Mass Transfer

Review
Mass Transfer in Micro- and Mesoporous Materials*
By Jörg Kärger* and Dieter Freude
Mass transfer in micro- and mesoporous materials is of crucial importance in their practical application for separation and
catalysis, since the mobility of adsorbed molecules ultimately limits the rate of the overall processes. Diffusion, i.e. the irregular
thermal motion of the molecules, is the dominating process. Diffusion measurements are therefore indispensable for the
evaluation of the quality parameters of porous materials. Due to their ability to directly follow the diffusion path of the
molecules, microscopic techniques are to the forefront, amongst the various methods of diffusion measurement. Besides
describing their fundamentals, this contribution describes the application of these techniques to investigate structure-mobilityrelations in zeolites and in mesoporous materials of type MCM-41. Some features of particular technological relevance, e.g., the
phenomena of correlated diffusion anisotropy, of single-file diffusion and of molecular traffic control, are discussed in detail.

1 Introduction
Diffusion, i.e., mass transfer generated by thermal motion,
is among the fundamental phenomena in nature and
technology [1]. It occurs in all types of matter and may be
observed over timescales from femtoseconds up to ages.
Current Contents (Physical, Chemical, and Earth Sciences)
reports over 8000 publications per year concerning this topic,
and the number of publications continues to steadily increase.
In micro- and mesoporous materials, diffusion is the
decisive mechanism of mass transfer. Therefore, diffusion is
of crucial relevance for a large number of technical
applications. Fig. 1 provides a survey of the development of
the number of publications in the field of molecular diffusion
in porous solids over the past 10 years. The continuous
increase in papers published, may also be considered as an
expression of the great relevance of diffusion in the important
fields of industrial application of porous solids, viz. mass
separation and mass conversion.
In mass conversion, diffusion becomes the rate determining
step as soon as the rate of adsorption of the reactant molecules
and/or the rate of desorption of the product molecules become
smaller than the intrinsic reaction rate. Prominent examples
are the hydroxylation of phenol with H2O2 on TS-1 [2] for
diffusion limitation of the reactants and the MTO reaction [3]
for diffusion limitation of the products. Substantial differences
in the diffusivities may give rise to dramatically enhanced
selectivities (as either reactant- or product-ªform selectivitiesº) and thus allow a sustainable productivity enhancement,
±
[*] Lecture on the occasion of the 518th DECHEMA-Kolloquium on
ªNanoporous Materials: Design ± Structure ± Applicationº, October 26,
2000 in Frankfurt/Main, Germany.
[**] Prof. Dr. J. Kärger (Lecturer, Corresponding author: e-mail: kaerger
@physik.uni-leipzig.de); Prof. Dr. D. Freude, Abteilung Grenzflächenphysik, Fakultät für Physik und Geowissenschaften, Universität Leipzig,
LinnØstr. 5, D-04103 Leipzig, Germany.
Chem. Eng. Technol. 25 (2002) 8,

Figure 1. Number of publications with reference to diffusion in porous materials
cited in Current Contents, Physical, Chemical and Earth Sciences, 1991±2000.

accompanied by the additional benefit of an efficient use of
raw materials and a reduced production of environmentally
questionable by-products. [4]
In contrast to catalysis and mass separation, where the
intimate correlation between effectiveness and mass transfer
has been well established with decades, it is only very recently
that the production of inorganic materials has opened a new
field of exploration into the practical consequences of mass
transfer in porous materials. Particularly, the application of
zeolite membranes, would revolutionize mass separation and
(by use of membrane reactors) technical catalysis [5]. Again,
efficiency enhancement in all of these applications implies an
optimization of mass transfer through the membranes and
hence an exact knowledge of the diffusivities of the involved
components.
A noted weak point in transferring the outcome of diffusion
research into industrial practice, is that there is still some
discrepancy between the results of different measuring

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

0930-7516/02/0808-0769$ $17.50+.50/0
17.50+.50/0
0930-7516/02/0808-0769

769

Review
techniques. Section 2 provides a classification of the different
measuring techniques. It emphasizes the particular relevance
of those techniques, which are able to directly monitor
molecular movement in the interior of the porous media
under investigation, and which are therefore referred to as
microscopic techniques. Section 4 provides examples of the
application of these techniques. The potential use of these
techniques for elucidating structural details is described in
Section 4. The particular correlations between structure and
diffusivity, which are brought about by the regular pore system
of zeolites, are discussed in Section 5.

2 Overview of the Measuring Techniques
Fig. 2 describes the possible situations which may arise in
diffusion experiments, under the condition of single-component adsorption [6].

jˆ

DT

@c
@z

(1)

defines the transport diffusivity DT. Here, j, c and z denote
respectively, the diffusion flux (density), the concentration of
the diffusing species, and the space coordinate in the direction
of the concentration gradient. This definition refers to nonequilibrium and thus, the diffusion flux is associated with mass
transfer. Therefore, the term transport diffusion has become
commonly used in the zeolite literature [6±8]. The index T
results from this terminology. Depending on the chosen
boundary conditions, one may distinguish between stationary
and non-stationary experiments.
Molecular redistribution under equilibrium is called selfdiffusion. This process does not lead to any macroscopically
observable phenomena. Hence, its observation necessitates
some labelling of the molecules involved. Fig. 2b schematically shows the usual procedure of using tracer techniques,
where a proportion of the diffusing species are distinguished
from the remaining ones, e.g. by use of isotopes, without
affecting their mobility. Analogous to Eq. (1), this leads to
j ˆ

