Indian Journal of Pure & Applied Physics
Vol. 44, October 2006, pp. 732-740
Molecular diffusion and confinement effect: Neutron scattering study
R Mukhopadhyay & S Mitra
Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085
Received 4 October 2005; accepted 3 July 2006
Confined fluids generally have properties different from those of adsorbed layer or bulk phases. Diffusion, translational
or rotational, is altered significantly on confinement. The molecular dynamics and its effect due to confinement in various
environments as studied by Quasi-elastic Neutron Scattering (QENS) technique have been reported. QENS offers a unique
possibility of analyzing spatial dimensions of atomic or molecular processes in their development over time. It is found that
dynamics of a fluid confined in porous medium depending on the nature of the medium and interaction, can have a variety
of modifications. The diffusivity of guest molecules embedded in zeolite or molecular sieves depends very much on the
combination of the guest and host structures, shape and specific interactions.
Keywords: Molecular diffusion, Neutron scattering, Zeolites, Quasi-elastic neutron scattering
IPC Code: G21K1/00, G01N21/47
1 Introduction
Neutron scattering is a powerful experimental tool
for studying the structure and dynamics of atoms and
molecules in condensed matter1,2. This is possible as
neutron has a wavelength matching with the interatomic spacings in solids and also its energy is close
to the excitations in the solid. It is particularly suited
for studying the dynamics of protons because of the
latter’s large scattering cross-section compared to that
of other elements (σH∼81 barns, σother element∼4-5
barns). The inelastic neutron scattering experiments
are used either for studying the periodic motions like
vibrations or for studying the thermally activated
single particle motions which show up in Dopplerbroadened elastic lines. The latter are studied by using
a technique known as the Quasi Elastic Neutron
Scattering (QENS). It offers a unique possibility of
analyzing spatial dimensions of atomic or molecular
processes in their development over time. The timescale of the dynamical motion, its geometry as well as
the nature of the hindering potential can all be
obtained from the neutron scattering experiments,
which are carried out using either a triple axis or a
time-of-flight spectrometer1. A few examples of
molecular motion studies in various solids and liquids
using QENS technique can be found in Ref. 3.
The problem of diffusion of molecules in restricted
geometries (confinement) can be found in a large
variety of physical situations. Common examples are
water in rock or sand stones and water in biological
membranes. Other examples of structures that impose
spatial restriction include polymer gels, clays, micells,
vesicles and micro-emulsions. Zeolites are also
porous crystalline materials that absorb a number of
molecules in relatively large quantities. This makes
them useful in a number of industrial proceses, mostly
in petroleum and petrochemical industries, as
catalysts and molecular sieves. An understanding of
the role of various factors in the characteristics of
molecular sieves could lead to tailor-made, highly
efficient molecular sieves. This is largely determined
by the diffusivity of the guest molecules within the
cavities of molecular sieves. The sorption, binding
and the transport characteristics of various adsorbents
in zeolitic pore systems have been investigated
extensively, both experimentally and theoretically,
with an objective to achieve the fundamental
understanding of the behaviour of guest molecules in
confined geometries. It is also widely reported that the
thermodynamic and transport properties of fluids are
considerably altered on their physical confinement in
well-defined channel and cavity system of porous
materials. Two competing effects seem to be the main
contributors to the modification of the dynamics of
the confined fluid with respect to the bulk phase: (i)
the geometric confinement and (ii) the interaction
with the host cage. In other words, the problem can be
addressed by asking how the properties of the porous
medium, such as size, surface area or the chemical
nature of the interface can modify the dynamical
behaviour.
