Correlating phase behaviour and diffusion in mesopores: perspectives revealed by pulsed field gradient NMR

Abstract

Porous solids represent an important class of materials widely used in different applications in the field of chemical engineering. In particular, mesoporous hosts attract special attention due to their fascinating match of transport, geometrical and chemical properties. Not only a very high specific surface area, accessible for adsorption and heterogeneous catalysis, but also their efficient transport properties are the key factors determining optimal use of these materials. Therefore, a fundamental understanding of the correlations between the phase state of confined fluids, their transport properties and the geometrical features of confinement are of particular importance. Among the different analytical techniques, nuclear magnetic resonance (NMR) is especially suited to cover various crucial aspects of the highlighted issues. In this work, we provide a short overview of recent advances related to the interrelations of phase behaviour and diffusion in mesoporous materials studied using various NMR techniques.

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Rustem Valiullin

Rustem Valiullin got his PhD degree from Kazan State University (Kazan, Russia) in 1997 and continued to work there as a Research Scientist. After two years postdoctoral work in the Royal Institute of Technology (Stockholm, Sweden), in 2003 he moved to the University of Leipzig, Germany as a fellow of the A. von Humboldt Foundation. Currently he is a Heisenberg fellow of the German Science Foundation at the Department of Interface Physics of Leipzig University.

Jörg Kärger

Jörg Kärger got his PhD in Physics in 1970 at Leipzig University, followed by the habilitation in 1978. As a member of Leipzig University since this time, research stays took him to Prague, Moscow, Fredericton and Paris. In 1994, he became Professor of Experimental Physics and head of the Department of Interface Physics. His research is dedicated to diffusion phenomena quite in general.

Roger Gläser

Roger Gläser has studied chemistry at the University of Stuttgart, Germany and got there his PhD degree from the Institute of Chemical Technology in 1997. After that he worked at the Georgia Institute of Technology, Atlanta, USA. In 1999 he returned to the Institute of Chemical Technology, Stuttgart where he completed his habilitation in 2007. Presently, he is a Professor of Chemical Technology at the University of Leipzig and is head of the Institutes of Chemical Technology and Non-Classical Chemistry.

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I. Introduction

Since the discovery of the M41S-family of molecular sieves by the researchers of Mobil Oil Corp.,^1,2 a plethora of ordered mesoporous materials with different compositions and surface properties was found including, for instance, the materials of the SBA-, HMS-, MSU-, TUD-, or CMK-type,^3–8 periodic mesoporous organosilicates (PMOs),⁹ mesocellular silica foams (MCFs)^10–12 or metal–organic frameworks (MOFs).^13–15 The continuing interest in these materials, especially in view of applications in sorptive separations and heterogeneous catalysis, relies, inter alia, on their high specific surface areas and their large pore diameters with respect to the conventionally applied zeolites and related microporous materials. As a consequence, sorptives and reactants with considerably larger molecular dimensions have access to the inner surface area of these molecular sieves.¹⁶

Moreover, a more rapid mass transfer into and egress of mesoporous materials offers obvious potential over their microporous counterparts.^17,18 Mesopores have often been invoked to serve as “transport pores” for higher molecular reactants and, thus, enable and accelerate their transport to the active sites of solid catalysts.¹⁹ Prominent examples are the alkylation of benzene with propene to cumene over dealuminated mordenite-type zeolites²⁰ or fluid catalytic cracking (FCC) of long-chain hydrocarbons over acidic faujasite-type zeolites.²¹ While in the sooner case, the mesopores are introduced into the catalyst by dealumination of the zeolite crystals, a mesoporous matrix around the zeolite crystallites provides access to the active surface sites in the latter. It was recently shown by pulsed field gradient (PFG) NMR that a transport optimization of FCC catalysts can be achieved by selecting a matrix for the active zeolite crystals with the appropriate mesopore size.²² The introduction of mesopores into the acidic faujasite catalyst by steam dealumination, however, does not have a pronounced effect on the hydrocarbon diffusivities within the micropores of the zeolite.²³ Besides catalysis, mesoporous materials are attractive for the adsorptive uptake of substances with larger molecules such as enzymes for biocatalysis²⁴ or pharmaceuticals for controlled drug release.^25,26 Other potential application areas for mesoporous molecular sieves include nanofluidics, photonics, and optical switches.^27–29

In addition to the apparent advantages regarding mass transfer, mesoporous materials exhibit most interesting properties with respect to the thermodynamic equilibria of the adsorbed phases. In addition to the dense “phase” adsorbed on the pore walls, there are solid, liquid, gaseous or supercritical phases within the pores with equilibrium and transition conditions significantly differing from those in the bulk.^30,31 This difference is a result of the confinement of the phases within the mesopores. In contrast, micropores are, for most substances, too small for another phase but a dense adsorbed phase to exist and in macropores the influence of the walls on the phase behaviour within the pores is almost negligible. Apparently, the conditions for phase transitions are especially well-defined for ordered mesoporous materials, i.e., for materials with a narrow pore size distribution as typical for molecular sieves.

The deviation of the conditions for phase equilibria or transitions under confinement, in particular by mesopores, from those of the bulk is a general phenomenon and not limited to particular phases. For instance, both the liquid/gas and the solid/liquid coexistence curves and, consequently, the critical and the triple point are shifted towards lower temperatures and pressures compared to the bulk. Moreover, a rich set of experimental data is available to date revealing, with a sufficiently high accuracy, proportionality of these shifts to the reciprocal mesopore size. Thus, the dependence of the solid/liquid phase transition (melting point) on the mesopore diameter is routinely used nowadays for the determination of the mesopore diameter by thermoporometry.^32–34 The same principle is exploited also in gas adsorption porometry.³⁵ It is worth noting that a better understanding of these phenomena may be of indispensable practical relevance in most diverse fields, ranging from material science to biophysics.^36–38

From the known phenomenon of sorption hysteresis, however, it is evident that the phase behaviour within mesoporous materials can be rather complex and that it depends on the way it is approached.³⁹ Although these basic features have been reported long ago,^40,41 a complete understanding of the phenomenon is still lacking. Beside fundamental interest, the practical importance of the problem is determined by the fact that the measurement of gas adsorption is widely used as a classical characterization technique³⁵ and, in turn, is of enormous importance in the development of novel porous materials.^42,43

While a wealth of studies is devoted to the nature and the thermodynamic equilibrium of phases within mesoporous solids, much less investigations were carried out to describe and understand the transport properties of mesopore-confined phases. It is obvious, of course, that the presence of different phase states or distributions will strongly affect the corresponding transport properties within the mesopores. This perspective focuses on diffusion within mesoporous materials in view of the behaviour of liquid and gas phases coexisting inside the mesopores. With respect to the theme of this special issue, the scope of this article is, thus, the treatment of the phases as “molecular assemblies”, rather than the behaviour of isolated molecular species in confined spaces of porous solids. The most important industrial applications of mesoporous materials, especially in sorption and heterogeneous catalysis, involve gas and/or liquid phases, not only in the bulk surrounding of the materials, but also within the mesopore space (vide supra). Other phase equilibria inside mesoporous solids such as those of solid/gas or solid/liquid systems will, thus, not be treated here.

Based on our previous experience, it is another goal of this article to highlight how transport measurements, especially by pulsed field gradient (PFG) NMR, can be used to infer on the phase behaviour within mesoporous solids. The information obtained from transport measurements is shown to be complementary to that typically extracted from sorption equilibrium data such as isotherms or isochores, i.e., from the loading of the sorptives as a function of the physical properties of the fluid phase outside the porous solid.

The present article is subdivided into five major sections. First, we describe the specific phase behaviour in mesopores and its partly significant differences from that in bulk phases. As mentioned above, this contribution will be restricted to the liquid/gas-coexistence and the related phenomena near the critical point. After a brief outline of how PFG NMR can be applied to measure different transport-related properties within porous solids (for more detailed description of this methods the readers may refer to ref. 44–49), two following sections will be focused towards the transport processes occurring in porous solids on the different porosity scales (micro- and macroporosity). Finally, we will summarize the current knowledge on the relation of transport properties and phase behaviour in mesoporous solids at conditions for which a transition of the pore-confined phases to the supercritical state occurs. We will give selected examples where this transition and the presence of a supercritical phase in the mesopores were utilized for applications in materials preparation, sorption, and heterogeneous catalysis. From the current knowledge, we will derive potential perspectives as to where and how the specific liquid/gas-phase behaviour and its effects on transport properties in mesoporous materials may be utilized in the future and outline the open scientific questions and challenges.

II. Fluid phase equilibria in mesopores

It is well known that the phase coexistence of fluids in porous solids is altered as compared to the bulk fluids. This is demonstrated by the sorption isotherms, a typical example of which is shown in Fig. 1. The shape of an isotherm reflects the interactions of the adsorbed molecules with the internal surface of the porous material as well as the effect of confinement on properties of the adsorbed fluid. At sufficiently low external gas pressures P the latter unequivocally determines the equilibrium between molecular ensembles adsorbed on the pore walls and the surrounding gas phase. This means that the sorption process, i.e., adding or releasing of molecules to or from the adsorbed (multi-)layers, is continued until the chemical potential equilibrates over the whole system. Notably, there are no imposed barriers of either type to prevent the equilibration under these conditions. At intermediate pressures, a sudden change of the amount adsorbed is observed, which is attributed to evaporation or capillary-condensation transitions. Importantly, both transitions occur at pressures well below the saturated vapour pressure P_s, manifesting the confinement effect upon the phase state. In addition, an important feature of mesoporous materials is the occurrence of hysteresis, where for a range of bulk gas pressures the sequence of states encountered on desorption does not coincide with that encountered on adsorption.

Fig. 1 Relative adsorption (open symbols) and desorption (full symbols) isotherms of nitrogen in channels of porous silicon (d_p≃ 6 nm), open at both ends (stars) and one end (circles), measured at 77 K. The inset shows a schematic phase diagram for a bulk liquid (solid line) and for a liquid confined within pores (dotted line).

