Transport phenomena in nanoporous materials.

CHEMPHYSCHEM REVIEWS DOI: 10.1002/cphc.201402340

Transport Phenomena in Nanoporous Materials Jçrg Krger*[a] Dedicated to the memory of Hellmut G. Karge

2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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CHEMPHYSCHEM REVIEWS Diffusion, that is, the irregular movement of atoms and molecules, is a universal phenomenon of mass transfer occurring in all states of matter. It is of equal importance for fundamental research and technological applications. The present review deals with the challenges of the reliable observation of these phenomena in nanoporous materials. Starting with a survey of the different variants of diffusion measurement, it highlights the potentials of “microscopic” techniques, notably the pulsed field gradient (PFG) technique of NMR and the techniques of microimaging by interference microscopy (IFM) and IR micros-

1. Introduction The advent of tailored nanoporous materials[1–4] has revolutionized matter-upgrading technologies and opened attractive new fields of fundamental research[5–7] by offering novel prospects for experimentally assessing such basic issues like the correlation and distinction between mass transfer under equilibrium and non-equilibrium conditions[8, 9] and between normal and anomalous diffusion.[10, 11] Intimate contact between guest molecules and host materials by a targeted matching of the pore sizes with the molecular dimensions has become key to manifold applications including mass separation,[12] catalytic conversion,[5, 13] selective adsorption,[14] sensing[15] and molecular ordering for generating optical functionality.[15, 16] The performance in many of these applications is closely related to the intrinsic guest mobilities and the rate of molecular exchange between the internal pore space and the surrounding of the material. This is a simple consequence of the fact that the rate of gaining value-added products can never be faster than allowed by the intrinsic mobility of the molecules under consideration. Traditionally, information about molecular mobility in nanoporous materials has been obtained “macroscopically”, that is, by recording the overall uptake or release, following a step change in the pressure of the surrounding atmosphere. Any prediction of the nature of the governing transport resistances and their quantitation had therefore to be based on model assumptions (such as, e.g., the implication of dominance by intracrystalline diffusion or by resistances on the outer crystallite surface (surface barriers)).[17, 18] In the last few decades, this type of analysis has been powerfully reinforced by the advent of several techniques of microscopic diffusion measurement. The combination of this new information with the evidence from molecular uptake and release measurements has led to a paradigm shift in our view of mass transfer in these materials. These changes include, in particular, a revision of the abso[a] Prof. J. Krger University of Leipzig Faculty of Physics and Geosciences Linnestrasse 5 04103 Leipzig (Germany) Tel.:(+49) 341-97-32-502 Fax: (+ 49) 341-97-32-549 E-mail: [email protected]


www.chemphyschem.org copy (IRM). Considering ensembles of guest molecules, these techniques are able to directly record mass transfer phenomena over distances of typically micrometers. Their concerted application has given rise to the clarification of long-standing discrepancies, notably between microscopic equilibrium and macroscopic non-equilibrium measurements, and to a wealth of new information about molecular transport under confinement, hitherto often inaccessible and sometimes even unimaginable.

lute values of intra-crystalline diffusivities, in some cases by several orders of magnitude. By serving as a reliable standard for comparison, this new view of intra-crystalline diffusion has become an indispensable prerequisite for the development of molecular dynamics simulations[19] stimulating, simultaneously, fine-tuning of the quantum-chemical determination of the interaction potentials.[20, 21] In the present contribution these novel approaches to diffusion measurement are reviewed and recent developments in our understanding of mass transfer in nanoporous materials are highlighted. Section 2 reviews the various options for observing guest diffusion in nanoporous materials. With these experimental possibilities in mind, Section 3 introduces the different mechanisms of mass transfer control and the parameters used for their quantitation (notably the coefficients of self- and transport diffusion and the surface permeability) with reference to the measurements. Moreover, it is illustrated how the microscopic techniques of diffusion measurement, notably the pulsed field gradient (PFG) technique of NMR and micro-imaging by IR microscopy (IRM) and interference microscopy (IFM), provide immediate experimental access to these quantities and how they have contributed to the establishment of a coherent, self-consistent view of mass transfer in nanoporous materials. Section 4 highlights the variety of phenomena of mass transfer thus observable in nanoporous materials, often associated with evidence on structural details which are inaccessible by other experimental techniques. Examples include the interrelation between mass transfer under equilibrium and non-equilibrium conditions considering both single-component and mixture adsorption, and various types of structuremobility correlations, including the phenomena of diffusion anisotropy and of diffusion enhancement by “transport” pores, as well as diffusion inhibition by barriers in the bulk and on the particle surfaces.

2. The Different Options of Recording Guest Diffusion Information about mass transfer of guest molecules in nanoporous materials may be provided by quite a number of different measuring techniques. These may be sub-divided into two broad categories, namely equilibrium and non-equilibrium experiments. In the first category, information on mass transfer is only accessible by following the diffusion paths of the individChemPhysChem 0000, 00, 1 – 29

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CHEMPHYSCHEM REVIEWS ual molecules. The category of non-equilibrium experiments may again be subdivided into experiments operating under transient conditions (where the information on mass transfer is attained by studying the rate of equilibration) and non-equilibrium experiments operating under boundary conditions that ensure stationarity. In this case, information on mass transfer is contained in the fluxes. The present section provides a short survey about typical representatives of both these classes of techniques. A more detailed description of those techniques which are the main focus of this review is given in Section 3, together with further details about the underlying theory. A more detailed overview of the various techniques of diffusion measurement and their modes of application may be found, for example, in the textbook in Ref. [18].

2.1. Measurement under Equilibrium: Tracing Molecules on their Diffusion Path Modern spectroscopic techniques provide us with much information characterizing the process of molecular propagation. This concerns, in particular, the elementary steps of diffusion. Being associated with changes in the interaction energy, the time constant of these changes may be readily taken as a measure of the mean lifetime between subsequent diffusion steps. It is in this way that dielectric spectroscopy[22] and NMR relaxation and exchange measurements[23] have been exploited for diffusion studies. The advent of dye molecules and their observation via absorption[24] and fluorescence excitation spectra[25] enabled the recording of their positions in subsequent intervals of time. Thus, by “connecting the dots”,[26] for the first time the construction of whole trajectories has become possible. Alternatively to the application of scanning tunneling microscopy to tracing the movement of “hot” atoms on metal surfaces by Ertl and coworkers,[27] now also dye molecules became the subject of single-particle tracking. This development, largely promoted by Bruchle and coworkers,[28] included manifold investigations Jçrg Krger was educated at the Leipzig University where, in 1994, he was appointed Professor of Experimental Physics and Head of the Department of Interface Physics. His activities include the establishment of the “Diffusion Fundamentals” conference series and online journal and the author/editorship of textbooks on “Diffusion in Nanoporous Materials” and “Diffusion in Condensed Matter: Methods, Materials, Models”. Exotics among his more than 500 publications are entries in the Guinness Book of Records for the largest bicycle bell orchestra and computer game attained during the Leipzig Physics Sunday Lectures. He was honored with the Donald W. Breck Award of the International Zeolite Association (1986), the Max Planck Research Award (1993) and the membership of the Saxony Academy of Sciences (2000).


www.chemphyschem.org of mass transfer in nanoporous materials[29] with emphasis on both the structural peculiarities of the host systems[30–32] and the location and spatial dependence of catalytic reactions.[33] In contrast to single-particle tracking, the information provided by the pulsed field gradient (PFG) technique of NMR[34–38] is taken as an average over essentially all molecules (of one type) within the sample. This information is contained in the so-called averaged propagator P(x,t) which denotes the probability (density) that, during a time interval t, an arbitrarily selected molecule is shifted over a distance x (with the x coordinate defined by the direction of the magnetic field gradient).[39] Space and time scales are typically in the range of micrometers and milliseconds. Conceptually related to PFG NMR is the information provided by (incoherent) quasi-elastic neutron scattering (QENS).[40] Also in this case, the primary data of experimental observation are directly related to the probability distribution of molecular displacements, but the space and time scales are typically in the range of nanometers and picoseconds.

2.2. Non-Equilibrium Techniques 2.2.1. Transient Uptake and Release In many technological applications the performance of nanoporous materials depends on the rate of molecular uptake and release. Following the time dependence m(t) of the amount adsorbed, in response to a pressure change in the surrounding atmosphere, is therefore among the key experiments used for characterization. Conventionally, sample weight (gravimetric methods) and gas pressure (piezometric methods) have been chosen as primary quantities of measurement for recording the time dependence of molecular uptake and release. The spectrum of parameters applicable as a measure of the amount adsorbed, however, is much larger (see, for example, Chapters 12 and 13 in Ref. [18]). Examples include the information provided by the application of various spectroscopic techniques such as IR,[41] NMR[42–45] and ESR[46] just as by any other method providing quantitative information about the amount of guest molecules in the sample. While in conventional gravimetric measurements[17, 47] sample amounts of tens of milligrams were typical, improved microbalances now allow measurements to be performed with only a few adsorbent particles/crystallites.[48] Variants of uptake and release measurements include the frequency response (FR) method[49] and the zero-length column (ZLC) technique.[17, 50] FR experiments represent a sequence of uptake and release events, generated by a periodic variation of the sample volume. All information about the intrinsic processes of mass transfer is contained in the variation of the gas pressure which is recorded as the primary quantity of measurement. In ZLC measurements, molecular desorption is followed by recording the time dependence of the decaying concentration of guest molecules in a purging flow. ChemPhysChem 0000, 00, 1 – 29

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2.2.2. Flux under Stationary Conditions Non-equilibrium experiments under stationary conditions are typically performed in flux-rate measurements with membranes where the nanoporous material can either be inserted as a single crystal in an otherwise impermeable membrane[51] or where it represents a continuous membrane by itself.[52] As a special representative of this type of experiments one may also refer to purposefully designed catalytic reactions in the nanoporous material.[53, 54] Here, under stationary conditions, the influx of the reactant molecules is counterbalanced by the efflux of the product molecules.

