over a particular slice extended in the y and z directions of Fig. 13 over all segments with a given number n x (which determines its distance from the crystal surface). However, if local concentrations are considered, i.e. the occupancy of each individual channel separately from the adjacent ones, the concentrations in the open channels close to the surface will clearly approach their equilibrium values c eq. Towards the crystal interior, the option of particle exchange between adjacent channels averages the concentrations to a value which is recorded as c surf. Hence, within a layer between the outer surface and total averaging of the concentrations, the local concentrations cover the whole range from c surf up to c eq. Having in mind that the local mobilities are, in general, a function of these local concentrations, therefore it is not unexpected, that the concentration dependence of the surface permeabilities (Figs. 11c and 12) may be based on the mean value of these concentrations, corresponding to a mean diffusivity in the layer of varying concentrations below the crystal surface. 5.3 A simple means for assessing the significance of diffusion and surface permeation in transient sorption experiments Conventionally, the relative weight of the two most important transport mechanisms controlling the rate of molecular uptake and release, namely intracrystalline diffusion and surface permeation, is assessed by comparing the respective time constants s diff = l 2 /(3D) for diffusion-limited uptake and s surf = l/a for uptake limited by transport resistances on the crystal surface. These relations result by assuming the crystals as plates of thickness 2l. The time constants are also known as the first statistical moments of the sorption curves. For other geometries they only vary in the numerical factor [6, 73]. By recording the transient concentration profiles during molecular uptake and release by IRM and IFM it has now become possible to correlate the relative boundary concentration with the total amount of uptake or release at each individual instant of time. As an example, Fig. 15b shows such a correlation plot which has been determined from the concentration profiles (Fig. 15a) recorded during transient uptake measurements with methanol by zeolite ferrierite along the crystallographic y direction (see Sect. 3.1). Following [74, 30] it will be shown that such representations allow a direct assessment of the rate-determining process. Figure 15. A novel option of assessing surface resistances: a) Knowledge of transient concentration profiles during molecular uptake yields boundary concentration (c surf ) and associated relative uptake (m, resulting as the area under the profile) at each of the considered instants of time. b) Plotting c surf as a function of m and extrapolating the long-time asymptote towards the ordinate yields a factor w by which, as a consequence of the surface barrier, molecular uptake is retarded in comparison with the limiting case of diffusion control (i.e. for infinitely large surface permeabilities). c) Examples of c surf vs. m plots where molecular uptake is controlled by mainly diffusion (la/d = 100), by diffusion and surface permeation to similar extents (la/d =1) and by mainly surface permeation (la/d = 0.01). (a) concentration, c t = 2250 s t = 1650 s 0.6 t = 1100 s 0.4 t = 230 s t = 110 s 0.2 t = 750 s t = 30 s t = 400 s 0.0 t = 0 s y / µm (c) (b) w Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

19 Chemie Ingenieur Technik Diffusion 797 With the simplifying assumption of constant diffusivities and surface permeabilities, the normalized concentration profiles during molecular uptake by a plate of thickness 2l are given by the relation [42] c y; t ˆ1 X 2L cos b n y=l exp b 2 n Dt=l2 n 1 b 2 n L2 L cos b n (11) where the ß n s are the positive roots of L ˆ la D ˆ b n tan b n (12) w ˆ 1 b2 1. (15) L Considering a large spectrum of quite different concentration dependences for both the diffusivities and surface permeabilities, such a dependence may be shown to hold quite generally, in excellent agreement with numerous examples of data analysis [75]. The significance of this finding shall be discussed in the context of the examples shown in Fig. 15c. These represent the correlation plots calculated analytically for constant diffusivities and permeabilities for ratios la/d = 100, 1 and 0.01, respectively [74]. With the above introduced time constants s diff = l 2 /(3D) for diffusionlimited uptake and s surf = l/a for uptake limited by transport resistances on the crystal surface [73], the expression la/d may be identified as a measure of the ratio s diff /s surf between the time constants of uptake or release, brought about exclusively by either diffusion or surface permeation. It is easily understandable that for prevailing surface resistances (i.e. for small values of la/d) the total amount adsorbed will increase essentially in parallel with the boundary concentration. On the contrary, large surface permeabilities will ensure that, essentially instantaneously with the beginning of molecular uptake, the boundary concentration will attain the equilibrium value. This is exactly