The pioneering theories of adsorption and hysteresis in pores are usually classified as capillary condensation theories.³⁹ They are typically constructed around the idea of a shifted capillary condensation and evaporation transitions due to curved menisci, i.e., by considering equilibrium between the condensed phase with a curved meniscus in the pores and its vapour just above it. As a result, this situation may often be described by the famous Kelvin equation for the transition pressure P_tr, ln(P_tr/P_s) ∝ (1/r₁ + 1/r₂), where r₁ and r₂ are the principal meniscus radii. As an example, an open-ended cylindrical capillary with a radius r yields an instructive idea of the hysteresis development through a difference in geometry of the liquid menisci on desorption (spherical concave, 1/r₁ + 1/r₂ = 2/r) and adsorption (cylindrical concave, 1/r₁ + 1/r₂ = 1/r).⁵⁰ Morphological details of the porous medium were also considered within this approach: In the so-called ink-bottle geometry, adsorption hysteresis was linked with a peculiar construction of the pore space.^51,52 It was argued, for example, that narrow interconnections between large pores could delay the evaporation of liquids from them. This is the so-called pore-blocking effect, which was later widely used to describe the hysteresis phenomenon in random porous materials.⁵³

Later on, density functional theories (DFT) and lattice gas models capturing microscopic features of the adsorption phenomena have been developed.^30,54 Applying DFT to uniform pores, with increasing external vapour pressure the molecular ensemble in pores is found to remain in a gas-like state beyond the point of the true thermodynamical transition, i.e., the system persists in a local minimum of free energy. This is continued until the barrier separating the local and the global energy minima becomes small enough. The same is true for the desorption process concerning a liquid-like state. Importantly, microscopic theories thus point out the metastable character of the transitions. To extend these theories to real porous materials, i.e., to allow the simultaneous coexistence of gas-like regions and regions with a capillary-condensed phase, one has to artificially incorporate effects of the geometrical heterogeneity of pore structure.^55,56

A more detailed analysis of realistic molecular ensembles under confinement using statistical thermodynamics has become possible due to the rapid progress of computer technologies. Exploiting quite different approaches, many details of phase equilibria in various model systems have been elucidated and rationalized.^30,57–61 In particular, the results obtained using non-local DFT pointed out that in sufficiently big pores condensation occurs at the vapour-like spinodal, while desorption takes place at the equilibrium.⁵⁷ The same conclusion has been drawn using molecular dynamics simulations of the molecular behaviour in one- and both-ends open pores.⁶² Moreover, this approach allowed the authors to address also the phenomenon of pore blocking, which is widely accepted to contribute to the development of hysteresis.^53,63 Thus, using an ink-bottle pore as confining geometry, it was found that liquid can evaporate from a large cavity even if the neck of the ink bottle remains filled with the capillary-condensed phase.⁶² On the other hand, other studies confirmed the relevance of the pore-blocking mechanism,⁶⁴ emphasizing the importance of the details of the pore structure and the involved interactions.

Progress in fabrication of nanoporous materials with well-defined structural properties made it possible to validate all these theoretical predictions by experimental means. Thus, using ordered mesoporous silica materials composed of spherical cage-like pores connected with each other by cylindrical pores and using different adsorptives, different regimes of desorption, namely controlled by pore-blocking or cavitation, have been identified.^65,66 In particular, cavitation-controlled evaporation in ink-bottle pores with the neck size smaller than a certain critical value has become evident. In pores with larger necks, evaporation was found to be percolation-controlled. Remarkably, the results obtained with seemingly simpler pore geometry revealed less consistency with the theoretical expectations. Thus, on the basis of recent experimental studies using MCM-41 silica material with well-defined open-ended cylindrical pores, it was concluded that capillary condensation rather than evaporation takes place near a thermodynamical equilibrium transition.^67,68 Note that this conclusion apparently contradicts the Cohan’s picture and poses important questions still to be answered. On the other hand, adsorption studies using porous alumina with well-defined cylindrical pores of diameters from 10–60 nm identified the desorption transition as the equilibrium one.⁶⁹

It has to be noted that, although such materials as MCM-41 or SBA-15 are generally considered to be ideally organized, their real pore structure is subject to indispensable defects which appear, e.g., in PFG NMR studies^70–74 and may affect the phase behaviour of the fluids therein. As a most demonstrative example, one may refer to mesoporous silicon. A number of adsorption studies using this material with channel-like, parallel pores have revealed very striking, sometime counterintuitive results.^75–78 Thus, this material exhibits similar behaviour, irrespective of whether both or only one end is open. It has already been suggested earlier that these effects may be caused by surface roughness inherent to this material.⁷⁷ Recent experiments⁷⁹ and theoretical considerations using mean-field theory of a lattice gas⁸⁰ confirmed the effects of a “quenched disorder” as the directing feature of adsorption hysteresis in pores of mesoporous silicon. Importantly, the disorder is thus found to be much more pronounced than expected based on the sole atomistic roughness. Thus, the linear (one-dimensional) pores in this material appear to exhibit all effects typically associated with materials with three-dimensional random pore networks. It is interesting to note that similar behaviour (e.g., irrelevance of closing of one of the channel ends) has recently also been obtained for self-ordered porous alumina, a material which is believed to be subject to less disorder effects.⁸¹ The origin of such a behaviour has still to be clarified.

For disordered mesoporous materials, the central questions about the interrelation between hysteresis and phase transitions has been recently addressed in a work based on mean field theory and Monte Carlo simulations for a lattice gas model.^82,83 The calculations indicated that the hysteresis could be understood in terms of the effects of the spatial disorder upon the density distribution in the material. In particular, it was anticipated that hysteresis is associated with the appearance of a very large number of metastable states, represented by minima in the local free energy corresponding to different spatial distributions of the adsorbed fluid within the void space of the porous material. In this respect these systems resemble the hysteresis encountered in disordered magnetic systems.⁸⁴ The major message of these studies was that slow dynamics associated with a rugged free energy landscape may greatly affect the phase equilibrium and prevent the system from equilibration.

The phenomena discussed above refer to temperatures below the bulk critical temperatures T_c. Plotting the average densities of the fluid at the loci of the hysteretic isotherm as a function of temperature, one may compile a hysteresis phase diagram, as shown by the inset in Fig. 1. Notably, these loci correspond (i) to coexisting multi-layered molecules on the surface and in the gaseous phase in the pore interior and (ii) to pores completely filled by the capillary condensate. The thus obtained hysteresis phase diagram is very much similar to the normal one of the bulk, but typically reveals a pore size-dependent temperature shift of the upper closure point.^85–87

For the confined fluids, this point at the temperature T_ch, called hysteresis critical point and corresponding to vanishing hysteresis, had earlier been considered as a counterpart of its bulk value T_c. This, however, has to be treated with caution. Indeed, in addition to the hysteresis critical temperature, one may as well define the pore critical point at T_cp.⁵⁴ The latter is defined as the temperature where a jump in the adsorption isotherm caused by capillary condensation just disappears. The first observations, suggesting a differentiation between these two temperatures, T_ch and T_cp, have again been made for MCM-41-type material. Pore uniformity allowed to measure the isotherms free of the disturbing effects of a pore size distribution, which, otherwise, would smear out the local condensation events over a certain pressure range. The obtained results unequivocally pointed out that T_ch < T_cp < T_c.⁸⁸ A simple explanation of the obtained relation T_ch < T_cp could be provided by recalling the metastable character of the condensation and evaporation transitions in the hysteresis regime. With increasing temperature, the local free energy minima may become sufficiently small to be bypassed by the thermal fluctuations.³⁰ Thus, the hysteresis loops would collapse at lower temperatures compared to T_cp.

III. Pulsed field gradient NMR

Nuclear magnetic resonance provides direct access to the key data of fluid behaviour in the porous space of interest: it is able to simultaneously record the number of molecules and their dynamical properties. Thus, measuring the NMR signal intensity in a sample of a porous solid as a function of the external gas pressure, one may compile the respective adsorption isotherm, as an indicator of the phase state in the pores. On the other hand, a number of other quantities may simultaneously be measured in the same sample under the same conditions. One of them, which will be in the focus of this review, is molecular translational mobility. Therefore, NMR may serve as a self-consistent experimental tool allowing to correlate phase state of a fluid in pores and its transport properties. The way in which this becomes possible may be rationalized using the classical interpretation of nuclear magnetism.^44–46,89

Under the influence of a magnetic field of intensity B₀, each nucleus (i.e., the nuclear “spin”) performs a precessional motion about the direction of B₀ with the angular frequency ω₀ = γB₀, where γ is the nuclear gyromagnetic ratio. By the application of a radio-frequency field of the same frequency ω₀ over a well-defined short interval of time, the net magnetization in the sample can be turned from the direction of the magnetic field (the equilibrium position) into the plane perpendicular to it. The net magnetization rotating in this plane induces a voltage in the receiver coil surrounding the sample under consideration. The thus measured voltage is monitored as the NMR signal. Obviously, the intensity of this signal is proportional to the number of respective nuclei and hence to the number of molecules in the sample (one should, however, take account of nuclear magnetic relaxation effects, which for fluids in pores may be significant and may lead to relaxation weighting of the measured signal). The different Larmor frequencies of different nuclei (e.g., ¹H, ¹³C or ²H) provide an additional option to separately track different species.

For rationalizing the way in which the translational mobility of the fluid molecules may be quantified, we consider the influence of field inhomogeneities on the Larmor frequencies ω₀ of the nuclear spin system under study. Applying a magnetic field linearly increasing in the z direction, B = B₀ + gz, ω₀ = γB₀ + γgz becomes a function of the z coordinate. With this inhomogeneous field held for a time δ, each spin will acquire a phase ϕ = γB₀δ + γ∫^δ₀g(τ)z(τ) dτ. This procedure is sometime called “position-dependent phase encoding”. In the basic version of pulsed field gradient (PFG) NMR the sample is subjected to two pulses of “field gradients” of constant amplitude g and of duration δ with separation t_d. With two such pulses of opposite gradient directions, to the end of the second pulse re-phasing may not be complete and the phase difference Δϕ is given by:

It now becomes obvious that, being able to detect Δϕ, one immediately gets access to the difference of the locations of the individual molecules at the instants of the gradient pulses. Here, we imply that the short-gradient approximation, namely δ≪t_d, is fulfilled and thus we may neglect the effects of molecular motion during the gradient pulse.^45–47 This is especially justified under our experimental conditions performed with the use of ultra-high gradient strength.⁹⁰ Implementing these field gradient pulses into an appropriate sequence of radio-frequency pulses, e.g., Hahn⁹¹ or stimulated-echo⁹² pulse sequences, one is able to determine the resulting NMR signal, the so-called spin echo. It may be shown that the attenuation S(q,t_d) (from now on we use the widely used notation q = γδg) of this signal intensity,⁹¹

S(q,t_d) = ∫P(z,t_d)e^iqzdz, (3.2)

is nothing else than the Fourier transform of the so-called mean propagator P(z,t_d), one of the key functions describing the internal dynamics in complex systems.⁹³P(z,t_d) denotes the probability density that, during time t_d, an arbitrarily selected molecule is shifted over a distance z in the direction of the applied field gradient. The range of molecular displacements observable by PFG NMR spans over a few orders of magnitude from about 100 nm, accessible by means of strong field gradients,⁹⁴ to a few mm. On the PFG NMR time scale from about 1 ms to 1 s, the former displacements can be observed in high molecular mass polymeric systems⁹⁵ or for strongly confined fluids,⁹⁶ while the latter displacements are typical for gas phase applications.⁹⁷ As an important point in comparison with other spectroscopic techniques, NMR has to operate with relatively big molecular ensembles of about 10¹⁸. Being a disadvantage, to require the presence of such a huge ensemble of molecules, on the other hand, well-averaged quantities are provided.

