www.rsc.org/pccp Volume 11 | Number 16 | 28 April 2009 | Pages 2781–2992
ISSN 1463-9076
Physical Chemistry Chemical Physics
Volum
e 11 | Num
ber 16 | 2009 PC
CP
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
RSC Publishing Online Services
www.rsc.org/publishingRegistered Charity Number 207890
Innovative systems for innovative science
User-friendly web-based manuscript submission, review and tracking www.rsc.org/resource
ReSourCe
Award-winning technology for enhancing html journal articles www.projectprospect.org
Connecting you to over 800 of the best chemical science eBookswww.rsc.org/ebooks
RSC eBook Collection
RSC Prospect
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: [email protected]
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
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2837
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.
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2839
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)
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2841
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.
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2845
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’’.
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2847
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.
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2849
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).
References
1 J. S. Beck, J. C. Vartuli, W. J. Roth, M. E. Leonowicz,C. T. Kresge, K. D. Schmitt, C. T. W. Chu, D. H. Olson,E. W. Sheppard, S. B. Mccullen, J. B. Higgins andJ. L. Schlenker, J. Am. Chem. Soc., 1992, 114, 10834–10843.
2 C. T. Kresge, M. E. Leonowicz, W. J. Roth, J. C. Vartuli andJ. S. Beck, Nature, 1992, 359, 710–712.
3 P. T. Tanev and T. J. Pinnavaia, Science, 1995, 267, 865–867.4 S. A. Bagshaw, E. Prouzet and T. J. Pinnavaia, Science, 1995, 269,1242–1244.
5 R. Ryoo, S. H. Joo and S. Jun, J. Phys. Chem. B, 1999, 103,7743–7746.
6 U. Ciesla and F. Schuth, Microporous Mesoporous Mater., 1999,27, 131–149.
7 G. J. D. Soler-Illia, C. Sanchez, B. Lebeau and J. Patarin, Chem.Rev., 2002, 102, 4093–4138.
8 F. Schuth, Angew. Chem., Int. Ed., 2003, 42, 3604–3622.9 F. Hoffmann, M. Cornelius, J. Morell and M. Froba, Angew.Chem., Int. Ed., 2006, 45, 3216–3251.
10 P. Schmidt-Winkel, W. W. Lukens, D. Y. Zhao, P. D. Yang,B. F. Chmelka and G. D. Stucky, J. Am. Chem. Soc., 1999, 121,254–255.
11 J. S. Lettow, Y. J. Han, P. Schmidt-Winkel, P. D. Yang, D. Y. Zhao,G. D. Stucky and J. Y. Ying, Langmuir, 2000, 16, 8291–8295.
12 D. T. On and S. Kaliaguine, J. Am. Chem. Soc., 2003, 125,618–619.
13 S. Kaskel, in Handbook of Porous Solids, ed. F. Schuth, K. S. W.Sing and J. Weitkamp, Wiley-VCH, 2002, vol. 2, p. 1190.
14 C. Janiak, Dalton Trans., 2003, 2781–2804.15 S. Kitagawa, R. Kitaura and S. Noro, Angew. Chem., Int. Ed.,
2004, 43, 2334–2375.16 A. Taguchi and F. Schuth, Microporous Mesoporous Mater.,
2005, 77, 1–45.17 M.-O. Coppens, J. Sun and T. Maschmeyer, Catal. Today, 2001,
69, 331–335.18 C. H. Christensen, K. Johannsen, E. Toernqvist, I. Schmidt and
H. Topsoe, Catal. Today, 2007, 128, 117–122.19 D. R. Rolison, Science, 2003, 299, 1698–1701.20 J. S. Beck and W. O. Haag, in Handbook of Heterogeneous
Catalysis, ed. G. Ertl, H. Knotzinger and J. Weitkamp,Wiley-VCH, Weinheim, 1997, vol. 5, pp. 2123–2136.
21 R. von Ballmoos, D. H. Harris and J. S. Magee, in Handbook ofHeterogeneous Catalysis, ed. G. Ertl, H. Knotzinger andJ. Weitkamp, Wiley-VCH, Weinheim, 1997, vol. 4, pp. 1955–1986.
22 P. Kortunov, S. Vasenkov, J. Karger, M. F. Elia, M. Perez,M. Stocker, G. K. Papadopoulos, D. Theodorou, B. Drescher,G. McElhiney, B. Bernauer, V. Krystl, M. Kocirik, A. Zikanova,H. Jirglova, C. Berger, R. Glaser, J. Weitkamp andE. W. Hansen, Chem. Mater., 2005, 17, 2466–2474.