3. Diffusion Fundamentals 3.1. Basic Experiments and Definitions Figure 1 a illustrates, in a schematic way, guest positions and guest trajectories (recorded over a certain time interval t) within the pore space of a nanoporous host material. We imply that the host–guest system is under macroscopic equilibrium. The mean square displacement of the guest molecules in any arbitrary direction is then found to increase linearly with time, following the Einstein relation given by Equation (1): hxðtÞ2 i ¼ 2 Dt

ð1Þ

with D denoting the self-diffusivities. In an isotropic host material, the mean square of displacement in space is thus easily seen to be given by Equation (2): hrðtÞ2 i ¼ 3hxðtÞ2 i ¼ 6 Dt

ð2Þ

With Equations (1) and (2) it is implicitly assumed that the mean displacement of the guest molecules is sufficiently large in comparison with the pore sizes and sufficiently small in comparison with the size of the nanoporous host particles. Moreover, under such conditions the probability density P(x,t) of finding a molecule, after time t, shifted over a distance x in x direction (referred to, in Section 2.1, as the averaged propagator), may be shown[18, 55] to be given by a Gaussian [Eq. (3)]: x2 Pðx; t Þ ¼ ð4 pDt Þ1=2 exp 4 Dt

ð3Þ

Exactly the same self-diffusivity D as appearing in Equations (1) to (3) may be determined by the arrangement shown in Figure 1 b. There is again a uniform guest concentration, as a characteristic feature of equilibrium measurements. Now, however, the guest molecules are assumed to exist in two species (achieved, for example, by the use of different isotopes) which, though being identical in their microdynamic properties, may be distinguished from each other. Now, the flux of each individual species may be noted as Equation (4): j* ¼ D

@c* @x


Figure 1. Definition of the various coefficients of molecular transport in nanoporous materials and schemes of measurement: a) self-diffusivity by the Einstein relation, b) self-diffusivity by tracer exchange and Fick’s first law, c) transport diffusivity, d) barrier permeability.

with the factor of proportionality, D, appearing between particle flux and concentration gradient, being exactly the self-diffusivity which we have just introduced via Equation (1). The equivalence of the two definitions (with a self-diffusivity invariable in time and space)[18, 55] is a criterion for the existence of “normal” diffusion. As a consequence of the equivalence of the definitions by Equations (1) and (4), the terms self- and tracer diffusivities are used synonymously. In homogeneous pore spaces, diffusion may in fact be expected to behave “normally” as long as the investigated molecular displacements significantly exceed the pore diameter and are still much smaller than the sizes of the nanoporous particle under study. Correspondingly, fluxes and concentrations must be defined with respect to unit areas and unit volumes that are large enough to comprise more than a single pore.

ð4Þ


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Equation (4) is immediately seen to be a close relative of Fick’s first law [Eq. (5)]: j ¼ DT

@c @x

ð5Þ

which is known to correlate the flux of a species with its concentration gradient (Figure 1 c). Whilst the two fluxes in Figure 1 b compensate each other, the situation shown in Figure 1 c gives rise to an overall flux, that is, to net mass transfer. In the adsorption community the factor of proportionality, DT, is commonly referred to as the transport or Fickian diffusivity. In other areas of science also the terms collective or chemical diffusivity are sometimes used.[56–58] Fick’s first law [Eq. (5)] is immediately seen to serve as the basis for the measurement of transport diffusivities in flux experiments (Section 2.2.2), with the obvious requirement that both the flux and the concentration gradient are accessible to measurement. Combination of Fick’s first law [Eq. (5)] with the equation of @c @j continuity @t ¼ @x yields Fick’s second law [Eq. (6)]: @c @ @c ¼ DT @t @x @x

ð6Þ

which simplifies to Equation (7): @c @2c ¼D 2 @t @x

ð7Þ

if the concentration dependence of the transport diffusivities and, hence, its dependence on the location x can be assumed to be negligibly small. Transient uptake and release experiments, dealing with the measurement of concentration changes, are analyzed on the basis Fick’s second law. Thus, for molecular uptake or release, induced by a pressure step in the surrounding atmosphere, one obtains, e.g., by solving Equation (7) with the relevant initial and boundary conditions and determining the total mass increase or decrease by integration over the whole particle, Equation (8):[18] 2 2 mðt Þ mð0Þ 6 X1 1 n p DT t gðt Þ ¼ 1 2 exp n¼1 n2 p R2 mð1Þ mð0Þ

ð8Þ

where it had been assumed that 1) uptake or release is controlled by intracrystalline diffusion, 2) the pressure step is small enough so that the diffusivity may be assumed to remain essentially constant and 3) the particle shape is close to that of a sphere of radius R. Even for other shapes, Equation (8) remains a good approximation if R is understood as the radius of the sphere with the same surface-to-volume ratio as the particle under study.[18] The time dependence of relative uptake and release is sometimes expressed by the short-hand notation g(t). 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Relations of the type of Equations (5) and (8) are entirely sufficient for determining intracrystalline diffusivities by flux and by uptake and release experiments, provided that the fluxes, uptake and release are in fact controlled by the diffusional resistance of the genuine pore system. However, today both external and internal resistances, acting in parallel to the diffusional resistance, are well known to control, in many cases, the overall phenomenon of molecular uptake and release.[17, 18, 59] Typical “external” resistances, appearing as a retardation of equilibration during molecular uptake and release, are related to the finite rates of heat release, of guest supply and/or of mass transfer through the particle assemblage. “Internal” resistances include transport barriers both on the surface of the individual particles and in their interior. In an idealized way (Figure 1 d), the influence of these barriers on mass transfer can be quantitated by Equation (9):[18, 60]

j ¼ aðcl cr Þ

ð9Þ

where a is referred to as the barrier permeability and cl(r) stands for the concentration on the left (right) side of the barrier which, in the given notation, is implied to be positioned within the particle. For surface barriers it is the difference between the actual boundary concentration c(x=0) and the concentration in equilibrium with the surroundings that now appears on the right-hand side of the defining equation [Eq. (10)]:

j ¼ a½cðx ¼ 0Þceq

ð10Þ

If surface barriers are dominant [opposite to the situation considered in Eq. (8)] molecular uptake and release follow the relation given by Equation (11): mðt Þ mð0Þ 3 at gðt Þ ¼ 1 exp R mð1Þ mð0Þ

ð11Þ

Nanoporous materials are typically available with particle sizes ranging from fractions of to hundreds of micrometers. Measuring techniques that cover displacements over exactly these distances are expected to contribute particularly effectively to the exploration of the transport properties of such materials. The present review deals with two ensemble techniques of diffusion measurement which ideally match this criterion; the pulsed field gradient (PFG) technique of NMR[35, 61, 62] and microimaging by interference microscopy (IFM) or IR microscopy (IRM).[63, 64] These novel techniques are presented in greater detail in the remainder of this section. Some of the wealth of information on diffusion and diffusional resistances that has been revealed by their application is then discussed in Section 4. ChemPhysChem 0000, 00, 1 – 29

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3.2. Pulsed Field Gradient (PFG) NMR Diffusion Measurements 3.2.1. Measuring Principle and Fundamental Relations Figure 2 introduces the fundamentals of diffusion measurement by NMR. It is based on the application of the Larmor condition (top right of Figure 2), given by Equation (12): w ¼ gB

ð12Þ

pulses of duration d. By the application of an appropriate program of RF pulses in combination with these gradient pulses, it becomes possible to record the distribution of molecular displacements in the time interval between the two pulses (rather than the molecular distribution itself as by MRT). In fact, the attenuation Y(g,t) = S(g¼ 6 0)/S(g=0) of the NMR signal (the “spin echo”)[67] with increasing gradient pulse intensity gd turns out to be nothing else than the Fourier transform of the mean propagator [Eq. (14)]:[39] yðg; t Þ ¼

Z

1

Pðx; t Þ cosðgdgx Þdx

ð14Þ

1

where the “observation” time t is the separation between the two gradient pulses. Details of the derivation may be found, for example, in Refs. [18, 34–38, 68]. For normal diffusion, by inserting the mean propagator as given by Equation (3) into Equation (14), the PFG NMR signal attenuation is easily seen to become an exponential [Eq. (15)]: Yðg,tÞ ¼ expðg2 d2 g2 DtÞ ¼ expðg2 d2 g2 ðx 2 ðtÞi=2Þ,

Figure 2. Schematics of diffusion measurement by the pulsed field gradient (PFG) technique of NMR.

correlating the resonance frequency w/(2 p) of the NMR signal with the externally applied magnetic field B. The gyromagnetic ratio g appearing as a factor of proportionality, is a characteristic quantity of the nuclei (mostly protons) under study (gproton = 2.67 108 T1 s1). By applying a space-dependent magnetic field B = B0 + gx, that is, by superimposing a constant magnetic field B0 and “magnetic field gradients” of intensity g, with Equation (12) the resonance frequency [Eq. (13)]: wðxÞ ¼ gB0 þ ggx

ð13Þ

is seen to become a linear function of the space coordinate x. Equation (13) serves as the governing relation of magnetic resonance tomography (MRT), the most powerful and ubiquitous imaging technique of medical diagnosis.[65] Since the intensity of the NMR signal is proportional to the number of resonating nuclear spins (and, hence, to the number of the spin-bearing molecules), with Equation (13) molecular spatial distributions are immediately seen to appear in the NMR spectra if recorded with appropriately chosen field gradients. An example of the early application of MRT in chemical engineering, namely for recording molecular uptake by beds of zeolite crystallites, may be found in Ref. [42] (see also Section 12.1.5 of Ref. [18] and the reviews in Refs. [44, 45, 66]). The PFG NMR experiment for observing molecular diffusion is based on the application of two magnetic field gradient 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

ð15Þ

with the second equation following from the Einstein relation, Equation (1). Equations (14) and (15) are strictly valid only for sufficiently short field gradient pulses but, in most cases, this is a good approximation. As a typical feature of nanoporous materials, diffusion cannot be considered to proceed in an infinitely extended region. Equation (15) is often replaced, therefore, by the more complex Equation (16):[69] yðg; t Þ¼ pintra ðt Þ exp g2 d2 g2 Deff intra t þplongrange ðt Þ exp g2 d2 g2 Dlongrange t ¼ pintra ðt Þ expðg2 d2 g2 hx 2 ðt Þiintra =2Þ þplongrange ðt Þ exp g2 d2 g2 hx 2 ðt Þilongrange =2

ð16Þ

Equation (16) takes into account that, after time t (i.e. at the instant of time of the second gradient pulse), a fraction of molecules [plong-range(t)] will have left the particles/crystallites in which they were initially located (during the first gradient pulse). The respective diffusivities (Deff intra , Dlong-range) and mean square displacements (hx2(t)iintra, hx2(t)ilong-range) are also correlated by the Einstein relation, Equation (1). pintra(t) 1plong-range(t) denotes the relative number of molecules which have remained within their crystallites. Their mean square displacement, hx2(t)iintra, has to remain, therefore, below an upper limit of order R2. Thus, with increasing observation time t, the effective diffusivity Deff intra becomes eventually independent of the genuine intracrystalline (nanopore) diffusivity, approaching a value of order R2/(5t). Only for sufficiently short observation eff times (given by hx 2 ðt Þi1=2 intra ! R), does Dintra approach the true intracrystalline diffusivity. Often, however, the separation of the PFG NMR signal attenuation into two exponentials as suggested by Equation (16) is not free from ambiguity. In such cases it is useful to concentrate, in data analysis, on only the very first part of the attenuaChemPhysChem 0000, 00, 1 – 29

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tion curve which may be shown to obey Equation (17) (see, for example, Refs. [18, 35, 70]): limgd!0 Yðg,tÞ¼ expðg2 d2 g2 Deff tÞ ¼ expðg2 d2 g2 hxðtÞ2 i=2Þ

ð17Þ

with Deff, once again, defined via the Einstein relation, Equation (1).