For unrestricted, normal diffusion, the propagator results as the solution of Fick’s second law with the initial concentration given by Dirac’s δ-function:

with D denoting the self-diffusivity. Inserting eqn (3.3) into (3.2) yields a spin-echo diffusion attenuation of exponential form:

S(q,t_d) = exp(−q²Dt_d). (3.4)

Thus, the diffusivity (and the mean square displacement, respectively) may be easily determined from the slope of the semi-logarithmic representation of the intensity of the NMR signal versus the squared pulse gradient “intensity”δg. Note that, for the sake of simplicity, S(q,t_d) in eqn (3.4) is given as normalized to S(0,t_d) to account for nuclear relaxation effects. Often, as a useful approximation, eqn (3.4) may be assumed to hold when molecular propagation deviates from normal, unrestricted diffusion. In such cases, the quantity D has to be interpreted as an “effective” diffusivity defined by Einstein’s equation D = 〈z²(t)〉/2t (which, clearly, coincides with the real diffusivity as soon as the prerequisites of normal diffusion are fulfilled). For anomalous diffusion D is not a constant and depends on the observation time.^98,99

Deviations of S(q,t_d) from the exponential shape of eqn (3.4) may also result from a distribution of molecular mobilities along the sample. An important example, in the context of the present review, is a sample composed of a powder of small porous particles. For such materials, two molecular ensembles characterized by different diffusivities may easily be distinguished, namely those in the inner pore space and those between the porous particles. Very generally, the spin-echo diffusion attenuation for such two-phase systems (here, by “phase” we refer to transport characteristics of the diffusing species rather than to the physical state) may be written as:^95,100

S(q,t_d) = ∫Ψ(τ_a,τ_b)exp{−q²(D_aτ_a + D_bτ_b)}dΩ. (3.5)

Here, Ψ(τ_a,τ_b) is the joint probability density that, during the observation time t_d = τ_a + τ_b, a molecule spent the times τ_a and τ_b in the phases a and b, respectively. Ω denotes a certain pair {τ_a,τ_b}. A particular solution of this problem under the assumption of an exponential lifetime distribution in the phases may be found in ref. 101.

It is instructive to consider two limiting cases of slow and fast molecular exchange between the phases. In the first case, all individual molecular trajectories are confined to either of the phases, i.e. there exist only two possible time pairs Ω₁ = {τ_a = t_d,τ_b = 0} and Ω₂ = {τ_a = 0,τ_b = t_d}. The relative fractions of such trajectories, or corresponding values of Ψ(τ_a,τ_b), ultimately coincide with the fractions p_a and p_b (p_a + p_b = 1) of molecules in the two phases a and b, respectively. Therefore, the respective diffusion attenuation function is given by the sum of two exponents:

S(q,t_d) = p_aexp{−q²D_at_d} + p_bexp{−q²D_bt_d}. (3.6)

Another limit of fast exchange corresponds to situations when all molecules during t_d experience many consecutive travels in both phases, i.e., if the molecular lifetimes in both phases are much shorter than the total diffusion time t_d. In this case, assuming that the ergodic hypothesis is fulfilled, namely τ_a/(τ_a + τ_b) = p_a, we may substitute τ_a and τ_b in eqn (3.5) by p_at_d and p_bt_d, respectively. Hence, Ψ(τ_a,τ_b) = 1 when τ_a = p_at_d and τ_b = p_bt_d and zero otherwise. Finally, in this limit eqn (3.5) becomes:

S(q,t_d) = exp{−q²t_d(p_aD_a + p_bD_b)}. (3.7)

In between these two regimes, S(q,t_d) can by described by eqn (3.5) with a continuous spectrum of the apparent diffusivities (τ_aD_a + τ_bD_b)t⁻¹_d, where τ_a varies from 0 to t_d. By performing inverse Laplace transform one may, in principle, access the function Ψ(τ_a,τ_b) containing all information about the exchange process between the two phases. This, however, is an ill-posed problem, which only recently is becoming of practical use by extending the experiments to the second dimension.^102–104 More practically, by performing the experiments with different observation times t_d one can, with a relatively high accuracy, measure the fraction p_sl of molecules which have not left a phase with the slowest diffusivity D_sl during t_d. Obviously, p_sl(t_d = 0) is the relative fraction of molecules in this phase. The measured p_sl(t_d) can be related to the molecular lifetime distribution function ψ(τ) in the corresponding phase via:⁹⁵In section VB, this relation will be shown to provide the basis for the so-called NMR tracer desorption technique. Alternatively, due to the short time scale of PFG NMR in comparison with “conventional” tracer exchange experiments, this method is as well referred to as “fast tracer desorption”.^96,105–107 The function ψ(τ) is a key function of the internal dynamics of stationary populations quite in general, and of particular relevance for catalytic processes involving porous solids. In ref. 108, 109, as part of a “network of characteristic function”, it is shown to directly yield such (further) important functions like the tracer exchange curve and the effectiveness factor of a chemical reaction between compounds of equal diffusivities.

The function ψ(τ) describes, in particular, how fast a molecular ensemble, initially residing in a particle, exchanges its molecules with the surrounding. Thus, for a spherical particle of radius R and an intra-particle diffusivity D, for diffusion-limited exchange (i.e., with no surface barriers, hindering the molecules to leave the particle at the interface with the surrounding phase) ψ(τ) is given by ref. 110

Note that the case of surface barriers may also be analyzed in this way.¹¹¹ With a known function ψ(τ) one may easily estimate the average time t_av after which the molecules leave the particle in which they just reside. For spherical particles, it immediately follows from eqn (3.9) that:⁹⁶On the other hand, by tracking the fraction of molecules which did not leave the porous particles during the experimental observation time t_d, one may estimate an average size of the particles. To do this, one may fit eqn (3.8) with ψ(τ)given by eqn (3.9) to the experimental data. In this case, the only fitting parameter is R while the diffusivity is known in advance from the analysis of the respective spin-echo diffusion attenuations.

To conclude this section, in recent years NMR progressed to provide a wealth of approaches to analyze different aspects of molecular dynamics in porous materials with inhomogeneities in the porous structure on very different length scales,^112–114 including its possibility to quantify molecular diffusivities in mesopores under different external conditions. The simultaneously measured NMR signal intensity provides the option to correlate the transport properties with the phase state in the pores. Moreover, by stepwise changing the external conditions, e.g., the vapour pressure, one may create a gradient of the chemical potential between the gas phase and the confined fluid allowing to follow its equilibration by means of NMR. In this way, the results of macroscopic and microscopic techniques may be compared to reveal information on the fluid behaviour, which so far, was inaccessible.^115,116 Altogether, a set of NMR approaches allow to address various aspects of molecular dynamics in mesoporous adsorbents of different pore architecture and macro-organization.

IV. Microscopic diffusivity in monoliths

In what follows, we are going to address different aspects of molecular dynamics in mesoporous materials. Referring, e.g., to heterogeneous catalysis involving porous catalysts, the whole molecular trajectories consist of diffusion paths within the porous particles intermitted by those in the free space between the particles (see Fig. 2). We will proceed by first considering the transport mechanisms within mesopores. Depending on various conditions, primarily those determined by the phase state of a fluid, these include surface diffusion, diffusion of molecules in multilayers, Knudsen diffusion and diffusion in the capillary-condensed phase. In disordered materials, e.g., random porous glasses, different molecular phases may coexist with each other in vastly different configurations, yielding, in their combination, very complex behaviours. The latter is evident even from a quick review of the literature data on diffusion in mesopores, reporting a wide spectrum of behaviour depending on pore size, adsorbate and surface properties, temperature, etc. refs. 117–124. Let us now start by considering some of these transport modes separately.

Fig. 2 Schematic representation of the transport paths during the process of conversion of a reactant to a value-added product by heterogeneous catalysis in a mesoporous solid. (a) and (b) show the two limiting cases of intra-particle transport along the mesopores, namely surface diffusion and Knudsen diffusion, respectively.

A Surface diffusion

Among the different mechanisms of molecular transport, surface diffusion plays a significant role, sometime being the dominating mechanism of the mass transfer through porous solids. Quite generally, it may be associated with a thermally-activated hopping motion of the guest molecules along the surfaces (see, e.g., the reviews^125–127). For experimental studies of surface diffusion, two different approaches are commonly used. One of them is based on the creation of a gradient of the chemical potential (e.g. via pressure or concentration gradients) and measuring the resulting fluxes. In this way, the so-called coefficient of transport diffusion is measured. It contains information about both single-molecule and cooperative modes of the molecular transport. Another approach is designed to operate under equilibrium conditions by, e.g., “labelling” a molecule and following its trajectory.^128–131 Such direct visualizations of the molecular trajectories on surfaces have illuminated many novel phenomena, like molecular clustering or mobility enhancement due to a specific molecule-substrate interaction. The analysis of the thus obtained single trajectories yields the so-called tracer diffusivity. Notably, exactly this quantity is as well probed by PFG NMR. The huge surface area of mesoporous adsorbents makes NMR applicable to probe surface diffusivity at sub-monolayer surface coverage.¹³² Importantly, this latter technique provides an ensemble-averaged quantity.

The required surface coverage c (also referred to here as concentration) may conveniently be prepared by tuning the external gas pressure in contact with the mesoporous material. Such a setup can easily be implemented to the in situ operation in an NMR spectrometer,¹²⁴ so that the adsorption (free induction decay (FID) signal intensity which is proportional to the number of spins in the sample) and diffusion (PFG NMR) could simultaneously be probed. Fig. 3 illustrates this possibility by showing the diffusivity of acetone in mesoporous silicon. The material was prepared by electrochemical etching of a (100)-oriented silicon wafer with a resistivity of 10 Ωcm using a current density of 30 mA cm² in an electrolyte containing 50% aqueous solution of HF and C₂H₅OH in a volume ratio of 2 : 1. In this way, a foam-like structure with an average pore diameter of about 4 nm is formed.¹³³ First, the adsorption isotherm has been compiled by measuring the FID signal intensity S as a function of the gas pressure. By using the BET equation,¹³⁴ the amount adsorbed corresponding to one monolayer surface coverage was estimated and used for transformation to the surface concentrations c. The surface diffusivities D_s, measured in parallel, are, thereafter, plotted via c. Note that the originally obtained diffusivity data have been corrected (although the correction was minor) for the diffusion through the gaseous phase in the pore interior. This procedure will be explained in more detail in the following section.