23 P. Kortunov, S. Vasenkov, J. Karger, R. Valiullin, P. Gottschalk,M. F. Elia, M. Perez, M. Stocker, B. Drescher, G. McElhiney,C. Berger, R. Glaser and J. Weitkamp, J. Am. Chem. Soc., 2005,127, 13055–13059.
24 M. Hartmann, Chem. Mater., 2005, 17, 4577–4593.25 W. Xia and J. Chang, J. Controlled Release, 2006, 110,
522–530.26 M. Vallet-Regi, M. Colilla and I. Izquierdo-Barba, J. Biomed.
Nanotechnol., 2008, 4, 1–15.27 D. Kovalev and M. Fujii, Adv. Mater., 2005, 17, 2531–2544.28 A. Holtzel and U. Tallarek, J. Sep. Sci., 2007, 30, 1398–1419.29 P. Barthelemy, M. Ghulinyan, Z. Gaburro, C. Toninelli,
L. Pavesi and D. S. Wiersma, Nature Photonics, 2007, 1, 172–175.30 L. D. Gelb, K. E. Gubbins, R. Radhakrishnan and M.
Sliwinska-Bartkowiak, Rep. Prog. Phys., 1999, 62, 1573–1659.31 K. Knorr, P. Huber and D. Wallacher, Z. Phys. Chem., 2008, 222,
257–285.32 M. Brun, A. Lallemand, J. F. Quinson and C. Eyraud, Thermochim.
Acta, 1977, 21, 59–88.33 C. L. Jackson and G. B. McKenna, J. Chem. Phys., 1990, 93,
9002–9011.34 J. Mitchell, J. B. W. Webber and J. Strange, Phys. Rep., 2008,
461, 1–36.35 K. S. W. Sing, F. Rouquerol and J. Rouquerol, Adsorption by
Powders and Solids, Academic Press, London, 1999.36 L. Bocquet, E. Charlaix, S. Ciliberto and J. Crassous, Nature,
1998, 396, 735–737.37 O. Beckstein andM. S. P. Sansom, Proc. Natl. Acad. Sci. U. S. A.,
2003, 100, 7063–7068.38 M. Nosonovsky and B. Bhushan, Phys. Chem. Chem. Phys., 2008,
10, 2137–2144.39 D. H. Everett, in The Solid-Gas Interface, ed. E. Alison Flood,
Marcel Dekker, Inc., New York, 1967, pp. 1055–1113.40 J. M. van Bemmelen, Z. Anorg. Allg. Chem., 1897, 13, 233–356.41 R. Zsigmondy, Z. Anorg. Allg. Chem., 1911, 71, 356–377.42 T. J. Barton, L. M. Bull, W. G. Klemperer, D. A. Loy,
B. McEnaney, M. Misono, P. A. Monson, G. Pez, G. W. Scherer,J. C. Vartuli and O. M. Yaghi, Chem. Mater., 1999, 11, 2633–2656.
43 Handbook of Porous Solids, ed. F. Schuth, K. S. W. Sing andJ. Weitkamp, Wiley-VCH, 2002.
44 J. Karger, H. Pfeifer and W. Heink, Adv. Magn. Reson., 1988, 12,2–89.
45 P. T. Callaghan, Principles of Nuclear Magnetic ResonanceMicroscopy, Clarendon Press, Oxford, 1991.
46 R. Kimmich, NMR: Tomography, Diffusometry, Relaxometry,Springer-Verlag, Berlin Heidelberg, 1997.
47 W. S. Price, Concepts Magn. Reson., 1997, 9, 299–336.48 W. S. Price, Concepts Magn. Reson., 1998, 10, 197–237.
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2851
49 G. H. Sørland and D. Aksnes, Magn. Reson. Chem., 2002, 40,S139–S146.
50 L. H. Cohan, J. Am. Chem. Soc., 1938, 60, 433–435.51 E. O. Kraemer, in Treatise on Physical Chemistry,
ed. H. S. Taylor, Van Nostrand, New York, 1931.52 J. W. McBain, J. Am. Chem. Soc., 1935, 57, 699.53 G. Mason, Proc. R. Soc. A: Math. Phys. Eng. Sci., 1988, 415,
453–486.54 R. Evans, J. Phys.: Condens. Matter, 1990, 2, 8989–9007.55 G. Mason, J. Colloid Interface Sci., 1982, 88, 36–46.56 P. C. Ball and R. Evans, Langmuir, 1989, 5, 714–723.57 A. V. Neimark, P. I. Ravikovitch and A. Vishnyakov, Phys. Rev.
