Unidirectional and single-file diffusion of molecules in one-dimensional channel systems. A quasi-elastic neutron scattering study

Article abstract of DOI:10.1021/jp970773i

Translational diffusion of molecules in one-dimensional channel systems has been measured by quasi-elastic neutron scattering. The scattering function for a single-file diffusion model has been derived in order to - differentiate this transport model from normal ID diffusion. In the AlPO₄-5 structure, ordinary ID diffusion is observed for methane and ethane, whatever the loading, which indicates that the molecules are able to pass each other. The diffusion coefficients for both molecules are on the order of 10^-9 m² s^-1. On the other hand, a quite different regime is obtained by varying cyclopropane concentration in the same structure. At low loading, normal ID diffusion is observed because the molecules can be considered as isolated, but at higher concentration cyclopropane is found to follow single-file diffusion. However, the mobility is too small to be determined. In the less open channel system of ZSM-48, ordinary ID diffusion is found for small methane concentration, the diffusion coefficient being 2.5 x 10^-9m²at 155 K.At medium loading, single-file diffusion is observed with a mobility factor of 2 x 10-¹² m² s^-1/2. This is the first time that this technique has provided experimental evidence of single-file diffusion in zeolites.

Full text of DOI:10.1021/jp970773i

5834
J. Phys. Chem. B 1997, 101, 5834-5841
Unidirectional and Single-File Diffusion of Molecules in One-Dimensional Channel Systems.
A Quasi-Elastic Neutron Scattering Study
Herve´ Jobic,*,† Karsten Hahn,‡ Jo1rg Ka1rger,‡ Marc Be´e,§ Alain Tuel,† Manfred Noack,|
Irina Girnus,| and Gordon J. Kearley
Institut de Recherches sur la Catalyse, CNRS, 2 AVenue Albert Einstein, 69626 Villeurbanne, France, Fakulta¨t
fu¨r Physik und Geowissenschaften, UniVersita¨t Leipzig, Linne´strasse 5, 04103 Leipzig, Germany, Laboratoire
de Spectrome´trie Physique, UniVersite´ Joseph Fourier, 38402 St. Martin d’He`res, France, Institut fu¨r
Angewandte Chemie, Rudower Chaussee 5, 12484 Berlin, Germany, and Institut Laue-LangeVin,
BP 156, 38042 Grenoble, France
ReceiVed: March 3, 1997; In Final Form: May 23, 1997X
Translational diffusion of molecules in one-dimensional channel systems has been measured by quasi-elastic
neutron scattering. The scattering function for a single-file diffusion model has been derived in order to
differentiate this transport model from normal 1D diffusion. In the AlPO4-5 structure, ordinary 1D diffusion
is observed for methane and ethane, whatever the loading, which indicates that the molecules are able to pass
each other. The diffusion coefficients for both molecules are on the order of 10-9 m2 s-1. On the other
hand, a quite different regime is obtained by varying cyclopropane concentration in the same structure. At
low loading, normal 1D diffusion is observed because the molecules can be considered as isolated, but at
higher concentration cyclopropane is found to follow single-file diffusion. However, the mobility is too
small to be determined. In the less open channel system of ZSM-48, ordinary 1D diffusion is found for
small methane concentration, the diffusion coefficient being 2.5 × 10-9 m2 s-1, at 155 K. At medium loading,
single-file diffusion is observed with a mobility factor of 2 × 10-12 m2 s-1/2. This is the first time that this
technique has provided experimental evidence of single-file diffusion in zeolites.
Introduction
channels run along the y axis, and sinusoidal channels run along
the x axis. Diffusion in the z direction can only be achieved
via alternating migration in the x and y directions. The
experimentally measured anisotropy factor (1/2)(Dx + Dy)/Dz
was found to be close to 5.10 The diffusion of methanol11 and
of a series of n-alkanes, methane, ethane, propane, butane, and
hexane,12-14 was studied by QENS in the MFI structure, but
only the mean diffusivity could be extracted from the data.
In Na-mordenite (structure type MOR4), the influence of one-
dimensional diffusion on the QENS profiles of diffusing benzene
was recognized,15 but the calculated curves were almost
indistinguishable from the usual Lorentzian profiles that are
observed for isotropic diffusion. This was due to the resolution
of the spectrometer, which was not high enough, or alternatively
to the slow mobility of benzene. Another attempt was made in
the MOR structure by measuring methane diffusion.16 However,
a large proportion of methane molecules were found to be
trapped in the side pockets, in agreement with Monte Carlo
simulations.17 The weak signal due to molecules diffusing in
the channels did not justify an analysis of the data in terms of
anisotropic diffusion.