Fig. 3 Surface diffusivity of acetone in porous silicon as a function of the surface coverage. The solid line is the function D*₀c^1/m−1 with D*₀ = 16 × 10⁻¹⁰ m² s⁻¹ and m = 0.26. The inset shows the surface coverage c as a function of the relative pressure z with the solid line being the best fit of eqn (4.14) to the data.

The most important message of Fig. 3 is that the diffusivity increases with increasing surface coverage. This is in contrast to what one expects for hard-core particles on homogeneous surfaces, where an increase of the number of particles leads to a decrease of the space available for diffusion. Due to this so-called site-blocking effect the surface diffusivity would rather decrease with increasing c. Remarkably, heterogeneity of the surface, e.g., site or saddle-energy disorder, inverts this trend. In a most simple way, this may be understood by considering the effect of site-energy disorder given by a distribution f_s(E) of site energies E. Due to the activated character of the jump process, this leads to a corresponding distribution of the jump rates, with the latter being W(E) ∝ exp{−E/RT}.

With such a condition, the solution of the diffusion problem for a single particle is obtained in the frame of the so-called random trap model, resulting in a diffusivity a²〈1/W〉⁻¹. Here, the brackets denote averaging over all surface sites and a is the inter-site distance.^135,136 It is worth noting that the solution does exist only, if there exists a finite average residence time 〈1/W〉 and if the model implies that there is no correlation between neighbour site energies, i.e., if a random energy topography is assumed. The particle ensembles may be treated similarly. However, the probabilities of site occupancy, p(E), have to be properly accounted for:

D_s = ∫f_s(E)p(E)W⁻¹(E)dE. (4.11)

Forbidding multiple site occupancies, p(E) has naturally to be chosen to follow the Fermi–Dirac statistics, p(E) = (1 + exp{(E−μ)/RT})⁻¹, where μ is the chemical potential of the surface ensemble. Such an occupancy factor explains increasing diffusivity with increasing surface coverage or, correspondingly, chemical potential: A new particle added to the system will occupy a site with a lower surface energy as compared to those already occupied. Therefore, the overall transition rate, obtained by averaging over the whole ensemble, will increase.

To illuminate the basic predictions of eqn (4.11), it is more convenient to use it in a modified form:^137,138

where E_av is the average site energy and D₀ is the diffusivity of a single particle on the surface with the only site energies E_av. Owing to the equilibrium conditions, the chemical potential of the surface ensemble is equilibrated with that of the external gas phase. In the first order, the latter may be approximated by the ideal gas, i.e., μ = μ₀ + RT ln(z), where z≡P/P_s. Thus, by absorbing all concentration-independent parameters into D*₀, eqn (4.12) becomesThis equation shows the direct interrelation between the surface diffusivity D_s and the adsorption isotherm c(z) (or, vice versa, z(c)). Let us now consider one of the typical isotherms used for the analysis of adsorption on heterogeneous surfaces, for example the generalized Freundlich isotherm¹³⁹In eqn (4.14)K is a constant and 0 < m < 1 is a parameter reflecting the degree of surface heterogeneity. The meaning of the parameter m may intuitively be understood by comparing eqn (4.14) to the Langmuir isotherm given by the same equation but with m = 1. The latter is valid for homogeneous surfaces, i.e. the addition of a new particle to the surface depends only on surface coverage. If, however, m < 1, the affinity of the surface decreases with increasing coverage, reflecting, thus, its heterogeneity. With an isotherm as given by eqn (4.14), the surface diffusivity will behave as demonstrated in Fig. 4 for different values of m. As expected, the diffusivity increases with increasing surface coverage. However, at intermediate coverage, D_s passes through a maximum. The latter is the result of two competing mechanisms-entropic (site-blocking) and energetic (increasing transition rates) ones.

Fig. 4 Surface diffusivity as a function of surface coverage calculated using eqn (4.13) assuming a generalized Freundlich isotherm (eqn (4.14)). Different curves refer to different surface heterogeneities represented by the parameter m.

Let us now analyze the experimental data shown in Fig. 3 in this way. The inset shows the low-pressure part of the adsorption isotherm for acetone in porous silicon. It may be fitted using eqn (4.14) with m = 0.26. Remarkably, the diffusivity data at low surface coverage nicely follow the pattern D*₀c^1/m−1 (low-c limit of eqn (4.13)), with m = 0.26 obtained from the isotherm. Note that this functional dependence is expected to hold for c < 0.2, while in our case the range is extended up to c≈ 0.8. However, we consider this to occur accidentally. More detailed understanding and description of the surface diffusivities at intermediate and high coverage requires a proper account of, e.g., particle distribution over the surfaces.

In this subsection we did consider transport along the surfaces only. If, however, the molecules posses sufficient kinetic energy to overcome the energetic barrier to get away from the surface, they may perform movements along the gas phase in the pore interior. Before discussing the combined effects of different transport modes, we would like to address some aspects of pure diffusion in the gaseous phase under confinement.

B Diffusion in the gas phase

The coefficient of gas diffusion D_gas in random pore networks is generally represented as the diffusivity in a well-defined reference system (denoted in what follows by the subscript 0), divided by a tortuosity factor τ.^140,141 For the two extreme cases, namely dominating molecule-molecule collisions (bulk diffusion, subscript b) and dominating molecule-wall collisions (Knudsen diffusion, subscript K), one thus may noteAs a reference system for Knudsen diffusion one generally considers an infinitely long pore with a diameter equal to the mean intercept length (d) of the real pore space, yielding:¹⁴²

D_K = ⅓ūd. (4.16)

with ū = (8RT/πM)^1/2 as the mean thermal speed and R, M and T denoting, respectively, the gas constant, molar mass and absolute temperature. For gas diffusion one has:

D_b = ⅓ūλ. (4.17)

with λ as the molecular mean free path. In the general case, by considering the overall diffusion resistance as the sum of the diffusion resistance by molecule-molecule and molecule-wall interaction, one has¹⁴³The limiting cases of Knudsen and of bulk diffusion as described by eqn (4.16) and (4.17) easily result from this relation by considering the respective inequalities λ≫d and λ≪d. The mean free path in the gas phase is generally notably larger than the mesopore dimensions. Therefore, diffusion in mesopores is generally subjected to Knudsen diffusion. Obviously, partial pore filling decreases the effective pore diameter, resulting in the relation^122,124Gas diffusion in the space of macropores, such as the inter-crystalline space in beds of nanoporous particles, may occur to cover the whole range between the limiting cases of Knudsen diffusion and bulk diffusion. Thus, under conditions where the intraparticle diffusion becomes negligibly small for overall mass transfer, the effective-diffusivity conception (see next section) allows a straightforward measurement of gas phase diffusion by PFG NMR. With an appropriately chosen loading, PFG NMR self-diffusion measurement with sealed samples (isochoric conditions) allows, by a corresponding temperature variation, to cover the regimes of both dominating Knudsen diffusion (low temperatures) and bulk diffusion (high temperatures).^144–146

Most importantly, in this way for the first time direct experimental evidence was provided¹⁴¹ that, on following the conception provided by eqn (4.15) to (4.18), the limiting cases of bulk diffusion and Knudsen diffusion have to be described by notably different tortuosity factors. In particular, in the Knudsen regime molecular propagation is found to be impeded by tortuosity effects much more intensely than under the conditions of bulk diffusion. Following refs. 147–149, this finding may be referred to an enhanced anti-correlation of subsequent molecular displacements during Knudsen diffusion in random pore networks. This means that during Knudsen diffusion-in addition to the same tortuosity-related enhancement of the diffusion path lengths as during bulk diffusion-there is an enhanced probability of backward-directed molecular displacements after collision with the pore wall. The findings of the PFG NMR diffusion measurements were nicely reproduced in mesoscopic kinetic Monte Carlo simulations with beds of zeolite NaX.¹⁵⁰

Anti-correlation of molecular displacements after wall encounters is expected to intensify with increasing wall roughness. Both kinetic Monte Carlo simulations^151–156 and analytical considerations¹⁵¹ confirm this assumption. Moreover, the kinetic Monte Carlo simulations^154–156 revealed coinciding effects on the transport diffusivity and on the self-diffusivity, i.e. on the molecular propagation rates under both non-equilibrium and equilibrium conditions.^157,158 Since, quite generally, for non-interacting diffusing particles transport diffusion and self-diffusion have to coincide and since during Knudsen diffusion, by its very nature, any interaction of the diffusants is excluded, this is exactly the behaviour to be expected.

C The effective diffusivity

With the two mechanisms of molecular transport in mesopores discussed, the overall one may be defined by both of them weighted appropriately. Before we will correlate these weights with the details of the phase equilibrium, let us consider two limiting situations: (i) at relatively low vapour pressures, before the onset of capillary condensation and (ii) at higher pressures, when the capillary-condensed bridges (small domains of the liquid extended over the whole pore cross-section) are formed. In the former case (i), the molecular displacements in both of the phases, the liquid one adsorbed on the pore walls and in the gaseous one in the pore interior, may proceed in parallel to each other. Thus, the problem becomes similar to that of a parallel connection of the resistivities. Therefore, the solution for the effective diffusivity D_eff is given by ref. 159 and 160

D_eff = p_gasD_gas + p_liqD_liq, (4.20)

where D_gas and D_liq are the diffusivities in two phases and p_gas and p_liq are the respective weights (p_gas + p_liq = 1). Note that the first term (in particular at sufficiently high temperatures) may notably exceed the second one. It is due to this reason that the effective diffusivity of liquids adsorbed in beds of porous materials may be significantly larger than in the real liquids. As to our knowledge, such a situation has been discovered for the first time in ref. 161 and has been reproduced in numerous subsequent publications, e.g., ref. 71,144 and 162.

In contrast, in the latter situation (ii) the propagation through a medium composed of two regions may require alternation between two regions. In the limiting case this is similar to the problem of sequential connection of resistivities. A corresponding general solution for the effective diffusivity D_eff, is ref. 160

where L_gas and L_liq are the typical extensions of the regions occupied by the gas and liquid volumes, respectively, and ρ_gas and ρ_liq are the respective densities.