MUKHOPADHYAY & MITRA: MOLECULAR DIFFUSION AND CONFINEMENT EFFECT
733
The QENS studies investigating localization and
dynamics of molecules and the role played by the
symmetry and the structure of the guest molecule in
addition to the physical characteristics of the sorbents
have been reported. The results of the investigation of
water dynamics in porous media (alumina gel and
hydrotalcite) and modifications in the diffusivity of
various
hydrocarbon
molecules
(benzene,
cyclohexane, methanol etc.) in various zeolites
(HZSM-5 and Na-Y) and a molecular sieve (MCM41) are discussed.
from the average leads to quasi-elastic part. L (Γ, ω)
is a Lorentzian and Γ is the half-width at halfmaximum (HWHM) of the Lorentzian function
inversely related to the time constant of the motion. It
is convenient to analyse the data in terms of elastic
incoherent structure factor (EISF), which provides the
information about the geometry of the molecular
motions. If Iel(Q) and Iqe(Q) are the elastic and quasielastic intensities respectively, the EISF is defined as1
2 Theoretical Details
In a neutron scattering experiment the scattered
intensity is analysed as a function of both energy and
momentum transfer. The quantity measured is the
double
differential
scattering
cross-section
representing the probability that a neutron is scattered
with energy change dE into the solid angle dΩ1
Therefore, A(Q) in Eq. (3) is nothing but the EISF.
d 2σ
∂ω∂Ω
∝
k
k0
[σ
coh
Scoh (Q, ω ) + σ inc Sinc (Q, ω ) ]
… (1)
S(Q,ω) is known as the scattering law and the
subscripts coh and inc denote the coherent and
incoherent components. k and k0 are the final and
initial wavevectors. Q = k-k0 is the wavevector
transfer [Q= 4π sinθ /λ, where 2θ is the scattering
angle] and ћω= E-E0 is the energy transfer. Incoherent
scattering involves the same nucleus at two successive
times, so there are no interference effects between the
amplitudes scattered by different nuclei. Whereas in
coherent scattering, the total intensity observed results
from the sum of the different intensities scattered
from the individual nuclei. In hydrogenous systems,
such as benzene, cyclohexane and methanol, total
scattering is dominated by the incoherent scattering of
protons as σcoh (any element) << σinc (proton). So in
the case of hydrogenous systems,
d 2σ
∂ω∂Ω
∝
k
k0
[σ
inc
Sinc (Q, ω ) ]
… (2)
In general, the incoherent scattering law can be
written as2
Sinc (Q, ω ) ∝ A(Q )δ (ω ) + [1 − A(Q ) ] L(Γ, ω )
… (3)
where the first term is the elastic part and second term
is the quasi elastic part. The contribution to the elastic
part comes from the average position and fluctuation
EISF =
I el(Q)
I el(Q)+ I qe(Q)
… (4)
3 Results
Water in porous materials such as vycor glass,
silica gel, nafion, lignite coal etc5. has been actively
under investigation because of its relevance in
catalytic and separation processes. Alumina
exemplifies the physio-chemical properties of this
class of porous materials in that firstly, it has high
surface area and in aqueous suspensions variable pH
values and secondly, that it undergoes reversible
chemical changes when exposed to water. These
properties make alumina very useful in adsorptive and
separation processes.
Hydrotalcite (HT), or the magnesium aluminium
hydroxycarbonate, Mg6Al2(OH)16(CO3).4H2O, one of
the representative materials belonging to the family of
layered double hydroxides (LDH), has attracted much
attention in recent times due to their practical
applications as catalysts, catalyst supports, ion
exchangers and composite materials6. Since these
materials provide three-dimensional structures
characterized by superior porosity and ion exchange
properties, these materials found some industrial
applications. LDHs containing carbonate anions have
been used as antacids7. Those containing chloride
anions are used as adsorbents of the phosphate ions
contained in the human intestine8.
The results found from the studies on water
confined on alumina gel and in hydrotalcite material
have been described.