3.2.2. The Various Messages of PFG NMR Diffusion Measurement in Nanoporous Materials 3.2.2.1. The Different Regimes of Diffusion Figure 3 provides an overview of the different types of information provided by PFG NMR diffusion studies with nanopo-

are, however, observed with the smaller crystallites above 50 8C. As to be expected, there is no further increase in the root mean square displacements of the water molecules (and, hence, of the effective diffusivities) in the blocked crystals while, in the loose bed of crystallites, the effective diffusivity is now seen to increase, with increasing temperature, much more strongly than the intracrystalline diffusivity. This type of dependence has been shown to be nicely represented by the two-region approximation[72] which, today, has also found widespread application in analyzing PFG NMR diffusion measurements of mass exchange in biological cells.[36, 73] At high enough temperatures when, in the smaller crystallites, the molecular mean lifetimes become much smaller than the observation time, the effective diffusivity simply becomes the weighted sum of the diffusivities for the different regions, as shown by Equation (18) (fast-exchange limit; see Refs. [74, 75] for an in-depth study of its applicability): Deff ¼ Deff intra þ pinter Dinter

ð18Þ

In the notation of Equation (16), the fast-exchange limit is equivalent to Plong-range(t) = 1 (and, correspondingly, Pintra(t) = 0) so that, in the given case, Deff = Dlong-range. Under the conditions of gas phase adsorption, generally considered in this review, pinter ! 1 so that the pre-factor in front of the effective intracrystalline diffusivity could have been set equal to 1. The diffusivity in the intercrystalline space can then be represented by Equation (19): 1 Dinter ¼ hnileff =ttort 3 Figure 3. Temperature dependence of the effective self-diffusivity of water in MFI-type zeolite crystals of different size: *, * H-ZSM-5, Si/Al 25, 7 4 3 mm3 ; &, & NaH-ZSM-5, Si/Al 40, 16 12 8 mm3 before (open symbols) and after (filled symbols) coating the crystals by waterglass, for observation time t = 1.2 ms. For comparison, the root mean square displacements, as following with Equation (2), are as well indicated. From Caro et al.[71] with permission.

rous materials. In the given example, the adsorbent is a bed of crystallites of MFI-type zeolites. Considered are the effective diffusivities of water in a bed of small (circles) and large (squares) crystallites, respectively. In one case, the crystals are applied as a loose assemblage (open symbols), in the other the crystals are embedded in waterglass,[71] prohibiting any exchange of water between different crystallites. The Arrhenius plots of the effective diffusivities determined with an observation time of 1.2 ms exhibit three characteristic patterns. We note, firstly, the regime of genuine intracrystalline diffusion exhibiting a straight line which is observed over the whole temperature range with the larger crystallites and up to about 50 8C for the smaller ones. The root mean square displacements as recorded during the measurements (right ordinate scale) are, as a prerequisite, indeed seen to be smaller than the mean crystal dimensions and the diffusivities remain, correspondingly, unaffected by any blockage of the intercrystalline space. Dramatic deviations from a linear Arrhenius dependence 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

ð19Þ

with leff, hni and ttort denoting, respectively, the effective mean free path, the mean thermal velocity and the tortuosity factor, which takes account of the increase in the length of the diffusion path in the gas phase due to the presence of the crystallites.[76, 77] For sufficiently low temperatures, the gas phase concentration within the closed NMR sample tube is so small that the mean free path of the molecules in the intercrystalline space is determined by their collisions with the outer surface of the crystallites (Knudsen regime).[78, 79] Dinter may exceed the intracrystalline diffusivities to such an extent that Dlong-range is, in general, exclusively given by the second term, pinterDinter, which may become even notably larger than the diffusivity in the neat liquid.[76, 80]

3.2.2.2. Pore Space Monitoring Figure 4 presents the results of a model experiment of diffusion in pore spaces,[81] based on the potentials of PFG NMR to record diffusional displacements over different times and, hence, also different space scales. The porous material (“Accurel” membranes, Akzo Faser AG, Wuppertal, produced by phase separation during cooling of a hot solution of polypropylene and subsequent evaporation of the solvent, with maximum and effective pore diameters of 200 and 650 nm, respectively) was purposefully chosen such that, using polydimethylsiloxane ChemPhysChem 0000, 00, 1 – 29

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Figure 4. Effective diffusivities of polydimethylsiloxane (PDMS, mw = 22 530 g mol1) at 293 K in a polypropylene host matrix for a pore-filling factor of 100 % as a function of the root-mean-square displacement (a) and for different pore fillings as a function of the observation time (b). The time exponents of the mean-square displacements are 0.83, 0.72 and 0.55 for the respective pore filling factors 100 %, 30 % and 15 %. From Appel et al.,[81] with permission.

(PDMS, mw = 22 530 g mol1) as a guest molecule, molecular displacements could be varied from, essentially, below the smallest pore diameters up to far above. The diffusivities recorded for the shortest and largest displacements are, correspondingly, found to be independent of the considered displacements and, hence, of the observation time, reflecting genuine “normal” diffusivities. The diffusivities recorded for small displacements are those of the neat liquid, while the “long-range” diffusivities are additionally retarded by the tortuosity of the pore network.

3.2.2.3. Anomalous Diffusion The range between these two limiting cases (traced with displacements between 100 nm and 1 mm) exhibits anomalous diffusion[7, 10, 82] (see also Section 2.6 of Ref. [18]). Since the mean square displacement increases less than linearly with time [Eq. (20)]: hx 2 i / t k

ð20Þ

with k < 1, it is referred to as sub-diffusion. By inserting Equation (20) into Equation (1), the effective diffusivity is immediately seen to decrease with increasing time and displacement [Eq. (21)]: 1k

Deff / t ð1kÞ / hx 2 i k

ð21Þ

This decrease may be understood by realizing that, in the short time range, the probability of molecular encounters with the pore wall increases with increasing diffusion path length. With increasing observation time, however, more and more molecules pass the constrictions between adjacent pores. Finally, when the displacements exceed the distances between 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

adjacent pores (or the pore-size correlation lengths, corresponding with the upper cut-off in pore space self-similarity as an upper boundary of fractality, see e.g. Ref. [83]), the diffusing molecules have experienced the complete scenario of possible transport resistances. Henceforth (in the given case, starting from about 1 mm) diffusion is, once again, “normal”. Figure 4 b illustrates that the deviation from normal diffusion (and, correspondingly, the decrease of the long-range diffusivities in comparison with the neat liquid) increases substantially with decreasing pore filling factor. This finding may, once again, be rationalized by realizing that the tortuosity exerted by the host system on the guest diffusion paths increases with decreasing pore filling. In nanoporous materials, with pore sizes of nanometers and displacements of micrometers, it is clearly the regime of normal diffusion (attained in the long time limit of Figure 4 a) that is generally observed by the application of microscopic measuring techniques (the focus of this review), provided that the influence of other transport resistances, such as intracrystalline barriers and/or surface barriers, can be neglected.

3.2.2.4. Single-File Confinement Guest diffusion may also become anomalous if it occurs in one-dimensional pores which are so narrow that adjacent molecules cannot pass each other. Random walk under the constraint that the molecules have to remain in the given order is referred to as single-file diffusion and leads, in infinitely extended channel pores, to mean square displacements increasing with only the square root of time rather than with the time itself as predicted by the Einstein relation, Equation (1).[7, 84, 85] PFG NMR diffusion studies revealing this type of anomalous time dependence for diffusion in zeolites (notably of type AlPO4-5) have been reported in the Ref. [86]. ChemPhysChem 0000, 00, 1 – 29

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CHEMPHYSCHEM REVIEWS The requirements for an unambiguous measurement of single-file diffusion, however, are far more stringent than for “normal” intracrystalline diffusion. While, with the example of Figure 3, for example, the information about (normal) intracrystalline diffusion for zeolite crystals with both “open” and “blocked” surfaces was seen to be still reasonably well provided, even for guest root mean square displacements approaching the crystal size, this is certainly not true for single-file systems, as a consequence of the long-range interaction mediated by the single-file confinement. In fact, the critical distance for diffusion measurementpunder single-file constraint is now ffiffiffiffiffi a quantity of the order of Ll, with L and l denoting, respectively, the file length (crystal size) and the distance between adjacent molecules, rather than the crystal pffiffiffiffiffisize L itself. For single-file diffusion in blocked channels, Ll (rather than L) represents the upper limit of root mean square displacements[87] while, with open channels, molecular diffusion with displacements exceeding this critical value is found to approach “normal” diffusion, but with an effective (“center-ofmass”) self-diffusivity. This diffusivity can be shown[87, 88] to be smaller than the diffusivity of a single molecule undergoing normal diffusion by a factor of the order of the total number of molecules in the file. Further details may be found in the reviews given in Chapter 5 of Ref. [18] and in Ref. [89]. It is important to emphasize that transport resistances due to single-file confinement do not exist for molecular uptake and release. The rate of these processes remains, as a matter of course, completely unaffected by whether adjacent molecules are able to mutually exchange their positions or not. Single-file confinement is (and has experimentally been shown[90] to be), however, of immediate relevance for the performance of transport-controlled catalytic reactions. Here, the rate of the overall process does clearly depend on the rate of displacement of the product molecules by the reactant molecules[91] which is dramatically impeded under single-file conditions. Following the conception of molecular traffic control[92] this impediment has been shown to be notably depleted in pore networks consisting of mutually intersecting channel sets offering different affinities to the reactant and product molecules where the resulting differences in the occupation probabilities notably reduce the transport resistances caused by counter-fluxes of the reactant and product molecules.[93] Quantitative predictions of the influence of single-file confinement on self-diffusion and tracer exchange as well as on catalytic reaction do imply an accurate knowledge of the pore architecture. Mainly due to the rich information of imaging and diffusion studies, the real pore structure of nanoporous materials is today known to notably deviate from textbook patterns[31, 32, 94] so that, in turn, it is sometimes the quantitative analysis of diffusion data which is used for structural exploration.[95, 96, 97] 3.2.2.5. Diffusion Anisotropy Being able to record molecular displacements in the direction of an externally applied gradient of the magnetic field (see Section 3.2.1), PFG NMR is particularly suitable for tracing diffu 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.chemphyschem.org sion anisotropy, that is, for determining the principle elements of the diffusion tensor. With sufficiently large particles (such as, e.g., crystals of natural chabazite), diffusion anisotropy may be determined directly from measurements with different sample orientations relative to the field gradient. In this type of measurement, the anisotropy of water diffusion was found to be much more pronounced than expected from a simple randomwalk estimate where molecular propagation is assumed to be controlled exclusively by the passage through the (“eightmembered”) windows between adjacent cavities.[98] The PFG NMR finding was confirmed by MD simulations[99] in which the population density profiles within one large cavity where found to exhibit two distinct maxima, separated by a minimum at the center.[100] With coffin-shaped MFI-type crystallites it has been possible to orient even large assemblages of crystallites by introducing them into a support material containing an array of parallel channels.[101] Thus, by performing the PFG NMR measurements with the field gradients directed either parallel or perpendicular to the channels (and, hence, to the longitudinal crystal extension) it has been possible to measure the diffusivities in the longitudinal crystal axis and perpendicular to this direction, independently from each other. Using methane as a probe molecule, the PFG NMR data were found to be in complete agreement with the correlation rule [Eq. (22)]: c2 =Dz ¼ a2 =Dx þ b2 =Dy