The catalytic and separation properties of zeolites are critically
dependent on diffusion and molecular sieving effects. In
particular, modeling the performance of zeolite membranes
requires a detailed description of gas adsorption and transport
at the microscopic level. Therefore, much effort is still devoted
to measuring diffusion coefficients in zeolites and to try to
understand the discrepancies between the different techniques.1
The two methods that are now available to elucidate the
mechanism of diffusion inside zeolite crystals are pulsed-field
gradient NMR (PFG NMR) and quasi-elastic neutron scattering
(QENS). These two methods are complementary in that
molecular migration is followed over a few unit cells with QENS
(time scale ≈ 1 ns) and over the whole crystal with PFG NMR
(time scale ≈ 1 ms).
The regular structure of zeolite frameworks has allowed
molecular dynamics (MD) simulations to be performed for
alkanes as long as n-hexane,2 and new methodologies have been
recently described3 to determine diffusivities of chains as long
as C20 in silicalite (structure type MFI4). These simulations
were found to predict diffusivities more in line with microscopic
experiments than with macroscopic measurements.
Up to now, the QENS technique has been mainly used to
study translational dynamics in 3D systems: methane,5 ethane,6
and benzene7 in NaX and NaY zeolites, and methane8 and
hydrogen9 in NaA. In the MFI structure, the two intersecting
channel system leads to diffusion anisotropy, the straight
In order to measure only one-dimensional diffusion and
possibly single-file diffusion, two 1D structures without pockets
were selected. Firstly, we have considered AlPO4-5 since PFG
NMR measurements have been carried out on this zeolite.18,19
In the literature, the value that is quoted for the free diameter
of the cylindrical channels in AlPO4-5 is 7.3 Å.4 It is based on
an oxygen radius of 1.35 Å. However, recent experimental20,21
and simulation22 studies indicate that up to six methane
molecules can be adsorbed per unit cell. The free diameter of
AlPO4-5 would then be much larger than the nominal value (it
is estimated as 8.2 Å in ref 21 in the presence of methane
molecules). The kinetic diameter of methane is 3.8 Å;23 it is
the intermolecular distance of closest approach for two mol-
* Author for correspondence.
† Institut de Recherches sur la Catalyse.
‡ Universita¨t Leipzig.
§ Universite´ Joseph Fourier.
| Institut fu¨r Angewandte Chemie.
Institut Laue-Langevin.
X Abstract published in AdVance ACS Abstracts, July 1, 1997.
S1089-5647(97)00773-6 CCC: $14.00 © 1997 American Chemical Society

Diffusion of Molecules in 1D Channel Systems
J. Phys. Chem. B, Vol. 101, No. 30, 1997 5835
ecules colliding with zero initial kinetic energy.23 Therefore,
methane molecules could pass each other in this structure.
Unidirectional but otherwise ordinary diffusion along the
channel axis has indeed been found in recent MD simulations
for methane in this structure.24 On the other hand, larger
molecules such as cyclopropane have no possibility at all to
change their relative order during the observation time. Trans-
port under these conditions has been termed single-file diffu-
sion,25,26 and it is expected to have a profound influence on
catalytic reaction rates since a molecule can only be desorbed
after all other molecules adsorbed up to the channel orifice have
diffused away.26-29 In order to test the concept that a tight fit
between the size of the molecule and the dimension of the
channel might enhance the mobility,30 a second structure was
chosen, ZSM-48. This high-silica zeolite has linear 10-ring
channels with smaller dimension: 5.3 × 5.6 Å.31
20H2O, was transferred into an autoclave and heated at 453 K
with stirring (300 rpm, magnetic stirrer) for 3 days. The
autoclave was then cooled and the solid recovered by centrifu-
gation. After washing with distilled water, the zeolite was dried
at 383 K overnight and calcined at 823 K in air for 10 h. X-ray
diffraction indicated that the zeolite was pure ZSM-48 without
any extra phases. Crystals were in the form of elongated needles
of dimension ca. 2 × 0.1 µm.
The samples were degassed to 10-3 Pa, at 700 K. They were
transferred, inside a glovebox, into slab-shaped aluminum
containers. Since the two zeolites do not contain protons, the
neutron transmission of the dehydrated samples was excellent,
≈99%. The cells were connected to a gas inlet system so that
adsorption could be performed in situ, even at low temperature.
The estimated accuracy on the loadings determined during the
neutron experiments is 20%.