Alumina gel is a porous material with average pore
size of 50 Å dia. QENS experiments were performed
on alumina gel heat treated at 700 oC with hydration
level of 30 wt% using the high resolution LAM-80ET
spectrometer (resolution 17 and 6.5 μeV) at KENS,
KEK, Japan9 and the low resolution QENS
INDIAN J PURE & APPL PHYS, VOL. 44, OCTOBER 2006
734
0.12
296 K
290 K
280 K
270 K
250 K
HWHM (meV)
I
0.08
II
0.04
0.00
0.0 0.05
0.5
1.0
1.5
2
2.0
2.5
3.0
-2
Q(Å )
Fig. 1⎯Variation of HWHM with Q2 in hydrated al-gel. Solid
lines are the fits with jump diffusion model
spectrometer10 (resolution 200 μeV) at Dhruva,
BARC, Mumbai. The translational motion is studied
using higher resolution instruments where the faster
dynamics (rotational) contribute only as a
background. After incorporating the information
about translational motion, the data from the low
resolution instrument were analysed to obtain the
information about the rotational dynamics11.
The high resolution data were analysed considering
a scattering law which consists of one elastic line (for
dry alumina), one elastic and one quasi-elastic line
(for translation motion of water) and a background.
An elastic line for translational motion of water
originates due the fact that the water molecules are
not allowed to leave the spherical pores and is used to
calculate the elastic incoherent structure factor
(EISF). The radii of these spherical pores known as
localisation radius a are obtained from the extracted
EISF. The variations of HWHM of quasi-elastic lines
at different temperatures with Q2 are shown in Fig. 1.
Γ(Q) versus Q2 shows a flat and constant value Γ0 at
small Q region and increases asymptotically
approaching another constant value Γ∞ at large Q. The
first feature is due to the confinement effect. Volino
and Dianoux12 predicted Γ(Q) to remain constant
(Γ0=4.33 [Dloc/a2]) from Q=0 to Q=Q0=π/a. The Dloc
represents the local diffusion constant within the
volume of confinement. At larger Q values, the line
widths follow the jump diffusion behaviour13 with
diffusion constant (Dt) and residence time in between
jumps (τ). From the constant value of Γ(Q) at low Q,
local diffusion constant, Dloc is estimated. The values
of Dloc, Dt, a and τ obtained for different temperatures
are presented in Table 1.
The data from the QENS spectrometer at Dhruva
have a contribution from both the translational and
rotational motions of the water molecules. A
scattering law comprising the translational and
rotational motions was used to analyse the data from
the low-resolution spectrometer. For the rotational
part of the scattering law, isotropic rotational
diffusion as derived by Sears14 was assumed. The
parameters corresponding to translational motion of
water as obtained from high-resolution experiment are
used as known parameters. Rotational diffusion
constant obtained as 0.1 meV at room temperature.
This is very similar to the bulk water diffusion
constant at room temperature15. So in the pores of the
alumina gel, water exists in two dynamically different
states, one attached to the surface and undergoes
localized motion, other in the remaining volume
undergoes hindered dynamics which approach the
bulk like at low temperatures.
Hydrotalcite
with
no
excess
water
(Mg6Al2(OH)16CO3. 4H2O) did not show any quasielastic broadening over the instrumental resolution
(ΔE=200 μeV). However, quasi-elastic broadening
was seen in 9, 21, 30 and 50% excess water sample.
The translational motion of water molecules can be
described by a Lorentzian function. Since, rotational
motion of a small water molecule (size ~ 2 Å) is not
expected to get affected due to confinement between
two layers with separation of about 8 Å, the rotational
diffusion constant is assumed to be the same as that of
bulk water15. The QENS spectra of excess water
sample are then fitted with a scattering function which
consisted of both rotational and translation motions of
water molecules. Only parameter to be found is the
HWHM of the translational component at each Q
value. QENS spectra can be fitted very well with this
assumption at all the Q values. Fig. 2 shows the
typical fitted QENS spectra with elastic (dotted) and
quasi-elastic (dashed) components separated. With
many other models for translational motion, obtained
Γ(Q) is found to follow the random translational jump
diffusion model13 which can be described by :
Γ (Q ) =
DT Q
2
1 + DT Q τ
2
… (5)
where DT and τ are the translational diffusion constant
and residence time in between jumps. Variation of
Γ(Q) with Q2 is shown in Fig. 3. Solid lines are the fit
with Eq. (5). Translational diffusion constants and
MUKHOPADHYAY & MITRA: MOLECULAR DIFFUSION AND CONFINEMENT EFFECT
735
Table 1⎯Extracted parameters for hydrated Al-gel at different temperatures11. Parameters for bulk
water are also given for comparison.
