www.rsc.org/pccp Volume 11 | Number 16 | 28 April 2009 | Pages 2781–2992

ISSN 1463-9076

Physical Chemistry Chemical Physics

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Pages 2781–2992

PERSPECTIVEMuylaert and van der VoortSupported vanadium oxide in heterogeneous catalysis

COVER ARTICLEValiullin et al.Correlating phase behaviour and diffusion in mesopores 1463-9076(2009)11:16;1-Y

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Physical Chemistry Chemical Physics

This paper is published as part of a PCCP Themed Issue on:

"Molecules in Confined Spaces: The Interplay between Spectroscopy and Theory to develop

Structure-Activity Relationships in the fields of Heterogeneous Catalysis, Sorption, Sensing and Separation Technology"

Guest Editor: Bert Weckhuysen

Editorial Highlight

Editorial Highlight: Molecules in confined spaces Robert A. Schoonheydt and Bert M. Weckhuysen, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b905015a

Perspectives

Nanoporous oxidic solids: the confluence of heterogeneous and homogeneous catalysis John Meurig Thomas, Juan Carlos Hernandez-Garrido, Robert Raja and Robert G. Bell, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819249a

Supported vanadium oxide in heterogeneous catalysis: elucidating the structure–activity relationship with spectroscopy Ilke Muylaert and Pascal Van Der Voort, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819808j

Correlating phase behaviour and diffusion in mesopores: perspectives revealed by pulsed field gradient NMR Rustem Valiullin, Jörg Kärger and Roger Gläser, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b822939b

Communications

Viscosity sensing in heated alkaline zeolite synthesis media Lana R. A. Follens, Erwin K. Reichel, Christian Riesch, Jan Vermant, Johan A. Martens, Christine E. A. Kirschhock and Bernhard Jakoby, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b816040f

A small molecule in metal cluster cages: H2@Mgn (n = 8 to 10) Phillip McNelles and Fedor Y. Naumkin, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819479c

Papers

Confinement effects on excitation energies and regioselectivity as probed by the Fukui function and the molecular electrostatic potential Alex Borgoo, David J. Tozer, Paul Geerlings and Frank De Proft, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b820114e

Rotation dynamics of 2-methyl butane and n-pentane in MCM-22 zeolite: a molecular dynamics simulation study Shiping Huang, Vincent Finsy, Jeroen Persoons, Mark T.F. Telling, Gino V. Baron and Joeri F.M. Denayer, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819334g

Reactivity in the confined spaces of zeolites: the interplay between spectroscopy and theory to develop structure–activity relationships for catalysis Mercedes Boronat, Patricia Concepción, Avelino Corma, María Teresa Navarro, Michael Renz and Susana Valencia, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b821297j

The influence of the chemical compression on the electric properties of molecular systems within the supermolecular approximation: the LiH molecule as a case study Anna Kaczmarek and Wojciech Bartkowiak, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819346k

Multinuclear gallium-oxide cations in high-silica zeolites Evgeny A. Pidko, Rutger A. van Santen and Emiel J. M. Hensen, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b815943b

Metal–organic frameworks as high-potential adsorbents for liquid-phase separations of olefins, alkylnaphthalenes and dichlorobenzenes Luc Alaerts, Michael Maes, Monique A. van der Veen, Pierre A. Jacobs and Dirk E. De Vos, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b823233d

Tetramethyl ammonium as masking agent for molecular stencil patterning in the confined space of the nano-channels of 2D hexagonal-templated porous silicas Kun Zhang, Belén Albela, Ming-Yuan He, Yimeng Wang and Laurent Bonneviot, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819872c

Photovoltaic activity of layered zirconium phosphates containing covalently grafted ruthenium tris(bipyridyl) and diquat phosphonates as electron donor/acceptor sites Laura Teruel, Marina Alonso, M. Carmen Quintana, Álvaro Salvador, Olga Juanes, Juan Carlos Rodriguez-Ubis, Ernesto Brunet and Hermenegildo García, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b816698f