D

@ c
@z

(2)

where j* and c* denote the diffusion flux and concentration of
the labelled molecules, and D stands for the coefficient of selfdiffusion. Fig. 2c illustrates how diffusivities are determined
by the pulsed field gradient NMR technique (PFG NMR) or
by Quasi-Elastic Neutron Scattering (QENS) [6]. These
techniques allow the observation of the mean value of the
displacements along the diffusion paths of all molecules under
study. Using Eq. (2), the quadratic mean <r2(t)> of molecular
displacements during the observation time t may be shown to
increase linearly with time, following the Einstein relation [6],
ár2(t)ñ = 6 D t

(3)

In classical diffusion experiments (Fig. 2a), the concentration gradient gives rise to the formation of a diffusion flux,
whose intensity, via Fick's first law,

Deviations from proportionality between mean square
displacement and observation time indicate anomalous
diffusion [9].
Later on, the diffusion measurement is characterized by the
relationship between the mean distances over which molecular propagation is observed, and the size of the porous
particles. Since zeolite crystallites are only available with
diameters between fractions of micrometers and several
hundred micrometers, this type of classification is particularly
critical for zeolites.
Tab. 1 provides an overview of the different techniques
developed for the measurement of intracrystalline zeolitic
diffusion and their classification. The differentiation between
macro-, meso-, and microscopic techniques concerns the
relation between the mean diffusion path and the crystal size.
The macroscopic techniques consider beds of crystallites or
compacted (pelletized) material. The diffusion paths covered
during the experiments are much larger than the individual
crystallites, so that information about the intracrystalline
processes may only be deduced indirectly. Mesoscopic

770

0930-7516/02/0808-0770 $ 17.50+.50/0

Figure 2. Microscopic situation during the measurement of transport diffusion
(2a) and of self-diffusion by molecule labeling (2b) or by monitoring the
individual molecules (2c), from [6].

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

Chem. Eng. Technol. 25 (2002) 8

Review
Table 1. Classification of the Techniques for Measurement of Intracrystalline Zeolitic Diffusion.
Non-Equilibrium
Stationary

Equilibrium

Macroscopic Sorption/Desorption[10]
Frequency Response (FR) [11]
Zero Length Column (ZLC) [12]
IR-FR [13]
Positron Emission Prof. (PEP) [14]
Temporal Product Analysis (TAP)
[15]
IR Spectroscopy [16]

Transient

MembranePermeation [5]
Effectiveness
Factor during
Catalytic
Reaction [17]

Tracer Sorption/Desorption [6]
Tracer ZLC [12]

Mesoscopic

IR Microscopy [18]

Single-Crystal
Permeation [19]

Tracer IR Microscopy [18]

Microscopic

Interference Microscopy [20]

Pulsed Field Gradient NMR (PFG
NMR) [1,6]
Stray Field Gradient NMR (SFG
NMR) [21]
Quasi-Elastic Neutron-Scattering
(QENS)[22]
Exchange NMR [23]

techniques consider the individual crystallites. However, they
also do not allow a resolution of transport phenomena in the
interior of the crystallites. Only microscopic techniques allow
the monitoring of sufficiently small diffusion paths, so that
intracrystalline diffusion may be directly measured.
Extensive descriptions of the different measurement
techniques may be found in the cited literature. Comprehensive representations are given in the respective monographs
[1,6±8], reviews [24,25], and handbooks [26,27].
Since transport diffusion and self-diffusion are observed
under different physical conditions one cannot imply coincidence of the corresponding coefficients. However, within
the limits of negligibly small concentrations and for inert
porous solids, as an immediate consequence of Eqs. (1) and
(2), the coefficients of transport diffusion and self-diffusion
can be expected to coincide. For a first estimate of the
relationship between transport and self-diffusion, one often
uses the relation [1,6±8,24±27],
DT ˆ D

@ ln p
@ ln c

(4)

where p denotes the gas pressure, which is necessary to
maintain the sorbate concentration c, under equilibrium
conditions. Eq. (4), which is sometimes referred to as the
Darken relation, is not generally valid. However, it may be
shown to be strictly valid in a number of model cases [6].
The first application of a microscopic method, i.e. the PFG
NMR technique, in studying intracrystalline zeolitic diffusion,
yields diffusivities which are five orders of magnitude greater
than the generally accepted values [6±8,24±27]. A critical
reconsideration of the previous data has helped to identify a
number of pitfalls in the interpretation of macroscopic
diffusion measurements. For a number of systems, their
exclusion yielded satisfactory agreement between the microscopic and macroscopic measurements [6]. Discovering the
Chem. Eng. Technol. 25 (2002) 8,

origin of remaining discrepancies, is one of the hot topics of
current diffusion research [28]. It is complicated by the
differing specifications of the measurement techniques
involved, as well as by the deficiency in our knowledge of
the real structure of the zeolites under study.