T (K)
a (Å)
Dloc
Dt
Dbulk
τ0 (ps)
τBulk (ps)
296
290
280
270
250
6.85±0.9
2.63
6.57±0.6
6.51±0.5
2.07
1.71
1.38± 0.02
1.26± 0.02
1.08± 0.04
1.04± 0.02
0.8± 0.2
2.3
2.0
1.3
0.9
0.4
3.77± 0.05
3.85± 0.08
4.34± 0.19
8.77± 0.13
20.27± 2.9
1.1
1.5
2.3
4.5
22.5
600
30 % water
500
400 Q=1.8 Å
-1
Q=1.8 Å
-1
200
HWHM (meV)
400
300
200
100
0
S(Q,ω) (arb. units)
0.12
50 % water
600
0
Data
Fit
Elastic
Quasielastic
500
400 Q=1.32 Å
-1
600
Q=1.32 Å
-1
0.08
0.04
50% excess water
30% excess water
21% excess water
9% excess water
400
300
200
200
100
0
0
500
600
0.00
0.0
0.5
1.0
1.5 2.0
2
-2
Q (Å )
2.5
3.0
3.5
400
300
Q=0.8 Å
-1
400 Q=0.8 Å
200
-1
Fig. 3⎯Variation of translational HWHM with Q2 for different
excess water samples. Solid lines are fit with random jump
diffusion model (Eq. (4))
200
100
0
-1.0
-0.5
0.0
0.5
1.0
0
-1.0
-0.5
0.0
0.5
1.0
Energy Transfer (meV)
Fig. 2⎯Typical fitted QENS spectra alongwith separated elastic
(dotted) and quasi-elastic (dashed) components for 30 and 50 %
excess water in hydrotalcite
Table 2⎯Dynamical parameters for the excess water in
hydrotalcite16
Water
content
(wt %)
9
21
30
50
Jump length
(<l>) (Å)
Residence
time (τ) (ps)
Diffusion
Constant=l2/6τ
(× 10-5 cm2/sec)
2.2
2.3
2.5
2.4
7.0
6.6
5.7
4.6
1.1
1.3
1.9
2.1
residence time obtained for 9, 21, 30 and 50% excess
water are given in Table 2. While the mean jump
lengths have not changed much with different water
content, residence time showed an increase when the
water content is less. This essentially indicates that
with the increase in water content, water molecules
move faster. These values can also be compared with
corresponding values obtained for bulk water. Mean
jump length and residence time obtained for bulk
water at 20oC were found to be 1.29 Å and 1.25 ps,
respectively15.
Hence, even in 50% water content sample, though
diffusion constant of excess water molecules is having
a similar value as that of bulk water, residence time is
still larger than that of bulk water. It is also found that
the translational diffusivity decreases with the
decrease of the amount of excess water. This is
mainly due to the fact that when the amount of excess
water is small they are mostly bound to the host layer.
With increase amount of the excess water more and
more of the water molecules are available away from
the layer surface and are relatively free. 1H T1
relaxation measurements by NMR have also clearly
indicated that the hydration water at lower
percentages (up to ~30%) shows that this water
appears to be in a more bound state.
INDIAN J PURE & APPL PHYS, VOL. 44, OCTOBER 2006
736
100
S(Q,ω) (arb. units)
50
Cyclohexane in HZSM-5
80
40
-1
Q=1.8 Å
60
30
40
20
20
10
0
200
-1
Q=1.8 Å
0
60
-1
150
Benzene in HZSM-5
-1
Q=1.32 Å
Q=1.32 Å
40
100
20
50
0
0
250
200
-1
75
Q=0.8 Å
150
-1
Q=0.8 Å
50
100
25
50
0
-1.0
-0.5
0.0
0.5
1.0
0
-1.0
-0.5
Energy Transfer (meV)
0.0
0.5
1.0
Fig. 4⎯Typical QENS spectra at some Q values for benzene and
cyclohexane adsorbed in HZSM-5. Spectra are separated into
elastic and quasi-elastic components. Solid lines represent the
fitted lines and dotted lines represent the instrument resolution
function. Dashed lines are the quasi-elastic components
1.0
Benzene
Cyclohexane
EISF
0.8
0.6
(a)
0.4
(b)
0.2
(c)
(d)
0.0
0.5
1.0-1
Q(Å )
1.5
2.0
Fig. 5⎯Variation of elastic incoherent structure factor with Q for
benzene and cyclohexane adsorbed in HZSM-5 zeolite. Lines are
the theoretical EISF using different models (see text)
The role played by the pore geometry and by the
frame-work acid sites of zeolites in the catalytic
activity and selectivity of various hydrocarbons has
been under extensive investigation. Dynamics of the
guest molecule depends on the shape and size of the
host zeolitic system. To understand the dynamics and
binding states of various hydrocarbon molecules
adsorbed in the different size of pores in different
zeolites, we have studied benzene, cyclohexane and
methanol in HZSM-5 zeolites and in MCM-41
molecular sieve under ambient conditions18-20 and
propane molecules in Na-Y zeolite at different
temperatures21.
First, we discuss the situation where the guest
molecule is of similar size to the pores (HZSM-5
zeolite) and then when pore size is larger (MCM-41
molecular sieve) than the molecular size. In ZSM-5
zeolites, there are two types of channels consisting of
10 membered oxygen rings. The straight elliptical
channels (5.7-5.2 Å) are interconnected by nearcircular channels (5.4 Å) in a zig-zag fashion and
there are four channel intersections per unit cell. In
the experiment the guest molecules were loaded to the
saturation. The MCM-41 sample (Si/Al ratio ~30)
used in this study was synthesized by a conventional
hydrothermal route; fumed silica and Al-isopropoxide
being the source of Si and Al, respectively. The N2
adsorption surface area, pore size and pore volume of
a calcined sample were evaluated to be around 720
m2g-1, 34 Å and 0.68 cm3 g-1, respectively.
QENS data from dehydrated HZSM-5 zeolite
showed presence of small amount of hydrogen but no
quasi-elastic broadening was observed indicating that
they are not mobile at least in the time window of the
spectrometer. However, quasi-elastic broadening was
observed for samples loaded with benzene as well as
cyclohexane indicating the presence of dynamical
motions associated with the guest molecules. Here
saturation loading was used for both the cases.
Experimental S(Q,ω) and the separated elastic and
quasi-elastic components at typical Q-values are
Table 3⎯Comparison of the dynamical parameters for rotational motion of various molecules adsorbed in
HZSM-5
Adsorbed
species
Model
Residence time
(ps)
Residence time for
bulk solid
(ps)
Residence time for
bulk liquid at RT
(ps)
Methanol
Benzene
Cyclohexane
No broadening observed
6-fold Jump rotation
3-fold Jump rotation
⎯
16.5
8.2
⎯
19.2 at 277 K
6.0 at 180 K
10.2
2.5
1.7
MUKHOPADHYAY & MITRA: MOLECULAR DIFFUSION AND CONFINEMENT EFFECT
0.16
0.12
HWHM (meV)
shown in Fig. 4. The QENS spectrum of the
dehydrated zeolite was subtracted from the spectrum
of the corresponding hydrocarbon loaded samples.
The presence of elastic line in the spectra of Fig. 4
reveals the presence of the rotational motion of the
molecules occluded in HZSM-5. The values of the
extracted EISF (using Eq. 3) are depicted in Fig. 5.
On comparing these EISF values with different
theoretical models, it is possible to delineate the
nature of the rotational motion of the guest molecules
inside the cages of HZSM-5 zeolite. Different models
that may account for the rotational motion of guest
molecules, benzene and cyclohaxene in the present
study, inside the cages of HZSM-5 zeolite are as
follows: (1) In the isotropic rotational diffusion
model, the molecules undergo small angle random
rotations. Then, on an average, no most probable
orientation exists. The incoherent scattering law for a
scattering particle undergoing isotropic rotational
diffusion on the surface of a sphere of radius a with
rotational diffusion constant Dr is reported in the
literature1, 2. In jump diffusion model among N
equivalent sites on a circle, it is assumed that the
motion is specified by jumps of certain angle with a
characteristic time τ called the residence time.
Barnes22 has worked out a scattering law for such a
model. The theoretical EISF for isotropic rotational
diffusion (curve d) and for a jump among N
equivalent sites on a circle (curve a: 2-fold, b: 3-fold
and c: 6-fold rotation), are shown in Figure 6.
The experimental EISF values for benzene
adsorbed in HZSM-5 closely follow the curve (c), it is
apparent that the benzene molecules most likely
perform a 6-fold jump rotation within the channels of
HZSM-5 (Fig. 4).
However, experimental EISF for cyclohexane
adsorbed in HZSM-5 follow the curve (b) indicating
the 3-fold jump for cyclohexane adsorbed in HZSM-5
zeolite. We may also point out that the isolated
benzene molecule has 6-fold symmetry. The lowest
energy conformation for isolated cyclohexane is the
chair conformation of D3d symmetry23. The 3-fold
jump rotation observed in case of cyclohexane is thus
in accordance with the chair conformation of
cyclohexane molecule. So, the molecular symmetry
seems to play a major role in deciding the geometry of
rotation. The value of the residence times
(HWHM= h / τ ) for the benzene and cyclohexane
molecules in the channel of HZSM-5 are given in
Table 3. Residence times obtained for corresponding
737
Data
Methanol
Benzene
Cyclohexane
0.08
0.04
0.00
0.0
Fit with C-E model
Methanol
Benzene
Cyclohexane
0.8
1.6
2.4
2
3.2
4.0
-2
Q (Å )
Fig. 6⎯Variation of HWHM with Q2 for various molecules
adsorbed in MCM-41. Lines are the fit with Chudley-Elliott
model
bulk solid or liquid are also given in Table 3. It can be
seen that residence times obtained for benzene and
cyclohexane adsorbed in HZSM-5 zeolite resemble
that of bulk solid of the corresponding molecule.
However, no quasi-elastic broadening was observed
for methanol adsorbed in HZSM-5 zeolite indicating
absence of motion in the time scale of 10-10-10-12 ps.
MCM-41 is a relatively large pore (~34 Å dia)
system. QENS spectra from the dehydrated MCM-41
did not show any quasi-elastic broadening. However,
significant broadening was observed from the sample
loaded with benzene, cyclohexane and methanol. In
order to understand the contribution of occluded
molecule, the QENS spectrum of dehydrated MCM41 was subtracted from the spectrum of the loaded
sample. The model scattering law given in Eq. (3) was
convoluted with the instrumental resolution and the
parameters (A(Q) and Γ ) were determined by least
squares fit to the data. It was found that the elastic
component (A(Q)) is almost zero at all the Q-values
suggesting that the observed dynamics does not
involve any elastic part and one Lorentzian function
explains the data quite well. The HWHM values of
the Lorentzian functions as obtained from the fit are
given in Fig. 6. To arrive at an understanding of the
dynamics of adsorbed molecules in the pores of
MCM-41 different models for the translational motion
of the guest molecules can be envisaged. The most
simple motion that can be thought of is the Brownian
motion where it is assumed that the particles move
under the influence of the forces arising from the
INDIAN J PURE & APPL PHYS, VOL. 44, OCTOBER 2006
738
Table 4⎯Comparison of the dynamical parameters for translational motion of various molecules adsorbed in MCM-41 according to
Chudley-Elliott model
2
Adsorption in MCM-41
Jump length
(<l>) (Å)
Residence time (τ)
(ps)
Diffusion constant (l /6τ)
-5
(×10 cm2/sec)
Bulk liquid Diffusion constant
-5
2
(×10 cm /sec) at RT
Methanol
Benzene
Cyclohexane
2.3 ± 0.6
3.1 ± 0.3
2.4 ± 0.7
5.8 ± 0.5
7.5 ± 0.6
5.7 ± 0.7
1.5± 0.1
2.2 ± 0.6
1.7 ± 0.4
2.6
2.2
1.7
collisions between them. In this case, the scattering
law shows a Lorentzian shape whose HWHM
increases with the momentum transfer according to a
DQ2 law and provides a direct method of determining
the diffusion constant. However, Half Width at Half
Maxima (HWHM) of the quasi-elastic component
deviates from the linear behaviour at high Q values
which suggests that simple Fick’s law is not adequate
in describing the experimental data (Fig. 6). The other
model formulated by Chudley and Elliott24 assumes
that for a time interval τ, the particle remains at a
given site, vibrating about equilibrium position,
building-up a thermal cloud and then jumps to another
site in a negligible time. The jump length l is assumed
to be the same for all such jumps under consideration.
This model is also called a fixed jump model. The
powder averaged scattering law in this case can be
written as a Lorentzian with Half Width at Half
Maxima (HWHM) can be written as:
Γ (Q ) =
1 ⎡ sin Ql ⎤
1−
τ ⎢⎣
Ql ⎥⎦
… (6)
One can also find out the diffusion constant, D from
Einstein’s relation
D=
l2
6τ
… (7)
In Fig. 6, the fit of the HWHM values of the
Lorentzians as per the Chudley-Elliott model is also
shown. As seen in Fig. 6, the experimental values fit
quite well with this model. The values of residence
time (τ), jump length (<l>) and the diffusion constant
(D) as obtained from the fitting are indicated in
Table 4 for various adsorbed molecules. On
comparing this D value with the value reported for the
diffusion constant of liquid methanol at 298 K (D =
2.4 × 10-5 cm2/sec (Ref. 25) or 2.6 × 10-5 cm2/sec
(Ref.26), we may conclude that the translational
diffusion constant of methanol in MCM-41 is much
less than that of bulk liquid methanol. However, the
diffusion constants obtained for benzene and
cyclohexane adsorbed in MCM-41 did not show any
change compared to the diffusion constant obtained
from its bulk liquid value27,28. This as well as absence
of rotational dynamics of methanol in HZSM-5 may
be attributed to the fact that methanol is a polar
molecule (dipole moment 1.7 D) whereas benzene and
cyclohexane are non-polar. Polar molecule while
passing through the cavities experiences forces from
different framework or extra-framework sites and in
the process diffusivity of the polar molecule reduces.
However, for non-polar molecules such as benzene
and cylohexane, molecular symmetry plays an
important role in deciding their diffusion parameters
such as jump lengths, residence time as indicated in
Table 4.
The dynamics of hydrocarbon molecule when the
size of the molecule is smaller than the pore size has
been investigated. Na-Y zeolite structure is made up
of a network of tetrahedrally connected pores (αcages) of diameter ~11.8 Å. The pores are
interconnected through windows of diameter ~8 Å.
Propane (‘size’∼ 6 Å) coverage of 4.5 molecules per
zeolitic supercage was used for the experiment.
Translational motion of propane in Na-Y zeolite was
studied using QENS spectrometer (ΔE=200 μeV) at
Dhruva. The data were interpreted in terms of the
translational motion of the propane molecule inside
the supercage of Na-Y. The analysis of the
experimental data was done using several available
models of the dynamics. But, the model of Hall and
Ross29 was found adequate and consistent with our
experimental data. in this model unlike Chudley and
Elliott model, the jump lengths s are not fixed but
Gaussian distributed Scattering law in this case is a
Lorentzian with HWHM,
Q
1⎡
Γ (Q ) = ⎢1 − exp( −
τ⎣
2
l
6
2
⎤
⎥ .
⎦
… (8)
here, τ is the residence time and <l2>0.5 is the root
mean square jump length. Diffusion constant, D can
be calculated from Einstein’s relation as defined in
MUKHOPADHYAY & MITRA: MOLECULAR DIFFUSION AND CONFINEMENT EFFECT
0.16
0.8
0.12
0.08
0.6
0.4
300 K
323 K
350 K
0.04
0.00
Expt.
Isotropic Rotational Diffusion
Uniaxial rotational Diffusion
1.0
EISF
HWHM (meV)
0.20
739
0.2
0.0
0
1
2
2
3
4
-2
Q (Å )
0
1
2
3
-1
Q (Å )
Fig. 7⎯Variation of HWHM of the quasi-elastic component with
Q for propane in Na-Y zeolite. Solid lines are the fit to the data
assuming Hall & Ross model (see text)
Fig. 8⎯Variation of EISF extracted from QENS spectra with Q.
Solid and dashed lines are the calculated lines for isotropic and
uniaxial model respectively
Eq. (7). Fig. 7 shows the variation of HWHM of the
quasi-elastic line with Q2. Solid lines are the fit with
Hall-Ross Model (Eq. 10). Residence time τ, root
mean square jump length, <l2>0.5 and diffusion
constant, D, are obtained for each temperature21. The
obtained diffusion constant, D (=<l2>/6τ), is 2.3×10-5
cm2/s at 300 K and found to increase with increase in
temperature. As molecular dynamics simulation
studies carried out by us have showed that the
propane molecules undergo very fast rotational
motion (6-8 times faster than the translational motion
in terms of the diffusion constant) and not expected to
be observed in the experiment on MARX
spectrometer10. In order to verify the MD prediction,
further experiments have been carried out using Triple
Axis Spectrometer (TAS) having a much wider
energy window (ΔE~3.3 meV, at fixed final energy30
of 20 meV). Data were recorded in the wavevector
transfer (Q) range of 0.8 -2.5 Å-1. Dehydrated zeolite
sample did not show any quasi-elastic (QE)
broadening, whereas propane loaded sample did show
significant QE broadening at room temperature
indicating the presence of dynamical motion of
propane molecules. To analyse the data, in the first
instant the elastic and quasi-elastic components in the
total spectrum were separated, which involves
convolution of the model scattering function S(Q,ω)
with the instrumental resolution function. Then, the
model parameters were estimated by least squares fit
with the experimental data. The fitting parameters
were the EISF and the half width at half maxima
(HWHM) of the Lorentzian function. The
contribution from the bare zeolite, towards elastic
scattering only, was subtracted from the data with the
propane-loaded sample. The extracted EISF is as
shown in Fig. 8. The theoretically calculated EISFs
for isotropic rotational diffusion and uniaxial
rotational diffusion1 are shown by solid and dashed
lines in Fig. 8. The isotropic rotational diffusion
model describes the experimental EISF quite well.
Thereafter, we have used the model S(Q,ω) for
isotropic rotational diffusion to fit with the
experimental QENS data and determine the rotational
diffusion constant DR. Very good fit was obtained at
all the Q-values with radius of gyration equal to 1.89
Å, which is also the average distance of the hydrogen
atoms from the center of mass of the propane
molecule. DR=1.05 (±0.09) × 1012/s, is the value
obtained from QENS experiment, is found to be close
to the value of 0.82 × 1012/s obtained from MD
simulation31.
4 Conclusion
Molecular dynamics and its effect due to
confinement in various environments as studied by
Quasi-elastic neutron scattering (QENS) technique are
reported. It is shown that diffusion, translational
and/or rotational, is altered significantly on
confinement in variety of system, like water in porous
media, hydrocarbon in zeolite or molecular sieves. It
is found that diffusivity of guest molecules depends
very much on the combination of the guest and host
INDIAN J PURE & APPL PHYS, VOL. 44, OCTOBER 2006
740
structures, their shape and mutual interactions. Gas
behaves as liquid and liquid behaves as solid due to
confinement effect.
Acknowledgement
We gratefully acknowledge our collaborators for
their contributions in the works reported here.
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