the behaviour reflected by Fig. 15c for la/d = 100. Moreover, quantitative analysis yields the reciprocal value of the ordinate intercept, w 1,asan estimate of the ratio s diff+surf /s diff. This ratio represents the factor by which the (actual) uptake time s diff+surf (or, quite generally, the time of equilibration) is retarded by the existence of the surface. [75] provides numerous examples where this type of analysis is shown to nicely reproduce the results of the corresponding procedure based on a separate determination of intracrystalline diffusivites and surface permeabilities. Integration over the system from -l to l yields m t X 2L 2 exp b 2 n ˆ 1 Dt=l2 m n 1 b 2 n L2 L b 2 n (13) for the relative uptake at time t. In the longtime limit one may confine to only considering the first terms in the sums in Eqs. (11) and (13) which then may be easily combined to c surf t ˆc y ˆ l; t ˆ1 b2 1 L b2 1 m t (14) L It is thus found that, with m 1, in fact any function c surf (m) will approach a straight line, yielding an ordinate intercept 5.4 Sticking probabilities In catalysis, sticking coefficients are defined as the probability that, on colliding with a plane surface, a molecule will be captured [76]. Experimental studies include hydrogen, nitrogen, oxygen, carbon monoxide and ethylene as probe molecules and a large spectrum of metals and metal oxides as targets, yielding sticking probabilities typically between 0.1 and 1 [77]. Chemical reactions involving nanoporous catalysts depend on the probability that, on encountering the surface of a catalyst particle, the reactant molecules penetrate into the particle interior, relevant for the desired conversion. Such probabilities have been estimated on the basis of both PFG NMR [78, 79] and IR [80-83] experiments, as well as by MD simulations [78]. Depending on the system under study, probabilities from as small as 10 7 [80-82] to close to one [78, 79] have been obtained. Monitoring the evolution of guest profiles in nanoporous materials provides a new and direct access to the sticking probabilities of nanoporous particles. It is worthwhile to refer, within this context, to the elementary mechanisms of sorption and diffusion. From the kinetic theory of gases, the flux of molecules encountering the particle surface is known to be given by the relation j Gas = N A p (2p RTM) 1/2, with N A and M denoting, respectively, the Avogadro number and the molar mass [84]. On the other hand, in the short-time limit the amount adsorbed during diffusion-controlled uptake scales with t 1/2 [6, 42, 85] which implies an influx proportional to t 1/2 which diverges in the limit of sufficiently short times. In the probability distribution of the particle life times in systems under diffusion exchange, this peculiarity is known to appear in an infinitely high probability density for zero life times [86]. The conflict between the finiteness of guest supply and infinitely large guest fluxes into the host system for short times exists, however, only in mathematical formalism. In reality one has rather to recollect Chemical reactions involving nanoporous catalysts depend on the probability that the reactant molecules penetrate into the particle interior, relevant for the desired conversion. Large surface permeabilities ensure that, essentially instantaneously with the beginning of molecular uptake, the boundary concentration will attain the equilibrium value. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

20 798 Übersichtsbeiträge Ch. Chmelik The diffusion equation only holds over time and space scales large enough to ensure that the superposition of the elementary steps of molecular displacement leads to a Gaussian probability distribution with the variance given by the Einstein relation. The benefit of nanoporous materials for mass conversion and mass separation comes at the cost of a decrease in molecular mobility. that the diffusion equation only holds over time and space scales large enough to ensure that the superposition of the elementary steps of molecular displacement leads to a Gaussian probability distribution with the variance given by the Einstein relation (Eq. (1)) [15, 87]. With the detection of surface barriers, the bulk phase of the nanoporous host materials is now known to be, in general, separated from the external space by a layer of dramatically reduced permeability. Knowledge of the permeability through this layer now makes it possible to determine directly the fraction of molecules which, after colliding with the external particle surface, are able to penetrate into this particle range where molecular propagation and, hence, the evolution of the guest profiles is controlled by intracrystalline diffusion. Fig. 16 provides an example of this type of analysis [32]. It shows the profiles of deuterated propane during tracer exchange with normal propane in a crystal of MOF Zn(tbip) as considered in Sect. 5.1 (Fig. 16a) and the permeabilities (Fig. 16b) resulting, via Eq. (10), for each individual time interval. The scattering in the permeability data with increasing time, i.e. with increasing tracer exchange, reflects their uncertainty since the overall concentration