The characterisation and catalytic properties of biomimetic metal–peptide complexes immobilised on mesoporous silica Gerhard D. Pirngruber, Lukas Frunz and Marco Lüchinger, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819678h

Physisorption and chemisorption of alkanes and alkenes in H-FAU: a combined ab initio–statistical thermodynamics study Bart A. De Moor, Marie-Françoise Reyniers and Guy B. Marin, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819435c

Accelerated generation of intracrystalline mesoporosity in zeolites by microwave-mediated desilication Sònia Abelló and Javier Pérez-Ramírez, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b819543a

Regio- and stereoselective terpene epoxidation using tungstate-exchanged takovites: a study of phase purity, takovite composition and stable catalytic activity Pieter Levecque, Hilde Poelman, Pierre Jacobs, Dirk De Vos and Bert Sels, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b820336a

Probing the microscopic hydrophobicity of smectite surfaces. A vibrational spectroscopic study of dibenzo-p-dioxin sorption to smectite Kiran Rana, Stephen A. Boyd, Brian J. Teppen, Hui Li, Cun Liu and Cliff. T. Johnston, Phys. Chem. Chem. Phys., 2009 DOI: 10.1039/b822635k

Correlating phase behaviour and diffusion in mesopores: perspectives

revealed by pulsed field gradient NMR

Rustem Valiullin,*a Jorg Kargera and Roger Glaserb

Received 19th December 2008, Accepted 18th February 2009

First published as an Advance Article on the web 16th March 2009

DOI: 10.1039/b822939b

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.

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 meso-

porous 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 meso-

pores 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

a Fakultat fur Physik und Geowissenschaften, Universitat Leipzig,Linnestr. 5, D-04103 Leipzig, Germany.E-mail: valiullin@uni-leipzig.de

b Institut fur Technische Chemie, Universitat Leipzig, Linnestr. 3,D-04103 Leipzig, Germany

Rustem Valiullin

Rustem Valiullin got his PhDdegree from Kazan State Uni-versity (Kazan, Russia) in 1997and continued to work there as aResearch Scientist. After twoyears postdoctoral work in theRoyal Institute of Technology(Stockholm, Sweden), in 2003he moved to the University ofLeipzig, Germany as a fellow ofthe A. von Humboldt Founda-tion. Currently he is aHeisenberg fellow of theGerman Science Foundation atthe Department of InterfacePhysics of Leipzig University.

Jorg Karger

Jorg Karger got his PhD inPhysics in 1970 at LeipzigUniversity, followed by thehabilitation in 1978. As amember of Leipzig Universitysince this time, research staystook him to Prague, Moscow,Fredericton and Paris. In 1994,he became Professor of Experi-mental Physics and headof the Department of InterfacePhysics. His research isdedicated to diffusion pheno-mena quite in general.

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2833

PERSPECTIVE www.rsc.org/pccp | Physical Chemistry Chemical Physics

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 pharma-

ceuticals 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 proper-

ties 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 super-

critical 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 meso-

porous 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 corres-

ponding 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 meso-

porous 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

Roger Glaser

Roger Glaser has studiedchemistry at the University ofStuttgart, Germany and gotthere his PhD degree fromthe Institute of ChemicalTechnology in 1997. After thathe worked at the GeorgiaInstitute of Technology,Atlanta, USA. In 1999 hereturned to the Institute ofChemical Technology,Stuttgart where he completedhis habilitation in 2007.Presently, he is a Professorof Chemical Technology atthe University of Leipzig and

is head of the Institutes of Chemical Technology andNon-Classical Chemistry.

2834 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

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 confine-

ment 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 inter-

mediate 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 Ps,

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.

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 Ptr, ln(Ptr/Ps)p (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 geo-

metry 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 simula-

tions 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

Fig. 1 Relative adsorption (open symbols) and desorption (full symbols)

isotherms of nitrogen in channels of porous silicon (dp C 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).