Gravimetric Adsorption. The measurements were made on
an electronic Sartorius microbalance, with a resolution of 10
µg. Gas pressure was recorded by MKS Baratron capacitive
gas pressure sensors. While recording the isotherms of cyclo-
propane in AlPO4-5, we tried to desorb cyclopropane at room
temperature over 15 h in the apparatus, at 10-3 Torr. This was
not possible; the reason is not known (slow diffusion or
residues). Therefore, a new calcination of the AlPO4-5 crystals
was performed out of the apparatus after each isotherm.
QENS results are reported here on the diffusivities of
methane, ethane, and cyclopropane in AlPO4-5. The scattering
function for ordinary 1D diffusion exists, but single-file diffusion
has not been yet studied theoretically in neutron scattering. We
have therefore derived the incoherent scattering function for the
single-file diffusion model. Additional results obtained on
methane diffusion in ZSM-48 are also presented. In this
structure, it is impossible for methane molecules to pass each
other. Since single-file diffusion has a larger sorbate concentra-
tion dependence compared to ordinary diffusion, it was preferred
to vary the loading rather than the temperature during the limited
allocated beam-time.
Theory of Neutron Scattering
For hydrogenous molecules, the incoherent scattering from
hydrogen dominates. The measured intensity is proportional
to the incoherent scattering function, Sinc(Q,ω), which is the
space and time Fourier transform of a self-correlation function,
Gs(r,t):33
Experimental Section
Neutron Spectrometer. The neutron experiments were
carried out at the Institut Laue-Langevin, Grenoble, using the
time-of-flight (TOF) spectrometer IN5. This spectrometer has
a good resolution, it covers the low-momentum transfer region,
and it has a reasonable neutron flux. Since the calculated energy
profile for single-file diffusion has a very narrow peak but very
broad wings (vide infra), a large dynamic range is also required.
The incident neutron wavelength was taken as 9.0 Å (1.01 meV).
This is a compromise between a long wavelength giving higher
resolution and a small wavelength giving higher flux. In our
case, the elastic resolution is 18 µeV (full-width at half-
maximum). After scattering by the sample, neutrons are
S (Q,ω) ) (1/2π)
dr dt exp[i(Q‚r - ωt)]G (r,t) (1)
s
∫∫
inc
In our case, the self-correlation function, also called propaga-
tor in NMR, is the probability density to find a hydrogen atom
at position r at time t if the same atom was at the origin at time
zero. The hydrogen atoms of the different adsorbates experience
several molecular motions: translation, rotation, and vibration.
The usual approximation is to consider that the molecular
motions are uncoupled and can be treated separately. The total
scattering function is then the convolution product of the
individual scattering functions:
3
analyzed in 1000 He detector tubes as a function of time and
angle. The TOF of the scattered neutrons is related to their
energy transfer and the scattering angle to the momentum
transfer. Spectra from different detectors are grouped to obtain
adequate counting statistics, excluding the Bragg peaks and the
small-angle scattering from the zeolite. The TOF spectra were
transformed to an energy scale after subtracting the scattering
of the bare zeolite.
Samples. Almost single crystals of AlPO4-5 (20 µm) were
prepared using microwave heating during synthesis.32 Larger
crystals of this aluminophosphate could have been prepared
using the same method, but it was found that crystallization
from a diluted gel yielded slightly smaller but very uniform
crystals low in defects. The crystals were calcined at 973 K.
For the neutron experiment, 2.2 g were used.
The ZSM-48 sample was synthesized hydrothermally using
hexamethonium hydroxide, HM(OH)2, as templating agent.
Typically, 0.3 mL of tetraethyl orthosilicate (TEOS, Aldrich)
was hydrolyzed with 60 mL of a 1 M solution of HM(OH)2 for
about 30 min with stirring. Distilled water (60 mL) was then
added, and the temperature was increased to 353 K. The
mixture was kept at this temperature for 2 h with stirring and
the precursor gel, with the composition SiO2-0.2HM(OH)2-
trans
inc
rot
inc
vib
inc
Sinc(Q,ω) ) S (Q,ω) X S (Q,ω) X S (Q,ω) (2)
where pQ is the neutron momentum transfer. It is defined as
Q ) k - k0, where k and k0 are the scattered and incident
wave vectors, respectively. Similarly, pω is the neutron energy
transfer; we limit our analysis to the range (0.4 meV. Since
this energy transfer is too small to excite the internal vibrational
modes of the molecules studied here, the vibrational term in eq
2 consists only in a Debye-Waller factor which has the effect
of decreasing the intensity of the spectra at increasing Q values.
To describe the rotational behavior of methane and cyclo-
propane, the isotropic rotational diffusion model is well adapted;
this model has already been used for methane adsorbed in ZSM-
5.12 For ethane, a less isotropic model, in which the molecules
perform continuous rotational diffusion about the main molec-
ular axis, is more adapted; this model was used in the case of
ethane adsorption in ZSM-5.13
The most interesting part is the translational scattering
function. The normal 1D diffusion and the single-file diffusion
will be examined separately.

5836 J. Phys. Chem. B, Vol. 101, No. 30, 1997
Jobic et al.
Normal 1D Diffusion. Let us consider a molecule diffusing
in a 1D channel, with a diffusion coefficient D. If this channel
makes an angle θ with the direction of Q, the scattering function
for this particular channel will be15
S1D(Q,ω) ) (1/π)DQ2 cos2 θ/[ω2 + (DQ2 cos2 θ)2] (3)
Since our samples are not oriented, one must perform a
powder average of S(Q,ω), which yields
π
S (Q,ω) ) (1/2π) dθ DQ2 cos2 θ sin θ/[ω2 +
∫
1D
0
(DQ2 cos2 θ)2] (4)
Integration leads to the following expression:
2
x
1
1 + y - 2y
S1D(Q,ω) )
ln
+ 2 arc tan(1 +
{ (
)
2
x
x
4π 2ωy
1 + y + 2y
x
x
2y) - 2 arc tan(1 - 2y)
}
with
y2 ) DQ2/ω
(5)
The same expression is obtained when considering a sample
of isotropically distributed 1D channels.34 In both cases, S1D
becomes infinite at ω ) 0. Experimentally however, there is
no discontinuity because of the finite energy resolution. The
simplest way to consider this influence is to convolute eq 3
with the instrumental resolution and then to perform the powder
average numerically. The resulting curve is far from the
Lorentzian found for 3D diffusion. However, in order to show
this effect, the experimental resolution must be high enough.
This is illustrated in Figure 1 for typical values of the
parameters: D ) 10-8 m2 s-1 and Q ) 0.3 Å-1
. It is
Figure 1. Simulated QENS spectra in the case of normal 1D diffusion
for the following parameters: D ) 10-8 m2 s-1, Q ) 0.3 Å-1. The
scattering function is convoluted with a triangular resolution function
of different full-width at half-maximum: (a) 0.2 µeV, (b) 18 µeV (solid
lines). The broken line in part b corresponds to a pure Lorentzian profile.
demonstrated in Figure 1a that for a sufficiently high energy
resolution (triangular, fwhm ) 0.2 µeV) a narrow component
is present. With the energy resolution corresponding to our
experimental conditions (triangular, fwhm ) 18 µeV), one
obtains the profile shown in Figure 1b as a solid line. It is
close to a Lorentzian convoluted with the same resolution,
Figure 1b, dotted line, so that good statistics are required to
differentiate 1D from 3D diffusion.
claiming single-file behavior, including QENS, should show this
strong loading dependence.
In order to derive the scattering function for single-file
diffusion, one has to take the Fourier transform of the self-
correlation function with respect to space and time. As in the
normal 1D diffusion case, a particular channel making an angle
θ with the direction of Q will be considered. Firstly, space
Fourier transformation leads to an intermediate scattering
function
Single-File Diffusion. The self-correlation function for a
particle following single-file diffusion has already been de-
rived.35,36 For a given channel, the self-correlation function was
found to be
1
r2
2 σ2
GS(r,t) )
exp -
(6)
(
)
2π σ2
x
I (Q,t) ) dr G (r,t) exp(iQ‚r)
∫
SF
s
The mean-square displacement of the particle, σ2 , varies in
proportion to t1/2
:
) exp(-FQ2 cos2 θ t1/2)
(8)
σ2 ) 2Ft1/2
(7)
An important difference with the case of normal diffusion
lies in the term t1/2 instead of t. After performing the powder
average, the Fourier transform of I(Q,t) with respect to time
gives the scattering function
where F is the single-file mobility factor.26 In the case of normal
diffusion, the mean-square displacement is proportional to t.
Comparisons have been made with PFG NMR measurements
and with MD simulations.37 A priori, the long-range mobility
should be reduced in single-file systems compared with normal
diffusion because of increased correlations between the mol-
ecules. For the same reason, a higher loading dependence is
expected for single-file diffusion; indeed, F was found to be
proportional to (1 - θ)/θ.26 Therefore, experimental results
SSF(Q,ω) )
π dθ sin θ ∞ dt exp(-FQ2 cos2 θ t1/2) cos(ωt) (9)
1
π
∫
∫
0
0
Integrating over t, one obtains38

Diffusion of Molecules in 1D Channel Systems
J. Phys. Chem. B, Vol. 101, No. 30, 1997 5837
FQ2ω-3/2
π
2
1
2
π
SSF(Q,ω) )
dθ cos2 θ sin θ cos z2
-
∫
[
(
)(
0
x
2π
π
1
2
C(z) + sin z2 - S(z)
)
(
)(
)
]
2
where
F2Q4 cos4 θ
2πω
z2 )
(10)
C and S are the Fresnel integrals as defined in ref 39 (7.3.3 and
7.3.4). Following ref 39, the term in brackets in eq 10 is denoted
as
π
2
1
2
π
2
1
2
g(z) ) cos z2 - C(z) + sin z2 - S(z) (11)
(
)(
)
(
)(
)
For g(z), the rational approximation39
1
g(z) )
+ ꢀ(z) (12)
2 + 4.142z + 3.492z2 + 6.670z3
can be used, where the absolute error of this approximation is
|ꢀ(z)| e 2 × 10-3
(13)
The scattering function then becomes
z3 sin θ dθ
π
2π
SSF(Q,ω) )
∫
F2Q4 cos4 θ(2 + 4.142z + 3.492z2 + 6.67z3)
0
(14)
Again, powder averaging and convolution with a resolution
function have a great influence on the spectral profile. With
the parameters F ) 10-11 m2 s-1/2 and Q ) 0.3 Å-1, one obtains
the QENS spectrum shown in Figure 2a, after convolution with
a narrow energy resolution function (triangular, fwhm ) 0.2
µeV). The peak is even sharper than in the normal 1D diffusion
case. With the actual energy resolution (triangular, fwhm )
18 µeV), the profile is still found to be sharp; cf. the solid line
in Figure 2b. In contrast to the case of ordinary 1D diffusion,
the more waisted peak shape cannot be fitted with a Lorentzian,
dotted line in Figure 2b. Analytical expressions of the scattering
function, for different resolution functions (triangular, Gaussian,
and Lorentzian) will be presented elsewhere.34
It should be stressed that the time scale of the measurement
has a prominent effect on the observed diffusion regime. In
PFG NMR, one is always concerned with the long-time limit
so that single-file diffusion as characterised by eq 7 is observed.
In our QENS experiment, broadenings as large as 1 meV or
down to 10% of the resolution can be resolved so that the time
scale is in the range 1-350 ps. On this time scale, the diffusion
can follow normal 1D diffusion at low loadings since correla-
tions between molecules can be neglected, but at high loadings
single-file diffusion could be measured.
Figure 2. Simulated QENS spectra in the case of single-file diffusion
for the following parameters: F ) 10-11 m2 s-1/2, Q ) 0.3 Å-1. The
scattering function is convoluted with a triangular resolution function
of different full-width at half-maximum: (a) 0.2 µeV, (b) 18 µeV (solid
lines). The broken line in part b corresponds to a pure Lorentzian profile.
1D than with 3D diffusion (the sum of residuals is smaller).
The spectra cannot be fitted with the single-file diffusion model.
The decrease of intensity of the spectra with increasing Q values
is due to the Debye-Waller factor. The diffusion coefficients
for methane are 1.6 × 10-9 m2 s-1 for a concentration of 0.7
molecule/unit cell and 1.2 × 10-9 m2 s-1 for a concentration
of 1 molecule/unit cell. The concentration dependence of the
diffusion coefficient is thus found to be rather moderate (the
experimental error is ≈50%). At the two loadings used, the
molecules cannot be considered to be isolated, especially at
small Q values corresponding to large diffusion paths. The first
grouping of detectors has a mean Q value of 0.19 Å-1 so that
translation is measured over a distance λ ) 2π/Q of 33 Å,
corresponding to 4 unit cells. It appears therefore that methane
molecules are able to cross each other in AlPO4-5 and single-
file diffusion is not observed.
Ethane in AlPO4-5. The diffusion of ethane was studied at
250 K and at loadings of 0.8, 1.4, and 1.6 molecules per unit
cell (uc).