The phase state is reflected by the adsorption isotherm θ(z), where θ is the amount adsorbed (also referred to as the pore filling factor) at the relative pressure z = P/P_s (with P_s denoting the saturation pressure). One may start to relate p_gas and p_liq to the given phase state with the set of equations:

N_gas + N_liq = N, (4.22)

V_gas + V_liq = V, (4.23)

where N_gas and N_liq are the number of molecules in the gaseous and liquid phases and V_gas and V_liq are the volumes occupied by them. The latter may be expressed as V_i = mN_i/ρ_i, where ρ_i is the density in a phase i, and m is the molecular mass. With θ = V_liq/V and taking account of ρ_liq≫ρ_gas, it immediately follows that¹²⁴The density ρ_gas may be related to the external gas pressure via the ideal gas lawwhere M is the molar mass. Thus, the effective diffusivity appears to be correlated with the isotherm, similarly to eqn (4.13) obtained for the surface diffusion.

It is clear that eqn (4.20) well holds for mesoporous materials with relatively big pore sizes and in the regimes where there are mono- and/or multi-layered molecules on the pore walls. To demonstrate the behaviour of D_eff under these conditions, we have prepared mesoporous silicon samples with an average pore diameter of 10 nm. The recipe for their fabrication was similar to that given in section IVA, with the only differences in the doping level of the silicon wafer (resistivity of 10 mΩ cm) and the anodization current density (80 mA cm²). The acetone diffusivities measured in this material are shown in Fig. 5 as a function of the amount adsorbed θ.¹⁶³ To do this, first the diffusivities had been measured as a function of the gas pressure and, using the simultaneously measured adsorption isotherm (not shown), the diffusivities were re-plotted versus θ. It is important to note that the isotherm reveals the onset of capillary-condensation being located at z≈ 0.7, corresponding to θ≈ 0.6. Thus, up to θ≈ 0.6, the multi-layered molecules on the pore walls coexist with the gaseous phase in the pore interior (parallel configuration), so that the use of eqn (4.20) is justified.

Fig. 5 Effective diffusivities for acetone in mesoporous silicon with an average pore diameter of about 10 nm as a function of the amount adsorbed. A monolayer is found at θ≈ 0.2. The dotted and dashed lines show the contributions by diffusion in the gaseous phase and in the phase adsorbed on the pore walls calculated viaeqn (4.20) and (4.24), respectively, with the solid line as their sum.

One immediate consequence of eqn (4.24) is that p_gas passes a maximum upon variation of θ. This is due to two competing effects, namely the increase of the gas density (controlled by z) and the decrease of the relative fraction of molecules in the gaseous phase (given by (1 −θ)/θ) with increasing P. Indeed, at low pore loadings (θ≲ 0.2), because of the adsorbate-surface interaction, the external pressure P must be appreciably decreased in order to release molecules from the surface. Thus, the term z in eqn (4.24) determines the behaviour of p_gas at low loadings. With increasing θ, the situation is reversed, and the contribution of the gas phase to overall diffusion becomes dominating. Finally, however, with θ→ 1 and thus p_gas→ 0 (eqn (4.24)), the gas phase contribution vanishes. Hence, irrespective of the fact that, with θ→ 0, the term (1 −θ)/θ goes to infinity, ρ_gas faster approaches zero, so that p_gas becomes negligibly small.

Let us quantify these statements. As shown by the dotted line in Fig. 5, at intermediate loadings (θ from about 0.2 to 0.6) the magnitude of the term p_gasD_gas in eqn (4.20), where D_gas is given by eqn (4.19), is calculated to be comparable to the diffusivity at full saturation. With the adsorbed phase, the situation is less obvious, since we do not have an equivalent model for molecular diffusion in multi-layers. As a first approximation, however, we assume that the diffusivity does not appreciably change with the pore loading and that it is equal to the diffusivity at full pore loading. At sub-monolayer coverage the diffusivity behaves as shown in Fig. 3. Thus, the calculated overall dependence of the diffusivity in the adsorbed (liquid) phase on pore loading (p_gasD_K + (1 −p_gas)D_s) results as that shown by the solid line in Fig. 5. It appears that it satisfactorily describes the experimental finding, except for a small difference in the region of the multi-layer adsorption. This reflects an underestimation of the diffusivity within the layer. With only these data, however, we do not yet feel ourselves totally confident that the diffusivity of the multi-layered molecules exceeds that of the capillary-condensed molecules, though they would clearly support such a conclusion.

The situation changes upon forming regions filled by the capillary-condensed liquid since, thus, the gaseous phase domains become spatially isolated (combination of parallel and sequential connections). Under such conditions, according to eqn (4.21), the overall diffusivity is predominantly controlled by the phase with the slowest diffusion. A nice exemplification of such a behaviour is provided by the diffusion behaviour in MAST activated carbon sample (MAST Carbon Ltd., Guildford, U.K.).^164,165 Its structure is schematically shown in the inset of Fig. 6. The material used in this work was composed of 1 mm-big particles with a highly-networked internal pore structure. In addition to the micropores, there are incorporated isolated spherical holes with diameters of about 30 nm. The pore volumes of the micro- and mesopores are roughly identical.

Fig. 6 Effective diffusivity for cyclohexane in a MAST carbon sample as a function of the external gas density measured at 298 K. The dotted line shows the dependence expected by the use of eqn (4.20) with all known parameters. Use of the same parameters, but the sum of eqn (4.20) and (4.21) with the weights 0.4 and 0.6, instead of eqn (4.20) yields the solid line. The inset demonstrates a schematic structure of an individual MAST carbon bed.

Fig. 6 shows the diffusivity of cyclohexane in this sample as a function of the external gas pressure P. In the figure the latter is presented by the gas density ρ_gas calculated viaeqn (4.25). The diffusivity increases with increasing gas pressure up to about ρ_gas = 0.25 kg m⁻³. This region corresponds to the mesopores filled by the gaseous phase. Further increase of ρ_gas is accompanied by the onset of capillary condensation in the mesopores, which nicely correlates with the deviation of the measured diffusivities from the dependence as expected viaeqn (4.20) for dominating gas phase diffusion (dotted line in Fig. 6). Now, due to a specific arrangement of the volumes occupied by the two phases, the transport should be described by a combination of eqn (4.20) and (4.21),¹⁶⁰ namely by the appropriately weighted sum of these two equations rather than by eqn (4.20) alone. A correspondingly designed dependency is shown by the solid line and yields satisfactory agreement with the experimental data.

V. Macroscopic transport

Typically, the majority of porous materials are available as powders, i.e., they are composed of solid particles with an internal pore structure. Moreover, the pore structure itself may have different levels of organization, i.e., different types of porosities and length-scales. Thus, the overall molecular transport ultimately involves molecular exchange between regions with different mobilities. The state of the fluid and, hence, its transport properties in these various regions depend in a complex way on the external conditions. The objective of this section is to demonstrate how the NMR technique may be used to address such problems. Among different examples, a particularly interesting one will be to demonstrate how NMR allows a self-consistent comparative analysis of the data on molecular transport provided by macroscopic and microscopic measurements.

A Molecular exchange between porous monoliths and their surroundings

We start our consideration with the probably most simple example—a mesoporous monolith in contact with the surrounding gas phase under equilibrium conditions. Here, one may clearly distinguish between two molecular ensembles with different transport properties, viz. molecules within the mesopores and in the surrounding phase. Thus, as discussed in section III, by recording their diffusivities one is able to attribute the molecules to either of these ensembles, and, moreover, one may monitor the exchange process between these two ensembles. Exchange times become directly accessible if they are within the time-scale of PFG NMR (1 ÷ 10³ ms). For sufficiently long observation times, i.e., for molecular displacements notably exceeding the typical pore size as the general case with mesopores, diffusion within the monoliths is characterized by a single, effective diffusivity D_eff (one has to be, however, aware of situations where it may not hold¹⁰⁰). Under such circumstances, the most efficient method to quantify the exchange process is to follow the time-dependent fraction of molecules having the diffusivity D_eff.

We illustrate this approach by considering a model sample with pre-defined structural characteristics. The experimental results may thus be easily compared with the theoretical predictions. The sample consists of mesoporous silicon chips prepared by electro-chemical etching of silicon crystals.¹³³ The parameters of the etching procedure and the initial silicon properties were chosen to yield pores with a diameter of about 6 nm.¹⁶⁶ After formation of a thin porous layer, they were removed from the substrate. Thus, the thus formed pores have immediate contact to the surroundings at both ends of the layer, rendering essentially one-dimensional uptake and release processes. Five different films with different thicknesses L of 10, 20, 29, 41 and 52 μm were prepared and mixed with the respective masses m_i = 29, 45, 50, 40 and 15 mg. The thus composed sample, consisting of porous films of different thicknesses, was, thereafter, outgassed in an NMR glass tube and brought into contact with a vapour of cyclohexane at 60 mbar. After equilibration the tube was sealed.

Fig. 7 demonstrates the normalized spin-echo diffusion attenuations for this model sample obtained using the 13-interval pulse sequence¹⁶⁷ at different observation times t_d from 5 to 300 ms. First of all, we note a multi-exponential decay and a significant influence of the diffusion time. With increasing t_d, there is a tendency to converge to a single-exponential shape shown in the figure by the dotted line. This is a clear indication of the exchange process between regions with different molecular diffusivities in the system under study, namely within the pores and through the surrounding free space, respectively.

Fig. 7 Normalized spin-echo diffusion attenuations for cyclohexane in porous silicon powder composed of small chips with different thicknesses. Different symbols refer to different diffusion times t_d from 5–300 ms (from top to bottom). The solid lines are shown as examples of the fit of the function p_sl exp{−q²D_slt_d} to the data in the high-q region. The dotted line shows the exponential function to which the data converge upon increasing t_d.

Let us now analyze the behaviour of the molecular ensemble characterized by the slowest diffusivity D_sl following the recipe described in section III. For the attenuation obtained with the shortest t_d = 5 ms, for which the effect of the exchange is minimal, D_sl and its respective fraction p_sl(t_d) in the spectrum of the diffusivities is easily found by fitting eqn (3.4) to the data in the high-q region. This is shown in the figure by the solid line. Thereafter, by fixing D_sl, the same procedure may be applied to the attenuation curves with larger diffusion times. The thus obtained dependence p_sl(t_d) is shown in Fig. 8.

Fig. 8 p _sl obtained from the data in shown Fig. 7. The solid line shows eqn (5.26) with all relevant parameters, which are known independently. The inset demonstrates the same data, but plotted versus t^1/2_d. Here, the solid line shows the fit of eqn (5.28) to the data at short times.