ð22Þ

between the diffusivities in the three different crystallographic directions in MFI-type zeolites, with a, b, c denoting the respective unit cell extensions. Equation (22) has been shown to hold rigorously[102] for mass transfer proceeding as a random walk between the intersections of the sinusoidal and straight channels, in x and y directions, respectively, and to serve, otherwise, as a reasonable approximation.[103] With the relevant data (a b 2 nm and c 1.34 nm), the diffusivity in the z direction (coinciding with the longitudinal crystal extension) is seen to be smaller than the mean of the other two diffusivities by a factor of at least 4.4, a reasonable result since diffusion in z direction has to proceed by interchanges between displacements along the straight and sinusoidal channels. These conclusions have been confirmed by MD simulations.[104] By analyzing the shape of the PFG NMR attenuation curve, information about diffusion anisotropy may even be obtained with powder samples. Here, however, in order to obtain sufficiently accurate data, one generally has to assume that the diffusion tensor is rotationally symmetric (Dx = Dy ¼ 6 Dz). As a typical example of such studies, Figure 5 shows the water diffusivities in MCM-41, jointly with the corresponding mean square displacements.[105] The measurements have been performed with water in excess so that, with measuring temperatures below zero, water diffusion is confined to the interior of the individual nanoporous particles. Plotting the mean square displacements as a function of the observation time (Figure 5 b) and comparison of the limiting values with the particle sizes may serve, therefore, as an informative internal standard, to confirm ChemPhysChem 0000, 00, 1 – 29

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Figure 5. Anisotropic self-diffusion of water in MCM-41 as studied by PFG NMR. a) Arrhenius plots of the parallel (&) and perpendicular (*) components of the axisymmetrical self-diffusion tensor at 10 ms observation time. The dotted lines are guides for the eyes. The full line represents the self-diffusion coefficients of super-cooled bulk liquid water. b) Dependence of the parallel (&, &) and perpendicular (*, *) components of the mean square displacement on the observation time at 263 K in two different samples. The horizontal lines indicate the limiting values for the axial (full lines) and radial (dotted lines) components of the mean square displacements for restricted diffusion in cylindrical rods of lengths l and diameters d. The oblique lines, which are plotted for short observation times only, represent the calculated time dependences of the mean square displacements for unrestricted (free) diffusion with Dpar = 1.0 1010 m2 s1 (full line) and Dperp = 2.0 1012 m2 s1 (dotted line), respectively. From Stallmach et al.,[105] with permission.


Figure 6. Correlation between the mean propagator R(z,t), the PFG NMR signal attenuation y((dg)2, t) and the PFG NMR tracer desorption curve (plong-range(t) g(t)), from Krger et al.,[18] with permission.

the transient (uptake, release or exchange) curve g(t) by the definition given by Equation (23):[18, 107, 108] tintra M1 ¼

Z

1

ð1 gðt ÞÞdt

which may be noted as the sum [Eq. (24)]: tintra ¼ tdiff þ tbar

data validity. It is interesting to note that diffusion in the direction of the axis of symmetry (assumed to coincide with the mean channel direction) is smaller by a factor of about 10 compared with the free liquid and that, moreover, diffusion is also possible perpendicular to this direction, indicating that the channel network of the sample under study must be far from perfect.


ð24Þ

with tdiff (tbar) denoting the mean life time if it is exclusively controlled by intracrystalline diffusion (permeation through a surface barrier). For spherical particles this is given by Equations (25) and (26):[18, 108]

tdiff ¼

R2 15 D

ð25Þ

tbar ¼

R 3a

ð26Þ

3.2.2.6. PFG NMR Tracer Desorption For long-range diffusivities notably exceeding the intracrystalline diffusivities, it is always possible to unambiguously separate the two exponentials appearing on the right-hand side of Equation (16) and to clearly distinguish, correspondingly, the two constituents of the mean propagator, standing for the molecules that have remained in one and the same crystallite over the total observation time and those that have exchanged between different crystallites. Such a situation is considered in Figure 6. It shows in a schematic way the increase of the intensity of the broad constituent of the propagator (left), corresponding to an increase of the intensity [plong-range in Eq. (16)] of the faster decaying exponential in the PFG NMR attenuation curve (center). Plotting this relative intensity as a function of time (right) is thus seen to be nothing else than a tracer exchange curve in a thought experiment where the molecules of a certain crystal have initially been labelled and are now going to be replaced by the unlabeled molecules, initially situated in the other crystallites.[106] Further analysis is facilitated by introducing the intracrystalline mean life time tintra as the first statistical moment M1 of

ð23Þ

0

with D and a denoting, respectively, the intracrystalline diffusivity and the surface permeability of the guest molecules. Since D can be directly measured by PFG NMR and tintra may be determined via Equations (23) from PFG NMR tracer desorption curves, a remains as the only unknown [after inserting Eqs. (25) and (26) into Eq. (24)] and is thus directly accessible by PFG NMR. In reality, however, the determination of both R2 tintra and the term 15 D is subject to substantial uncertainty. As a consequence, reliable information about surface permeabilities is only obtained if the influence of surface barriers significantly exceeds that of intracrystalline diffusion. This condition was found to be fulfilled in numerous cases, notably for samples of zeolites that had been used in industrial separation and catalytic processes.[109] The option to determine surface permeabilities, even under conditions such that diffusional resistance is dominant, is among the many advantages of the techniques of microimaging which are presented in some more detail in the next section. ChemPhysChem 0000, 00, 1 – 29

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CHEMPHYSCHEM REVIEWS 3.3. Diffusion Measurement by Microimaging 3.3.1. Measuring Principle and Fundamental Relations Whilst in PFG NMR measurements molecular displacements (Figure 1 a) are directly recorded, diffusion studies by microimaging via interference microscopy (IFM) and IR microscopy (IRM) are based on monitoring the evolution of guest concentration profiles (Figure 1 b–d) during transient uptake and release experiments or, essentially, with any other experiments with a suitably chosen variation of the external pressure. The efficacy of these techniques became apparent over only the last few years.[63, 64, 110] Figure 7 provides an overview of the measuring principles of the techniques of microimaging, jointly with the information thus attainable. Microimaging by IFM (top right) exploits the fact that the refractive index n1 of a nanoporous crystal depends on the local guest concentration. IFM provides an image of the crystal in terms of the differences between the

www.chemphyschem.org R optical path length 0L n1 ðx; y; z; t Þdz through the crystal (of thickness L) in the observation direction and through the surroundings (of refractive index n2), as recorded in the corresponding phase shifts Df(L). Since, in the first-order approximation, changes in local concentration c(x, y, z; t) and refractive index n1(x, y, z; t) are proportional to each other, the recorded interference patterns lead directly to plots of the concentration R integrals 0L cðx; y; z; t Þdz over the crystal in the observation direction. The type of information accessible in this way is illustrated in the bottom of Figure 7 which shows how the interference patterns (observed during uptake of propene by a crystal of zeolite AlPO-LTA induced by a pressure step from 0 to 25 kPa, see Ref. [111] for more details) are, with a corresponding color code (Figure 7 right), immediately transferred into the transient profiles of the concentration integrals. The concentration profiles shown above on the right-hand side are cross sections through the two-dimensional representations in vertical (y) direction.

Figure 7. Measuring principle of microimaging by IR microscopy (IRM, top left) and interference microscopy (IFM, top center) and accessible information illustrated with 2-D plots of the concentration integrals during uptake of propene by a crystal of type AlPO-LTA by a pressure step from 0 to 25 kPa at room temperature, recorded by IFM (bottom, taken from Hibbe et al.,[111] with permission) and selected cross sections through these profiles in vertical (y) direction (centre, right)



&11&




In micro-imaging by IRM (Figure 7 top center) it is the absorbance of a characteristic IR band of the guest molecule under study which, as the primary quantity of measurement, can be taken to be directly proportional to the integral over guest concentration in observation direction. Absorbance can be recorded both spatially resolved [by using a focal plane array (FPA) detector] and integrated over the whole sample by means of a single-element (SE) detector. With respect to its spatial resolution (ideally 2.7 mm by use of the FPA detector but, as a consequence of the finite sample thickness, in reality often not better than 5 to 10 mm), IRM is inferior to IFM (allowing spatial resolutions of up to 0.5 mm). However, it offers the unique option of recording different species simultaneously. This makes IRM particularly powerful for the observation of multicomponent diffusion and self-diffusion by tracer exchange. As a common feature of both techniques, concentrations are only recorded in relative units. Absolute concentrations become accessible, however, by comparison with the adsorption isotherm for the given system. For one- and two-dimensional pore systems and observation perpendicular to the pore system (see Sections 3.3.2.1, 4.1 and 4.3 and Figures 8 and 15), the fact that microimaging records concentration integrals RL 0 cðx; y; z; t Þdz rather than local concentrations does not impose any restriction. This is also true in general for observation parallel to one (or several) crystal faces as long as the fluxes stemming from the different faces do not interfere with each other. In contrast to other imaging techniques,[66, 112] the very nature of IFM and IRM does not allow simultaneous determination of the concentration integrals for different directions. However, in some exceptional cases, researchers have managed to achieve this by repeating uptake and release experiments with one and the same crystal in different orientations.[113, 114] Section 3.3.2.2 deals with such experiments.

quential concentration profiles which are, generally, subject to substantial uncertainties. Examples of such procedures are given in Figure 9[117] which shows the result of uptake measurements with ethane on a crystal of zeolite ZSM-58 of framework type DDR. Zeolites of this type have rotational symmetry, with mass transfer occurring essentially only in the radial direction.[118] Their potential for mass separation[21, 119, 120] make them an attractive system for both fundamental and application-oriented studies. Following the Boltzmann–Matano approach,[18, 57, 122] in Figure 9 b the concentration profiles have been plotted as a funcpffiffi tion of r/ t rather than of the spatial coordinate alone. The impressive coincidence of the profiles in a master plot indicates that, for the chosen observation times, molecular uptake proceeds as if taking place in an infinitely extended medium (with no interference by fluxes stemming from different crystal faces). Moreover, the diffusional resistance is seen to be dominant. Under such conditions, the diffusivity may be derived directly from the master plot by using Equation (27):[18, 57, 122]