remains constant and, hence, also a constant permeability should be expected. The (net) flux through the surface as accessible in our experiments is the difference between the counter-directed fluxes leaving and entering the crystal through the surface boundary (Fig. 16c). The flux of molecules entering from the gas phase in total is thus easily found to be equal to j in = a c eq with c eq given by the loading at equilibrium with the gas phase, since under equilibrium it has to be balanced by the flux of the leaving molecules. Combining this information yields a sticking probability of j in /j gas We have thus found that, out of 20 million molecules colliding with the surface of the MOF crystal under study, only a single one is able to enter into the intracrystalline bulk phase where the evolution of the guest profiles is exclusively controlled by intracrystalline diffusion in channel direction! 6 Peculiarities of Diffusion in Mesopores As a prerequisite for the application of nanoporous materials in shape-selective catalysis and mass separation, the critical pore diameters are generally of the order of the diameters of the investigated molecules. The benefit of these materials for mass conversion and mass separation, therefore, comes at the cost of a decrease in molecular mobility. Today, for performance enhancement under technological application, the overall diffusivity is often increased by using materials of hierarchical pore structure [88-92] in which the micropore bulk phase is traversed by mesopores for transport acceleration. NMR monitoring of the guest molecules during transient sorption experiments in mesoporous materials has revealed that, with the onset of capillary condensation in these pores, the time constants for guest equilibration after an external pressure change may dramatically exceed the estimates based on the PFG NMR diffusion measurements. 6.1 Correlating uptake and diffusion in mesopores Figure 16. a). Evolution of the concentration profiles of deuterated propane in MOF Zn(tbip) during tracer exchange with non-deuterated propane and b) the resulting surface permeabilities a. c) With the knowledge of the flux j in of molecules entering the crystal and the flux j gas of molecules colliding with the external surface it is possible to determine the sticking probability j in /j gas that a molecule encountering the crystal surface is able to continue its trajectory into the interior of the crystal. Fig. 17a illustrates the novel options of PFG NMR when applied to a sample of nanoporous material in contact with a guest atmosphere of variable pressure: Simultaneously with the variation of the external pressure, the relative amount adsorbed (lower part, right ordinate) and the self-diffusivities (upper part, left ordinate) can be determined [93]. The data have been obtained with cyclohexane in Vycor porous glass. The adsorption data reproduce the wellknown phenomenon of hysteresis [3], which are now, for the first time, combined with the self-diffusivities as a measure of the translatio Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

21 Chemie Ingenieur Technik Diffusion 799 nal mobility of the guest molecules. In addition to the (quasi-)equilibrium values of Fig. 17a, Figs. 17b and c show the two uptake curves following a pressure step both below (Fig. 17b) and within (Fig. 17c) the range of hysteresis. In addition, both figures also show the time dependence of molecular uptake which one would obtain from the analytical expression for diffusion-limited uptake (i.e. a relation of the type of Eq. (8), taking account of the given sample geometry) by using the self-diffusivity as determined by PFG NMR as a first-order approximation for the transport diffusivity relevant for molecular uptake (see Sect. 2). Outside the range of hysteresis, i.e. before the onset of capillary condensation, the measured uptake is in perfect agreement with the behaviour expected from the molecular mobilities, but the onset of capillary condensation leads to a dramatic slowing down of equilibration. After a fast initial part following, up to some degree, the pattern of diffusion limitation, molecular uptake is progressively retarded. It is interesting to note that, in Sect. 4.2, we have referred already to a situation where self-diffusion was also found to be faster than the approach to equilibrium. In that case the retardation was shown to be caused by the attractive forces between the guest molecules resulting in a permanent reduction of the driving force for transport diffusion in comparison with self-diffusion. Uptake retardation under the conditions of capillary condensation is caused by a fundamentally different mechanism which is associated with the rearrangement of molecular ensembles (e.g. between adjacent constrictions in the pore space) rather than with the rearrangement of individual molecules [94]. It is this difference which makes the retardation of the approach to equilibrium in mesopores, in the range of hysteresis, so pronounced that one is able to attain essentially completely reproducible data even before the (final) equilibrium state is established. It is for this reason that sorption hysteresis may be considered to proceed through a series of quasi-equilibrium states. 