3 Microscopic Techniques: The Fundamentals
of PFG NMR and of Interference Microscopy
3.1 The PFG NMR Technique
Self-diffusion measurements by NMR are based on the fact
that the resonance frequency of a nucleus under consideration
is a function of the intensity of the applied magnetic field.
Thus, by applying an inhomogeneous magnetic field, the
spatial position of a nucleus in the sample may be unambiguously attributed to the position of the signal, stemming from
this nucleus, in the spectrum. This correlation is the basis of
NMR tomography, which meanwhile has become the most
powerful imaging method in medical diagnosis [29]. In the
NMR pulsed field gradient technique (PFG NMR), by the
subsequent application of two ªfield gradient pulsesº the
magnetic field is made inhomogeneous over two short time
intervals. In this way, the NMR signal becomes sensitive to the
differences in the locations of the nuclei (and hence of the
respective atoms and molecules) between the two field
gradient pulses. These differences are revealed by analyzing
the attenuation of the NMR signal as a function of the
intensity of the field gradient pulses [6]. Differences in
molecular positions at two subsequent intervals, however,
represent nothing other than their displacements. As an
analytical expression of the attenuation of the NMR signal
one obtains [6,9,27],
W (t,cdg) = ò P(z,t)cos(cdgz)dz

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

0930-7516/02/0808-0771 $ 17.50+.50/0

(5)
771

Review
where d, g and t denote the width, amplitude and separation
of the gradient pulses. The ªgyromagnetic ratioº c is a
characteristic quantity of the given nucleus. z denotes the
coordinate in the direction of the applied field gradients.
P(z,t) stands for the probability (density) that a nucleus
contributing to the NMR signal has been shifted over a
distance z during the time interval t. It is commonly referred to
as the propagator. In quasi-homogeneous systems, i.e. if the
diffusion paths are large in comparison with the pore sizes but
small in comparison with the particle sizes, the propagator
coincides with the solution of Fick's second law,
@ c…z;t†
@ 2 c…z;t†
ˆD
@t
@ z2

(6)

placement as accessible by the Einstein relation (3) has to be
much smaller than the crystallite diameter. It is only in this
case that any interference of the diffusion data by the finite
size of the crystallites can be excluded. Fig. 3 shows the
dependence of the coefficients of intracrystalline self-diffusion on the sorbate concentration for some systems.
Depending on the system under study, the observed
concentration dependence follows completely different patterns [6,30,31]. The decreasing mobility with increasing
concentration (cases (i) and (ii)) obviously indicates a mutual
inhibition of the diffusants, while the reverse effect (cases (iii)
to (v)) is most likely to occur, if there are adsorption sites,
which ± with increasing concentrations ± are able to
accommodate a continuously decreasing fraction of the
molecules.

with the initial condition,
c(z,t = 0) = d(z)

(7)

With the given initial condition and negligible boundary effects, i.e. with c(z = ¥,t) = 0,
the solution of Eq. (6) is the Gaussian
expression,
1
z2
P…z; t† ˆ p exp…
†
4Dt
4pDt

(8)

Inserting Eq. (8) into Eq. (5) yields
W (t,cdg) = exp(±c2d2g2Dt)
or, by use of the Einstein relation (3),


1 2 2 2 2
c d g hz …t†i
W (t,cdg) = exp
2

(9)

(10)

In first approximation Eq. (10) is still valid
if the propagator deviates from the shape of
Eq. (8), i.e. if there is no ªnormalº diffusion.
Owing to their large gyromagnetic ratio,
protons offer the best measuring conditions
with respect to both the minimum concentration of the diffusants and their minimum
displacements. However, PFG NMR diffusion studies have also been performed with
other nuclei including 13C, 15N, 19F, and 129Xe
[6,27]. The lower limit of observable displacements is of the order of 100 nm. With an
observation time of a few 100 ms as a typical
value of the maximum observation time, this
corresponds to minimum diffusivities of
about 10±14 m2s±1.
As a prerequisite of an unambiguous
measurement of intracrystalline diffusion in
zeolites, the mean value of molecular dis772

Figure 3. The different patterns for concentration dependence of molecular self-diffusion in zeolites,
from [30,31].

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

0930-7516/02/0808-0772 $ 17.50+.50/0

Chem. Eng. Technol. 25 (2002) 8

Review
3.2 Interference Microscopy
The application of interference microscopy to diffusion
measurement is based on the fact that the refractive index of a
zeolite crystallite is generally a linear function of its
concentration. Thus, information about intracrystalline concentration profiles may be provided by determining the
optical density over a zeolite crystallite. In interference
microscopy, this is achieved by observing interference
between the light beam passing the crystallite and an adjacent
one, which remains unaffected by intracrystalline concentration changes. This procedure is schematically shown on the
right side of Fig. 4. The left side shows this central part as a
constituent of the whole device.

Figure 4. Experimental arrangement for the measurement of intracrystalline
transport diffusion by interference microscopy, from [20].

RL

n…x; y; t†dz

4 Structure Analysis by Diffusion Measurement
4.1 Pore Architecture of MCM-41 Investigated by PFG NMR

The optical path length results as the integral,
s…x; y; t† ˆ

such cases the direction of observation (i.e. the z coordinate) is
chosen to be perpendicular to the channel direction,
molecular concentrations, and hence the respective indexes
of refraction do not depend anymore on the z coordinate and
there is no need for integration.
An alternative possibility to determine intracrystalline local
concentrations exists with cubic crystallites. Owing to the
internal symmetry of the concentration profiles during
adsorption, the ªinverseº problem of calculating the refractive
indexes n(x,y,z;t) and hence the concentrations c(x,y,z;t) from
the integrals s(x,y,t) may be exactly solved [20]. As an
example, Fig. 5 shows the concentration profiles during
uptake of methanol on a zeolite crystallite of type NaCaA,
which have been determined in this way.
As expected, the new equilibrium values of concentration
over a plane close to the external crystallite faces (Fig. 5a) are
much faster than those attained in a central plane (Fig. 5c).
Knowing the spatial and temporal dependence c(x,y,z;t) of
intracrystalline diffusion, Fick's second law (Eq. (6)) allows
the direct determination of the coefficient of transport
diffusion. In the present case, quantitative analysis yielded a
value of (8±2)”10±14 m2s±1, i.e. essentially independent of the
concentration.