3.3.2. The Messages of Microimaging

DT ¼

1 dx DT ðc ¼Þ 2 t dc

Z

c

ð27Þ

xdc 0

In contrast to the short-time observation of the concentration profiles in Figure 9 b, Figure 9 c shows the merging of the concentration fronts and concentration increase in the middle. Here, Fick’s 2nd law (noted in polar coordinates) becomes [Eq. (28)]:[117] @c @2c ¼ 2 DT 2 @t @r

ð28Þ

The experimentally obtained profiles are found to be of parabolic shape c(r) = c(0) + ar2 so that, by insertion into Equation (28), the diffusivity results as Equation (29): @c =ð 4 a Þ @t

ð29Þ

3.3.2.1. From Profiles to Diffusivities Figure 8, which shows a variety of differently shaped profiles recorded in micro-imaging studies of the methanol–ferrierite system, provides an impression of the wealth of information that can be derived concerning the evolution of concentration profiles during uptake and release. Hitherto such profiles have been derived theoretically (from the solution of the transient diffusion equation) but experimental profiles have been measured for only a few special cases.[60, 115] Since such profiles reveal both the respective fluxes (namely from the area between the profiles recorded at subsequent instants of time, divided by the time interval between their recording) and the concentration gradients, the transport diffusivity is seen to immediately result, via Fick’s first law, Equation (5), as the factor of proportionality between these two quantities. In tracer exchange experiments it is, correspondingly, the self-diffusivity which results via Equation (4). For accuracy enhancement, it is clearly advisable to base data analysis on larger data sets rather than on only two se 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Figure 9 d provides an impression of the accuracy of the diffusivities determined by these different approaches. Also indicated is the concentration-dependent diffusivity obtained analytically by fitting the solution of the diffusion equation to the experimental data. The overall agreement is astonishingly good considering that, due to substantial scattering in the dx/dc data, the Boltzmann–Matano approach is not applicable anymore for small concentrations. In addition, also the conventional way of analysis as represented, for example, by relations of the type of Equations (8) and (23)–(26), is clearly applicable. This is in particular true if the particles under study are of irregular shape so that IRM measurements have to be performed with the single-element detector. Even under such conditions, however, it can be of substantial advantage to operate with a single particle rather than with a bed of particles since, in the latter case, the influence of both bed diffusion and heat release[123] are likely to complicate the analysis. ChemPhysChem 0000, 00, 1 – 29

&12&




Figure 8. Uptake and release of methanol by ferrierite along the eight-ring channels: Comparison of simulated and experimental profiles for pressure steps 0 to 5 mbar (a), 5 to 0 mbar (b), 0 to 10 mbar (c), 10 to 0 mbar (d), 0 to 40 mbar (e), 40 to 0 mbar (f), 0 to 80 mbar (g), and 80 to 0 mbar (h). The points refer to experimental measurements, the lines are simulated from the 2-D finite difference solution of Fick’s 2nd law with concentration dependent transport diffusivities and surface permeabilities. From Kortunov et al.,[116] with permission.

3.3.2.2. Orientation-Dependent Diffusivities By monitoring, upon uptake and release, the evolution of concentration profiles in the plane of observation, microimaging ideally complements the potentials of PFG NMR for tracing diffusion anisotropy. Figure 10 gives an example of such studies. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

It shows a “rounded-boat”-shaped zeolite crystallite[124] of type MFI (Figure 10 a), together with a schematic of the three different crystal orientations (Figure 10 b) under which uptake of 2methyl-butane was observed.[114] The particle coordinate system is deliberately represented by capital letters (X, Y, Z) and ChemPhysChem 0000, 00, 1 – 29

&13&




Figure 9. Ethane uptake by zeolite ZSM-58 (DDR) and the different options of determining diffusivities: a) One-dimensional concentration profiles. b) Plotting pffiffi the concentration profiles of (a), in the short-time regime, as c-vs-x/ t reveals master plot behavior. c) Center region of the mid-time adsorption profiles of (a). Solid lines represent parabolic fits using c(r) = c(0) + ar2 (d) Variation of diffusivity with concentration based on the Boltzmann-Matano approach [Eq. (27), open symbols], mid-time regime analyzed by center-line approach (full stars) and the best fit to the complete set of profiles applying the diffusivity-concentration-dependence given by Fujita’s model (solid line).[121] From Binder et al.,[117] with permission.

must not be confused with the crystallographic directions determining the directions of the sinusoidal (x) and straight (y) channels and the direction perpendicular to them (z) (see Section 3.2.2.5). MFI “crystals” are, in reality, intergrowths, so that the crystallographic directions are known to vary within a given particle, following quite different patterns.[31, 125] Figure 10 c shows the evolution of the guest profiles over three crystal cross sections in the crystal orientation shown on the left of Figure 10 b. Since the uptake patterns are found to be independent of the distance from the upper and lower ends of the crystal (in the Z direction), mass transfer in the Z direction is seen to be notably reduced so that the particle Z direction can be assumed to coincide closely with the crystallographic z direction. Since observation in X direction (crystal orientation in the bottom of Figure 10 b) gives rise to profiles essentially identical to those shown in Figure 10 c,[114] it follows that the crystallographic x and y directions must be oriented along the particle X and Y directions with equal probabilities. 3.3.2.3. Surface Barriers, the “Heinke plot” Being able to determine the evolution of the boundary concentration c(x=0) till equilibration with the surrounding atmosphere (see, for example, the transient profiles in Figure 8), microimaging yields all information necessary for the determination of surface resistances via Equation (10). Most importantly, this option is not connected with the requirement of strong 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Figure 10. SEM picture of a rounded-boat shaped crystal of silicalite-1 (a), schematics of the three different observation directions (b) perpendicular to the X-Z-plane (left), the Y–Z-plane (bottom) and the Y–X-plane (right) and (c) one-dimensional concentration profiles during 2-methyl-butane uptake at room temperature by a pressure step from 0 to 1 mbar pressure in X direction at three different positions Z as indicated by blue lines in (b). From Gueudr et al.,[114] with permission.


&14&




barrier resistances as implied by the application of the NMR tracer desorption technique (Section 3.2.2.6) and “macroscopic” techniques like ZLC.[59, 126] Detailed investigations with propane in MOFs of type Zn(tbip)[110, 127] have, moreover, shown that the thus determined surface permeabilities may reasonably well be represented as Figure 11. “Heinke plots” correlating actual boundary concentration (csurf) and uptake (m); a) measured for methaa function of the mean value be- nol along the ferrierite eight-ring channels for a pressure step from 0 to 10 mbar at room temperature (two symtween the given boundary con- bols for two crystal sides, data from Figure 8 c) and b) calculated for a plate of thickness 2 l considering mass barriers (la/D = 0.01), and by centration and the equilibrium transport controlled by, essentially, intracrystalline diffusion (la/D = 100), by surface both intracrystalline diffusion and surface barriers (la/D = 1). From Heinke et al.,[128] with permission. concentration. The information provided by with NA and M denoting, respectively, the Avogadro constant microimaging makes it possible to plot the increase in bounand the molar mass. The fraction of molecules which, upon dary concentration (as a fraction of the equilibrium value) as [18, 128, 129] colliding with the surface of a (metal) catalyst, is not reflected a function of the relative uptake. As an example, Figinto the gas phase, is referred to as the sticking coefficient. ure 11 a shows the data provided by the profiles shown in FigSince the desired catalytic reaction can only take place if, on ure 8 c. Towards its final point (corresponding to complete moencountering the catalyst surface, the molecule remains under lecular uptake and equilibrium concentration) the plot is seen its influence, this is a key parameter of heterogeneous catalyto approach a straight line which, in Refs. [128, 129], is demon[131] [110] sis. Figure 12 shows schematically how microimaging strated to be a common feature of such presentations. Imporcan provide the analogous information for porous catalysts. tantly, the intercept w with the ordinate attained by backward The fraction of molecules which, upon colliding with the exterextrapolation of this straight line can be shown to provide, via [18, 128, 129] nal surface of a nanoporous catalyst, are able to surmount the Equation (30): surface barrier and to propagate into the genuine pore space, t is given by the ratio jin/jgas so that the flux jin entering the genw 1 intra ð30Þ tdiff an estimate of the increase in the intracrystalline mean life time arising from the presence of surface barriers. Special cases of such plots are shown in Figure 11 b, by which one may easily rationalize the message of Equation (30): when surface barriers are dominant (i.e. flat profiles), relative uptake increases linearly with the boundary concentration, leading to vanishing w and, with Equation (30), to infinite enhancement of the mean life time in comparison with diffusion limitation. For diffusion limitation, from the very beginning of molecular uptake, the boundary concentration assumes the equilibrium value so that w is always 1 and tintra = tdiff. From Figure 11 a, for the uptake experiments considered in Figure 8 c (methanol uptake 0–10 mbar, along the eight-ring channels) w 2, indicating that diffusion and surface permeation have similar influence on the molecular uptake rate.

3.3.2.4. Sticking Probabilities From the kinetic theory of gases[130] the flux of molecules colliding with a plain surface is known to be given by Equation (31): jgas ¼ NA p ð2 pRTMÞ1=2


ð31Þ

Figure 12. Estimating “sticking” probabilities by analyzing transient guest profiles: a) Evolution of the concentration profiles of deuterated propane in MOF Zn(tbip) during tracer exchange with non-deuterated propane; b) The resulting surface permeabilities a. c) With the knowledge of a, the flux jin of molecules entering the crystal may be determined. With the known relation for the flux jgas of molecules colliding with the external surface, the “sticking” probability jin/jgas that a molecule from the gas phase may continue its trajectory inside the crystal can be calculated. From Chmelik et al.,[110] with permission.