6.2 A multitude of states The existence of a stable adsorption and desorption branch in the adsorption isotherm, distinctly separated from each other, indicates the existence of at least two local minima in the free energy profile that determines the respective states of the guest molecules. There is, clearly, no reason to assume that these two states represent the only local minima in the free energy landscape. Again, diffusion measu- The existence of a stable adsorption and desorption branch in the adsorption isotherm indicates the existence of at least two local minima in the free energy profile that determines the respective states of the guest molecules. (a) D / 10-9 m 2 s pore filling relative pressure, z = p/p s (b) amount adsorbed (c) time, t / s 8000 amount adsorbed time, t / s Figure 17. a). Diffusivity of cyclohexane in Vycor porous glass measured as a function of the relative pressure z on the adsorption (open circles) and the desorption (filled circles) branches at T = 297 K. Squares show the respective isotherms. b) and c) demonstrate the typical adsorption kinetic data (points) obtained upon stepwise change of z from b) to and c) to In the insets the long-time part of the same data are shown. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

22 800 Übersichtsbeiträge Ch. Chmelik During desorption, cohesion between the guest molecules leads to smaller densities of the liquid phase in the pore space. rements prove to be a most sensitive tool for probing these different states [94]. As an example, Fig. 18 represents similar data as those shown in Fig. 17a. However, in addition to the data characterizing the situation of conventional hysteresis, Fig. 18 also displays the results of so-called scanning sorption experiments. As an example of this type of measurement, Fig. 18a also shows the guest concentrations on the desorption branch when adsorption has been switched to desorption already before complete saturation, namely in one series of experiments at a relative pressure of p/p s = 0.68 (squares) and in the other at p/p s = 0.65 (triangles). Fig. 18b shows the diffusivities measured at each individual point of the scanning curves. There is a clear tendency that, at a given pressure, the diffusivities decrease with increasing loading. This is exactly the expected situation since, under the given conditions, the contribution of the pore gas phase to overall diffusion (i.e. the product of the diffusivity (in fact, the Knudsen diffusivity) and the relative number of molecules in the pore free space) may be shown to notably exceed the diffusivities in the liquid phase [94,95]. In Fig. 19, the diffusivities of Fig. 18b are redrawn via the correlations given by Fig. 18a as a function of the amount adsorbed. Fig. 19 contains further diffusivities which have been measured during a complete adsorption-desorption cycle of scanning curves [94]. It is worthwhile to emphasize once again that the diffusivities shown in this figure are attained under quasi-equilibrium conditions. This means: repetition of the measurements after a couple of hours (clearly, under strict maintenance of temperature and pressure) leads to exactly the same diffusivities. Thus, PFG NMR provides immediate and unequivocal evidence that, depending on the history, one and the same number of molecules (since nothing else means a fixed value of the relative pore loading) may be subjected to quite different rates of molecular propagation. Since it is the actual state of the guests within the host system, which determines their diffusivities, differences in the observed diffusivities indicate differences in the states. Since the diffusivities remain constant in the course of (at least) hours, also the respective states are found to be stable over such long intervals of time - irrespective of the fast internal movement of each individual guest molecule. Again, the trend in the diffusivities may be rationalized by the microscopic situation in the host-guest system. During desorption, cohesi- D / m 2 s amount adsorbed Figure 19. The diffusivities from Fig. 18b plotted versus the relative amount adsorbed using the data of Fig. 18a. Also diffusivities obtained by more complex cycles of the pressure variation are shown. (a) 1.0 (b) 5.0 amount adsorbed D / m 2 s relative pressure, z = p/p S relative pressure, z = p/p S Figure 18. a). The relative amount adsorbed and b) the corresponding diffusivities of cyclohexane in Vycor porous glass at T = 279 K as a function of the relative pressure z. The open and filled circles show the data obtained on the complete adsorption and desorption branches by a pressure increase from zero to p s and from p s to zero, respectively. The triangles and squares give the results of desorption scanning experiments, where pressure was first increased from zero to z = 0.68 (squares) and to z = 0.65 (triangles) and then the corresponding data were measured upon reducing pressure to zero. The lines are shown to guide the eye Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chemie Ingenieur Technik 2010, 82, No. 6