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2835

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, cavita-

tion-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 experi-

mental 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 studies70–74 and may affect

the phase behaviour of the fluids therein. As a most demon-

strative example, one may refer to mesoporous silicon. A

number of adsorption studies using this material with

channel-like, parallel pores have revealed very striking, some-

time 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 considera-

tions 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 rough-

ness. 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 transi-

tions 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 Tc. 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 Tch,

called hysteresis critical point and corresponding to vanishing

hysteresis, had earlier been considered as a counterpart of its

bulk value Tc. 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 Tcp.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, Tch and Tcp, 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 Tch o Tcp o Tc.88

A simple explanation of the obtained relation Tch o Tcp 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 Tcp.

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 adsorp-

tion isotherm, as an indicator of the phase state in the pores.

On the other hand, a number of other quantities may simul-

taneously 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

2836 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

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

o0 = gB0, where g is the nuclear gyromagnetic ratio. By the

application of a radio-frequency field of the same frequency o0

over a well-defined short interval of time, the net magnetiza-

tion 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 frequen-

cies o0 of the nuclear spin system under study. Applying

a magnetic field linearly increasing in the z direction,

B = B0 + gz, o0 = gB0 + ggz becomes a function of the

z coordinate. With this inhomogeneous field held for a time d,each spin will acquire a phase f= gB0d+ g

R d0g(t)z(t) dt. 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 d with

separation td. 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 Df is given by:

Df ¼ ggZ d

0

zðtÞdt�Z tdþd

td

zðtÞdt !

: ð3:1Þ

It now becomes obvious that, being able to detect Df, oneimmediately 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

d { td, 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,td) (from now on we use the

widely used notation q = gdg) of this signal intensity,91

S(q,td) =RP(z,td)e

iqzdz, (3.2)

is nothing else than the Fourier transform of the so-called

mean propagator P(z,td), one of the key functions describing

the internal dynamics in complex systems.93 P(z,td) denotes the

probability density that, during time td, 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 d-function:

Pðz; tÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffi4pDtp exp � z2

4Dt

� �; ð3:3Þ

with D denoting the self-diffusivity. Inserting eqn (3.3)

into (3.2) yields a spin-echo diffusion attenuation of

exponential form:

S(q,td) = exp(�q2Dtd). (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’’ dg. Note

that, for the sake of simplicity, S(q,td) in eqn (3.4) is given as

normalized to S(0,td) 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 = hz2(t)i/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,td) 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,td) =RC(ta,tb)exp{�q2(Data + Dbtb)}dO. (3.5)

Here, C(ta,tb) is the joint probability density that, during the

observation time td = ta + tb, a molecule spent the times taand tb in the phases a and b, respectively. O denotes a certain

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pair {ta,tb}. 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

O1 = {ta = td,tb = 0} and O2 = {ta = 0,tb = td}. The

relative fractions of such trajectories, or corresponding values

of C(ta,tb), ultimately coincide with the fractions pa and pb(pa + pb = 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,td) = paexp{�q2Datd} + pbexp{�q2Dbtd}. (3.6)

Another limit of fast exchange corresponds to situations when

all molecules during td experience many consecutive travels in

both phases, i.e., if the molecular lifetimes in both phases are

much shorter than the total diffusion time td. In this case,

assuming that the ergodic hypothesis is fulfilled, namely

ta/(ta + tb) = pa, we may substitute ta and tb in eqn (3.5)

by patd and pbtd, respectively. Hence, C(ta,tb) = 1 when

ta = patd and tb = pbtd and zero otherwise. Finally, in this

limit eqn (3.5) becomes:

S(q,td) = exp{�q2td(paDa + pbDb)}. (3.7)

In between these two regimes, S(q,td) can by described by

eqn (3.5) with a continuous spectrum of the apparent diffusivities

(taDa + tbDb)t�1d , where ta varies from 0 to td. By performing

inverse Laplace transform one may, in principle, access

the function C(ta,tb) 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 td one can, with a relatively

high accuracy, measure the fraction psl of molecules which

have not left a phase with the slowest diffusivity Dsl during td.