&15&


CHEMPHYSCHEM REVIEWS uine pore space is given by the relation jin = a ceq.[110, 132] As experienced with “conventional” sticking coefficients,[131] also the “sticking” probability of nanoporous materials depends on both the nature of the guest molecules and the host material, with values ranging from 0.01 (representing the present upper limit of accessibility), observed, e.g., with isobutane on specially pretreated crystals of MFI-type zeolites,[133] down to 5 108. The latter result was obtained for propane on MOFs of type Zn(tbip),[110] as illustrated in Figure 12. Here, out of about 20 million molecules colliding with the surface, only a single one enters the genuine pore space!

www.chemphyschem.org molecules. By exploiting the beneficial properties of nanoporous glasses (notably the homogeneity of their internal surface) as a host material and by choosing Atto532 as a (fluorescing) guest molecule (ATTO532-COOH, ATTO-TEC, Siegen, Germany, for structural details see the Supporting Information of Ref. [141]) this theorem could, eventually, be proved as illustrated in Figure 13.[141] It is true that there still remains a gap

3.4. Checks of Consistency 3.4.1. Microscopic vs. Macroscopic Measurements Extensive diffusion studies over the last few decades entailed a general consensus that the differences generally observed on comparing the results of macroscopic and microscopic measurements[18, 134–136] resulted simply from differences in the diffusion path lengths. Since these studies revealed that mass transfer in nanoporous materials is often controlled by processes other than diffusion in the genuine pore space, differences in the scale of observation were immediately understood to give rise to differences in the recorded resistances and, hence, in their magnitudes. The fabrication of a homogeneous nanoporous material with all additional transport resistances excluded, yielding coinciding results in micro- and macroscopic measurements, however, turned out to be quite a challenge. Specially prepared nanoporous glasses which are statistically homogeneous over macroscopic distances[137] eventually provided a practically reasonable host system for such studies. Ref. [138] reports a concerted study of the application of different (both microscopic and macroscopic) techniques to diffusion measurements with these host materials, yielding satisfactory agreement between the measured diffusivities. 3.4.2. Single-Particle Tracking vs. Ensemble Measurement Whilst with reference to QENS (Section 2.1) and PFG NMR (Sections 2.1 and 3.2.1), the mean square displacement in the Einstein relation, Equations (1) and (2), is understood as the average over all molecules over a certain time interval, the average considered in single-particle tracking (Section 2.1) is taken by considering a certain molecule over subsequent time intervals. The equality of these two types of averages is postulated by the famous theorem of ergodicity.[139] The specification of its range of applicability, however, is far from trivial.[11, 85, 140] Covering comparable diffusion path lengths, single-particle tracking and PFG NMR are, potentially, well suited for an experimental proof of this theorem. However, the problem with such studies becomes obvious when it is recognized that, to obtain sufficiently large signal intensities in PFG NMR, concentrations greater than one hydrogen atom per cubic nanometer are typically required while, in single-particle tracking, concentrations of less than one fluorescing molecule per cubic micrometer are necessary to avoid the overlapping of signals from different 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Figure 13. Diffusivity of Atto532, dissolved in deuterated methanol at different concentrations, in nanoporous glass (pore size 3 nm) determined by single-particle tracking (circles) and in ensemble measurements (PFG NMR, squares), from Feil et al.,[141] with permission.

between the considered ranges of concentration. However, the diffusivities are seen to remain essentially constant, reflecting a high degree of sample homogeneity. Any significant influence of concentration may therefore be excluded so that the values from the two techniques evidently coincide.

3.4.3. Equilibrium vs. Non-Equilibrium Measurement IR microimaging makes it possible to make measurements under both non-equilibrium and equilibrium conditions and to determine the relevant diffusivities, namely the transport diffusivity [by recording molecular net fluxes (Figure 1 c)] and the self- or tracer diffusivity [by recording counter fluxes (Figure 1 b)] with essentially the same experimental arrangement. In this way, self- and transport diffusivities can be correlated with unprecedented ease and reliability. As an example of such a comparison, Figure 14 a shows the coefficients of self- and transport diffusion of ethanol in a metal organic framework (MOF) of type ZIF-8.[142] In the range of low concentrations, most surprisingly, the transport diffusivities are found to be smaller than the self-diffusivities. This finding means, with reference to Figure 1, that a molecular flux (Figure 1 c) is enhanced rather than diminished by the presence of a second (counter-directed) flux (Figure 1 b). To understand this finding we refer to a second remarkable feature observed in these studies, namely the fact that the experimentally determined diffusivities are correlated by Equation (32): D ¼ DT

d ln c d ln p

ð32Þ

with c(p) denoting the adsorption isotherm, that is, the equilibrium concentration c as a function of the external pressure p (Figure 14 d). ChemPhysChem 0000, 00, 1 – 29

&16&




Figure 14. Concentration dependence of the self- (or tracer) diffusivities D and transport (or Fickian) diffusivities DT of ethanol (a), methanol (b), and ethane (c) in ZIF-8 at 298 K as determined experimentally by IRM, plotted together with the corrected (or Maxwell-Stefan) diffusivities D0 calculated on the basis of the measured adsorption isotherms (d) and transport diffusivities. From Chmelik et al.,[142] with permission.

The term on the right-hand side of Equation (32) is referred d ln c to as the corrected or Maxwell–Stefan diffusivity D0 DT d ln p. It is seen to coincide with the transport (or Fickian) diffusivity in the range of linearity of the adsorption isotherm. Equation (32), that is, equating of the self-diffusivity with the corrected diffusivity, was suggested by Richard Barrer for correlating self- and transport diffusion.[108, 143] Owing to its similarity to an expression applied by Darken[144] for studying inter-diffusion in binary metal alloys, Equation (32) has become known as Darken’s equation. In terms of irreversible thermodynamics, it is applicable if there is no cross-correlation between the fluxes and the gradients of the chemical potentials of differently labelled molecules.[79, 145, 146] This requirement is, in particular, fulfilled if the guest molecules have to overcome, on their diffusion path through the pore space, pronounced constrictions such as, e.g., narrow “windows” between adjacent cavities. “eight-ring” zeolites,[120, 147] including the structure types LTA[146, 148] and DDR,[21, 117, 149] have proven to be ideal candidates for this type of analysis for normal alkanes and alkenes. Now also in ZIF-8 diffusion of short-chain length hydrocarbons is found to follow this pattern. Never before has the validity of Equation (32) been fully confirmed by independent measurements of all three terms.[142] Equation (32), moreover, makes it possible to rationalize the experimentally observed inversion in the relation between the self- and transport diffusivities on passing from small to high concentrations. The special pattern of the logarithmic deriva 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

tive d ln c/d ln p of the adsorption isotherm (yielding values > 1 up to intermediate pore loadings and < 1 at higher loadings) is a consequence of the strong intermolecular attraction leading to the (S-shaped) isotherms of type III or V.[1] With reference to the scenarios of Figure 1 b, c, one may thus reiterate that, at sufficiently low concentrations, the molecules “prefer” to remain in the region of higher concentrations, that is, on the left-hand side in Figure 1 c. The flux without a counter-flux is thus found to be diminished in comparison with Figure 1 b where a counter flux of labelled molecules ensures a constant concentration all over the sample. At sufficiently high concentrations, however, the effect of steric confinement makes the molecules “more willingly” propagate to regions of lower concentrations. This leads to an enhancement of the flux without a counter-flux and, thus, to transport diffusivities that exceed the self-diffusivities. This chain of reasoning is nicely confirmed by the finding that for methanol (Figure 14 b, here intermolecular forces are notably stronger) the self-diffusivities exceed the transport diffusivities over the entire concentration range, while the behavior of the non-polar ethane molecules (Figure 14 c, with notably weaker intermolecular forces) was dominated by the host– guest interaction, giving rise to a common Langmuir-type isotherm (see Figure 14 d) and, thus, to transport-diffusivities that progressively exceed the self-diffusivities with increasing loading. In all cases, self- and transport diffusivities were observed to approach each other at very low concentration. This is ChemPhysChem 0000, 00, 1 – 29

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CHEMPHYSCHEM REVIEWS a general consequence of vanishing molecular interaction in the low concentration limit[9] and can be understood intuitively from Figure 1 b, c by recognizing that, at sufficiently low loadings, the marked molecules are “unaware” of the existence of the other molecules. The measurement of diffusion under equilibrium and non-equilibrium conditions has thus, eventually, been found to be in complete mutual agreement.

www.chemphyschem.org 4. Microscopic Diffusion Measurements: Highlights and Vistas 4.1. Orientation-Dependent Fluxes: Impact of Preadsorption Zeolites of type ferrierite contain a network of mutually intersecting eight- and ten-ring channels, shown schematically in Figure 15 a. Thus, by observation perpendicularly to the crystal plane (and, hence, to the two types of channels), micro-imag-

Figure 15. Diffusion anisotropy in zeolite ferrierite: Schematics of a zeolite crystal accommodating two sets of nanoporous channels (a), and transient guest profiles during methanol (b) and ethanol (c) uptake, as well as during methanol uptake (d) as the second step in a two-step experiment, preceded by ethanol uptake over 1.8 h. In Figure 15 d use has been made of the fact that the ethanol mobilities are negligibly small in comparison with methanol. Variations in the refractive index can exclusively be referred, therefore, to changes in the methanol concentration. Adapted from Hibbe[152] and Krger et al..[64]



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CHEMPHYSCHEM REVIEWS ing makes it possible to record separately the concentration profiles during transient uptake and release along the directions of both channel types. The guest profiles shown in Figure 8 were, e.g., recorded in the direction of the (shorter) eight-ring channels, under conditions where molecular uptake along the ten-ring channels was essentially inhibited due to a blockage of the entrances to these channels.[116, 150] These blockages could be removed by special sample pretreatment.[122, 151] Figure 15 shows the concentration profiles recorded with these samples during transient molecular uptake of methanol (Figure 15 b) and ethanol (Figure 15 c). With both guest molecules, molecular uptake is seen to follow the expected pattern, namely to be notably faster along the larger ten-ring channels (in the horizontal direction) than along the eight-ring channels (vertical direction). A most remarkable reversal in this pattern, however, is observed in a two-step experiment in which methanol uptake is preceded by a period of ethanol uptake.[64, 152] Now, as illustrated by Figure 15 d, methanol uptake along the eight-ring channels is found to proceed

www.chemphyschem.org notably faster than along the ten-ring channels. This behavior is, most likely, a consequence of the fact that the amount of molecules, having entered the ten-ring channels during the “preparatory period” of ethanol uptake, significantly exceeds those having entered the eight-ring channels. The presence of these ethanol molecules evidently gives rise to a significant enhancement of the transport resistance experienced by the methanol molecules, due to the higher loading of ethanol molecules in the ten-ring channels.

4.2. Guest Separation by Pressure Enhancement Whilst IFM is able to selectively record transient concentration profiles in mixtures only under specific conditions (constancy of all other concentrations as considered in the preceding section), IRM is able to fingerprint each individual component. This opportunity has been exploited in a series of experiments shown in Figure 16.[153]

Figure 16. Two-component uptake of n-hexane (nC6)–2-methylpentane (2MP) mixtures (with p1 = p2 = pt/2) at room temperature. a) Comparison of the IRM experimental data for transient uptake (points) with simulation results based on the Maxwell-Stefan model (continuous solid lines). The broken line shown for Run 1 represents the simulation result if uptake of the two components is assumed to occur independently from each other (i.e. for vanishing off-diagonal elements of the diffusion matrix). b) IRM equilibrium data and CBMC simulation results of guest loadings as a function of the external total pressure pt From Titze et al.,[153] with permission.