Obviously, the diffusivity D_sl, leading to the decrease of the number of molecules which have not left the pore segments during t_d, has to be associated with the diffusivity of cyclohexane in the porous silicon. Because all geometrical characteristics of the porous silicon in the sample under study are known, we may try to model the time dependence of p_sl(t_d) using eqn (3.8). Because the lengths L_i are different for the channels of different porous silicon plates, one has to sum over all i, yielding:

where for the one-dimensional case (diffusion out of a region bounded by two plates) p_sl(L) is given by:Eqn (5.26), calculated with all relevant parameters known, is shown in Fig. 8. It is found to nicely reproduce the experimental data. In addition, the perfect agreement suggests that the molecules may leave the initial particles without any imposed restrictions, in contrast to situations often encountered with microporous adsorbents.

In reality, the distribution of particle sizes is often not know. In this respect, it is instructive to analyze the short-time behavior p_sl(t_d). Applying Poisson’s summation formula to eqn (5.27) one may show that in this limit p_sl(t_d) is given by:

The inset in Fig. 8 shows the fit of eqn (5.28) to the data at short times, yielding an effective average particle size of about 23 μm. This is again in excellent agreement with the expected value of 22 μm. The result of this subsection demonstrates that the described method may be satisfactorily applied to study the molecular exchange between macroparticles and the surrounding phase in the case of smooth particle exchange at the interface. Moreover, if the exchange is notably influenced by additional transport resistances at the interface between the crystal and the surroundings, significant deviations from the dynamics predicted by eqn (5.28) occur. These deviations may, in turn, be used for a quantification of surface barriers, as to be shown in the next section.

B Surface barriers

Most techniques for studying mass transfer in porous media are based on following the rate of molecular uptake or release by the individual particles. They are referred to as macroscopic or mesoscopic techniques of observation,^168–170 depending on whether the whole bed of nanoporous particles or an individual particle is in the focus of observation. A model-free discrimination between internal and external transport resistances, i.e., an unambiguous separation between the influences of intraparticle diffusion and of surface barriers, however, necessitates the measurement over path lengths notably below the particle sizes which is provided by the microscopic techniques only.

As explained in the preceding section, PFG NMR is able to determine the relative amount of molecules (p_sl(t_d)/p_sl(0) in eqn (3.8) and (5.26)), which have not yet left the particles which they accommodated at time t_d = 0. Under the conditions of gas phase adsorption, the signal of the gas phase is negligibly small so that in this case p_sl(0) = 1 and there is no need for normalization by dividing p_sl(t_d) by p_sl(0). For variable t_d, this information is exactly that of a tracer exchange curve γ(t_d), with γ(t_d)(≡ 1 −p_sl(t_d)/p_sl(0)) denoting the relative amount of (labelled) molecules which, after time t_d, have left a particular porous particle, being replaced by other (unlabelled) ones from the surroundings and/or other particles. This special ability of PFG NMR is exploited by the so-called NMR tracer desorption technique, to which we have referred to already in section III.

The experimentally accessible tracer exchange curve γ(t_d) may most conveniently be correlated with the governing transport mechanisms via the method of moments.^96,171,172 Thus, e.g., the “first” moment

∫^∞₀(1 −γ(t))dt≡∫^∞₀p_sl(t)/p_sl(0)dt = ∫^∞₀tψ(t)dt = t_av (5.29)

is easily identified as the molecular mean life time within a particular porous particle (which, in eqn (3.10), has been referred to as t_av and where, in addition, we have made use of eqn (3.8)). Most importantly, different transport resistances simply appear in the first moment as a sum of the terms corresponding to each individual resistance on its own.^96,171,172 Thus, assuming spherical particles, with eqn (3.10) for diffusion-limited exchange and with the corresponding expression for exchange processes governed by the surface permeability (t_av,barr = R/3α), the general case may be noted aswhere the surface permeability α^96,173 is defined as the ratio between the flux through the particle surface and the difference between the actual boundary concentration of the diffusing species under study and the concentration, which eventually, would be established in equilibrium with the surrounding atmosphere.

Thus, with the value of D directly determined by PFG NMR with sufficiently short observation times (diffusion path length during observation time much shorter than the particle size), the value of t_av following viaeqn (5.29) from the “NMR tracer desorption curve” and the particle radius R accessible by microscopic inspection, the surface permeability follows directly from eqn (5.30). This option has been widely exploited for the exploration of the conditions under which either the production of zeolitic host systems^174,175 or their technological use for mass separation and chemical conversion^175–177 give rise to the formation of surface barriers.

The accuracy of this way of analysis is limited by the fact that the uncertainty of t_av as resulting viaeqn (5.29) is typically of the order of a factor of 2. Hence, reliable data for the surface permeability are accessible only, if the second term in the right-hand side of eqn (5.30) notably exceeds the first one. The inclusion of the information provided by an analysis of the time dependence of the PFG NMR signal attenuation for large gradient intensities (see Fig. 7) has recently been shown to notably enhance this accuracy.^178,179 It should be mentioned that also the thus attainable accuracies are still inferior to those reachable by the application of interference microscopy.^180,181 Being able to directly monitor transient concentration profiles, this technique is essentially totally unaffected by the rate of intra-crystalline diffusion and allows the determination of surface permeabilities with unprecedented accuracy. As a big disadvantage, however, the measurements can only be performed with single crystallites with dimensions (favourably some tens of micrometers) which must notably exceed the minimum sizes still accessible by PFG NMR studies.

C Overall transport in powders and agglomerates

Many porous materials are available in the form of macroscopically big particles with a complex internal porous structure. Such structures may result from, for example, different levels of agglomeration leading to very different types of porosities in the sample. Depending on the external parameters, such as pressure or temperature, the local phase state in different regions may be quite different. In this section we are going to address how transport properties are correlated with the external conditions in materials with a hierarchy of typical length scales.

As a representative example, we consider a commercially available MCM-41 material.^1,2Fig. 9 shows the internal texture of an individual particle, where one may identify typical building blocks of sufficiently different dimensions. According to the literature data,^182,183 the basic building units are individual, mesoporous quasi-crystallites with MCM-41 structure, i.e., traversed by cylindrical channels with a diameter of a few nanometres. These channels are hexagonally arranged resulting in an internal porosity of about 40%. As a next level of organization, the crystallites form clusters or grains with a dimension of a few hundred nanometres. Finally, the grains agglomerate to particles with a broad distribution of sizes from tens to hundreds of micrometers.

Fig. 9 Scanning electron micrograph of the internal texture of an MCM-41 particle comprising the individual MCM-41 crystallites and the secondary macroporous space.

The diffusion attenuation functions for cyclohexane measured in the sample under study have been found to be slightly non-exponential and to depend on the observation time t_d which was varied from 3 to 300 ms. Such a behaviour is an indication of a macroscopic heterogeneity with the time-dependence being the consequence of the molecular exchange between different regions in the sample. In what follows, we are going to analyze the effective diffusivities D_eff deduced from the attenuations in the low-q limit. The thus obtained values of D_eff represent the ensemble-average over the sample and are shown in Fig. 10 as a function of the external gas pressure. With a value of 10⁻⁷ m²/s using the Einstein relation 〈r²〉 = 6D_efft_d, the extension of the regions with a uniform diffusivity may be estimated to be . The latter appears to be of the order of the particle sizes. Therefore, the heterogeneity may be associated with the textural differences in the different particles.

Fig. 10 The effective diffusivity D_eff for cyclohexane in MCM-41 agglomerates as a function of the relative vapor pressure z measured on the desorption branch. The solid line shows the best fit of eqn (5.31) to the experimental data. The inset shows the desorption isotherm normalized to the volume of the mesopores.

To understand the pressure dependence of the diffusivity one may perform an analysis similar to that given in section IVC. In this case, also equilibrium between the mesopores in crystallites and the macropores formed between the grains and the particles has to be taken into account.¹⁸⁴ In this way, one finally gets

where λ and v̄ denote the mean free path and mean molecular velocity in the gas phase, η is the ratio of the meso- and macropore volumes, d_g is the effective inter-grain distance, and α is the ratio of the intra-particle macropore volume to the total macropore volume, respectively. The best fit of eqn (5.31) to the experimental data is attained with α = 0.16 and d_g = 0.61 μm (all other parameters are known) and is shown in Fig. 10.

A more detailed analysis of eqn (5.31) suggests that the obtained dependency of D_eff on the gas pressure is predominantly controlled by the equilibrium conditions between the gaseous and adsorbed phases in the mesopores. Thus, the slight decrease of D_eff observed for decreasing pressures from z = 1 till the onset of the evaporation transition in mesopores may be referred to the decrease in the gas density ρ_gas in the macropores. At z≈ 0.3, the evaporation of the liquid out of the mesopores leads to a sudden increase of the relative fraction of the molecules in the macropores. In the low-pressure region the equilibrium is determined by the interaction of the liquid with the walls of the mesopore. Thus, the equilibrium conditions reasonably well explain the obtained results. Quantitative agreement between theory and experiment is clearly obtained only, if the characteristic geometrical features of macroporous space are taken into account. These are predominantly reflected by the diffusivity of the gas phase in the macropores. In the present case, it has been simply accounted for by summing up the transport resistivities due to intermolecular collisions and collisions with the grain walls.

Being able to record the rate of molecular displacements over different length scales, PFG NMR is able to assess the limiting processes of mass transfer in agglomerated catalyst particles formed by nanoporous crystals dispersed in a (macroporous) binder. As a prominent example, in Fluid Catalytic Cracking (FCC)^21,185 catalyst particles with radii R_p of typically about 30 μm are applied which contain small zeolite crystals of type NaY with radii R_c of about 0.5 μm. In such materials it is not obvious whether overall exchange with the surrounding atmosphere is controlled by mass transfer through the binder or by mass exchange between each individual zeolite crystal and the surrounding binder. In fact, following an old tradition in overestimating the transport resistance of the genuine zeolite pore network,¹⁶⁹ reactant and product transport within the zeolites is generally considered to be the rate-limiting process of overall mass transfer.¹⁸⁶ However, recent detailed PFG NMR^22,23,187 studies revealed exactly the opposite. In fact, by choosing the observation time t_d of the PFG NMR experiment in such a way that the experimentally covered molecular displacements are either much larger or much smaller than the zeolite crystallites, one is able to directly measure the diffusivities for molecular transport both within the individual zeolite crystals (D_c) and through the whole catalyst particle (D_p). With these diffusivities and the corresponding radii, eqn (3.10) allows a straightforward estimate of the respective time constants of molecular exchange. As a result, molecular mean life times within the individual zeolite crystals turned out to be much smaller than the mean life times within the catalyst particles, indicating that molecular transport through the particle rather than through the zeolitic component is the rate-controlling process in the overall mass transfer.^22,23,187 During technical application, inter-crystalline exchange may clearly become impeded by the formation of surface resistance. In this case, replacing eqn (3.10) by eqn (5.30), one also has to take account of surface resistance (i.e., a finite value of the surface permeability γ).¹⁸⁸

VI. Phase state and transport

A Dynamics in disordered pores under sub-critical conditions

So far, we have clearly demonstrated that the diffusion behaviour of guest molecules in mesoporous solids is strongly coupled to their phase state in the pores. The latter can be associated with a certain point on the sorption isotherm. As a very essential property of these materials, however, one has to be aware of the development of hysteresis. Thus, under identical external conditions, the phase state becomes dependent on the way how these conditions have been attained, i.e., on the “history”. It, thus, becomes quite interesting to compare the intrinsic diffusivities in the hysteresis region.