Obviously, psl(td = 0) is the relative fraction of molecules in

this phase. The measured psl(td) can be related

to the molecular lifetime distribution function c(t) in the

corresponding phase via:95

pslðtdÞpslðtd ¼ 0Þ ¼ 1�

Z td

0

cðtÞdt: ð3:8Þ

In section VB, this relation will be shown to provide the basis

for the so-called NMR tracer desorption technique. Alter-

natively, due to the short time scale of PFGNMR in comparison

with ‘‘conventional’’ tracer exchange experiments, this method

is as well referred to as ‘‘fast tracer desorption’’.96,105–107 The

function c(t) 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 c(t) 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) c(t) is given by ref. 110

cðtÞ ¼ 6D

R2

X1m¼1

exp �m2p2DtR2

� �: ð3:9Þ

Note that the case of surface barriers may also be analyzed in

this way.111 With a known function c(t) one may easily

estimate the average time tav after which the molecules leave

the particle in which they just reside. For spherical particles, it

immediately follows from eqn (3.9) that:96

tav ¼Z 10

tcðtÞdt ¼ R2

15D: ð3:10Þ

On the other hand, by tracking the fraction of molecules which

did not leave the porous particles during the experimental

observation time td, one may estimate an average size of the

particles. To do this, one may fit eqn (3.8) with c(t) 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 simul-

taneously 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 condi-

tions, 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

2838 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

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.

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 reviews125–127). 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 mole-

cular 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 PFGNMR.

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 Ocmusing 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 Ds, 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.

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, hetero-

geneity of the surface, e.g., site or saddle-energy disorder,

inverts this trend. In a most simple way, this may be under-

stood by considering the effect of site-energy disorder given

by a distribution fs(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) p 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 a2h1/Wi�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

h1/Wi and if the model implies that there is no correlation

between neighbour site energies, i.e., if a random energy

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.

Fig. 3 Surface diffusivity of acetone in porous silicon as a function of

the surface coverage. The solid line is the function D*0c1/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.

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topography is assumed. The particle ensembles may be treated

similarly. However, the probabilities of site occupancy, p(E),

have to be properly accounted for:

Ds =Rfs(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 � m)/RT})�1, where m 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

Ds ¼ D0ð1� cÞ2

cexp

m� Eav

RT

� �; ð4:12Þ

where Eav is the average site energy and D0 is the diffusivity of

a single particle on the surface with the only site energies Eav.

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., m= m0 + RT ln(z), where z� P/Ps. Thus, by

absorbing all concentration-independent parameters into D*0,

eqn (4.12) becomes

Ds ¼ D�0ð1� cÞ2

cz: ð4:13Þ

This equation shows the direct interrelation between the

surface diffusivity Ds 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 hetero-

geneous surfaces, for example the generalized Freundlich

isotherm139

c ¼ Kz

1þ Kz

� �m

: ð4:14Þ

In eqn (4.14) K is a constant and 0 o m o 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 o 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, Ds passes through a maximum. The

latter is the result of two competing mechanisms-entropic

(site-blocking) and energetic (increasing transition rates) ones.

Let us now analyze the experimental data shown in Fig. 3 in

this way. The inset shows the low-pressure part of the adsorp-

tion 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*0c1/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 o 0.2, while in our case the range is

extended up to c E 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 Dgas 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 t.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 note

DK ¼D0K

tK; Db ¼

D0b

tb: ð4:15Þ

As 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

DK = 13�ud. (4.16)

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.