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CHEMPHYSCHEM REVIEWS Figure 16 a shows a sequence of two-component uptake experiments performed with a binary nC6/2MP mixture with partial pressures p1 = p2 = pt/2 in the bulk gas phase, starting at zero pressure with subsequent stepwise pressure increases. The uptake of both components is recorded separately by application of a SE detector (see Section 3.3.1). The first uptake experiment (0–2.6 Pa) reveals a peculiar but well documented pattern of behavior:[154] the faster species [in this case nhexane (nC6)] is seen to attain, during the process of molecular uptake, a concentration that exceeds the final equilibrium value and, hence, also the concentration at the crystal boundary (which is equal to the equilibrium value under diffusion limitation and even smaller otherwise). The “overshoot” in nC6 concentration must have been accomplished, therefore, by an nC6 flux in the direction of increasing nC6 concentration! For rationalizing this at first sight counter-intuitive observation we may follow the reasoning of Section 3.4.3. and the fact that, within the frame of irreversible thermodynamics, the flux of a given component is driven by the gradient of the chemical potential of this component (rather than by only the gradient of the concentration) and that, under mixture adsorption, the chemical potential of each component is, in general, a function of the concentrations of all components. As a result, during the adsorption of a mixture, a negative gradient of concentration (of a given component) may correspond to a positive gradient of its chemical potential. As a brand-new result of these studies, however, with further increasing total pressure (and, hence, with also increasing partial pressures) the amount of branched hydrocarbons is found to decrease rather than to increase.[153]An explanation of this remarkable behavior is offered by Figure 16 b. The overall loading at which (for equal pressures in the gas phase) overall pressure enhancement is found to give rise to a decrease in 2MP concentration is seen to correspond to about four molecules per unit cell. Four, however, is also the number of channel intersections per unit cell which are known to be the favorite sites of accommodation. The need for molecular rearrangement at higher loadings leads to a dramatic gain in configurational entropy of nC6[155] leading, with further increasing total pressure, to essentially complete expulsion of the bulkier 2MP molecules, that is, to their de- rather than adsorption with increasing pressure! 4.3. The Nature of Surface Barriers Although the NMR tracer desorption technique (see Section 3.2.2.6) has been shown to provide the first definitive experimental evidence of the existence of surface resistances, the permeability of these barriers could not be determined with accuracies high enough to provide any deeper insight into their nature. The advent of microimaging led to a significant enhancement in the accuracy with which the surface permeabilities could be measured (Section 3.3.2.3). These permeabilities are determined together with the intracrystalline diffusivities (Section 3.3.2.1) from the transient concentration profiles recorded during molecular uptake or release. Surprisingly, for a series of nanoporous host–guest systems (see, for example, 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim


Figure 17. Remarkable similarities in the dependencies of intracrystalline diffusion and surface permeation. a) Transport diffusivities (squares) and surface permeabilities (triangles) of propylene in AlPO-LTA at 295 K as a function of loading. b) Surface permeability (filled symbols) and intracrystalline diffusivity (open symbols) of propane in MOF Zn(tbip) at vanishing loading as a function of temperature, recorded with two different crystals. From Hibbe et al.,[111, 156] with permission.

Figure 17 a, b), surface permeabilities and intracrystalline diffusivities were found to follow very similar patterns. In fact, for a given crystal, the ratio a/Dintra turned out to be essentially constant, independent of loading, temperature, the type of guest molecule and even (on comparing equilibrium with nonequilibrium measurements) on the type of experiment.[111, 127, 156, 157] Surface permeation and intracrystalline diffusion must therefore be attributed to the same elementary processes; this means that the surface barrier cannot be a quasihomogeneous surface layer of dramatically reduced diffusivity and/or permeability, uncorrelated with the nature of the genuine pore space. The crystal surface must rather be assumed to be essentially impermeable, with a few dispersed “holes”. For circular holes of radius a, e.g., effective medium theory[158] predicts the relation[159] (see also Sections 2.3.4 and 19.6 of Ref. [18]) [Eq. (33):

2 popen a ¼ Dintra pa

ð33Þ

with popen denoting the fraction of unblocked surface area. For the Zn(tbip) crystals considered in Figure 17 b, for example, one thus finds that, on the average, only one out of 45 45 channel entrances is unblocked. ChemPhysChem 0000, 00, 1 – 29

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4.4. Intracrystalline Barriers Though the detection of surface barriers gave a perspicacious explanation of the observation that, in many cases, molecular uptake and release was notably retarded in comparison with the behavior expected from the NMR diffusivities, the time dependence of uptake and release was quite often found to follow the pattern of diffusion limitation [Eq. (8)] rather than barrier limitation [Eq. (11)].[135, 136, 160] As a consequence, transport barriers acting in addition to the diffusional resistance of the genuine pore network were assumed to be present not only on the crystal surface but also in the interior. Assuming their random distribution, macroscopic observation would, once again, comply with the pattern of diffusion limitation, now, however, with an effective diffusivity Deff which may be estimated by implying a series connection of the corresponding resistances, yielding Equation (34):[18, 58] 1 1 1 þ Deff D al

ð34Þ

where l stands for the mean distance between the barriers. Direct experimental evidence for the existence of such additional, intracrystalline transport resistances was provided by Vasenkov et al. in PFG NMR diffusion studies with MFI type zeolites[95, 96] and confirmed in a series of subsequent studies[161] in which, moreover, by combination with the information of high-resolution transmission electron microscopy the occur-

rence of these resistances could be correlated with the existence of stacking faults in the crystal structure.[162] Assuming that, just as considered in Section 4.3. for surface barriers, the intracrystalline barriers are also formed by impermeable layers with dispersed “holes”, combination of Equations (33) and (34) yields proportionality between the genuine intracrystalline diffusivity and the effective one as observable in uptake and release measurements. This proportionality would explain why, although differing by orders of magnitude, the diffusivities for many different systems deduced from “macroscopic” and “microscopic” experiments differ from each other only by an approximately constant factor, exhibiting, e.g., similar activation energies.[17, 163] Recent application of high-intensity PFG NMR to hydrated zeolite Li-LSX enabled (by using 7 Li NMR) the first PFG NMR measurements of cation diffusion in zeolites.[164, 165] As a remarkable secondary result of these studies, the structural details of the particles were found to be nicely reflected in the propagation patterns, that is, in the time dependence of the effective diffusivities of both the cations and the water molecules as shown in Figure 18. The first, steep decay is referred to the resistance “experienced” by the diffusants on the boundaries between the individual crystallites while the second decay is correlated with the external particle surface, representing an ultimate limit for all displacements. Data analysis leading to the indicated particle and crystallite radii[165] is based on the formalism introduced by Mitra et al.[166]

Figure 18. Effective diffusivities of water at 25 8C (a) and of the lithium cations at 100 8C (b) in hydrated zeolite Li-LSX. The straight lines represent the best fit to an analytical approach of the influence of the particle surface and of the boundaries between the individual sub-units (“crystallites”) of the zeolite particles shown in (c) as an SEM picture and in (d) in a simplified schematic representation. From Beckert et al.,[165] with permission.



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CHEMPHYSCHEM REVIEWS 4.5. Guest-Induced Host Changes Figure 19 summarizes Tomas Binder’s surprising observation made on recording, as part of his Ph.D. work, molecular uptake of benzene on MFI-type crystallites by IFM microimaging.[167]

www.chemphyschem.org NMR[169, 171] and simulated with flexible zeolite lattices,[172] referred to a re-distribution of the molecules for loadings above 4 molecules per unit cell.[173] The same phenomenon of molecular redistribution was already mentioned in Section 4.2 and is found, in both cases, to be associated with a dramatic increase in guest diffusivity.[153, 167, 174] Now, owing to IFM, the location of such changes can even be monitored. They are found to evolve perpendicularly to the direction of preferred molecular propagation![64] Coincidence of the profiles after 8365 s and 23 065 s indicate final equilibration. Changes in optical density upon phase transitions depend on molecular orientations which may be different in different crystal regions, leading to the eventual differences in phase shift. 4.6. Pore Hierarchies: Interpenetrating Micro- and Mesopores

Figure 19. Two-dimensional maps of the phase shift Df(L) recorded by IFM during benzene uptake by a crystal of silicalite-1 by a pressure step from 5.0 to 10 mbar. Following the first 265 s of adsorption (a) a second process (associated with a phase transition of the sorbate phase) becomes visible (b). Note, the legend color was changed for better visualization. From Ref. [167], with permission.

Following the messages of both the PFG NMR studies (Section 3.2.2.5) and of IFM microimaging (Section 3.3.2.2) on diffusion anisotropy in MFI, Figure 19 a tells the expected story: With the straight and sinusoidal channels directed perpendicular to the crystal’s longitudinal extension (i.e. in X and Y directions), molecular uptake is seen to proceed, essentially, in the direction of these channels. After about 200 s, uniform concentration over the crystal seems to indicate that molecular uptake is concluded. With continued observation, however, the host–guest system is found to undergo further changes, but now with time constants in the order of thousands of seconds. Starting from the two far ends, the changes are seen to evolve in the direction of the crystal’s longitudinal extension, that is, along the direction of lowest molecular diffusivities. The observed changes must therefore be correlated with phase transitions in the host–guest system rather than with molecular transport. Guest-induced changes in the lattice structure have indeed been recorded experimentally by X-ray diffraction analysis,[168, 169] adsorption measurements[170] and solid-state 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

The functionality of nanoporous materials for application in separation and catalysis is generally based on the similarity between the pore diameters and the sizes of the involved molecules, ensuring a particular intimate interaction with the internal surface. On the other hand this similarity impedes molecular diffusion and, hence, the technological performance of such materials, since the output of value-added product can never be faster than allowed by the diffusivities. The fabrication of materials with interpenetrating networks of micro- and mesopores is among the most promising routes for overcoming these limitations.[4, 175, 176] The interpenetration of the two pore networks is, in fact, a prerequisite for any substantial transport enhancement. With mesopores isolated from each other, molecular trajectories are always interrupted by periods of “lowspeed” transport in the micropores,[62] giving rise to only moderate, if any, transport enhancement[74, 177] such as has been observed, for example, with dealumination by steaming of zeolite Y for FCC catalysis.[178] Corresponding with the complexity of bi-porous materials, their transport properties depend on a number of influences, including the diffusivities in and the population of the different pore systems and the rate of their mutual exchange, as well as the possibility of additional transport barriers on the outer surface of the individual particles. This complexity enhances the benefits attainable by the application of microscopic measuring. Since such particles are generally available in only irregular shape, most of these studies have been performed using PFG NMR. Figure 20 shows, as an example, the results of PFG NMR selfdiffusion measurements with bi-porous (“hierarchical”) zeolite LTA.[176] For this purpose, three samples of LTA zeolite were hydrothermally synthesized at the gel compositions of 100 SiO2/ 333 Na2O/67.0 Al2O3/20 000 H2O/n 3-(Trimethoxysilyl)propylhexadecyldimethylammonium chloride (TPHAC), where n was varied from 0 to 2 and 5.[176] The zeolite samples thus synthesized were denoted as Na-LTA-0, Na-LTA-2 and Na-LTA-5, respectively. Here, “Na” means the cationic form of LTA zeolite, and the numbers following “LTA” refer to the TPHAC mole numbers and are a measure of the mesopore volume fraction. Figure 20 a, b shows typical SEM images of the thus produced ChemPhysChem 0000, 00, 1 – 29