(11)

0

over the refractive index n along the direction of the beam
(viz. along the z coordinate) within a pixel with the
coordinates x and y. L denotes the extension of the crystallite
in z-direction. Therefore, the optical density n(x,y,z;t) and
hence the concentration c(x,y,z;t) at position x,y,z are not
directly accessible. There is a trivial possibility of determining
the local concentrations if ± as in the case of zeolites ZSM-12,
-22, -23, 48 und AlPO4-5, -8, -11 ± the intracrystalline pore
system consists of one-dimensional channels, which permit
molecular propagation to occur only in one dimension. If in

With the synthesis of ordered mesoporous materials [32],
sorption research and technology had to take notice of a
further important field of application. Though for most
systems the principal structure is well known, numerous
structural features are still unclear, in particular the pore
architecture of mesoporous materials of type MCM-41. They
consist of hexagonally arranged channels with diameters in
the range of nanometers. There is essentially nothing known,
however, about the longitudinal extension of the channels and
about the permeability of the channel walls. Recent PFG
NMR diffusion studies using water as a probe molecule
yielded initial information [33]. Fig. 6 shows typical examples
of the measured attenuation curves of the PFG NMR signals.

Figure 5. Concentration profiles c(x,y,z;t) at
times t = 0, 40, 80 and 160 s (from bottom to
top) after onset of adsorption of methanol on a
zeolite crystallite of type NaCaA in different
planes through the crystallite, from [20].

Chem. Eng. Technol. 25 (2002) 8,

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

0930-7516/02/0808-0773 $ 17.50+.50/0

773

Review

Figure 6. Examples of measured PFG NMR attenuation curves for water in
MCM-41 and comparison with the dependencies to be expected for onedimensional diffusion (broken lines in Fig. 6a) and for diffusion anisotropy with
rotational symmetry (Fig. 6b) [33].

Figure 7. Arrhenius plots of the components of the diffusion tensor of water in
MCM-41 in the radial (d) and axial (j) directions, and comparison with the
dependence of free water (full line) [33].

According to Eq. (9), in the case of normal isotropic
diffusion, a semi-logarithmic plot of the signal amplitude vs.
the squared gradient intensity should yield a straight line. As
expected, Fig. 6 does not display such behavior. However,
neither do the measured dependencies reflect the case of onedimensional diffusion, as would be expected for infinitely
extended straight channels with impenetrable walls (curves I
and II in Fig. 6a). Rather, they are perfectly compatible with
the behavior expected for a diffusion tensor of rotational
symmetry (cf. Fig. 6b), where the component in the axial
direction represents the diffusivity in the mean channel
direction and the radial component stands for the mean
diffusivity vertical to the mean channel direction. Comparison
of these two components with the diffusivity of free water
(Fig. 7) yields the following conclusions:
(i) Though the channel diameters in the studied samples
(~ 3 nm) exceed the diameter of the water molecules by more
than one order of magnitude (hence the rate of molecular
propagation in channel direction should be expected to be
close to that in the free liquid), the diffusivities are found to be
smaller by one order of magnitude. This indicates the presence
of constrictions in the channel system, whose separations
should be notably smaller than the shortest diffusion paths
(~ 1 lm) observed in these experiments.
(ii) Mass transfer may also occur perpendicular to the mean
direction of the channel axes, even if at a rate reduced by an
additional order of magnitude. There essentially exists three
different explanations for this behavior: Firstly, the channel
walls might be penetrable for the water molecules. Secondly,
propagation in the direction perpendicular to the channel
direction could also be possible, if the channels consist of
segments, which allow molecular exchange between different

channels on their margins. Finally, the observed behavior may
even be expected for infinitely extended, perfect channels, if
the channel axis deviates from a straight line. The present data
is not sufficient to discriminate between these different
explanations.

774

0930-7516/02/0808-0774 $ 17.50+.50/0

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

4.2 Intracrystalline Transport Barriers in Zeolites of Type MFI
The existence of internal transport resistances has often
been referred to, in order to explain the discrepancy between
the diffusion results of different techniques [6±8]. Their
influence should be negligibly small for microscopic techniques, if their separation were much larger than the
monitored molecular displacements. In macro- and mesoscopic measurements, however, such barriers might lead to
substantial rate reductions. It should be possible to confirm
the validity of such an assumption by means of PFG NMR,
since according to Eq. (9), in this case the effective diffusivity
should decrease with increasing observation time. Most recent
progress in the instrumentation of PFG NMR [34] enabled the
observation of such a behavior. For example, Fig. 8 shows the
dependence of the effective diffusivity of n-butane in silcalite1, on the magnitude of the covered diffusion path. As is
expected for unrestricted diffusion, the diffusivities are seen to
be independent of the length of the diffusion path, for the
highest temperatures. Obviously, possible transport resistances may still easily be overcome owing to the thermal
energy of the diffusants. With decreasing temperature (and for
the resultant smaller diffusion paths), the diffusivity is found
to drop notably with increasing diffusion path lengths. This
Chem. Eng. Technol. 25 (2002) 8