2840 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

with �u = (8RT/pM)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:

Db = 13�ul. (4.17)

with l 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 has143

1

Dgas¼ 1

DKþ 1

Db: ð4:18Þ

The 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 l c d and

l{ 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 relation122,124

DK ¼d

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8RTð1� yÞ

pM

r: ð4:19Þ

Gas 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 enhance-

ment 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 simulations151–156

and analytical considerations151 confirm this assumption.

Moreover, the kinetic Monte Carlo simulations154–156 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 Deff is given

by ref. 159 and 160

Deff = pgasDgas + pliqDliq, (4.20)

where Dgas and Dliq are the diffusivities in two phases and pgasand pliq are the respective weights (pgas + pliq = 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

Deff, is ref. 160

Lgas þ Lliq

Deff;s¼ Lgas

rgasDgasþ Lliq

rliqDliq; ð4:21Þ

where Lgas and Lliq are the typical extensions of the regions

occupied by the gas and liquid volumes, respectively, and rgasand rliq are the respective densities.

The phase state is reflected by the adsorption isotherm y(z),where y is the amount adsorbed (also referred to as the pore

filling factor) at the relative pressure z = P/Ps (with Ps

denoting the saturation pressure). One may start to relate pgasand pliq to the given phase state with the set of equations:

Ngas + Nliq = N, (4.22)

Vgas + Vliq = V, (4.23)

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whereNgas andNliq are the number of molecules in the gaseous

and liquid phases and Vgas and Vliq are the volumes occupied

by them. The latter may be expressed as Vi = mNi/ri, where riis the density in a phase i, and m is the molecular mass. With

y = Vliq/V and taking account of rliq c rgas, it immediately

follows that124

pgas ¼1� yy

rgasrliq

: ð4:24Þ

The density rgas may be related to the external gas pressure via

the ideal gas law

rgas ¼zPsM

RT; ð4:25Þ

where 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 Deff 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 mO 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 y.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 y. It is important to note

that the isotherm reveals the onset of capillary-condensation

being located at z E 0.7, corresponding to y E 0.6. Thus, up

to y E 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.

One immediate consequence of eqn (4.24) is that pgas passes

a maximum upon variation of y. 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 � y)/y) with increasing P. Indeed,

at low pore loadings (y t 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 pgas at low

loadings. With increasing y, the situation is reversed, and the

contribution of the gas phase to overall diffusion becomes

dominating. Finally, however, with y - 1 and thus pgas - 0

(eqn (4.24)), the gas phase contribution vanishes. Hence,

irrespective of the fact that, with y - 0, the term (1 � y)/ygoes to infinity, rgas faster approaches zero, so that pgasbecomes negligibly small.

Let us quantify these statements. As shown by the dotted

line in Fig. 5, at intermediate loadings (y from about 0.2 to 0.6)

the magnitude of the term pgasDgas in eqn (4.20), where Dgas 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 (pgasDK + (1 � pgas)Ds) 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 mole-

cules 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, accord-

ing 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 micro-

pores, 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 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 rgas calculated via

eqn (4.25). The diffusivity increases with increasing gas

pressure up to about rgas = 0.25 kg m�3. This region

corresponds to the mesopores filled by the gaseous phase.

Further increase of rgas is accompanied by the onset of

capillary condensation in the mesopores, which nicely correlates

with the deviation of the measured diffusivities from the

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 y E 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 via eqn (4.20) and

(4.24), respectively, with the solid line as their sum.

2842 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

dependence as expected via eqn (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 trans-

port 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 acces-

sible if they are within the time-scale of PFGNMR (1C 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 Deff (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 Deff.

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 mm were prepared and mixed with

the respective masses mi = 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 tdfrom 5 to 300 ms. First of all, we note a multi-exponential

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. 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 td from

5–300 ms (from top to bottom). The solid lines are shown as examples

of the fit of the function psl exp{�q2Dsltd} to the data in the high-q

region. The dotted line shows the exponential function to which the

data converge upon increasing td.

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2843

decay and a significant influence of the diffusion time. With

increasing td, 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.