For all loadings, the QENS spectra could be fitted simulta-
neously with normal 1D diffusion convoluted with uniaxial
rotation and with the instrumental resolution. Experimental and
calculated spectra are shown in Figure 4 for the medium loading;
the agreement is quite good. As in the case of methane, the
spectra cannot be fitted with the single-file diffusion model and
the agreement is better with 1D than with 3D diffusion. The
Results
Methane in AlPO4-5. QENS measurements were performed
at 155 K, for the two methane concentrations of 0.7 and 1
molecule per unit tube length (8.417 Å40). Higher loadings were
not considered to avoid signal from the gas phase.
At both loadings, all the QENS spectra could be fitted
simultaneously with normal 1D diffusion convoluted with
isotropic rotation and with the instrumental resolution. The
agreement between experimental and calculated spectra is
reasonable, as shown in Figure 3. The agreement is better with

5838 J. Phys. Chem. B, Vol. 101, No. 30, 1997
Jobic et al.
Figure 5. Adsorption isotherms for cyclopropane in AlPO4-5: T )
250 K (O), 273 K (3), and 306 K (+).
Figure 3. QENS spectra, obtained at different values of the momentum
transfer Q, for methane adsorbed in AlPO4-5 (concentration of 1
molecule per unit cell, T ) 155 K): (9) experimental points, (s) profile
calculated with the normal 1D diffusion model.
Cyclopropane in AlPO4-5. QENS spectra were measured
at 273 K, at concentrations of 0.6 and 1.3 molecule/uc.
Isotherms obtained at different temperatures are shown in Figure
5.
Only the spectra obtained at low loading showed a significant
broadening. All the spectra could be fitted simultaneously with
normal 1D diffusion, convoluted with isotropic rotation and with
the instrumental resolution. The fit is better with 1D than with
3D diffusion. A comparison between the experimental and
calculated spectra for Q ) 0.39 Å-1 is shown in Figure 6a. At
this Q value, there is a large broadening due to normal 1D
diffusion, and only 14% of the signal is due to rotation. The
spectrum cannot be fitted with the single-file diffusion model.
The diffusion coefficient for cyclopropane at this loading is 1.8
× 10-9 m2 s-1
.
At higher loading, the spectrum corresponding to the same
Q value shows only an elastic peak, as can be seen from Figure
6b. In the case of ethane, for an even larger number of
molecules per unit cell, a broadening was observed and the
diffusion was found to follow normal 1D diffusion. For
cyclopropane, the elastic response at high loading can only be
explained by single-file diffusion, indicating that the molecules
cannot pass each other in AlPO4-5. We believe that single-file
diffusion can only be inferred if there is such a large loading
dependence of the diffusivity. However, the mobility is too
small to be determined with this resolution. Fits of the spectra
with eq 14 indicate that the mobility factor F is smaller than
10-12 m2 s-1/2. A broadening is measured at higher Q values,
but one has to then take into account the contribution due to
the rotation.
Methane in ZSM-48. The previous system gives some
evidence that single-file diffusion is occurring, but it is
impossible to extract a mobility factor. Methane in ZSM-48
was selected to obtain additional results on single-file diffusion
and to test whether the mobility of methane would be enhanced
compared with AlPO4-5, due to the better fit between the size
of the molecule and the size of the channel. QENS measure-
ments were first made at 100 K, close to saturation capacity
(≈1.2 molecule per unit length tube). Under these conditions,
no long-range mobility could be observed. The temperature
was then raised to 155 K, and three different loadings were
studied. The isotherm at 155 K was measured volumetrically
Figure 4. QENS spectra, obtained at different values of the momentum
transfer Q, for ethane adsorbed in AlPO4-5 (concentration of 1.4
molecule per unit cell, T ) 250 K): (9) experimental points, (s) profile
calculated with the normal 1D diffusion model.
diffusion coefficients of ethane are 1.4 × 10-9 m2 s-1 for a
concentration of 0.7 molecule/uc, 1.1 × 10-9 m2 s-1 for 1.4
molecule/uc, and 9.0 × 10-10 m2 s-1 for 1.6 molecule/uc,
respectively. Again, the concentration dependence of the
diffusion coefficient is only moderate. As for methane, the
evidence of ordinary 1D diffusion indicates that ethane mol-
ecules are able to pass each other in AlPO4-5. In view of the
similarity of the kinetic diameters of these two molecules, this
finding does not seem to be unreasonable.

Diffusion of Molecules in 1D Channel Systems
J. Phys. Chem. B, Vol. 101, No. 30, 1997 5839
Figure 6. Comparison of experimental (9) and calculated (s) QENS
spectra obtained at 273 K for cyclopropane adsorbed in AlPO4-5: (a)
0.6 molecule per unit cell, profile calculated with the normal 1D
diffusion model; (b) 1.3 molecule per unit cell, profile calculated with
the single-file diffusion model (Q ) 0.39 Å-1).