Review
concentration profiles are referred to, are situated in the
centre of the crystallite (left column, Figs. 9A, 9C and 9E) and
at about one quarter (right column, Figs. 9B, 9D and 9F),
respectively. It turns out that the measured profiles are in
satisfactory agreement with the simulations, if they are based
on the existence of internal transport resistances (Figs. 9C and
9D). Simulations under the assumption of fast mass transfer
along the interfaces, however, leads to completely different
patterns (Figs. 9E and 9F). Therefore, this possibility may
definitely be ruled out for the investigated system.

5 Structure-Mobility Relations
5.1 Correlated Diffusion Anisotropy

Figure 8. Dependence of PFG NMR diffusivities on the mean diffusion path for
n-butane in two different samples of silicalite-1 (open and filled symbols) at
213 K (s,d), 297 K (,,.), and 383 K (h,j) [34].

may only be explained by assuming the existence of transport
resistances in the interior of the MFI crystals. The exploration
of their structural origin is one of the important tasks of
current zeolite research.

4.3 Diffusion Measurements by Interference Microscopy
for Elucidating Structural Features of MFI-Type Crystallites
PFG NMR is able to record the probability distribution of
molecular displacements over the entire sample. In principle,
by combination with NMR imaging [29], it is even possible to
attain a spatial resolution of these probability distributions.
However, this resolution is far too insufficient, to attribute this
distribution to different locations within the individual
crystallites. Such an attribution, however, is possible in the
case of interference microscopy.
Irrespective of the fact that apparently ideally shaped
crystallites of type MFI are well known to occur as twins [5],
the role of the internal interfaces on mass transfer has so far
remained unclear. Two extreme cases are feasible: Firstly, the
diffusants may be expected to propagate very fast along the
internal interfaces, similar to grain boundary diffusion in
polycrystalline solids [1]. However, on the other hand, these
interfaces may represent internal transport resistances.
Eventually, by monitoring concentration profiles during
uptake by such crystallites, it has now become possible to
discriminate between these two cases [35]. As an example,
Fig. 9 compares the time dependence of the recorded
concentration profiles of iso-butane with the results of
simulations in dependence on a time parameter t. The crystal
extensions in the plain of projection are 108 lm and 18 lm in z
and y direction, respectively. Therefore, the cuts to which the

Chem. Eng. Technol. 25 (2002) 8,

The well-defined pore structure of crystalline nanoporous
solids generates a number of unique properties of molecular
mass transfer, which can be of great relevance for the technical
application of these materials.
Most of these nanoporous crystalline solids are of non-cubic
symmetry so that the diffusivity in different directions
assumes different values. For a number of structure types, it
may be shown that the given pore geometry leads to
correlations between the diffusivities in different directions.
This is particularly true for MFI zeolites. Being among the
most promising candidates for the fabrication of zeolite
membranes, their transport properties are of special practical
relevance [5]. With respect to an optimum crystal orientation
within the membranes, diffusion anisotropy is one of the key
factors for their practical use. Zeolites of type MFI consist of
a network of mutually intersecting ªstraightº channels in the
y- and ªzig-zagº channels in the x-direction. There is no
channel system in z-direction. Mass transfer in the z-direction
has to proceed, therefore, by mutually alternating periods of
displacements along elements of the straight and zig-zag
channels. This is due to the fact that, in addition to
displacements in the x-direction, the elements of the zig-zag
channels also allow alternating shifts in the (+z)- and (±z)directions.
Under the assumption that molecular passages from
channel intersection to channel intersection are completely
uncorrelated, i.e. that the correlation time of molecular
propagation (the particle ªmemoryº) is smaller than the
mean migration time from channel intersection to channel
intersection, the described coupling mechanism of displacements in different directions yields the relation [6,36,37],
c2
a2
b2
ˆ
‡
Dz Dx Dy

(12)

where a, b and c denote the extensions of the unit cell in the x-,
y- and z-directions, and Dx, y, z represents the corresponding
diffusivity. For a number of systems, the correlation formula
(12) is found to be in satisfactory agreement with the results of
MD simulations [36].

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

0930-7516/02/0808-0775 $ 17.50+.50/0

775

Review

Figure 9. Concentration profiles in the longitudinal extension of a zeolite crystallite of type silicalite-1 during
adsorption of iso-butane, as observed by interference microscopy (A, B) and comparison with the
corresponding results of dynamic Monte Carlo simulations by assuming that the internal interface (C, D)
exclusively acts as a transport barrier or (E, F) allows a fast uptake of sorbate from the surrounding atmosphere.
The left column (A, C, E) displays the values along a plane in the middle of the crystallite, the plane considered
in the right column (B, D, F) is shifted by about one fourth towards the external face [35].