Let us now analyze the behaviour of the molecular ensemble

characterized by the slowest diffusivity Dsl following the recipe

described in section III. For the attenuation obtained with the

shortest td = 5 ms, for which the effect of the exchange is

minimal, Dsl and its respective fraction psl(td) 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 Dsl, the same procedure

may be applied to the attenuation curves with larger diffusion

times. The thus obtained dependence psl(td) is shown

in Fig. 8.

Obviously, the diffusivity Dsl, leading to the decrease of the

number of molecules which have not left the pore segments

during td, 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 psl(td)

using eqn (3.8). Because the lengths Li are different for the

channels of different porous silicon plates, one has to sum over

all i, yielding:

pslðtdÞ ¼P

i mipslðLi; tdÞPi mi

; ð5:26Þ

where for the one-dimensional case (diffusion out of a region

bounded by two plates) psl(L) is given by:

pslðLi; tdÞ ¼8

p2X1m¼1

1

ð2mþ 1Þ2exp �ð2mþ 1Þ2p2Dsltd

L2i

( ):

ð5:27Þ

Eqn (5.26), calculated with all relevant parameters known, is

shown in Fig. 8. It is found to nicely reproduce the experi-

mental 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 psl(td). Applying Poisson’s summation formula to

eqn (5.27) one may show that in this limit psl(td) is given by:

limtd!0

pslðtdÞ ¼ 1� 4

ffiffiffiffiffiffiffiffiffiffiffiDsltd

p

r1

L

� �: ð5:28Þ

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 mm. This is again in excellent agreement with the expected

value of 22 mm. 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 resis-

tances, 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 (psl(td)/psl(0) in

eqn (3.8) and (5.26)), which have not yet left the particles

which they accommodated at time td = 0. Under the condi-

tions of gas phase adsorption, the signal of the gas phase is

negligibly small so that in this case psl(0) = 1 and there is no

need for normalization by dividing psl(td) by psl(0). For

variable td, this information is exactly that of a tracer exchange

curve g(td), with g(td)(� 1 � psl(td)/psl(0)) denoting the relative

amount of (labelled) molecules which, after time td, 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 g(td)may most conveniently be correlated with the governing

Fig. 8 psl 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 t1/2d . Here, the solid line shows the fit of eqn (5.28) to the data at

short times.

2844 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

transport mechanisms via the method of moments.96,171,172

Thus, e.g., the ‘‘first’’ momentRN

0 (1� g(t))dt�RN

0 psl(t)/psl(0)dt=RN

0 tc(t)dt= tav (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 tav 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 corres-

ponding expression for exchange processes governed by the

surface permeability (tav,barr = R/3a), the general case may be

noted as

tav ¼ tav;diff þ tav;barr ¼R2

15Dþ R

3a; ð5:30Þ

where the surface permeability a96,173 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 ofD directly determined by PFGNMR

with sufficiently short observation times (diffusion path length

during observation time much shorter than the particle size),

the value of tav following via eqn (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 systems174,175 or their technological

use for mass separation and chemical conversion175–177 give

rise to the formation of surface barriers.

The accuracy of this way of analysis is limited by the fact that

the uncertainty of tav as resulting via eqn (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 para-

meters, 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.1,2 Fig. 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.

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 td 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 Deff deduced

from the attenuations in the low-q limit. The thus obtained

values of Deff 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

hr2i = 6Defftd, the extension of the regions with a uniform

diffusivity may be estimated to beffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6Deff tdp

� 40 mm. The

latter appears to be of the order of the particle sizes. There-

fore, the heterogeneity may be associated with the textural

differences in the different particles.

To understand the pressure dependence of the diffusivity

one may perform an analysis similar to that given in section IVC.

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.