This type of experiments can be easily carried out by performing PFG NMR diffusion measurements using NMR samples with a porous material connected to a reservoir with the vapour of a liquid under study. Thus, either by decreasing or increasing the vapour pressure in the reservoir, the effective diffusivities can be measured along with the sorption isotherm.¹²⁴Fig. 11 shows one of the typical examples, demonstrating a well-pronounced hysteresis loop when the diffusivity is plotted versus the external gas pressure.¹³² It should be noted that such an approach has earlier been used to demonstrate that nuclear magnetic relaxation properties of adsorbed fluids may also differ for adsorption and desorption.¹⁸⁹ We could show that this type of diffusion hysteresis is not limited to systems exhibiting adsorption hysteresis,¹⁹⁰ but also holds for other types of hysteretic behaviour, for example freezing/melting hysteresis.¹⁹¹

Fig. 11 The effective diffusivity D_eff (circles) for cyclohexane and its relative amount adsorbed (triangles) in Vycor porous glass (d_p = 6 nm) measured on the adsorption (open symbols) and desorption (filled symbols) branches as a function of the relative gas pressure.

Recalling the discussion of section IVC, the development of hysteresis in diffusivity-pressure coordinates might appear to be natural. Indeed, the external gas pressure defines the amount adsorbed θ, which is, in turn, different on adsorption and desorption as given by the sorption isotherms. Thus, one may simply attribute the difference in the diffusivities D_eff to the difference in θ. D_eff may clearly also be plotted directly as a function of the pore loading θ. Remarkably, upon such a transformation, hysteresis is still preserved as shown in Fig. 12.⁷⁴ Moreover, for one and the same loading, the diffusivities are also found to differ for the various “scanning isotherms”,³⁹i.e., for isotherm branches which result from incomplete cycles of adsorption and desorption. The thus obtained diffusivities lie inbetween the two main loops and it becomes obvious that, by a corresponding “tuning” of the history of pressure variation, in this way one may obtain a whole “spectrum” of arbitrary diffusivities.

Fig. 12 The effective diffusivity D_eff for cyclohexane in Vycor porous glass (d_p = 6 nm) of Fig. 11 as a function of the amount adsorbed θ. The filled rectangles show D_eff measured during various “scanning isotherms”.

As to the most important feature of the results in Fig. 12 we refer to the existence of several different diffusivities at one and the same pore loading θ. In ref. 74, two mechanisms which may contribute to the observed behaviour, have been discussed. The first one refers to the fact that, on desorption, the liquid in the regions with the capillary-condensed phase occurs in a somewhat stretched, lower-density state. This stretching effect means that, for one and the same θ, the liquid phase occupies a notably larger part of the pore space during desorption than during adsorption. The second mechanism is related to the differences in the microscopic distribution of the liquid phase within the porous solid.^192–194 In our case, the PFG NMR data indicate that, on the micrometer length scale, such a distribution is homogeneous. The origin of different distributions on adsorption and desorption is, thus, attributed to an interplay between cavitation and pore-blocking effects, leading to the formation of more extended mesoscale regions of the capillary-condensed and gaseous phase during adsorption.⁷⁴

The multiplicity of the diffusivities in Fig. 12 may thus be related to differences in the (history-dependent) density distribution over the pore space. This conception is in good agreement with the idea of a dynamically prohibited equilibration in random systems.^82,83 Notably, being separated by large barriers in the system’s free energy, these out-of-equilibrium states, associated with different fluid distributions, are found to remain stable over very long intervals of time. Quite importantly, the non-invasive and non-perturbative access to the intrinsic molecular diffusivities in the pores using PFG NMR technique allows to provide further experimental evidence of such a scenario.

In this respect, it is interesting to note that while there is a great deal of experimental data for adsorption isotherms including hysteresis, much less attention has been given to the dynamics of systems exhibiting sorption hysteresis. Notably, there were a few experimental reports indicating that equilibration kinetics may slow down in the hysteresis region.^195–198 An analytical description of the observed behaviour was undertaken by involving a priori assumptions on diffusion-controlled sorption. Thus, all features of the experiments, in particular a slowing down of uptake in the hysteresis region, have been attributed to a corresponding change of the diffusivity.

To address this issue in more detail, a comparative study of sorption kinetics and genuine diffusivities in one and the same material under almost the same external conditions have recently been performed using NMR.^115,116 The experimental procedure included a small stepwise change of the external gas pressure. Thereafter, the resulting density relaxation, i.e., sorption kinetics, has been followed. After giving a sufficiently long time for the equilibration, the effective diffusivity D_eff has been measured using PFG NMR. With the thus determined value of D_eff and the geometry of the used mesoporous material (rod-shaped Vycor porous glass), the sorption kinetics under the condition of diffusion control can be easily predicted. Thus, with θ₀ and θ_eq as the amounts adsorbed before the pressure steps and after equilibration, the uptake by a cylinder of radius a is given by ref. 110

where α_n are the positive roots of the equation J₀(αa) = 0, and J₀ is the Bessel function of the first kind.

Fig. 13 shows the experimental data for the adsorption kinetics of cyclohexane into Vycor porous glass with a pore diameter of about 6 nm in two regions of the adsorption isotherm, namely out of and in the hysteresis region. In addition, it shows the predictions of eqn (6.32). Note that (i) D_eff has been measured independently (see Fig. 11) and (ii) eqn (6.32) does not have any fitting parameters. Beyond the hysteresis region, the experimental data are found to be nicely reproduced. This confirms that under this condition mass transfer is in fact limited by diffusion. Remarkably, however, in the hysteresis region the model predicts much faster equilibration. This discrepancy between the observed time dependence of uptake and the prediction on the basis of the measured diffusivities unequivocally points out that only the early-stage uptake is controlled by diffusion.¹¹⁵ At later stages, however, the system equilibration is governed by extremely slow relaxation due to the thermally activated process of the liquid redistribution within the porous material. This process is reminiscent of a very slow process of phase separation of critical binary mixtures in porous media and successfully treated in the frame of random-field Ising systems.^199,200

Fig. 13 The adsorption kinetics for cyclohexane in Vycor porous glass (d_p = 6 nm) out of (triangles) and within (circles) the adsorption hysteresis region following a small (5 mbar) pressure step. The solid lines show theoretical predictions using eqn (6.32) with all known parameters.

B Cavitation-governed diffusivity enhancement

The experiments described in the preceding section have been performed under isothermal conditions. However, one may also affect and change the phase state of a liquid in pores using further experimental schemes, e.g. by varying the temperature and keeping the system volume constant. This experimental procedure is easily achieved by using sealed NMR samples containing a porous material and some volume available for the vapour above it. The impact of the external parameters on the molecular diffusivities under these conditions may most illustratively be demonstrated by a comparative study of two different samples. In one of them, in the whole temperature range probed in the experiments, the liquid under study should always completely cover the porous particles. That means that the mesopores will always be filled by the capillary-condensed phase. This sample will provide a reference diffusivity of the liquid in pores as a function of temperature. The second sample has to be prepared in such a way that only at the lowest temperature the mesopores are completely filled by the liquid. Thus, by changing the temperature, one may affect the equilibrium between the vapor phase above the porous material and the adsorbed phase.

The results of such comparative experiments are shown in Fig. 14. The figure displays the diffusivities of n-pentane within the pore system of Vycor porous glass as a function of inverse temperature as obtained by PFG NMR method.²⁰¹ In the over saturated sample, molecular diffusion is found to follow an Arrhenius dependence with an activation energy for diffusion typical of the bulk liquid.

Fig. 14 The effective intra-pore diffusivity for n-pentane in Vycor porous glass (d_p = 6 nm) for samples with (filled circles) and without (filled triangles) excess liquid phase as a function of inverse temperature.

In the other sample, however, starting from about 300 K, the diffusivities deviate from those in the reference sample. Let us consider what could be the impact of the temperature on the phase state within the mesopores. Recall, first, that this sample contains an amount of guest molecules ensuring complete filling of the mesopores only at the lowest temperature. Therefore, with increasing temperature, there is a competing process between either maintaining the vapour pressure in the gas phase surrounding the porous particles saturated or maintaining the liquid phase in the mesopores intact. At lower temperatures, up to about 300 K, under saturation (P/P_s) is found to be relatively small. This is provided by the evaporation of the liquid from the pores. At temperatures above 300 K, however, the latter process cannot compensate any more the strong decrease of the relative pressure P/P_s due to the dramatically increasing value of P_s. Thus, P/P_s becomes sufficient to trigger cavitation in the sufficiently large pores of the Vycor glass.

The enhancement of molecular propagation in the sample containing no excess liquid, thus, originates from the contribution of fast molecular diffusion in such cavitated regions of the porous structure. Molecular transport therein is considered to follow Knudsen diffusion, the simplest model one would imply under the given conditions. Exactly such a change of the phase state of the liquid in the pores, resulting in the formation of internal gas-filled regions with the establishment of extremely fast diffusion paths therein, leads to the increase in the slope of the Arrhenius plot of the diffusivity seen in Fig. 14.

C Dynamics in pores near the critical point

1 Transition to the supercritical state. At elevated pressures a transition to the supercritical state can occur. Due to their tunable properties between those of typical liquids and gases, supercritical fluids (SCFs) have widely been explored, e.g., as innovative media for chemical conversions.²⁰² In particular, the gas-like diffusivities of SCFs represent an attractive advantage for heterogeneously catalyzed reactions. In fact, an accelerated mass transfer from reactants to or of products away from the catalyst surface have been invoked in numerous cases to rationalize the rate increase upon a transition from the (bulk) liquid (or biphasic) to a single supercritical reaction phase.²⁰³ Unfortunately, however, a clear correlation of mass transfer properties to phase behaviour, especially within the pores of nanoporous catalysts, has not been attempted so far (see also the examples from heterogeneous catalysis given in section VIC2). So far, only indirect experimental evidence on transport properties within mesoporous solids at higher pressures is provided as obtained, for instance, from the fluid densities within the pores measured by in situ FT-IR spectroscopy.²⁰⁴ In the following, the first results of a direct measurement of diffusion properties during a phase transition from the liquid into the supercritical state in mesoporous materials are summarized. These results provide a quantitative basis for the rational design of processes utilizing SCFs for extraction, sorption or heterogeneous catalysis with mesoporous solids.