rather than in proportion to t, as
required by Eq. (3) for normal
diffusion. The parameter F is the
analogue of the self-diffusivity in the
Einstein relation (3) and is commonly referred to as the mobility
factor of single-file diffusion. The
validity of Eq. (13) has been confirmed for a series of probe molecules in zeolites of type AlPO4-5
[38].
From a practical point of view, the
molecular mean lifetime in the interior of the zeolite crystallites deserves
particular interest. Molecular life
times should be as small as possible
to guarantee a sufficiently fast exchange between the product molecules in the crystallite interior and the
reactant molecules in the surrounding, or to allow the shortest possible
residence times, for the separation of
the different mixture components in
multi-component systems. Again, the
dependence of this important parameter on the crystallite extension under
the conditions of normal diffusion is
completely different from that for
single-file systems. In the case of
normal diffusion, this dependence
may quite easily be derived by realizing that the only parameters responsible for the rate of molecular exchange are the crystal extension L,
and the diffusivity D. Therefore, by
reasons of dimensionality, the intracrystalline mean life time has to obey
the relation,
sintra µ

5.2 Single-File Diffusion
In contrast to zeolites of the type MFI, which possess two
systems of mutually intersecting channels, the pore systems
of a number of zeolites consist of only one system of parallel
channels. As soon as the molecular diameters exceed the
channel radii, adjacent molecules cannot further mutually
exchange their positions. Therefore, the sequence of their
positions has to be preserved. Mass transfer under such
conditions has been termed single-file diffusion [1]. Molecular mean square displacement in an infinitely extended
single-file system increases with the square root of the
observation time,

2 
p
z …t† ˆ 2F t
776

(13)

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

L2
D

(14)

For single-file systems, analogous
reasoning would predict the mean
residence time to increase with the fourth power of the crystal
size. In this reasoning, however, one would disregard a second
mechanism of molecular displacement, which with increasing
observation times soon becomes the dominant factor [39].
This mechanism is based on the particle exchange on the
marginal sites of the system. Each elementary act of
adsorption or desorption on these sites may give rise to a shift
of the whole particle chain, and hence of each of its elements.
Since subsequent processes of adsorption and desorption are
uncorrelated, so also are the associated chain displacements.
This means, that the latter have to obey the laws of normal
diffusion, yielding total displacements increasing linearly with
increasing observation times. In the infinitely extended single0930-7516/02/0808-0776 $ 17.50+.50/0

Chem. Eng. Technol. 25 (2002) 8

Review
file system, however, subsequent displacements are correlated, since they are performed with a higher probability in
opposite directions than in the same direction. As a
consequence of this correlation, the mean square displacement increases only with the square root of the observation
time. Therefore, in finite single-file systems, molecular
displacements are very soon dominated by the other
mechanism. It may be described by an effective diffusivity
[39], which is inversely proportional to the length L of the
single-file system. With Eq. (14), this yields
sintra µ L3

(15)

Molecular exchange in porous materials of single-file
structure is thus found to be even more strongly inhibited
with increasing particle size (and hence the transport
resistance to mass conversion and separation even more
enhanced) than it would be in the case of the conditions
imposed by ordinary diffusion.

5.3 Molecular Traffic Control
The first syntheses of zeolites with different channel systems
(MFI ± cf. Sec. 5.1) gave rise to a controversial discussion
about the possibility of enhancing the efficiency of catalytic
processes, by offering different diffusion paths to the reactant
and product molecules [40]. A first quantitative evaluation of
this possibility has recently been proposed by considering
quadratic networks of single-file systems [41]. It has been
assumed that the reactant and product molecules may either
be located in all channel systems with equal probability,
(ªreferenceº system) or that, e.g. the reactant molecules are
exclusively accommodated by one set of parallel channels,
while the product molecules can only be accommodated by
the other, perpendicularly oriented, channel system (ªmolecular traffic control (MTC)º system). In both cases, reactions
may only take place at the channel intersections. For simplicity
it is assumed that all reactions are irreversible and that the
properties of the reactant and product molecules are otherwise identical.
Fig. 10 displays the ratio of the reaction rates in the MTC
and the reference system within a network of 5 ” 5 single-file
channels at a mean site occupancy of 0.1 and a reaction
probability of 0.01 per jump attempt at the channel intersections. This ratio increases dramatically with the increasing
number l of lattice sites between the channel intersections. In
this way, for the first time, it could be quantitatively confirmed
that a mutual exclusion of diffusion paths during catalytic
reactions in nanoporous materials may in fact give rise to an
increase in the overall reactivity.
The increase in the effective reactivity in the MTC system
may be rationalized by the peculiarities of mass transfer under
single-file conditions. According to Eq. 15, the mean life time
in single-file systems increases with the third power of its
extension. One may use this relation to estimate the mean
Chem. Eng. Technol. 25 (2002) 8,

Figure 10. Dependence of the ratio of the effective reaction rates in a system
subjected to ªMolecular Traffic Controlº and in the corresponding reference
system, on the number l of lattice points between adjacent intersections [41].

time during which a diffusant in the network propagates from
intersection to intersection. This relation does only hold in the
case of tracer or self-diffusion, or for exchange between two
otherwise identical types of molecules, as in the case of
catalytic reaction in the reference system. The mutual
exclusion of accommodation in the MTC system, however,
gives rise to concentration gradients, i.e. into the system along
one array of parallel channels for the reactant molecules, and
out of the system along the other array of parallel channels for
the product molecules. As a consequence, mass transfer in the
MTC system is subjected to the conditions of transport
diffusion rather than of self- or tracer diffusion. Under the
conditions of transport diffusion, however, single-file systems
do not exhibit any peculiarities, since under such conditions,
the fact that molecules are able to exchange their positions is
irrelevant. As a consequence, the mean exchange time
between adjacent intersections then increases with the square
of the distance l (cf. Eq. 14) rather than with the third power.
Since the mean exchange time between adjacent intersections
also determines the mean exchange time between the network
and the surrounding atmosphere, increasing separations l
between adjacent channel intersections must be expected to
lead to much more pronounced transport inhibition in the
reference system, rather than in the MTC system. It is due to
this reason that the ratio of the reaction rates displayed in Fig.
10 experience such a dramatic change in favor of the MTC
system with increasing separation l between adjacent channel
intersections.