Figure 7. Comparison of experimental (9) and calculated (s) QENS
spectra obtained at 155 K for methane adsorbed in ZSM-48: (a) θ )
0.11, profile calculated with the normal 1D diffusion model; (b) θ )
0.48, profile calculated with the single-file diffusion model (Q ) 0.35
Å-1).
since the rotational contribution has been neglected, the value
of the mobility factor is only an estimation. It is an upper limit.
Larger values would be easier to measure since the quasi-elastic
broadening would be larger. One may note that only large
single-file diffusivities can be measured with microscopic
techniques: QENS or PFG NMR.
during the neutron experiment so that the relative pore occupan-
cies could be derived.
At low methane loading, θ ) 0.11, a large broadening was
observed. All the spectra could be fitted simultaneously with
normal 1D diffusion, convoluted with isotropic rotation and with
the instrumental resolution. The fit is better with 1D than with
3D diffusion. An example for the comparison between an
experimental and calculated spectrum, for Q ) 0.35 Å-1, is
shown in Figure 7a. This spectrum cannot be fitted with the
single-file diffusion model. At this Q value, only 5% of the
signal comes from the rotation so that one can be confident
that the measured broadening is due to diffusion. For this
loading, the diffusion coefficient of methane is 2.5 × 10-9 m2
s-1, which is not much larger than the value obtained in AlPO4-
5. Therefore, there is no indication of supermobility. In ZSM-
5, the value of the diffusion coefficient extrapolated at 155 K
At θ ) 0.8, the signal is perfectly elastic, like in Figure 6b.
At this loading, the mobility factor is therefore less than 10-12
m2 s-1/2
.
Discussion
In the present study, QENS has been successfully applied
for the first time to confirm the occurrence of single-file
diffusion in zeolites with unidimensional channel structure.
Although the experimental error is large, best fits of the QENS
data for methane in ZSM-48, at medium loading, have been
obtained with the single-file model of proportionality between
the mean-square displacement of the molecule and the square
root of the observation time, in complete agreement with the
theoretical predictions, eq 7. On the other hand, at low loading,
the measurements indicate proportionality between the mean-
square displacement and the observation time corresponding to
normal 1D diffusion.
The crossover between the two cases may be easily attributed
to the change in the relation between the mean free distance l
between the adsorbed molecules and the space scale λ of the
present QENS experiments (λ ) 18 Å for the spectra shown in
Figure 7).
is close to 10-9 m2 s-1 12
so that the diffusivity of methane in
,
these three zeolites is similar and does not depend on the size
of the channel.
For a higher loading, θ ) 0.48, the spectrum obtained at the
same Q value is completely different, as can be seen in Figure
7b. The large loading dependence of the diffusivity of methane,
which was not observed in AlPO4-5, indicates that single-file
diffusion is observed for this molecule in ZSM-48. The
evidence is provided by the quasi-elastic foot in Figure 7b, which
cannot be fitted adequately by a Lorentzian function. The
mobility factor, obtained by fitting the spectra with eq 14, is 2
× 10-12 m2 s-1/2. Since the quasi-elastic intensity is weak and

5840 J. Phys. Chem. B, Vol. 101, No. 30, 1997
Jobic et al.
The mean free distance l between methane molecules can be
et al.24 for a slightly larger molecule (σ ) 5 Å) showing pure
single-file diffusion in AlPO4-5.
derived from the relation41
The explanation we offer for the experimental discrepancies
is that the observation or nonobservation of single-file diffusion
decisively depends on the real structure of the adsorbent under
study. In all these studies, different AlPO4-5 crystals have been
used. It is mentioned in ref 20 that some samples adsorbed 4
molecules/uc, but others 6 molecules/uc. It is also known that
the crystallinity of this material deteriorates with time and water
moisture44 and that the calcination procedure must be slow
enough to avoid defects or partial collapse of the structure.
To explain the discrepancy between our QENS results for
ethane and the MD investigation,24 one may speculate that the
results of MD simulations in this particular structure will be
affected by subtle changes of the potential, as shown in ref 22.
Furthermore, the simulations were performed with a rigid
framework, neglecting the influence of lattice vibrations, which
might have an effect for such a limiting case.
1 - θ
θ
l )
σ
(15)
where σ is the kinetic diameter of the methane molecule.