For two controlled pore glasses (CPGs) with different mean pore diameters in the mesoporous range (d = 6 or 15 nm), the diffusivities in the bulk and within mesopore space were measure by PFG NMR. Starting from a liquid phase around and inside the porous materials, the diffusivities for both pore glasses follow the Arrhenius dependence on increasing temperature. At a distinct temperature below the bulk critical temperature, however, upon a small temperature increase the diffusivities in both materials increase significantly. As an example, Fig. 15b shows such a diffusivity jump for a Vycor porous glass with d_p = 6 nm. This jump in diffusivity is an indication of a phase transition from the liquid to a supercritical fluid within the pores of the material. The occurrence of the liquid/supercritical-phase transition is corroborated by the fact that the meniscus of the liquid phase above the pore glasses in the NMR tube rises strongly when reaching this temperature, as one may see in Fig. 15a. Upon reaching supercritical conditions, the density of the pore-confined phase decreases rapidly. A corresponding part of the liquid is, thus, expelled from the pores, eventually leading to the rising liquid meniscus.

Fig. 15 Temperature dependence of (a) the meniscus level in an NMR glass tube containing Vycor porous glass (d_p = 6 nm) and cyclohexane in excess and of (b) the cyclohexane diffusivity in the bulk liquid phase (triangles) and in the Vycor porous glass particles. The solid lines show the model-based calculations.²⁰⁵ The vertical dashed and dotted lines indicate the positions of T_c and T_cp, respectively.

Expectedly, the increase of the diffusivities in the CPGs is related to their pore diameter: the jump is more pronounced and its temperature, i.e., T_cp, is closer to that of the bulk critical temperature T_c for the material with the larger pores. The diffusivity change can, thus, be directly assigned to the shift of the phase transition caused by confinement of the phases in the mesopores. Notably, the temperature shift ΔT = T_c−T_cp is linearly correlated to the reciprocal pore diameter as earlier mentioned in other studies.^30,85 Above the pore critical point, the diffusivity on the mesopores of the CPGs remains essentially constant at the value given by the mean free path in the mesopore according to the Knudsen diffusion limit.¹⁴³ Evidently, we find this value to be higher for the material with the larger mean mesopore diameter.²⁰⁵

It is worth mentioning that the experimental diffusivity data in Fig. 15 can be quantitatively described by a simple analytical model based on gas-kinetic arguments.^205,206 Here, the effective diffusivities in the pores are calculated by a weighted superposition of the diffusivities in the adsorbed phase (assumed to have a liquid density) and in the fluid phase in the interior of the pore space. The full lines in Fig. 15 denote the results of these calculations. Obviously, the experimentally observed data are well described by this simple model.

2 Selected application cases.
Materials preparation. While a supercritical phase itself is an attractive medium to produce nanomaterials,²⁰⁷ in the mesopores of a solid host it may be used to introduce guests into the pores or to chemically modify the walls within the pores of a mesoporous material. Due to the high diffusivity in the supercritical state, these processes can be achieved in significantly shorter times and with a better controllable loading than with conventional liquids. If the bulk phase surrounding the solid is also in the supercritical state, the virtually negligible surface tension of supercritical fluids may lead to a further acceleration. Two examples will illustrate these advantages.

Holmes et al.²⁰⁸ have demonstrated that nanostructured metals and semi-conductors within ordered mesoporous materials may be prepared with the aid of supercritical fluids. For instance, nanowires of Silicon or Germanium can be generated within MCM-41-type molecular sieves by a controlled decomposition of metal organic precursors in supercritical n-hexane. As opposed to conventional gas-phase procedures such as chemical vapour deposition (CVD), a complete filling of the internal pore volume can be achieved. Moreover, the metal deposition is completed within 15–30 min, whereas durations up to days are needed for the same deposition efficiency in liquid solvents. The resulting silicon-containing composite materials show a UV-photoluminescence with an emission maximum depending on the pore diameter of the mesoporous host. They are, thus, attractive as sensors or for other optical applications.

The reactive deposition of metals from supercritical fluid solutions (Supercritical Fluid Deposition, SFD) is particularly attractive for the preparation of solid catalysts.²⁰⁹ For this procedure, a metal organic complex is dissolved, for instance, in supercritical carbon dioxide in the presence of a solid support material. Through addition of hydrogen to the supercritical complex solution, the metal is directly deposited onto the support. Thus, both the solubilizing power of supercritical fluids for high-boiling compounds and the miscibility with gases like hydrogen are utilized in this application. First results for the deposition of platinum by reduction of the precursor complex cyclooctadienyldimethylplatinum(II) in supercritical carbon dioxide indicate that the SFD can be achieved in mesoporous materials such as MCM-41 (d = 2.3 nm), but not in microporous zeolites like silicalite-1 or beta.²¹⁰

Sorption. Similar to the application for materials preparation described above, the sorption of functional compounds into mesoporous materials may benefit from the presence of a supercritical pore phase. Only very few investigations on the adsorption of substances from supercritical solutions onto mesoporous molecular sieves are reported in the literature so far. However, an application of sorption on a large-pore zeolite-type molecular sieve VPI-5 from supercritical solution may serve as an example. This molecular sieve was loaded within a short time with benzoic acid, salicylic and acetyl salicylic acid.²¹¹ The resulting composite materials are of potential interest for the controlled release of pharmaceutically active compounds.
Heterogeneous catalysis. The ample opportunities arising from the application of SCFs in heterogeneous catalysis were already referred to above.²⁰³ It should be emphasized here again that several studies in the literature invoke the improved transfer from a bulk supercritical phase to the catalysts surface or within the pores of a catalyst for the rationalization of high reaction rates under SCF conditions. So far, however, higher diffusivities in mesoporous materials achieved by “pressure-tuning” of the bulk phase have been measured indirectly only, e.g., by inference from density measurements via vibrational spectroscopy.²⁰⁴ Another, most interesting way to experimentally determine the effective diffusivity is from catalytic conversions under conditions for which the diffusion inside the mesopores of a catalyst is the rate-limiting step. Such a study was systematically conducted by Arunajatesan et al.²¹² for the double-bond isomerization of 1-hexene over a commercially available catalyst consisting of 0.6 wt% Pt on mesoporous γ-Al₂O₃ (average pore diameter d = 5 nm) close to critical point of the reactant (T_c = 231 °C, P_c = 31.7 bar). Varying the reaction temperature and pressure over a wide range of conditions, the effective diffusivities were calculated as a function of reduced density applying the Thiele concept to the mesopore diffusion-limited conversion rates at steady state.

Remarkably, the effective diffusivity in the mesopores of the 0.6 Pt/γ-Al₂O₃-catalyst can be tuned close to the critical point by two orders of magnitude. This range of diffusivities is clearly broader than that observed in by PFG NMR over the CPG with a similar pore diameter (vide supra). Note, however, that the corresponding experiments were not performed for systematically varied density, but just by heating a pure liquid hydrocarbon in the presence of the mesoporous solid enclosed in an NMR tube. Most interestingly and in perfect agreement with the findings from PFG NMR measurements on the CPGs, the maximum diffusivity in the mesoporous catalyst is observed for a reduced density slightly below that of the bulk phase. This, again, indicates that particularly high diffusivities can be achieved when the pore-confined phases reach the supercritical state. It is quite challenging to see whether this transport optimization via control of the phases inside the pore space can also be fruitfully exploited for other heterogeneously catalyzed conversions over (especially ordered) mesoporous materials.

VII. Conclusions and perspectives

Pulsed field gradient NMR has been established as a powerful tool to explore mass transfer in porous media. It allows the direct monitoring of the various constituents of overall mass transfer and the quantification of the relevant parameters, including the diffusivities of the guest molecules in different sample regions as well as their permeability through internal transport barriers and at the interface between the pore space and the surrounding atmosphere. Most importantly, this information is provided non-invasively, without any interference with the intrinsic processes of the sample under study. NMR spectroscopy has thus proven to be particularly qualified to explore the influences of the different phase states of the guest molecules on overall transport, opening novel routes for the transport-optimized design and performance of porous materials.

Particular concern deserve those cases where, initiated by very small changes in temperature or pressure, phase transitions may lead to dramatic changes in the transport properties. This is especially evident for hierarchically structured pore systems. In this way, by transferring the fluid in the mesopores from sub- to supercritical state, the accessibility of the micropores in hierarchically organized porous materials, and hence, e.g., the effectiveness factor of heterogeneously catalyzed chemical reactions, may be dramatically enhanced. A specially interesting route for process design is offered by the experimentally demonstrated fact that a fluid in pores may already transit to the supercritical phase, while the surrounding, excess fluid is still in the liquid state. Thus, properly designed mesoporous materials can be used to spatially localize regions (possibly hierarchically organized), where by only a tiny change of the external parameters the fluid can be turned into a supercritical one, with all unique properties inherent to it, and vice versa. The design of processes based on such phenomena is notably facilitated by the fact that, irrespective of huge changes in the guest propagation rates within the mesopores, diffusion in the microporous space, i.e. in the region of relevance for the elementary processes of molecular separation and/or conversion, remains essentially unaffected.

Challenges of future research, which most notably will benefit from the potentials of PFG NMR, include (i) the establishment of a comprehensive view on diffusion in mesopores over the whole range of loadings and phases, including the supercritical state, (ii) experiments where transitions into the supercritical state are initiated by changing either temperature or pressure (rather than by coupled changes of both of them as so far generally considered) and (iii) the consideration of all these phenomena for multicomponent guest phases. A close correlation with the ample potentials of research in this field is a prime prerequisite of the success of these efforts. In fact, also in this respect, namely in its ability to be performed under the conditions of conventional sorption experiments, PFG NMR has proven to offer ideal conditions. Consequently, a deeper understanding of the correlation between phase behaviour and transport properties may lead to promising options in the future use of mesoporous materials in a very diverse range of applications including sorption, catalysis, sensors, life sciences, biochemistry and bio-systems.

Acknowledgements

The work has been done in the frame of the DFG Research Unit 877 and the International Research Training Group "Diffusion in porous materials", jointly supported by the Science Foundations of Germany (DFG, GRK 1056) and the Netherlands (NWO).

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