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The temperature dependence of the second term is, essentially, given by pmeso which (in the closed PFG NMR sample tubes used in this study) increases with the pressure, thus following an Arrhenius dependence with an activation energy given by the isosteric heat of adsorption. We note that, even at the highest temperatures considered, the effective diffusivities in the mesoporous samples increase, with increasing temperatures, much more slowly. This is due to the high ethane diffusivities in the purely microporous zeolite, leading to a significant contribution of Deff intra to Deff. The situation becomes totally different with propane whose diffusivity in purely Figure 20. PFG NMR diffusion measurement with hierarchical zeolite LTA: SEM images of a) the purely micropomicroporous zeolite NaCaA rous (NaA-) and b) the mesoporous (NaA-5) LTA zeolite sample. c) Intracrystalline diffusivities of ethane in the drops by several orders of magpurely microporous specimen (NaCaA-0, circles) and in mesoporous NaCaA (NaCaA-2, triangles and NaCaA-5, diamonds), both with only ethane adsorbed (open symbols) and with co-adsorbed cyclohexane, (filled symbols) and nitude.[180] The propane diffusivia guest loading of about three molecules per supercage. Full lines represent the best Arrhenius fit to the diffusivities in mesoporous zeolite ties in the purely microporous specimen and their parallel shifts to lower values. The broken line is a guide for NaCaA are, correspondingly, the eyes representing the variation of the contribution of diffusion in the (unblocked) mesopores to the overall infound to exceed those in the tracrystalline diffusivity. d) Diffusivities of n-propane in purely microporous zeolite NaCaA-0 (squares), and in mesoporous zeolites NaCaA-2 (circles) and -5 (triangles) at room temperature as a function of the observation time. purely microporous species by From Mehlhorn et al.,[69, 179] with permission. more than two orders of magnitude (Figure 20 d). This difference would further increase significantly with increasing temperature, given the big difference zeolite crystallites. Figure 20 c illustrates the variety of the thus between the activation energy of intracrystalline diffusion accessible information, with ethane as a guest molecule. The (11 kJ mol1)[180] and the heat of adsorption (34 kJ mol1),[181] diffusivity data for the purely microporous specimen (circles) and for two mesoporous specimens with increasing mesowhich, differing from the situation with ethane shown in Figporosity (open triangles and diamonds) are complemented by ure 20 c, is now essentially exclusively controlling the activation the data obtained with the mesoporous samples preloaded energy of Deff. with (deuterated) cyclohexane for ensuring mesopore blockage With Equation (35), overall mass transfer is seen to be affect(filled symbols). We note that the diffusivities in the samples ed by the relative population pmeso of and the diffusivity Dmeso in the mesopores. Phase transitions, between both the solid with blocked mesopores result by simply shifting the Arrhenius and liquid[182] and the liquid and gaseous states,[183] are known plot of the diffusivities in the purely microporous samples to to give rise to substantial changes in Dmeso. Such changes may lower values, a behavior that is easily rationalized by the inlead to pronounced hysteresis effects in the diffusivities[184] and creased tortuosity. At sufficiently low temperatures, the diffusivities in the blocked and unblocked zeolites coincide. To raoffer great potentials for a performance-related triggering of tionalize this behavior we adopt Equation (18), now in the notransport rates in nanoporous host–guest systems during their tation [Eq. (35)]: technological use.[185] In addition to providing “shortcuts” in intracrystalline diffusion, under the conditions of barrier-limitation, the incorporað35Þ Deff ¼ Deff tion of networks of mesopores has been found to accelerate intra þ pmeso Dmeso uptake and release by a simple “perforation” of surface resistances, appearing in a clear shift in the pattern of their time dependence from Equation (11) to (8).[186] Simultaneously, in with the second term on the right-hand side referring to mass view of the decreased diffusivities shown in Figure 20 c for transfer in the mesopores. Since, for sufficiently low temperasamples with mesopores blocked, the mere presence of mesotures, pmeso drops to arbitrarily small values, the contribution of pores must not be considered to necessarily give rise to diffuthe mesopores to overall transport is also seen to become sivity enhancement. As a prerequisite of transport enhancenegligibly small, just as in the samples with blocked ment, exchange between the micro- and mesopore systems mesopores. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim


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CHEMPHYSCHEM REVIEWS must be sufficiently rapid. In this context, molecular simulations of the rate of mass transfer through the interface between the genuine micropore space and the gas phase attain particular attention.[187]

5. Summary and Outlook With the introduction of microscopic measuring techniques, diffusion in nanoporous materials has been recognized as far more complex than previously assumed. In addition to the diffusional resistance of the genuine pore space, molecular transport was found to be affected also by transport resistances both in the intracrystaline space and on the external surface. The magnitudes of these resistances depend sensitively on the material synthesis, storage and pretreatment. Taking account of these influences and of possible differences in the diffusion pathways considered during the experiments, different measuring techniques are found to lead to compatible results. A long-standing controversy between the results of microscopic and macroscopic techniques of diffusion measurement has thus, eventually, been clarified. The microscopic techniques of diffusion measurement have opened up a wealth of phenomena for experimental exploration and assessment. Once primarily developed to investigate zeolitic diffusion, today they refer to essentially all classes of nanoporous materials. Examples of the attainable information include the quantification of surface barriers and of sticking probabilities, the determination of orientation-dependent diffusivities, the selective measurement of single-component diffusivities under the conditions of multi-component adsorption and the spatially resolved observation of two-stage adsorption, with the second step initiated by phase transitions. They provide, simultaneously, an impression of the rich options of their further exploitation, with the investigation of diffusion and reaction[188] and of structure-mobility correlations with flexible (“breathing”) host frameworks[189] as particularly promising examples. Microscopic techniques of diffusion measurement focus, by their very nature, on fluxes and concentrations within the nanoporous particles. They are recorded, therefore, in exactly the sense in which they have been introduced by Fick’s laws, Equations (4) to (7). This marks a decisive difference in comparison with the conventional determination of reactivities and diffusivities in nanoporous materials which, originally, was based on analyzing the composition of the surrounding gas or fluid phase, associated with the convention of introducing “effective” diffusivities and reactivities referred to concentrations in the surroundings rather than to the genuine intracrystalline ones[190] , with the inherent risk of misinterpretations (see, for example, Refs. [53, 54] and Sections 4.2.2.3 and 21.2 of Ref. [18]). The advent of novel synthesis strategies has provided us, over the last decade, with an increasing spectrum of nanoporous materials including well-shaped crystals with dimensions in the range of tens of micrometers. Their availability was quintessential for the development and application of the techniques of microimaging, as a prerequisite of our present un 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.chemphyschem.org derstanding of the phenomena of mass transfer in nanoporous materials. Similar restrictions also exist for the application of PFG NMR. Though allowing, in principle, the measurement of diffusion path lengths down to the range of hundreds of nanometers, reliable in-depth studies of intracrystalline diffusion by PFG NMR require crystallites of the order of 10 mm or larger. These requirements are in conflict with the trend of many technological applications based on the preferential use of nanosized nanoporous particles.[3, 4, 191] Acquiring direct information about mass transfer in such particles remains an attractive task of future methodological development.

Acknowledgements On preparing this review, I was reminded of many highly esteemed colleagues and dear friends and fellow-combatants who were decisive for my career, starting with Harry Pfeifer, Leipzig, who directed, within the frame of my Ph.D. work, my interest to NMR and to diffusion quite in general, and with Sergey Petrovich Zhdanov, Leningrad, who supplied me with the possibly largest zeolite crystals available at this time, up to the present generation of post-docs and Ph.D. students, notably to Christian Chmelik, Alexander Lauerer and Tobias Titze whom I, in addition, have to thank for their support with the technical work for this review. I am grateful for the possibility of having, from the very beginning of my career, contacts all over the world and for their ample extension after the breakdown of the iron curtain. Further progress was based on the generous financial support by European Community, Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie and, even more importantly, on the support and the personal engagement of many individuals. With my sincerest thanks to all of them, I dedicate this paper to the memory of Hellmut G. Karge[192] for his particular merits and his friendship. Keywords: diffusion · mass transfer · nanoporous materials · nmr spectroscopy

microimaging

·

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REVIEWS Transport phenomena in nanoporous materials are, in the most direct way, explored by recording molecular fluxes (picture, left) and diffusion pathways (picture, right) within the individual particles/crystallites of the material. The review introduces into the fundamentals and most recent developments of these (microscopic) measuring techniques and into the wealth of information thus attainable for fundamental research and technological application.


J. Krger* && – && Transport Phenomena in Nanoporous Materials


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Water confinement in nanoporous silica materials.

Optimizing nanoporous materials for gas storage.

Transport phenomena in laminar flow of blood.

Monitoring Transport Across Modified Nanoporous Alumina Membranes.

Predicting Product Distribution of Propene Dimerization in Nanoporous Materials.

SAXS investigation of nanoporous structure of thermal-modified carbon materials.

Characterization of Nanoporous Materials with Atom Probe Tomography.

Advanced structural analysis of nanoporous materials by thermal response measurements.

Reversible assembly of tunable nanoporous materials from "hairy" silica nanoparticles.

Mechanical properties and interface characteristics of nanoporous low-k materials.

Quantum Simulator for Transport Phenomena in Fluid Flows.

Modeling of transport phenomena in concrete porous media.

The role of the microvascular tortuosity in tumor transport phenomena.

Transport Phenomena of Water in Molecular Fluidic Channels.

Transport and flow phenomena in a microchannel membrane oxygenator.

Hydrogen transport in non-ideal crystalline materials.

Efficient solar photoelectrolysis by nanoporous Mo:BiVO4 through controlled electron transport.

Nanoporous Cyanate Ester Resins: Structure-Gas Transport Property Relationships.

Microimaging of transient concentration profiles of reactant and product molecules during catalytic conversion in nanoporous materials.

On the importance of a precise crystal structure for simulating gas adsorption in nanoporous materials.

The Influence of Intrinsic Framework Flexibility on Adsorption in Nanoporous Materials.

Microimaging of transient guest profiles to monitor mass transfer in nanoporous materials.

Behavior of the Enthalpy of Adsorption in Nanoporous Materials Close to Saturation Conditions.

chemical molecules.