Acknowledgement
The novel concepts of mass transfer in porous media
presented, have been decisively promoted by the contributions of NMR spectroscopy to zeolite research. One of the
pioneers of this development is Professor Dr. Dr. Harry
Pfeifer, with whom we had the honour and pleasure to work
over many years. We are deeply obliged to him for advice,

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

0930-7516/02/0808-0777 $ 17.50+.50/0

777

Review
encouragement and sharing of ideas. Financial support of the
Deutsche Forschungsgemeinschaft (SFB 294 and Graduiertenkolleg ªPhysikalische Chemie der Grenzflächenº), Fonds
der Chemischen Industrie und Max-Buchner-Stiftung, is
gratefully acknowledged.
Received: April 24, 2002 [B 6092]

References
[1] J. Kärger, P. Heitjans, R. Haberlandt, Diffusion in Condensed Matter,
Vieweg, Braunschweig/Wiesbaden 1998.
[2] G. Bellussi, C. Perego, Phenol Hydroxylation and Related Oxidations, in
Handbook of Heterogeneous Catalysis (Eds: G. Ertl, H. Knözinger,
J. Weitkamp) Wiley-VCH, Weinheim 1997, pp. 2329±2334.
[3] K. P. Möller, W. Böhringer, A. E. Schnitzler, E. van Stehen, C. T.
O'Conner, The Use of a Jet Loop Reactor to Study the Effect of Crystal
Size and of the co-Feeding of Olefins and Water on the Conversion of
Methanol over HZSM-5, Microporous Mesoporous Mater. 1999, 29, 127.
[4] C. Song, J. Garces, Y. Sugi, Shape-Selective Catalysis, ACS Symp. Ser.
2000, 738.
[5] J. Caro, M. Noack, P. Kölsch, R. Schäfer, Zeolite Membranes ± State of
their Development and Perspectives, Microporous Mesoporous Mater.
2000, 38, 3.
[6] J. Kärger, D. M. Ruthven, Diffusion in Zeolites and Other Microporous
Solids, Wiley, New York 1992.
[7] N. Y. Chen, T. F. Degnan, C. M. Smith, Molecular Transport and
Reaction in Zeolites, Wiley, New York 1994.
[8] F. Keil, Diffusion und Chemische Reaktion in der Gas/Feststoff-Katalyse,
Springer, Berlin 1999.
[9] J. Kärger, G. Fleischer, U. Roland, PFG NMR Studies of Anomalous
Diffusion, in [1], pp. 144±168.
[10] M. Bülow, A. Micke, Determination of Transport Coefficients in
Microporous Solids, Adsorption 1995, 1, 29.
[11] Y. Yasuda, Frequency Response Method for Investigation of Gas/
Surface Dynamic Phenomena, Heterog. Chem. Rev. 1994, 1, 103.
[12] D. M. Ruthven, S. Brandani, in Recent Advances in Gas Separation by
Microporous Ceramic Membranes, (Ed: Molecular Transport and
Reaction in Zeolites N. Kanellopoulos), Elsevier, Amsterdam 2000, 187.
[13] V. Bourdin, A. Germanus, P. Grenier, J. Kärger, Application of the
Thermal Frequency Response Method and of Pulsed Field Gradient
NMR to Study Water Diffusion in Zeolite NaX, Adsorption 1996, 2, 205.
[14] R. R. Schumacher, B. G. Anderson, N. J. Noordhock, F. J. M. M. de
Gauw, M. de Jong, M. J. A. de Voigt, R. A. van Santen, Tracer
Exchange Experiments with Positron Emission Profiling: Diffusion in
Zeolites, Microporous Mesoporous Mater. 2000, 35, 301.
[15] O. P. Keipert, M. Baerns, Determination of the Intracrystalline
Diffusion Coefficients of Alkanes in H-ZSM-5 Zeolite by a Transient
Technique Using the Temporal-Analysis-of-Products (TAP) Reactor,
Chem. Engin. Science 1998, 53, 3623.
[16] H. G. Karge, W. Niessen, A New Method for the Study of Diffusion and
Counter Diffusion in Zeolites, Catal. Today 1991, 8, 451.
[17] W. O. Haag, R. M. Lago, P. B. Weisz, Transport and Reactivity of
Hydrocarbon Molecules in a Shape-Selective Zeolite, Faraday Discuss.
1982, 72, 317.
[18] M. Hermann, W. Niessen, H. G. Karge, in Catalysis by Microporous
Materials (Eds: H. K. Beyer, H. G. Karge, I. Kiricsi, J. B. Nagy), Elsevier,
Amsterdam 1995, 131.
[19] D. T. Hayhurst, A. R. Paravar, Diffusion of C1 to C5 Normal Paraffins in
Silicalite, Zeolites 1988, 8, 27.