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In this case, also equilibrium between the mesopores in crystal-

lites and the macropores formed between the grains and the

particles has to be taken into account.184 In this way, one

finally gets

Deff ¼rgasl�v

rliqZyaþ ð1� aÞ dg

dg þ l

� �; ð5:31Þ

where l and �v denote the mean free path and mean molecular

velocity in the gas phase, Z is the ratio of the meso- and

macropore volumes, dg is the effective inter-grain distance, and

a 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 a = 0.16 and

dg = 0.61 mm (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 Deff on the gas pressure is pre-

dominantly controlled by the equilibrium conditions between

the gaseous and adsorbed phases in the mesopores. Thus, the

slight decrease of Deff 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 rgas in the

macropores. At z E 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 inter-

action of the liquid with the walls of the mesopore. Thus, the

equilibrium conditions reasonably well explain the obtained

results. Quantitative agreement between theory and experi-

ment 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 Rp

of typically about 30 mm are applied which contain small

zeolite crystals of type NaY with radii Rc of about 0.5 mm. 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 NMR22,23,187 studies revealed exactly

the opposite. In fact, by choosing the observation time td of

the PFG NMR experiment in such a way that the experimen-

tally 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 (Dc) and through

the whole catalyst particle (Dp). With these diffusivities

and the corresponding radii, eqn (3.10) allows a straight-

forward 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 g).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.124 Fig. 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

Fig. 10 The effective diffusivity Deff 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.

2846 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

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

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 y, 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 Deff to

the difference in y. Deff may clearly also be plotted directly as a

function of the pore loading y. 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’’,39 i.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.

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 y. 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 y, the liquidphase 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 experi-

ments, 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 Deff has

been measured using PFG NMR. With the thus determined

value of Deff 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 y0 and yeq as the amounts adsorbed

Fig. 11 The effective diffusivity Deff (circles) for cyclohexane and

its relative amount adsorbed (triangles) in Vycor porous glass

(dp = 6 nm) measured on the adsorption (open symbols) and

desorption (filled symbols) branches as a function of the relative gas

pressure.

Fig. 12 The effective diffusivity Deff for cyclohexane in Vycor porous

glass (dp = 6 nm) of Fig. 11 as a function of the amount adsorbed y.The filled rectangles show Deff measured during various ‘‘scanning

isotherms’’.

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before the pressure steps and after equilibration, the uptake by

a cylinder of radius a is given by ref. 110

yðtÞ ¼ y0 þ ðyeq � y0Þ

� 1� 4

a2

X1n¼1

1

a2nexpf�a2nDeff tg

!; ð6:32Þ

where an are the positive roots of the equation J0(aa) = 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) Deff 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

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 NMRmethod.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.

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/Ps) 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/Ps due to the dramatically

increasing value of Ps. Thus, P/Ps 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

Fig. 13 The adsorption kinetics for cyclohexane in Vycor porous

glass (dp = 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.

Fig. 14 The effective intra-pore diffusivity for n-pentane in

Vycor porous glass (dp = 6 nm) for samples with (filled circles) and

without (filled triangles) excess liquid phase as a function of inverse

temperature.

2848 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

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 dp = 6 nm. This jump in diffusivity is an

indication of a phase transition from the liquid to a super-

critical 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.

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., Tcp, is closer to that of the bulk

critical temperature Tc 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

DT = Tc � Tcp 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 meso-

pores 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 super-

critical 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

Fig. 15 Temperature dependence of (a) the meniscus level in an

NMR glass tube containing Vycor porous glass (dp = 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 Tc and Tcp, respectively.

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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 deposi-

tion 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 super-

critical 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 prepara-

tion 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 experi-

mentally 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

g-Al2O3 (average pore diameter d = 5 nm) close to critical

point of the reactant (Tc = 231 1C, Pc = 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/g-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 NMRmeasure-

ments 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 super-

critical 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 transi-

tions 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 micro-

pores 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 experi-

mentally demonstrated fact that a fluid in pores may already

2850 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009

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 pheno-

mena 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 meso-

pores 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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