E, 2000, 62, R1493–R1496.58 F. Restagno, L. Bocquet and T. Biben, Phys. Rev. Lett., 2000, 84,
2433–2436.59 A. V. Neimark, P. I. Ravikovitch and A. Vishnyakov, Phys. Rev.
E, 2002, 65, 031505.60 P. A. Monson, Adsorption, 2005, 11, 29–35.61 I. Brovchenko and A. Oleinikova, inHandbook of Theoretical and
Computational Nanotechnology, ed. M. Rieth and W. Schommers,American Scientific Publishers, 2005.
62 L. Sarkisov and P. A. Monson, Langmuir, 2001, 17, 7600–7604.63 N. A. Seaton, Chem. Eng. Sci., 1991, 46, 1895–1909.64 B. Coasne and R. J. M. Pellenq, J. Chem. Phys., 2004, 121,
3767–3774.65 P. van der Voort, P. Ravikovitch, K. De jong, M. Benjelloun,
E. Van Bavel, A. Janssen, A. Neimark, B. Weckhuysen andE. Vansant, J. Phys. Chem. B, 2002, 106, 5873–5877.
66 M. Thommes, B. Smarsly, M. Groenewolt, P. Ravikovitch andA. Neimark, Langmuir, 2006, 22, 756–764.
67 K. Morishige and Y. Nakamura, Langmuir, 2004, 20, 4503–4506.68 E. A. Ustinov, Langmuir, 2008, 24, 6668–6675.69 F. Casanova, C. E. Chiang, C.-P. Li, I. V. Roshchin, A.M. Ruminski,
M. J. Sailor and I. K. Schuller, Nanotechnology, 2008, 315709.70 F. Stallmach, J. Karger, C. Krause, M. Jeschke and
U. Oberhagemann, J. Am. Chem. Soc., 2000, 122, 9237–9242.71 F. Stallmach, A. Graser, J. Karger, C. Krause, M. Jeschke,
U. Oberhagemann and S. Spange, Microporous MesoporousMater., 2001, 44, 745–753.
72 L. Gjerdaker, D. W. Aksnes, G. H. Sørland and M. Stocker,Microporous Mesoporous Mater., 2001, 42, 89–96.
73 D. W. Aksnes, K. Førland and M. Stocker, MicroporousMesoporous Mater., 2005, 77, 79–87.
74 S. Naumov, R. Valiullin, J. Karger, R. Pitchumani andM.-O. Coppens,MicroporousMesoporousMater., 2008, 110, 37–40.
75 B. Coasne, A. Grosman, N. Dupont-Pavlovsky, C. Ortega andM. Simon, Phys. Chem. Chem. Phys., 2001, 3, 1196–1200.
76 B. Coasne, A. Grosman, C. Ortega and M. Simon, Phys. Rev.Lett., 2002, 88, 256102.
77 D. Wallacher, N. Kunzner, D. Kovalev, N. Knorr and K. Knorr,Phys. Rev. Lett., 2004, 92, 195704.
78 A. Grosman and C. Ortega, Langmuir, 2008, 24, 3977–3986.79 S. Naumov, A. Khokhlov, R. Valiullin, J. Karger and
P. A. Monson, Phys. Rev. E, 2008, 78, 060601–4.80 P. A. Monson, J. Chem. Phys., 2008, 128, 084701.81 L. Bruschi, G. Fois, G. Mistura, K. Sklarek, R. Hillebrand,
M. Steinhart and U. Gosele, Langmuir, 2008, 24, 10936–10941.82 E. Kierlik, P. A. Monson, M. L. Rosinberg, L. Sarkisov and
G. Tarjus, Phys. Rev. Lett., 2001, 87, 055701.83 H. J. Woo and P. A. Monson, Phys. Rev. E, 2003, 67, 041207.84 J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B.W. Roberts
and J. D. Shore, Phys. Rev. Lett., 1993, 70, 3347–3350.85 C. G. V. Burgess, D. H. Everett and S. Nuttall, Pure Appl. Chem.,
1989, 61, 1845–1852.86 M. Thommes and G. H. Findenegg, Langmuir, 1994, 10,
4270–4277.87 W. D. Machin, Langmuir, 1999, 15, 169–173.88 K. Morishige and M. Shikimi, J. Chem. Phys., 1998, 108,
7821–7824.89 M. H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic
Resonance, Wiley & Sons, Chichester, 2001.90 P. Galvosas, F. Stallmach, G. Seiffert, J. Karger, U. Kaess and
G. Majer, J. Magn. Reson., 2001, 151, 260–268.91 E. O. Stejskal and J. E. Tanner, J. Chem. Phys., 1965, 42,
288–292.
92 J. E. Tanner, J. Chem. Phys., 1970, 52, 2523–2526.93 J. Karger and W. Heink, J. Magn. Reson., 1983, 51, 1–7.94 F. Stallmach and P. Galvosas, Spin Echo NMR Diffusion
Studies, Annual Reports on NMR Spectroscopy 61,ed. G. Webb, Academic Press, 2007, pp. 52–131.
95 A. I. Maklakov, V. D. Skirda and N. F. Fatkullin, in Encyclopediaof Fluid Mechanics, ed. N. M. Chremisinoff, Gulf-Publishing Co.,Houston, 1990, vol. 9.
96 J. Karger and D. Ruthven, Diffusion in Zeolites and OtherMicroporous Solids, Wiley & Sons, New York, 1992.
97 R. W. Mair, G. P. Wong, D. Hoffmann, M. D. Hurlimann,S. Patz, L. M. Schwartz and R. L. Walsworth, Phys. Rev. Lett.,1999, 83, 3324–3327.
98 D. ben Avraham and S. Havlin, Diffusion and Reactions inFractals and Disordered Systems, Cambridge University Press,Cambridge, 2000.
99 J. Karger and F. Stallmach, in Diffusion in Condensed Matter,ed. P. Heitjans and J. Karger, Springer, Heidelberg, 2005.
100 R. Valiullin, V. D. Skirda, S. Stapf and R. Kimmich, Phys. Rev.E, 1997, 55, 2664–2671.
101 J. Karger, Annalen der Physik, 1969, 24, 1.102 Y. Q. Song, L. Venkataramanan, M. D. Hurlimann, M. Flaum,
P. Frulla and C. Straley, J. Magn. Reson., 2002, 154, 261–268.103 P. T. Callaghan, S. Godefroy and B. N. Ryland, Magn. Reson.
Imaging, 2003, 21, 243–248.104 P. T. Callaghan and I. Furo, J. Chem. Phys., 2004, 120, 4032–4038.105 J. Karger, AICHE J., 1982, 28, 417–423.106 J. Karger, W. Heink, H. Pfeifer, M. Rauscher and J. Hoffmann,
Zeolites, 1982, 2, 275–278.107 D. Ruthven, Principles of Adsorption and Adsorption Processes,
Wiley, New York, 1984.108 C. Rodenbeck, J. Karger and K. Hahn, Ber. Bunsen-Ges., 1998,
102, 929–944.109 C. Rodenbeck, J. Karger, H. Schmidt, T. Rother and
M. Rodenbeck, Phys. Rev. E, 1999, 60, 2737–2740.110 H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids,
Clarendon Press, Oxford, 1946.111 A. Yekta, J. Duhamel and M. A. Winnik, J. Chem. Phys., 1992,
97, 1554–1561.112 J. Karger, F. Stallmach, R. Valiullin and S. Vasenkov, in NMR
Imaging in Chemical Engineering, ed. S. Stapf and S. Han,Wiley-VCH, Weinheim, 2005, p. 622.
113 R. Valiullin and M. Dvoyashkin, Adsorption, 2007, 13, 239–245.114 Y.-Q. Song, H. Cho, T. Hopper, A. E. Pomerantz and P. Z. Sun,
J. Chem. Phys., 2008, 128, 052212.115 R. Valiullin, S. Naumov, P. Galvosas, J. Karger, H. J. Woo,
F. Porcheron and P. A. Monson, Nature, 2006, 443, 965–968.116 R. Valiullin, S. Naumov, P. Galvosas, J. Karger and
P. A. Monson, Magn. Reson. Imaging, 2007, 25, 481–484.117 F. D’Orazio, S. Bhattacharja, W. P. Halperin and R. Gerhardt,
Phys. Rev. Lett., 1989, 63, 43–46.118 N. K. Dvoyashkin, V. D. Skirda, A. I. Maklakov, M. Belousova
and R. Valiullin, Appl. Magn. Reson., 1991, 2, 83–91.119 K. P. Tzevelekos, E. S. Kikkinides, A. K. Stubos,
M. E. Kainourgiakis and N. K. Kanellopoulos, Adv. ColloidInterface Sci., 1998, 76–77, 373–388.
120 F. Courivaud, E. W. Hansen, S. Kolboe, A. Karlsson andM. Stocker, Microporous Mesoporous Mater., 2000, 37, 223–232.
121 D. W. Aksnes, L. Gjerdaker, L. Kimtys and K. Forland, Phys.Chem. Chem. Phys., 2003, 5, 2680–2685.
122 I. Ardelean, C. Mattea, G. Farrher, S. Wonorahardjo andR. Kimmich, J. Chem. Phys., 2003, 119, 10358–10362.
123 C. Mattea, R. Kimmich, I. Ardelean, S. Wonorahardjo andG. Farrher, J. Chem. Phys., 2004, 121, 10648–10656.
124 R. Valiullin, P. Kortunov, J. Karger and V. Timoshenko,J. Chem. Phys., 2004, 120, 11804–11814.
125 R. Gomer, Rep. Prog. Phys., 1990, 53, 917–1002.126 J. V. Barth, Surf. Sci. Rep., 2000, 40, 75–149.127 T. Ala-Nissila, R. Ferrando and S. C. Ying, Adv. Phys., 2002, 51,
949–1078.128 B. G. Briner, M. Doering, H. P. Rust and A. M. Bradshaw,
Science, 1997, 278, 257–260.129 T. Mitsui, M. K. Rose, E. Fomin, D. F. Ogletree and
M. Salmeron, Science, 2002, 297, 1850–1852.130 G. Antczak and G. Ehrlich, Phys. Rev. Lett., 2004, 92, 166105.
2852 | Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 This journal is �c the Owner Societies 2009
131 A. Zurner, J. Kirstein, M. Doblinger, C. Brauchle and T. Bein,Nature, 2007, 450, 705–708.
132 R. Valiullin, P. Kortunov, J. Karger and V. Timoshenko, J. Phys.Chem. B, 2005, 109, 5746–5752.
133 V. Lehmann, R. Stengl and A. Luigart, Mater. Sci. Eng. B, 2000,69, 11–22.
134 W. Rudzinski and D. Everett, Adsorption of Gases onHeterogeneous Surfaces, Academic Press, New York, 1992.
135 J. W. Haus and K. W. Kehr, Phys. Rep., 1987, 150, 263–406.136 J. P. Bouchaud and A. Georges, Phys. Rep., 1990, 195, 127–293.137 R. Kirchheim, Defect Diffus. Forum, 1997, 143, 911–925.138 L. M. Cameron and C. A. Sholl, J. Phys.: Condes. Matter., 1999,
11, 4491–4497.139 R. Sips, J. Chem. Phys., 1950, 18, 1024–1026.140 P. Levitz, J. Phys. Chem., 1993, 97, 3813–3818.141 S. Vasenkov, O. Geir and J. Karger,Eur. Phys. J. E, 2003, 12, 35–38.142 M. Knudsen, Annalen der Physik, 1909, 28, 75.143 W. G. Pollard and R. D. Present, Phys. Rev., 1948, 73, 762–774.144 J. Karger and P. Volkmer, J. Chem. Soc., Faraday Trans. I, 1980,
76, 1562–1568.145 J. Karger, M. Kocirik and A. Zikanova, J. Colloid Interface Sci.,
1981, 84, 240–249.146 F. Rittig, C. G. Coe and J. M. Zielinski, J. Phys. Chem. B, 2003,
107, 4560–4566.147 B. V. Derjaguin, Dokl. Acad. Sci. URSS, 1946, 7, 623.148 V. N. Burganos, J. Chem. Phys., 1998, 109, 6772–6779.149 J. M. Zalc, S. C. Reyes and E. Iglesia, Chem. Eng. Sci., 2004, 59,
2947–2960.150 G. Papadopoulos, D. N. Theodorou, S. Vasenkov and J. Karger,
J. Chem. Phys., 2007, 126, 094702.151 K. Malek and M. O. Coppens, Phys. Rev. Lett., 2001, 8712,
125505.152 K. Malek andM. O. Coppens, Colloid Surf. A: Physicochem. Eng.
Asp., 2002, 206, 335–348.153 K. Malek and M. O. Coppens, J. Chem. Phys., 2003, 119,
2801–2811.154 S. Russ, S. Zschiegner, A. Bunde and J. Karger, Phys. Rev. E,
2005, 72, 030101.155 S. Zschiegner, S. Russ, A. Bunde and J. Karger, EPL, 2007, 78.156 S. Zschiegner, S. Russ, R. Valiullin, M. O. Coppens, A. J. Dammers,
A. Bunde and J. Karger,Eur. Phys. J.-Spec. Top., 2008, 161, 109–120.157 A. Bunde, J. Karger, S. Russ and S. Zschiegner, in Diffusion
Fundamentals, ed. J. Karger, F. Grinberg and P. Heitjans,Leipziger Universitatsverlag, Leipzig, 2005.
158 J. Karger, Leipzig, Einstein, Diffusion, Leipziger Universitatsverlag,Leipzig, 2007.
159 J. Karger, Adv. Colloid Interface Sci., 1985, 23, 129–148.160 A. Herrmann, L. Schimmele, J. Mossinger, M. Hirscher and
H. Kronmuller, Appl. Phys. A, 2001, 72, 197–208.161 J. Karger, H. Pfeifer, E. Riedel and H. Winkler, J. Colloid
Interface Sci., 1973, 44, 187–188.162 S. Vasenkov and J. Karger, Magn. Reson. Imaging, 2005, 23,
139–145.163 R. Valiullin, P. Kortunov, J. Karger and V. Timoshenko, Magn.
Reson. Imaging, 2005, 23, 209–213.164 S. R. Tennison, Appl. Catal. A: Gen., 1998, 173, 289–311.165 M. Melillo, V. M. Gun’ko, S. R. Tennison, L. I. Mikhalovska,
G. J. Phillips, J. G. Davies, A. W. Lyoyd, O. P. Kozynchenko,D. J. Malik, M. Streat and S. V. Mikhalovsky, Langmuir, 2004,20, 2837–2851.
166 A. Khokhlov, R. Valiullin, J. Karger, F. Steinbach andA. Feldhoff, New J. Phys., 2007, 9, 272.
167 R. M. Cotts, M. J. R. Hoch, T. Sun and J. T. Markert, J. Magn.Reson., 1989, 83, 252–266.
168 J. Karger and D. Freude, Chem. Eng. Technol., 2002, 25, 769–778.169 J. Karger, Adsorption, 2003, 9, 29–35.170 J. Karger, in Handbook of Heterogeneous Catalysis, ed. G. Ertl,
H. Knotzinger, F. Schuth and J. Weitkamp, Wiley-VCH,Weinheim, 2008, vol. 3, pp. 1714–1727.
171 R. Barrer, Zeolites and Clay Minerals as Sorbents and MolecularSieves, Academic Press, London, 1978.
172 M. Kocirik and A. Zikanova, Ind. Eng. Chem. Fund., 1975, 13,347–350.
173 J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford,1975.
174 J. Karger, M. Bulow, B. Millward and J. Thomas, Zeolites, 1986,6, 146–150.
175 J. Karger, H. Pfeifer, R. Richter, H. Furtig, W. Roscher andR. Seidel, AICHE J., 1988, 34, 1185–1189.
176 J. Karger, H. Pfeifer, J. Caro, M. Bulow, H. Schlodder,R. Mostowicz and J. Volter, Appl. Catal., 1987, 29, 21–30.
177 J. Volter, J. Caro, M. Bulow, B. Fahlke, J. Karger andM. Hunger, Appl. Catal., 1988, 42, 15–27.
178 M. Krutyeva, S. Vasenkov, X. Yang, J. Caro and J. Karger,Microporous Mesoporous Mater., 2007, 104, 89–96.
179 M. Krutyeva, X. Yang, S. Vasenkov and J. Karger, J. Magn.Reson., 2007, 185, 300–307.
180 J. Karger, P. Kortunov, S. Vasenkov, L. Heinke, D. R. Shah,R. A. Rakoczy, Y. Traa and J. Weitkamp, Angew. Chem., Int.Ed., 2006, 45, 7846–7849.