23 Chemie Ingenieur Technik Diffusion 801 on between the guest molecules leads to smaller densities of the liquid phase in the pore space. For identical loadings this, in turn, results in a larger liquid-phase volume fraction and, hence, in a reduction of the contribution of the free pore space to overall molecular transport. Novel options for diffusion measurement under the conditions of transient sorption have provided unprecedented insight into the elementary mechanisms of molecular transport in nanoporous materials. They include a first detailed comparison of diffusivity data determined under equilibrium and non-equilibrium conditions which are found to be in excellent agreement with the behaviour expected from fundamental principles. Systematic studies of this type are indispensable for the establishment of reliable routes towards the determination of multi-component diffusivities as the key quantities for the design of transportoptimized technologies for mass separation and mass conversion based on such materials. In addition to the influence of intracrystalline diffusion, in many cases molecular uptake and release are found also to be affected by transport resistances at the interface between the porous bulk phase and the surrounding atmosphere, the so-called surface barriers. With the novel experimental possibilities, for the first time an in-depth study of the permeabilities through these resistances has become possible. For the investigated systems (short-chain length n-alkanes in MOF Zn(tbip)) these barriers are shown to be caused by a total blockage of most of the pore mouths connecting the intracrystalline pore space with the surroundings, rather than by a quasi-homogeneous layer of dramatically reduced diffusivities. When the host materials are traversed by mesopores, the onset of capillary condensation is accompanied by a significant retardation in the final period of equilibration, following a pressure step in the surrounding atmosphere. This dramatic slowing down in comparison with the behaviour estimated from the guest mobilities indicates that, in this situation, equilibration necessitates the re-arrangement of molecular ensembles rather than of individual molecules. The self-consistency of the diffusivity data accessible by these new techniques sets a final point under a decade-long discussion of the origin of the discrepancies which have been observed in numerous experiments: Implying identical host materials, identical sample preparation and complete equivalence in the experiments performed (including, in particular, identical diffusion path lengths observed) there is no reason to doubt the possibility of correlating the results of the measurements of selfdiffusion and of transport diffusion by firstprinciples approaches of diffusion theory. 7 Conclusion (from left to right) Jörg Kärger, Rustem Valiullin, Lars Heinke, Christian Chmelik Christian Chmelik got his PhD in 2008 at Leipzig University. He continued to work at the Faculty of Physics and Earth Science in the Department of Interface Physics as a postdoc, chairing the development of the techniques of diffusion measurement in nanoporous materials by interference microscopy and IR micro-imaging. He is a member of the editorial board of the online journal Diffusion Fundamentals and representative of Leipzig University in the European Nanoporous Materials Institute of Excellence. Lars Heinke studied physics in Greifswald and Leipzig. In 2009, he got his PhD degree at Leipzig University on the analysis of the concentration profiles, which evolve during mass transfer of guest molecules in nanopores. Currently, he works as a postdoc at the Fritz-Haber-Institute in Berlin with Prof. Dr. Hans-Joachim Freund and investigates oxide surfaces by means of atomic force and scanning tunnelling microscopy. Rustem Valiullin got his PhD degree from Kazan State University (Kazan, Russia) in 1997 where he continued to work as a Research Scientist. After two years postdoctoral work in the Royal Institute of Technology (Stockholm, Sweden), in 2003 he moved to the University of Leipzig, Germany as a fellow of the A. von Humboldt Foundation. Currently he is a Heisenberg fellow of the German Science Foundation at the Faculty of Physics and Earth Science of Leipzig University. Dr. Valiullin s research interests concern phase transitions and dynamics of molecules in confined spaces. Jörg Kärger got his PhD in Physics in 1970 at Leipzig University, followed by the habilitation in Research stays took him to Prague, Moscow, Leningrad, Fredericton and Paris. He was awarded together with colleagues the Donald-W-Breck Award of the International Zeolite Association for NMR Investigations of Acidity and Molecular Transport in Zeolites. In 1994, he became Professor of Experimental Physics and head of the Department of Interface Physics. His research is dedicated to diffusion phenomena quite in general. His activities include the establishment of the Diffusion Fundamentals - Conference Series and the author/editorship of textbooks and an Online- Journal. He also was chair of the German Zeolite Association. Chemie Ingenieur Technik 2010, 82, No Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim