JOURNAL OF CHEMICAL PHYSICS
VOLUME 120, NUMBER 24
22 JUNE 2004
Concentration-dependent self-diffusion of liquids in nanopores:
A nuclear magnetic resonance study
Rustem Valiullina)
Fakultät für Physik und Geowissenschaften, Universität Leipzig, 04105 Leipzig, Germany
and Department of Molecular Physics, Kazan State University, 420008 Kazan, Russia
Pavel Kortunov and Jörg Kärger
Fakultät für Physik und Geowissenschaften, Universität Leipzig, 04105 Leipzig, Germany
Victor Timoshenko
Physik Department E16, Technische Universität München, 85747 Garching, Germany
and Physics Department, Moscow State M. V. Lomonosov University, 119992 Moscow, Russia
共Received 20 January 2004; accepted 31 March 2004兲
Nuclear magnetic resonance has been applied to study the details of molecular motion of
low-molecular-weight polar and nonpolar organic liquids in nanoporous silicon crystals of straight
cylindrical pore morphology at different pore loadings. Effective self-diffusion coefficients as
obtained using the pulsed field gradient nuclear magnetic resonance method were found to pass
through a maximum with increasing concentration for all liquids under study. Taking account of a
concentration-dependent coexistence of capillary condensed, adsorbed and gaseous phases a
generalized model for the effective self-diffusion coefficient was developed and shown to
satisfactorily explain the experimental results. An explicit use of the adsorption isotherm properties
within the model extends its applicability to the mesoporous range and highlights the role of surface
interaction for the transport of molecules in small pores. The problem of surface diffusion and
diffusion of multilayered molecules is also addressed. © 2004 American Institute of Physics.
关DOI: 10.1063/1.1753572兴
I. INTRODUCTION
The understanding of the details of molecular motion in
small pores is an important problem having direct application to many technological processes.1–3 The effect of molecular confinement by a solid matrix and by the interaction
of intraporous fluids with a surface may lead to different
patterns of molecular dynamics. However, elucidating of the
different contributions to the diffusion process in small pores
is still far from trivial. Fabrication of new nanoporous materials with controlled porous structure reinforced the scientific
activity in this field. First of all, the enormous specific surface area dramatically enhances the contribution of the surface interaction to the overall dynamical properties of the
ensemble of guest molecules. Second, with the known structural properties of the porous space and/or the possibility to
modify it in a desired way, one can selectively study the
influence of the different modes on the mechanisms of molecular transport.
Variation of the molecular concentration in pores may
change the character of the diffusion process in an appreciable extent. It depends on the number of phenomena involved, including capillary condensation transitions, different types of adsorption, and molecular exchange between
coexisting phases. Each of these phenomena has a complex
behavior itself which is, consequently, reflected in the
mechanism of the diffusion process. At low concentrations,
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2004/120(24)/11804/11/$22.00
corresponding to coverages of one surface monolayer or less,
the molecular diffusion consists of two contributions. The
first one is the molecular diffusion in the vapor phase which
is a sequence of collisions either with the pore walls, known
as Knudsen diffusion,4 or between the molecules. The second
mode is the surface diffusion, proceeding, for example, by a
hopping of molecules along the surface. When the concentration is increased to the range of multilayer adsorption, the
character of surface diffusion changes significantly. This case
is usually described as a diffusion of multilayered
molecules.5 At further increased concentrations, eventually
the range of capillary-condensed area is attained. It is worth
to note that all these transport mechanisms follow very
specific, in general different dependencies on the concentration. These dependencies can be further modified by the details of the porous space geometry, such as pore-size distribution, interconnectivity of the porous space, and surface
roughness.5– 8
The pulsed field gradient nuclear magnetic resonance
共PFG NMR兲 technique has been widely used to probe the
dynamics of molecules in porous materials.1,9 In contrast to
the transient methods yielding the transport diffusion coefficient, PFG NMR provides information about the molecular
displacements under equilibrium conditions, i.e., about selfdiffusion. Concerning the concentration dependence of the
effective self-diffusion coefficients in meso- and macroporous materials as registered by PFG NMR, three different
patterns have been reported: the self-diffusion coefficient 共I兲
increases, 共II兲 passes a minimum, or 共III兲 decreases as the
11804
© 2004 American Institute of Physics
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J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
Self-diffusion of liquids in nanopores
11805
liquids in small pores. The earlier proposed two-phase exchange model10,20 will be generalized for the case of small
pores. Thereafter, the experimental data obtained using PFG
NMR for some polar and nonpolar liquids in single crystals
of a porous silicon of 10 nm pore diameter will be analyzed
in the frame of the developed theoretical model.
II. THREE-PHASE MODEL
A. Aspects of PFG NMR
FIG. 1. The different patterns of the concentration dependence of the selfdiffusion coefficients in mesoporous and macroporous materials as obtained
using the pulsed field gradient NMR method.
concentration is increased 共for a schematical representation,
see Fig. 1兲.10–20
The dependence of type I was attributed to the fact that
with an increasing total amount of molecules both the steric
restrictions 共owing to reflective boundary conditions at the
liquid–vapor interface兲 and the relative amount of molecules
with a reduced mobility is increased.12,14,19,20 The dependence of type III was first explained 10,11,14 and later experimentally proved17,19 to be caused by the reduction of molecular exchange between the adsorbed and the vapor phases
with increasing concentration. Complex studies of the diffusion of hydrocarbons in different mineral clays have shown
that the concentration dependence of the self-diffusion coefficient can undergo a transition from type I to II with variation of temperature.13–16 This was attributed to the opposite
trends in the contribution of the vapor phase to the apparent
diffusivities with increasing concentration and temperature.
While the general features of the concentration dependencies in Fig. 1 are understood, there are still some uncertainties in their relative importance. D’Orazio et al. have observed that the effective self-diffusion coefficients of water
have opposite tendencies with respect to changes in the water
content in porous glasses with 95⫼240 nm 共type III兲11 and
3.5 nm 共type I兲12 pore diameters. It was anticipated that the
smaller pore size leads to a suppression of vapor-phase diffusion proceeding by the Knudsen mechanism since the coefficient of Knudsen diffusion is proportional to the pore
size. However, recently20 it was found that cyclohexane in
similar porous glasses of 4 nm pore diameter exhibits an
enhancement of the apparent self-diffusion coefficient at low
loadings 共type III兲, while water in the same material behaves
as reported earlier 共type I兲.12 Therefore, since the water diffusivity has to surpass that of cyclohexane, the explanation
given in the Ref. 12 cannot be valid. In our opinion, this
discrepancy results form the underestimation of the role of
the surface interactions in mesopores. Indeed, the latter may
lead to a change in the coexistence between the vapor and
the adsorbed phases as compared to that in macroporous materials which was not taken into account in the previous theoretical models.10,20
In the present study we address the role of the pore size
and the surface interactions in the translational properties of
Depending on the particular features of the used experimental technique 共probed time and length scales, etc.兲 molecular motion can be displayed in the recorded experimental
data in different ways. Therefore, we proceed with the theoretical model of self-diffusion of inhomogeneous fluids in
porous systems and give some general description of the
experimental basics relevant to our study.
The quantity originally registered in PFG NMR experiments is the attenuation of the NMR signal intensity under
the influence of magnetic field gradient pulses of magnitude
g and duration ␦, which may formally be addressed as an
incoherent scattering function S(q,t d ), where q⫽ ␥ ␦ g is the
wave number, ␥ denotes the nuclear gyromagnetic ratio, and
t d is the effective diffusion time. The diffusion attenuation of
the normalized spin–echo amplitude S(q,t d ) in a pure liquid
with the self-diffusion coefficient D 0 is expressed as
S 共 q,t d 兲 ⬅
A 共 q,t d 兲
⫽exp共 ⫺q 2 D 0 t d 兲 ,
A 共 0,t d 兲
共1兲
where A(q,t d ) and A(0,t d ) are the spin–echo signal intensities with and without magnetic field gradient applied, respectively. If the system consists of nuclei having different diffusion coefficients, S(q,t d ) is the sum of terms given by
Eq. 共1兲 with the appropriate weighting factors p i ,
S 共 q,t d 兲 ⫽
兺 p i exp兵 ⫺q 2 t d D i 其 .
共2兲
The fraction p i is given by the relation p i ⫽N i / 兺 N i , where
N i is the number of nuclei 共or molecules, in case of a single
liquid兲 characterized by D i . In our case, the molecules in the
spatially separated thermodynamical phases have different
mobilities, thus by p i we can understand the fraction of molecules in a phase i. Note, that the values of p i may be additionally affected if the nuclear relaxation properties are notably different in the different phases.
In the fast-exchange limit, i.e., when during the observation time t d the molecules experience translational displacements in essentially all phases, Eq. 共2兲 is transformed into
the single-exponential form Eq. 共1兲 with the effective selfdiffusion coefficient
D e⫽
兺 p iD i .
共3兲
It is important to note that Eq. 共3兲 is valid even if the fastexchange condition is not fulfilled. In this case, D e is obtained by fitting Eq. 共1兲 to S(q,t d ) in the small-q range. This
can be seen from the equality:
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11806
Valiullin et al.
J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
⫺ lim
q 2 →0
1 ⳵ S 共 q,t d 兲
⫽
td ⳵q2
兺 p iD i ,
共4兲
which directly follows from Eq. 共2兲. In the following, we are
going to develop our model for the effective self-diffusion
coefficient D e by using Eq. 共3兲.
As discussed in the introduction, we shall consider the
following three phases present in the pores at different concentrations: the capillary condensed or liquid phase, the vapor phase, and the phase adsorbed on the surface. We refer to
them by the subscripts l, g, and s, respectively. The parameters p i and D i , in general, depend on the concentration
which is represented as the relative pore filling ␪ ⫽N/N tot ,
where N and N tot are the actual number of molecules and the
total number of molecules at full pore saturation, respectively. We start to consider the concentration dependence of
the effective self-diffusion coefficient in the simplest possible situation in a single pore of cylindrical morphology.
␳ g⫽
B. Single cylindrical pore
At low concentrations, molecules exist in the adsorbed
surface layer and in the gas phase, which are in equilibrium
with each other. In order to evaluate the relative fraction of
the molecules in both phases we start with the two equations
stating the mass and the volume balances,
N g ⫹N s ⫽N,
共5a兲
mN g mN s
⫹
⫽V p ,
␳g
␳s
共5b兲
where m is the molecular mass, V p is the total pore volume,
and ␳ g and ␳ s are the bulk densities in the vapor and the
adsorbed phases, respectively. Consequently, the fraction of
molecules in the vapor phase is found to be
p g⫽
冉
␳ SV p
⫺1
mN
冊冉 冊
␳s
⫺1
␳g
⫺1
.
共6兲
From the definition of the concentration ␪ ⫽N/N tot and using
the physically reasonable relation ␳ s Ⰷ ␳ g it follows that
mN
⫽␪.
␳ sV p
共7兲
It should be pointed out, however, that the density of the
adsorbed phase may change with the concentration. The
most pronounced difference can be expected in the range of
less than one monolayer surface coverage where ␳ s is determined by the details of the surface interactions, including the
nature of the adsorption centers and their surface density.
With Eq. 共7兲 and relying on the huge difference in the densities in the adsorbed and vapor phases, Eq. 共6兲 can be rewritten as
1⫺ ␪ ␳ g
p g⫽
.
␪ ␳s
able for macroporous systems, in our opinion such a procedure is not applicable for porous materials down to the mesoporous range. In small pores, the increasing influence of
the surface interactions and the effects of the pore curvature
notably modifies the conditions of phase equilibrium in comparison with that in big pores.
Indeed, the amount of physically adsorbed molecules is
associated with the relative pressure z⫽ P/ P s , where P and
P s are the partial vapor pressure of a component and the
saturated vapor pressure at the same temperature T. The particular shape of the adsorption isotherm, which relates the
amount of the adsorbed molecules to the relative pressure z,
depends on many features including the type of adsorption
共chemisorption, physisorption兲, the pore size and the morphology. Thus, within our model, we express the vapor density using the ideal gas relation, in accordance with the earlier approaches,10,20 but with a concentration-dependent
relative vapor pressure z( ␪ ),
共8兲
In the previous models 共see, for example, Refs. 10 and
20兲 the density of the vapor phase has been assumed not to
change with concentration. It has been set equal to the value
corresponding to the saturated vapor pressure. Being reason-
z共 ␪ 兲PsM
,
RT
共9兲
where M is the molar mass, and R is the universal gas constant. In this concentration range below the capillary condensation point, as a good approximation one can use the wellknown BET equation:
␪⫽
cz ␪ 1
,
共 1⫺z 兲共 1⫺ 共 1⫺c 兲 z 兲
共10兲
where ␪ 1 ⫽N m /N tot , with N m being the number of molecules
required to cover a monolayer, and where c is a characteristic constant for a given liquid.
Diffusion in the vapor phase is governed by the collisions with the pore walls or surface layers, respectively, and
by intermolecular collisions. For the small pores considered,
because of the low vapor densities 共the molecular mean freepath in a bulk vapor is much longer than the pore diameter兲
the influence of the latter mechanism is negligibly small.21
Thus, as a good approximation, the Knudsen self-diffusion
coefficient D K can be used. For a cylindrical pore of diameter d, D K along the tube axis is given by4,21
D K⫽
d
3
冑
8RT
.
␲M
共11兲
The multilayer character of the adsorption may lead to a
lowering of the coefficient of Knudsen diffusion because of
the reduced effective space available for vapor diffusion. Using Eqs. 共5兲 and 共7兲 it is straightforward to show that for a
cylindrical pore geometry the effective tube diameter d eff is
given by
d eff⫽d 冑1⫺ ␪ ,
共12兲
so that d in Eq. 共11兲 has to be replaced by this quantity.
Thus, for a single cylindrical pore of diameter d, at concentrations below the capillary condensation point the effective self-diffusion coefficient D e may be expressed as
D e ⫽D s ⫹
冉冑
共 1⫺ ␪ 兲 z 共 ␪ 兲 P S M d
␪
␳ s RT
3
冊
8RT 共 1⫺ ␪ 兲
⫺D s ,
␲M
共13兲
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J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
Self-diffusion of liquids in nanopores
where the identity p g ⫹p s ⫽1 has been applied. Note, that we
have not further specified the self-diffusion coefficient D s in
the adsorbed phase. Surface diffusion and diffusion of the
multilayered molecules are complex functions of the concentration depending on many parameters of the system under
study and we have desisted from arbitrarily selecting a special pattern from the various dependencies presented in the
literature 共see, for example, Ref. 5 and references therein兲.
At higher concentrations, when capillary condensation
takes place, D e ⫽D l . Under equilibrium conditions, the selfdiffusion coefficient of the capillary condensate D l is a function of only the strength of the molecule–solid interaction
and the details of the pore-space geometry. These, usually,
lead to a reduced self-diffusion coefficient as compared to
that in the bulk liquid, D 0 .
Finally, the transition between the low- and highconcentration regions is determined by the Kelvin equation.
The relative pressure z cr 共or, correspondingly, because of
their unambiguous interrelation through the adsorption isotherm, the concentration ␪ cr ) at this transition is related to
the pore curvature by
ln共 z cr 兲 ⫽⫺
2 ␴M
,
d ␳ l RT
共14兲
where ␴ is the surface tension. When the experiments are
performed with decreasing concentration 共i.e., under desorption兲, the factor 2 in Eq. 共14兲 has to be replaced by 4.22,23
All porous materials usually exhibit a certain pore-size
distribution. Thus, variation of the relative pressure may lead
to a situation where some pores 共those of smaller diameter兲
will be completely filled by the liquid, while others 共those of
bigger diameter兲 will consist of the vapor and the adsorbed
phases. The equilibrium between these two types of pores at
a certain pressure is governed by the pore-size distribution
function. Moreover, because the Knudsen diffusion coefficient and the fraction of molecules in the vapor phase are
directly proportional to the pore size, its existence may also
render the concentration dependence of the effective selfdiffusion coefficient different from that given by the single
pore model. The extension of the model to pore-size distributions is given in the Appendix 关Eq. 共A12兲 with Eqs. 共A10兲
and 共A11兲兴.
11807
TABLE I. Physical properties of the liquids under study at T⫽297 K.
Liquid
Acetone
n-hexane
Cyclohexane
P sa
␴b
Mc
␳ Ld
D 0e
29.3
19.3
12.4
23.3
17.9
25
58
86
84
0.79
0.66
0.78
4.5
4.2
1.4
a
Saturated vapor pressure (103 N/m2 ).
Surface tension (10⫺3 N/m).
c
Molar mass (10⫺3 kg/mol).
d
Liquid density (103 kg/m3 ).
e
Bulk self-diffusion coefficient (10⫺9 m2 /s).
b
In order to determine the pore-size distribution in the PSi
used, the NMR-cryoporometry method has been
employed.27,28 The PSi crystals in the NMR tube were oversaturated by cyclohexane having a bulk melting temperature
T m ⫽279.6 K. The probe was initially cooled to T⫽220 K in
order to freeze the intraporous cyclohexane. 1H NMR cryoporometry experiments were performed using the
90° – ␶ – 180° spin–echo pulse sequence with a time interval
␶ ⫽3 ms on an NMR spectrometer operating at a 400 MHz
resonance frequency for protons. The intensity S SE of the
spin–echo signal was recorded as a function of the slowly
increasing temperature T 共the heating rate between two subsequent temperature points at which measurements were performed was 0.1 K/min and the equilibration time before
measurements was 10 minutes兲. With S SE(T) the pore-size
distribution function f (d) was obtained as
f 共 d 兲⫽
⳵ S SE K
,
⳵T d2
共15兲
where the characteristic constant K⫽180 K nm for cyclohexane was used 共for details, see Ref. 27兲.
The organic liquids n-hexane, cyclohexane, and acetone
have been investigated. Some of their physical properties
relevant to our study are given in Table I. The principal
scheme of the experimental setup is shown in Fig. 2. Adsorption and desorption of the liquids into PSi was performed by
use of a big reservoir connected to an NMR glass tube where
the PSi crystals were placed 共the initial PSi crystal wafers
with a radius of 2 cm have been crashed and filled into the
NMR tube兲. The volume of the reservoir exceeded that of the
III. MATERIALS AND METHOD
Porous silicon 共PSi兲24 was prepared by electrochemical
etching 共anodization兲 of single-crystalline 共100兲-oriented
p-type Si wafers with resistivity of 10–15 m⍀ cm. The electrolyte contained HF 共50%兲 and C2 H5 OH in a ratio of 2:1.
The anodization current density was 80 mA/cm2 and the anodization time was 30 min. For removing the porous silicon
film from the substrates we applied an electropolishing step
with a current density of 500 mA/cm2 for 2–3 s. The film
thickness is in the range of 100 ␮m and the porosity determined gravimetrically is about 0.69. According to the literature data, the described procedure yields the mean pore diameter in the prepared PSi to be about 6 – 8 nm.25 Moreover,
the pores are expected to be separated from each other and of
close to cylindrical morphology.26
FIG. 2. Schematic of the experimental setup: 共1兲 NMR tube with porous
silicon crystals; 共2兲 NMR spectrometer; 共3兲 buffer reservoir; 共4兲 flask with
liquid; 共5兲 manual multiturn valves; 共6兲 pressure detector; 共7兲 turbomolecular pump.
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11808
Valiullin et al.
J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
probe substantially. The vapor pressure of the liquids in the
reservoir was regulated either through a valve connected to a
tube with the liquid under study 共the adsorption branch兲 or a
valve connected to a vacuum pump 共the desorption branch兲.
The vapor pressure was registered using a digital manometer.
The temperature in the external reservoir was kept equal to
T⫽297 K.
All experiments have been performed using the same
NMR spectrometer equipped with a home-built pulsed field
gradient NMR probe.29 The protocol was as follows. First,
the pressure in the reservoir was set to a desired value. Then
it was connected to the NMR tube and some time was given
for equilibration. The latter was controlled by the NMR free
induction decay 共FID兲 signal S FID following a single 90°
radio-frequency pulse: we waited until the intensity of the
eq
FID signal attained its equilibrium value S FID
. After that, the
eq
intensity S FID and the corresponding pressure magnitude
were recorded and used as the dataset for the adsorption or
the desorption isotherms. Thereafter, the relevant NMR studies were carried out and, as a last step, we checked that the
FID signal had not been changed.
For the diffusion study, the 13-interval stimulated echo
pulse sequence with two gradient pulses of opposite signs
was used.30 Typical values for the gradient pulse duration ␦
and the distance between the first and second 90° radiofrequency pulses were 0.4 and 1.2 ms, respectively. During
the experiments, the wave number q⫽ ␥ ␦ g was scaled by
changing the magnitude g and keeping ␦ constant. Other
details concerning the experimental setup and the data preprocessing relevant to the present diffusion measurements
can be found in Ref. 29. Additionally, T 2 -relaxation measurements for n-hexane in PSi using the CPMG pulse
sequence31,32 90° – 关 ␶ – 180° – ␶ 兴 N with an interpulse delay ␶
⫽150 ␮ s and a number N⫽64 of 180° pulses have been
performed.
IV. RESULTS AND DISCUSSION
A. Pore-size distribution
NMR-cryoporometry uses the fact that the melting point
of intraporous liquids decreases with decreasing pore radii in
a well-defined way. It has been become a well-established
method for the characterization of nanoporous materials.27,33
Figure 3共a兲 shows the experimentally obtained signal intensity S SE(T). It is proportional to the number of molecules in
the liquid state, since the signal of the crystalline phase is
removed by choosing a sufficiently long spin–echo time ␶.
The figure reveals two distinct regions for the melting of the
cyclohexane molecules: first one in the temperature range
from T⬇232 K to T⬇267 K, originating from the suppressed melting point in the small pores of PSi and a second
region at T⬇280 K corresponding to the melting of the bulk
cyclohexane. Recall that the sample was oversaturated by the
liquid and that the melting point of cyclohexane is T m
⫽279.6 K. The normalized pore-size distribution f (d) 共i.e.,
the relative volume of pores of diameter d) resulting by use
of Eq. 共15兲 is shown in Fig. 3共b兲. Note, that normalization
means 兰 ⬁0 f (x)dx⫽1.
The determined distribution function f (d) is found to
FIG. 3. 共a兲 NMR spin–echo intensity S SE of cyclohexane in the porous
silicon as a function of temperature 共the line is given as an eye guide兲. 共b兲
Normalized pore-size distribution function f (d) 共the relative volume fraction of pores with a given diameter兲 obtained from the data in 共a兲. The line
represents Eq. 共A2兲 with ␻ ⫽0.27 and d a ⫽9.6 nm.
have a log-normal shape as given by Eq. 共A2兲. The fit of Eq.
共A2兲 to the experimental data is shown in Fig. 3共b兲 by the
full line. The resulting mean pore diameter d a is equal to 9.6
nm which is slightly greater than the value of 6 – 8 nm as
expected from the preparation procedure. As a possible reason for this difference we consider the significantly longer
etching time, used in our case, in comparison with Herino’s
original procedure.25 In this case one may thus expect a lowering of the real HF concentration inside a thick PSi layer
during the etching process and it is known that the pore size
increases with decreasing HF content in the solution.
B. Adsorption–desorption isotherms
As a typical example, Fig. 4 shows the adsorption isotherm for acetone in PSi. The data points for concentration
eq
(z)
were obtained by dividing the FID signal intensity S FID
eq
(z
by that registered at the saturated vapor pressure S FID
⫽1) since the FID signal is proportional to the total number
of molecules in the sample. The sharp increase of the concentration in the range of relative pressures z between ⬃0.6
and ⬃0.9 is caused by capillary condensation. At low vapor
pressures in the adsorption isotherms for all liquids under
study a monolayer step is observed. Taking also account of
the displayed hysteresis loop, the isotherms can be classified
to be of type IV of the 1985 IUPAC classification.34 In the
concentration region below capillary condensation (z⬇0
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J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
Self-diffusion of liquids in nanopores
FIG. 4. Adsorption 共circles兲 and desorption 共stars兲 isotherms for acetone in
porous silicon at T⫽297 K. The line shows ␪ r given by Eq. 共A7b兲 with the
experimentally determined parameters for the pore-size distribution function
and the isotherm as given by Table II.
⫼0.6) the isotherms can be approximated by the BET equation 关Eq. 共10兲兴. The parameters of the fit of Eq. 共10兲 to the
experimental data are given in Table II.
The BET constants c in Table II are usually expressed as
c⫽exp兵(q1⫺qliq)/RT 其 , where q 1 and q liq are the heat of adsorption in the first monolayer and the molar heat of condensation, respectively. Their small values indicate that in the
systems under study physisorption is the prevailing sorption
mechanism, most likely as a consequence of the van der
Waals forces and the weak dipole interactions. This is also
confirmed by the observed fast equilibration 共⬃ few minutes兲 of the NMR signal intensity S FID with changing the
external pressure and by the reversibility of the isotherms in
the region of low relative pressures z⬍0.6. Being a polar
liquid, acetone is characterized by a larger value of c than
the nonpolar liquids n-hexane and cyclohexane.
In order to capture the shape of the isotherms for the
whole range of relative pressures, capillary condensation has
to be taken into account. Within our model, this was accounted for by relying on the known pore-size distribution
function f (d) and the isotherm as given by Eq. 共A7b兲. The
calculated isotherms are shown in Fig. 4 by the full line.
They satisfactorily describe the experimental data. Note that,
following Cohan,22 in Eq. 共A1兲 we used the factor 2 for the
adsorption branch and the factor 4 for the desorption branch.
The observed slight discrepancy between the calculated and
experimental adsorption isotherms in the region of high relative pressures may be a consequence of the fact that the
thickness of the adsorbed layers prior to condensation was
neglected. However, this discrepancy has a minor effect on
the following discussion.
TABLE II. The data on ␪ 1 and c obtained by fitting Eq. 共10兲 to the experimental isotherms. z 1 is the relative pressure corresponding to ␪ ⫽ ␪ 1 .
Liquid
Cyclohexane
Hexane
Acetone
␪1
c
z1
0.17
0.13
0.14
4.3
9.7
21.5
0.32
0.24
0.18
11809
FIG. 5. Spin–echo attenuations for n-hexane in porous silicon at the relative
pressures z⫽0.91 共circles兲 and z⫽0.27 共triangles兲. The lines show fits to the
experimental data with D e ⫽2.23⫻10⫺9 m2 /s and ␻ ⫽0.30 for z⫽0.92, and
D e ⫽3.64⫻10⫺9 m2 /s and ␻ ⫽0.32 for z⫽0.27. The dashed line represents
Eq. 共16兲 with D e ⫽4.2⫻10⫺9 m2 /s.
C. NMR spin–echo diffusion attenuations
The spin–echo diffusion attenuations S(q,t d ) are found
to have a strong nonexponential character for all liquids under study. As an example, S(q,t d ) for n-hexane in PSi is
shown in Fig. 5 for two different relative pressures 共the experiments have been performed on the desorption branch of
the isotherm兲 for the diffusion time t d ⫽10 ms. It was found
that in the range t d ⫽5⫼200 ms, within the experimental
precision, the shape of S(q,t d ) does not depend on the diffusion time.
Formally, such a behavior of S(q,t d ) can be observed
either under slow-exchange multiphase conditions or if the
diffusion process is non-Gaussian.1,9 We can rule out the
former reason referring to the curve obtained at the relative
vapor pressure z⫽0.92. Indeed, in this case all pores are
completely saturated by the condensed phase which is
proved by the plateau region of the desorption isotherm at
this relative pressures. The vapor phase surrounding the PSi
crystals does not contribute to the spin–echo signal because
of its low density. At the shortest diffusion time used, t d
⫽5 ms, the root-mean-square molecular displacements are
much larger than the pore diameter (⬃10 nm) and much
smaller than the minimal crystal dimension (⬃100 ␮ m).
Therefore, along the pore axis, molecular diffusion can be
well described as a one-dimensional diffusion process with a
Gaussian propagator. Thus, these reasons cannot account for
the nonexponential character of the observed spin–echo diffusion attenuations, and the real reason has to be associated
with the particular macrostructure of the samples under
study.
Indeed, the used sample consists of PSi wafers with
straight nonintersecting channels perpendicular to their surface. Assuming that the wafers in the NMR tube are randomly oriented, we arrive at a one-dimensional diffusion
problem in randomly oriented tubes. Under these conditions,
the spin–echo attenuation is given by1,9
S 共 q,t d 兲 ⫽
1
2
冕
1
⫺1
exp兵 ⫺q 2 t d D e cos2 共 ␾ 兲 其 d 共 cos共 ␾ 兲兲 ,
共16兲
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11810
Valiullin et al.
J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
FIG. 6. Effective self-diffusion coefficients 共circles兲 vs the relative pressure z for cyclohexane 共a兲, n-hexane 共b兲, and acetone 共c兲, and vs the concentration ␪
for acetone 共d兲 in porous silicon. The data given by the filled and open symbols were obtained on the desorption and the adsorption branches, respectively. The
calculated self-diffusion coefficients within the single-pore model using Eq. 共13兲 and the condition D s ⫽D l are shown by the lines. Taking account of the
pore-size distribution using Eq. 共A12兲 leads to the dashed (D s ⫽D l ) and the dotted (D s ⫽0) lines, respectively. The vertical lines show the relative pressure
z 1 corresponding to monolayer coverage as given in Table II.
where ␾ is the angle between the tube axis and the direction
of the magnetic field gradient, and D e is the effective selfdiffusion coefficient along the tube axis. As an example, Eq.
共16兲, with D e ⫽D 0 for the bulk n-hexane, is shown in Fig. 5
by the dashed line. The figure obviously demonstrates that
Eq. 共16兲 does not at all capture the shape of the experimental
attenuation functions.
However, because the thickness of the PSi wafers
(⬃100 ␮ m) is much less than their length, orientation of the
wafers in the NMR tube probably cannot be assumed to be
completely isotropic, so that there is a preferential packing
parallel to the flat tube bottom. This corresponds to a preferential orientation of the pore axes parallel to the direction of
the field gradient. In order to take this into account, we have
modeled their orientation using a normal distribution of the
angles ␾ between the field gradient direction and the pore
axis:
f o 共 cos共 ␾ 兲兲 ⫽
1
冑2 ␲␻ o
exp
冉
冊
⫺ 共 cos共 ␾ 兲 ⫺1 兲 2
.
2␻o
共17兲
Equation 共16兲 weighted with the orientation distribution
function f o (cos(␾)) gives
S 共 q,t d 兲 ⫽F q erf共 ⫺&/ ␻ o 兲 exp共 ⫺q 2 D e t d F 2q 兲
⫻ 关 erf共共 & ␻ o q 2 D e t d ⫹&/ ␻ o 兲 F q 兲
⫹erf共 & ␻ o q 2 D e t d F q 兲兴 ,
D e t d ␻ 2o ) ⫺1/2.
共18兲
Equation 共18兲 共the full
where F q ⫽(1⫹2q
lines in Fig. 5兲 provides an excellent fit to the experimental
data. Importantly, the fits yield the same orientation distribution parameter ␻ o ⯝0.31 for all relative pressures, which
2
confirms the validity of the proposed model. It is also worth
to note that the effective diffusion coefficients in PFG NMR
experiments are usually obtained using Eq. 共4兲. However,
when one deals with anisotropic diffusion, an appropriate
model 关in our case, Eq. 共18兲兴 has to be used in order to get
the correct absolute magnitude for D e .
D. Effective self-diffusion coefficients
Figures 6共a兲– 6共c兲 show the effective self-diffusion coefficients D e , obtained using Eq. 共18兲 with ␻ o ⫽0.31, as a
function of the relative pressure z. The data given by the
filled symbols were measured on the desorption branch. For
comparison, in the case of acetone those measured on the
adsorption branch are given by the open symbols. As we
have discussed above, it is equivalent to consider D e as a
function of z or of ␪ because they are in a unique fashion
interrelated through the adsorption isotherm. As an example,
the concentration dependence of D e for acetone in PSi is
shown in Fig. 6共d兲. For the transformation z→ ␪ the experimental data of Fig. 4 were used. Pressure and concentration
dependencies are obviously found to be of essentially the
same shape.
When, at relatively high pressures, all pores are completely filled by the liquid the value of D e 共which is equal to
D l according to the notation used in the theoretical section兲
is slightly smaller than the bulk self-diffusion coefficient D 0 .
The ratios D l /D 0 are 0.7, 0.53, and 0.42 for cyclohexane,
n-hexane, and acetone, respectively. It should be noted that
the data evaluation procedure yields the self-diffusion coefficient along the pore axis, and one expects D l ⫽D 0 in the
limit of weak pore wall-molecule interactions. The slight de-
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J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
viation of the diffusivities from their bulk values can be explained by a lower mobility of the molecules at the pore
walls due to the dispersive forces18 or/and the surface
roughness.35 Importantly, it is anticipated that the pores in
PSi crystals of similar preparation do not have pore
intersections26 which may lead to an additional suppression
of the diffusion coefficient due to increased tortuosity.
In the high-pressure region, with decreasing loading an
enhancement of D e , following the type-III pattern 共see Fig.
1兲, is observed. The absolute magnitudes of D e in this pressure range depend on the adsorption direction, leading to the
exhibition of a hysteresis loop. The example given in Fig.
6共c兲 refers to the acetone–PSi system. Interestingly, the hysteresis collapses in the diffusivity-concentration coordinates
关Fig. 6共d兲兴. Qualitatively, the same behavior has been found
for the T 2 -relaxation times 共not shown兲: They exhibit a hysteresis on the adsorption–desorption branches upon variation
of the pressure, but coincide into one curve in the
T 2 -concentration coordinates. Earlier, the analogous finding
was reported by stating that the NMR parameters 共longitudinal and transverse NMR relaxation times and the diffusivity兲
of water in porous silica gels depend only on the liquid content but not in the direction that this saturation level has been
reached.36 It was suggested that at a given concentration of
water in pores the geometry of the liquid network is the same
for the adsorption and the desorption branches. This explanation is reasonably valid for the PSi used in the present
work, consisting of a set of parallel, nonintersecting cylindrical pores, opened at both ends. However, recent molecular
dynamics simulations of adsorption and desorption in an ink
bottle pore suggests that this picture can be of general relevance for porous materials.37
The common feature of all dependencies D e (z) in Figs.
6共a兲– 6共c兲 is the exhibition of a maximum with decreasing
relative pressure. Note, that for the effective diffusion coefficients of liquids imbibed into nanoporous materials such a
behavior has not yet been reported. A similar tendency observed in Ref. 20 for cyclohexane in nanoporous silica glass
Vycor was attributed to traces of residual water on the surface. The relative position of the maximum depends on the
liquid under study and, in general, does not coincide with the
relative pressure z 1 corresponding to monolayer surface coverage 关the vertical lines in Figs. 6共a兲– 6共c兲兴. Moreover, the
similarity of the curves suggests that there is no strong correlation of their shapes with the liquids’ polarity.
E. Comparison with the theoretical models
In the developed theoretical models the self-diffusion coefficient in the adsorbed phase, denoted by D s , is an unknown parameter. The calculation of the surface selfdiffusion is a nontrivial task and involves many assumptions,
concerning, e.g., the character of the adsorbate–adsorbate
and adsorbate–adsorbent interactions and the surface geometry. Depending on these assumptions, quite different dependencies of the surface diffusivity on the surface coverage
may be obtained. The choice of the correct model requires
additional experimental studies. Therefore, in this paper, we
confine ourselves to two simple cases.
Self-diffusion of liquids in nanopores
11811
In the first case we assume that molecular interaction
with the solid surface weak enough so that diffusion in the
adsorbed phase is the same as at full saturation, i.e., D s
⫽D l . It should be pointed out that this assumption does not
represent the upper limit because self-diffusion may be enhanced due to the free volume effects.38 In the second case,
as a lower limit, we shall consider negligible surface diffusion, i.e., D s ⫽0. This is often assumed to hold for microporous adsorbents such as zeolites.1
The effective self-diffusion coefficients calculated in
frame of the single-pore model 共i.e., with Eq. 共13兲 for z
⬍z cr and D e ⫽D l for z⬎z cr ) by assuming D s ⫽D l are
shown in Fig. 6 by the solid lines. In the calculations a bulk
density ␳ s in the adsorbed phase equal to that in the bulk
liquid ( ␳ l ) was used. It is seen that the single-pore model
describes the shape of the experimental curves in a qualitatively satisfactory way, with the most pronounced deviation
in the low-pressure 共or -concentration兲 region. With decreasing pressure, at z⫽z cr a step change in D e is observed. This
may be referred to when a break of the liquid meniscus is
observed. Taking account of the pore-size distribution
smoothes the step change 共dashed lines in Fig. 6兲 and thus
provides a better correspondence between the experimental
and the calculated values in the high-pressure region. At low
pressures, the results of both models almost coincide. Assuming that the diffusivity in the adsorbed phase is negligibly small gives a much poorer description of the experimental data 共dotted lines in Fig. 6兲. This clearly shows that the
contribution of the adsorbed phase to the effective selfdiffusion coefficient is substantial.
Comparison of the experimental data with the calculation results shows that there are two mechanisms leading to
the observed maxima in the effective diffusivities D e . The
first mechanism is associated with the fact that there is a fast
exchange between the adsorbed molecules and the vapor
phase in the pores. The diffusivity in the vapor phase is much
higher, being of the order of 10⫺6 m2 /s. The fraction of molecules in the vapor phase is determined by the competition of
two mechanisms. First, on lowering the pressure in more and
more pores, the liquid meniscus will break giving rise to a
volume of the vapor phase and, consequently, a fraction of
molecules therein. On the other hand, its density is directly
proportional to the pressure. Thus, the fraction of molecules
p g passes a maximum. This is shown in Fig. 7 where the
pressure dependence of the quantity p g calculated using Eq.
共A8b兲 is presented for all liquids under study.
Comparison of Fig. 7 and Figs. 6共a兲– 6共c兲 points out that
there is a good correlation in the positions of the maximum
for p g and the effective self-diffusion coefficients. Within the
single-pore model, the position of the maximum for both
pressure (z max) and concentration ( ␪ max) dependencies of the
fraction of molecules in the vapor phase is readily obtained,
yielding
z max⫽
c 共 1⫺ ␪ 1 兲 ⫺2
,
2 共 c⫺1 兲
共19a兲
2 ␪ 1 c 共 1⫺ ␪ 1 兲 ⫺2
.
c
1⫺ ␪ 21
共19b兲
␪ max⫽
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11812
J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
FIG. 7. Relative fractions p g of molecules in the vapor phase obtained using
Eq. 共A8b兲 vs the relative pressure z for cyclohexane 共dashed line兲, n-hexane
共solid line兲, and acetone 共dotted line兲 at T⫽297 K.
Equation 共19兲 shows that the stronger the interaction of molecules with the pore walls 共corresponding to larger values of
c), the higher is the pressure or concentration where the
maximum is observed. The upper bonds for z max and ␪ max are
0.5 and 2 ␪ 1 , respectively.
According to Eq. 共8兲, the absolute magnitude of p g at a
given temperature is directly proportional to the liquid’s
saturated-vapor pressure P s and the molar mass M . For acetone and n-hexane the products P s M are almost equal,
however p g for n-hexane is higher. This is determined by the
interaction of molecules with the pore surface. At a certain
relative pressure z, the adsorption ␪ will be higher for liquids
stronger interacting with the surface 共those having larger c
values兲. Consequently, the term (1⫺ ␪ )/ ␪ in Eq. 共8兲 will lead
to lower values of p g for such liquids. Similarly, in order to
reach a certain concentration ␪, lower relative pressures are
required for liquids stronger interacting with the surface.
A second mechanism related to the observed maxima is
connected with the peculiarities of diffusion in an adsorbed
phase and is of importance at low concentrations. Indeed, it
is known that the surface diffusivity generally increases with
increasing concentration.5,39 This may be also seen from a
comparison of the experimental data with the calculated diffusivities provided by Fig. 6. For cyclohexane and n-hexane,
in the concentration region corresponding to less than monolayer coverage, a deviation of the experimental data from the
calculated values is observed. For acetone, however, this deviation is prolonged to coverages of about two monolayers,
or even into the region of multilayer adsorption. In our opinion, this finding has to be attributed to the polar nature of
acetone.
V. CONCLUSIONS
The pulsed field gradient nuclear magnetic resonance
method has been applied to study the self-diffusion process
of polar and nonpolar liquids in nanopores at different pore
loadings. Many experimental studies reported in the literature for random mesoporous and macroporous systems revealed different patterns for the concentration dependence of
the effective self-diffusion coefficient.10–20 They were attributed to the interplay of two processes: 共a兲 modification of the
morphology of the liquid and vapor phases and 共b兲 change of
Valiullin et al.
the vapor-phase contribution to the effective self-diffusion
coefficient upon variation of the concentration. In the present
work, a new phenomenon is described. It was found, that the
effective self-diffusion coefficient passes a maximum with
increasing concentration. Importantly, the behavior was the
same for polar 共acetone兲 and nonpolar (n-hexane, cyclohexane兲 liquids.
The use of a porous material with simple pore geometry,
namely silicon crystals having straight nonintersecting channels of about 10 nm pore diameter, allowed us to concentrate
on the details of the interphase coexistence with changing
concentration and its influence on the molecular selfdiffusivity. In the frame of the interphase exchange, an analytical treatment of the effective self-diffusion coefficient of
liquids in such pores has been performed. The details of the
interphase equilibrium as given by the adsorption isotherm
were taken into account. The obtained analytical results well
explained the obtained concentration dependencies of the effective diffusivity.
The formation of a maximum in the diffusivities is associated with the fact that the fraction of molecules in the vapor phase also passes a maximum upon variation of the concentration. In the low-concentration region the surface
interactions strongly influence the coexistence between the
adsorbed and vapor phases. Thus, in this region, the effective
diffusivity is determined by an increasing fraction of molecules in the vapor phase and an increasing surface diffusivity with increasing pore loading. As soon as a certain concentration, corresponding to about one to two monolayers
surface coverage is reached, the situation changes. Now, surface interactions have a minor influence on the molecular
density in the vapor phase, and the effective volume for the
vapor phase decreases. Together with the assumption that
there is no appreciable change of the diffusion coefficient of
the multilayered molecules, this leads to a decrease of the
effective diffusivities with increasing concentration. Finally,
when capillary condensation takes place, the effective selfdiffusion coefficient attains its plateau value corresponding
to the diffusivity of the liquid phase in the pores.