[20] U. Schemmert, J. Kärger, J. Weitkamp, Interference Microscopy as a
Technique for Directly Measuring Intracrystalline Transport Diffusion
in Zeolites, Microporous Mesoporous Mater. 1999, 32, 101.
[21] R. Kimmich, W. Unrath, G. Schnur, E. Rommel, NMR Measurements of
Small Self-Diffusion Coefficients in the Fringe Field of Superconducting
Magnets, J. Magnet. Reson. 1991, 91, 136.
[22] H. Jobic, in Recent Advances in Gas Separation by Microporous Ceramic
Membranes (Ed: N. Kanellopoulos), Elsevier, Amsterdam 2000, 109.
[23] O. Isfort, B. Boddenberg, F. Fujara, R. Grosse, Molecular Dynamics of
Benzene in Zeolite NaY Studied by 2D Deuteron NMR, Chem. Phys.
Lett. 1998, 288, 71.
[24] D. N. Theodorou, R. Q. Snurr, A. T. Bell, in Comprehensive
Supramolecular Chemistry (Eds: G. Alberti, T. Bein), Pergamon, Oxford
1996, 507.
[25] D. M. Ruthven, M. F. M. Post, in Introduction to Zeolite Science and
Practice (Eds: H. van Bekkum, E. M. Flanigen, J. C. Jansen), 2nd Ed.,
Elsevier, Amsterdam 2001.
[26] J. Kärger, in Handbook of Heterogeneous Catalysis (Eds: G. Ertl,
H. Knözinger, J. Weitkamp) Wiley-VCH, Weinheim 1997, 1252.
[27] J. Kärger, D. M. Ruthven, in Handbook of Porous Solids (Eds: F. Schüth,
K. S. W. Sing, J. Weitkamp), Wiley-VCH, Weinheim 2001.
[28] J. Kärger, in Proc. of the 12th Int. Zeolite Conf. (Eds: M. M. J. Treacy,
B. K. Marcus, M. E. Bisher, J. B. Higgins), Material Research Society,
Warrandale 1999, 35.
[29] K. H. Hausser, H. R. Kalbitzer, NMR für Medizin und Biologie,
Springer, Berlin 1989.
[30] J. Kärger, H. Pfeifer, NMR-Self-Diffusion Studies in Zeolite Science
and Technology, Zeolites 1987, 7, 90.
[31] F. J. Keil, R. Krishna, M. O. Coppens, Modeling of Diffusion in Zeolites,
Rev. Chem. Engin. 2000, 16, 71.
[32] 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, J. L. Schlenker, A New Family of Mesoporous Molecular Sieves
Prepared with Liquid Crystal Templates, J. Am. Chem. Soc. 1992, 114,
10834.
[33] F. Stallmach, J. Kärger, C. Krause, M. Jeschke, U. Oberhagemann,
Evidence of Anisotropic Self-Diffusion of Guest Molecules in Nanoporous Materials of MCM-41 Type, J. Am. Chem. Soc. 2000, 122, 9237.
[34] S. Vasenkov, W. Böhlmann, P. Galvosas, O. Geier, H. Liu, J. Kärger,
PFG NMR Study of Diffusion in MFI-Type Zeolites: Evidence of the
Existence of Intracrystalline Transport Barriers, J. Phys. Chem. B 2001,
105, 5922.
[35] O. Geier, S. Vasenkov, E. Lehmann, J. Kärger, U. Schemmert, R. A.
Rakoczy, J. Weitkamp, Interference Microscopy Investigation of the
Influence of Regular Intergrowth Effects in MFI-Type Zeolites on
Molecular Uptake, J. Phys. Chem. B 2001, 105, 10217.
[36] R. Haberlandt, J. Kärger, Molecular Dynamics under the Confinement
by the Host Lattice in Zeolitic Adsorbate-Adsorbent Systems, Chem.
Eng. J. 1999, 74, 15.
[37] J. Kärger, Random Walk through Two-Channel Networks: A Simple
Means to Correlate the Coefficients of Anisotropic Diffusion in ZSM-5
Type Zeolites, J. Phys. Chem. 1991, 95, 5558.
[38] V. Kukla, J. Kornatowski, D. Demuth, I. Girnus, H. Pfeifer, L. V. C.
Rees, S. Schunk, K. K. Unger, J. Kärger, NMR Studies of Single-File
Diffusion in Unidimensional Channel Zeolites, Science 1996, 272, 702.
[39] C. Rödenbeck, J. Kärger, Length and Occupancy Dependence of the
Tracer Exchange in Single-File Systems, J. Phys. Chem. 1999, 110, 3970.
[40] E. G. Derouane, Z. Gabelica, A Novel Effect of Shape Selectivity:
Molecular Traffic Control in Zeolite ZSM-5, J. Catal. 1980, 65, 486.
[41] N. N. Neugebauer, P. Bräuer, J. Kärger, Reactivity Enhancement by
Molecular Traffic Control, J. Catal. 2000, 194, 1.
This paper was also published in German in Chem. Ing. Tech. 2001, 12, 1517.

_______________________

778

Ó WILEY-VCH Verlag GmbH & Co. KG aA, Weinheim, 2002

0930-7516/02/0808-0778 $ 17.50+.50/0

Chem. Eng. Technol. 25 (2002) 8