For a medium concentration, θ ) 0.48, l is equal to 4 Å, and
thus l < λ. In this case, the mutual encounters between adjacent
diffusants become relevant, ensuring the necessary condition
for single-file behavior.
For a small concentration, θ ) 0.11, l is equal to 31 Å, and
thus l > λ. Molecular diffusion is then essentially unaffected
by the other molecules. Depending on the rate of momentum
and energy exchange with the zeolite lattice, molecular propaga-
tion can be described by the ballistic stage or, for sufficiently
fast momentum exchange, by ordinary diffusion.37
It is possible to relate the single-file mobility factor, F, with
a diffusivity that could be measured at infinitely small concen-
tration, Di.41,42 The expression derived by Hahn and co-
workers41 is simply
Conclusion
Two different diffusion regimes have been measured with
the quasi-elastic neutron scattering technique, by varying the
concentration of molecules in 1D channel structures. Ordinary
1D diffusion is observed at low loading for cyclopropane in
AlPO4-5 and for methane in ZSM-48. At higher loadings,
single-file diffusion is found for both systems. The crossover
between the two regimes is attributed to the variation of the
mean free distance between molecules with respect to the space
scale of the present QENS experiments. Ordinary 1D diffusion
is observed for methane and ethane in AlPO4-5, whatever the
loading. It is probable that the different origin of the samples
explains the discrepancies found with PFG NMR results in this
aluminophosphate. Comparative studies with identical sample
materials are thus required. In ZSM-48, the mobility factor of
methane at medium concentration is 2 × 10-12 m2 s-1/2. This
yields a diffusivity for an isolated molecule more than 4 orders
of magnitude larger than the diffusivity derived at low loading,
which seems surprising. It is clear that the characterization of
single-file diffusion is a fairly recent topic and that it deserves
further studies.
F2
Di ) π
(16)
l2
For methane in ZSM-48, at medium concentration, F was
found to be 2 × 10-12 m2 s-1/2. Using eq 16, this gives a value
of 7 × 10-5 m2 s-1 for Di, which would be in striking contrast
to the diffusion coefficient D ) 2.5 × 10-9 m2 s-1, directly
measured for θ ) 0.11. The diffusivity of an isolated molecule
would then be more than 4 orders of magnitude larger than the
diffusivity derived for θ ) 0.11. This discrepancy could be
due to the fact that mutual collisions of the molecules occur
already for θ ) 0.11. Additional measurements are planned,
with a better elastic resolution, to measure single-file diffusion
with a better accuracy.
In the more open 1D channel system of AlPO4-5, best fits
between the experimentally observed scattering behavior and
transport models have been obtained with normal 1D diffusion,
for methane and ethane. It appears that the experimental and
simulation results reported so far in the literature about the
transport properties of the light alkanes in AlPO4-5 are in
disagreement. For methane, the PFG NMR measurements of
Kukla et al. were interpreted by a single-file diffusion model,18
whereas according to another study methane is found to conform
to normal 1D diffusion.19 Quite recently, ordinary 1D diffusion
was found by Martin et al. using QENS.43 The diffusion
coefficient reported by Nivarthi et al. is 2.9 × 10-9 m2 s-1 at
300 K, which is in keeping with our value of 1.6 × 10-9 m2
s-1, at 155 K, for the same concentration of 0.7 molecule/uc.
For larger loadings, the diffusion coefficient derived from the
other QENS experiment43 is 1.0 × 10-9 m2 s-1, at 97 K, for
1.2 molecule/uc, in good agreement with our value of 1.2 ×
10-9 m2 s-1, obtained at 155 K for 1 molecule/uc. We note
that recent MD simulations for methane in this structure24 are
also in favor of ordinary 1D diffusion. However, it is also found
in this work24 that ethane molecules cannot pass each other
easily and exhibit a mobility in between single-file and normal
1D diffusion. Experimentally, the same group has found by
PFG NMR that ethane molecules follow single-file diffusion,42
while our present QENS results suggest ordinary 1D diffusion.
For cyclopropane, we find that this molecule follows single-
file diffusion. This could be the reason for the slow desorption
rates observed during the gravimetric measurements. This
molecule would correspond to the simulation made by Keffer
Acknowledgment. The neutron experiments were performed
at the Institut Laue-Langevin, Grenoble, France. Two of us, K.H.
and J.K., are obliged to the Deutsche Forschungsgemeinschaft
(Sonderforschungbereich 294) for financial support. We also
thank Dr. J. Caro, Institut fu¨r Angewandte Chemie-Berlin, for
discussions.
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