The experimental data and the theoretical analysis presented in this work highlight the role of the surface interactions and of the pore size on molecular self-diffusion in
pores. First, surface interactions seem to be of importance
only at very low pore loadings, up to a few monolayers. In
this concentration region, adsorption of molecules by active
sites of the surface strongly influences the density of the
vapor phase. This makes the contribution of the molecules in
the gas phase to the overall diffusion process in the mesopores relatively small. At the same time, surface self-diffusion
strongly depends on the surface coverage itself. This is
closely related to the features of self-diffusion on energetically heterogeneous surfaces.40 Experimental studies of this
phenomenon are in progress.
At higher concentrations, the influence of the moleculepore wall interactions becomes screened by the multilayered
molecules. The enhancement of the effective diffusion coefficient is determined by the pore size, which in turn affects
the relative volume of the vapor phase in the pores and the
coefficient of Knudsen diffusion therein. Clarification how
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J. Chem. Phys., Vol. 120, No. 24, 22 June 2004
Self-diffusion of liquids in nanopores
the morphology of the porous space may change the picture
presented above needs further experimental investigations.
ACKNOWLEDGMENTS
R.V. and V.T. are obliged to the Alexander von Humboldt Foundation for financial support. Stimulating discussions with Douglas M. Ruthven and Sergey Vasenkov are
gratefully acknowledged.
APPENDIX: DISTRIBUTION OF PORE SIZES
With the given pore-size distribution function f (d), at
the relative pressure z pores with diameters up to
d cr ⫽
2␴M
,
␳ l RT ln共 1/z 兲
共A1兲
f 共 d 兲⫽
1
& ␲␻ d
再
exp ⫺
ln共 d/d a 兲
2␻2
冎
The following results will be given for this particular distribution function. The results for other distribution forms can
be obtained in the same manner.
We start with the mass and the volume balance equations,
N l ⫹N g ⫹N s ⫽N,
共A3a兲
mN l mN g mN s
⫹
⫹
⫽V p .
␳l
␳g
␳s
N l⫽
Vp
␷0
冕
d cr
0
f 共 x 兲 dx⫽
冋 冉
ln共 d cr /d a 兲
Vp
1⫹erf
2␷0
&␻
冊册
冉
N g ⫹N s ⫽n s 4V p
␪a
␪1
冕
⬁
d cr
冊
f 共x兲
dx ,
x
冉
冊
2n s V p ␪ a ␻ 2 /2
ln共 d cr /d a 兲 ⫹ ␻ 2
e
erfc
.
N g ⫹N s ⫽
da ␪1
&␻
By introducing
共A6兲
冊
共A7b兲
共A8a兲
␳ g 1⫺ ␪ r
,
␳s ␪r
共A8b兲
f l ␳ g 1⫺ ␪ r
⫺
,
␪r ␳s ␪r
共A8c兲
respectively.
In this model, the coefficient of Knudsen diffusion also
appears to be pressure-dependent because the effective tube
diameter changes with the relative pressure. In this case Eq.
共12兲 has the form
d eff⫽d
冑
1⫺ ␪ r
.
1⫺ f l
共A9兲
Consequently, D K is obtained by averaging over the number
of molecules contained in the pore of diameter d eff :
D K⫽
冑
⬁
8RT 1⫺ ␪ r 兰 d cr x f 共 x 兲 dx
.
9 ␲ M 1⫺ f l 兰 d⬁ f 共 x 兲 dx
共A10兲
cr
The ratio I of the integrals in Eq. 共A10兲 is equal to
1⫹erf
I⫽d a e
共A5兲
where n s is the surface number density, and the term in
brackets gives the total surface of the condensate-free pores.
Here we used the notation ␪ a to distinguish between the
adsorption in the pores containing the vapor and the adsorbed phases ( ␪ a ) and the total adsorption 共which further
will be referred to as ␪ r ). If doing so, one may use, for
example, Eq. 共10兲 for ␪ a .
With the pore-size distribution Eq. 共A2兲 the last equation
leads to
冉
共A7a兲
,
fl
,
␪r
p s ⫽1⫺
, 共A4兲
where ␷ 0 is the effective molecular volume. The number of
molecules in the rest of the pores containing the vapor phase
and surface layers is
␪a 2
ln共 d cr /d a 兲 ⫹ ␻ 2
N␷0
⫽ f l ⫹ e ␻ /2 erfc
,
Vp
2
&␻
p g⫽
共A3b兲
Note, that by N l we denote the total amount of molecules in
the liquid-filled pores. It may be found to be
␪ r⬅
p l⫽
共A2兲
.
冊册
ln共 d cr /d a 兲
N l␷ 0 1
⫽ 1⫹erf
Vp
2
&␻
i.e., the fraction of molecules in the condensate-filled pores
with respect to the total filling, and the total pore loading, the
representation of the relevant equations may be simplified.
Here, the identity n s ⫽d a ␪ 1 /4␷ 0 was used. It is important to
note that ␪ r describes the isotherm in the whole range of
relative pressures z⫽0⫼1, also including the region of capillary condensation. It is equal to ␪ a in the low-pressures
range without capillary condensation. Finally, using Eqs.
共A3兲 and 共A7兲, the fractions of molecules in the liquid, the
vapor, and the adsorbed phases can be written as
will contain the liquid phase, and bigger ones the vapor
phase. As we have shown, the pore-size distribution function
f (d) in the porous silicon crystals under study is well described by the log-normal distribution function
2
冋 冉
f l⬅
11813
␻ 2 /2
冉
␻ 2 ⫺ln共 d cr /d a 兲
1⫺erf
冉
&␻
ln共 d cr /d a 兲
&␻
冊
冊
.
共A11兲
Thus, in the frame of the considered model with a poresize distribution, the effective self-diffusion coefficient is
given by the equation
D e⫽ p lD l⫹ p gD K⫹ p sD s ,
共A12兲
with all parameters given above.
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