181 L. Heinke, P. Kortunov, D. Tzoulaki and J. Karger, Phys. Rev.Lett., 2007, 99, 228301.
182 C. G. Sonwane and S. K. Bhatia, Langmuir, 1999, 15, 2809–2816.183 P. Selvam, S. K. Bhatia and C. G. Sonwane, Ind. Eng. Chem. Res.,
2001, 40, 3237–3261.184 R. Valiullin, M. Dvoyashkin, P. Kortunov, C. Krause and
J. Karger, J. Chem. Phys., 2007, 126, 054705.185 J. Weitkamp and L. Puppe, Catalysis and Zeolites, Springer,
Berlin Heidelberg, 1999.186 A. M. Avila, C. M. Bidabehere and U. Sedran, Chem. Eng. J.,
2007, 132, 67–75.187 J. Karger and S. Vasenkov, Microporous Mesoporous Mater.,
2005, 85, 195–206.188 J. Karger, Chem. Eng. J., 2009, 145, 522–524.189 P. Porion, A. M. Faugere, P. Levitz, H. Van Damme, A. Raoof, J. P.
Guilbaud and F. Chevoir, Magn. Reson. Imaging, 1998, 16, 679–682.190 S. Naumov, R. Valiullin, P. Galvosas, J. Karger and
P. A. Monson, Eur. Phys. J. Special Topics, 2007, 141, 107–112.191 M. Dvoyashkin, A. Khokhlov, R. Valiullin and J. Karger,
J. Chem. Phys., 2008, 129, 154702.192 J. H. Page, J. Liu, B. Abeles, H. W. Deckman and D. A. Weitz,
Phys. Rev. Lett., 1993, 71, 1216–1219.193 E. Hoinkis and B. Rohl-Kuhn, J. Colloid Interface Sci., 2006, 296,
256–262.194 L. Xu, S. Davies, A. B. Schofield and D. A. Weitz, Phys. Rev.
Lett., 2008, 101, 094502.195 P. Rajniak, M. Soos and R. Yang, AICHE J., 1999, 45, 735–750.196 F. Stepanek, M. Kubicek, M. Marek, M. Soos, P. Rajniak and
R. T. Yang, Chem. Eng. Sci., 2000, 55, 431–440.197 J. W. Lee, W. G. Shim and H. Moon, Microporous Mesoporous
Mater., 2004, 73, 109–119.198 M. Soos, P. Rajniak and F. Stepanek, Colloid Surf. A:
Physicochem. Eng. Asp., 2007, 300, 191–203.199 D. S. Fisher, Phys. Rev. Lett., 1986, 56, 416–419.200 D. A. Huse, Phys. Rev. B, 1987, 36, 5383–5387.201 M. Dvoyashkin, R. Valiullin and J. Karger, Phys. Rev. E, 2007,
75, 041202.202 Chemical Synthesis Using Supercritical Fluids, ed. P. G. Jessop
and W. Leitner, Wiley-VCH, Weinheim, 1997.203 R. Glaser, in Handbook of Heterogeneous Catalysis, ed. G. Ertl,
H. Knotzinger, F. Schuth and J. Weitkamp, Wiley-VCH,Weinheim, 2008, vol. 4, pp. 1987–2007.
204 M. S. Schneider, J. D. Grunwaldt and A. Baiker, Langmuir, 2004,20, 2890–2899.
205 M. Dvoyashkin, R. Valiullin, J. Karger, W.-D. Einicke andR. Glaser, J. Am. Chem. Soc., 2007, 129, 10344–10345.
206 M. Dvoyashkin, R. Valiullin and J. Karger, Adsorption, 2007, 13,197–200.
207 E. Reverchon and R. Adami, J. Supercrit. Fluid, 2006, 37, 1–22.208 J. D. Holmes, D. M. Lyons and K. J. Ziegler, Chem.–Eur. J.,
2003, 9, 2144–2150.209 J. J. Watkins, J. M. Blackburn and T. J. McCarthy, Chem.
Mater., 1999, 11, 213–215.210 R. Glaser, M. Schmidt, S. Dapurkar, M. Turk and G. Upper, in
Proceedings of 10th European Meeting on Supercritical Fluids,ed. M. Perrut, Lorraine, 2005.
211 C. Domingo, J. Garcia-Carmona, J. Llibre and R.Rodriguez-Clemente, Adv. Mater., 1998, 10, 672–676.
212 V. Arunajatesan, K. A. Wilson and B. Subramaniam, Ind. Eng.Chem. Res., 2003, 42, 2639–2643.
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 2833–2853 | 2853
Top Related