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J. Phys. Chem. B 2009, 113, 8066–8072
Dynamics of Adsorbed Hydrocarbon in Nanoporous Zeolite Framework V. K. Sharma,† S. Gautam,† S. Mitra,† Mala N. Rao,† A. K. Tripathi,‡ S. L. Chaplot,† and R. Mukhopadhyay*,† Solid State Physics DiVision, Chemistry DiVision, Bhabha Atomic Research Centre, Mumbai 400 085, India ReceiVed: February 17, 2009; ReVised Manuscript ReceiVed: April 23, 2009
We report quasielastic neutron scattering (QENS) and molecular dynamics (MD) simulation study of the dynamics of propylene molecules adsorbed in Na-ZSM5 zeolite. MD simulation studies suggest that rotational motion is almost an order of magnitude faster than translational motion. Therefore, spectrometers having different energy resolutions were used to determine the translational and rotational contributions. Translational motion, being slower, was distinctly observed in a narrower window spectrometer while both contribute to the wider one. Diffusion of propylene in the channels of Na-ZSM5 zeolite was found to follow jump diffusion model. Dynamical parameters corresponding to translational diffusion obtained from experiment are found to be consistent with MD simulation. Variation of elastic incoherent structure factor (EISF) suggests that rotational motion of propylene is isotropic. Although at short times the rotational motion was found to be anisotropic, as indicated in the MD simulation depicting restricted channel framework, but at long time it results in isotropic rotational motion. I. Introduction Diffusion of hydrocarbons in zeolites has attracted a great deal of attention in research because of both fundamental and applied interests. Well-ordered porous structure of zeolites makes them an ideal tool to experimentally study the phenomenon of molecular dynamics in confined geometries. Zeolites offer a wide range of cation exchange and catalytic property that can be used in many practical applications such as water purification by exchanging cations, petroleum industries for separation of hydrocarbons, and so forth.1,2 One of the most unique cation exchange properties of zeolites that is used for purification of water is that of removing dissolved cations (e.g., NH4+) that are not favorable to human or animal health by exchanging with biologically acceptable cations such as Na+, Mg2+, and so forth. Because of their acid strength and shape selectivity, zeolites offer a wide range of catalytic applications also in the petrochemical industry. One particular reaction, which is of immense interest to the automobile industry, is the selective catalytic reduction of NO. In a review on this topic, Amiridis et al.3 have highlighted the fact that unsaturated hydrocarbons (e.g., propylene) under excess oxygen condition inside a ZSM5 zeolite pore have a tremendous potential in reducing NO. Diffusion of adsorbed hydrocarbons in zeolites, determines the performance of zeolitic materials in catalytic, sorption, separation, and ion exchange applications. This depends upon many factors that include the host-zeolite interaction, the shape of the host, the volume of the cages and their potential energy landscape, temperature, and so forth.4 All of this makes the study of diffusion of hydrocarbons in zeolites interesting and essential. A diffusion anomaly as a function of molecular length in hydrocarbons adsorbed in zeolite was reported earlier.5 Quasielastic neutron scattering (QENS)6-17 and pulsed field gradient NMR18,19 are the two main experimental techniques to study self-diffusivity of hydrocarbons confined in zeolite pores. During * To whom correspondence should be addressed. E-mail: mukhop@ barc.gov.in. † Solid State Physics Division. ‡ Chemistry Division.
the past three decades, molecular dynamics (MD) simulations have been used to study diffusion of guest molecules confined in zeolites.20-29 Experimental and MD simulation studies are even more powerful when they are used in conjunction with each other. The QENS study provides an experimental test of simulation model whereas the latter not only provides insight on the details of the possible different types of motions but also does not suffer instrumental limitations. Dynamics of different hydrocarbons such as propane,9,11 acetylene,12,13,21 and 1,3 butadiene14,22 confined in Na-Y zeolite have been studied using QENS and MD simulation techniques by us. A comparison of diffusivity with respect to different guest molecules in Na-Y zeolite was reported.14 We have also studied diffusivity of benzene in HZSM5,7,8 cyclohexane confined in ZSM5, and MCM-41 zeolites.10 To probe the effect of host structure in this work, we report studies of the dynamics of propylene, an unsaturated hydrocarbon, adsorbed in Na-Y and Na-ZSM5 zeolites.15,16 Na-ZSM5 zeolite structure has two different channel systems, straight as well as sinusoidal, unlike the spherical cages of Na-Y zeolite. In ZSM5 zeolite, there are straight channels with an elliptical cross section of approximately 5.7-5.2 Å and are parallel to the crystallographic axis b. The sinusoidal channels having nearly circular cross section of 5.4 Å run along the crystallographic axes a and c. The resulting intersections are elongated cavities up to 9 Å in diameter. In general, a molecular system can have several types of motions, namely, whole molecule translation, rotation, and internal motions. A convenience in the study of molecular motion is provided by the fact that not all of these motions occur at the same time scales. The vibrational motion involves energies that are much larger than the other two types of motion and hence can be easily decoupled from them. However, decoupling of rotational and translational motion is not always possible since for some cases these might occur at the same time scales. In that case, it becomes imperative to consider the contribution of rotational motion along with the center of mass or translational motion. In many of our earlier studied systems,9,11,30 it was
10.1021/jp9014405 CCC: $40.75 2009 American Chemical Society Published on Web 05/15/2009
Adsorbed Hydrocarbon in Nanoporous Zeolite Framework possible to separate the contributions from translation and rotation by using two different energy resolutions either in different instruments or in the same instrument in different configurations. The slow translational motion could be investigated with high resolution whereas the faster rotational motion would contribute as background. The data with the lower resolution instrument will have contribution from both slow and fast motions, but in the analysis one could use the information as obtained from the high resolution instrument for the slower motion and obtain information pertaining to that of the faster one. Here we report the dynamics of propylene adsorbed in NaZSM5 zeolite as studied by quasielastic and molecular dynamics simulation. The studies were carried out in parallel. Results are consistent with each other. MD simulations corroborate the experimentally observed properties and predict more detailed insight that is confirmed by further experiments. II. Experiment The Na-ZSM5 samples used in the QENS study have composition Na2.08Al2.08Si93.92O192 · 16H2O and were obtained from Sud-Chemie. For ascertaining the crystalline nature of the samples, we have carried out X-ray diffraction, which revealed that the sample is crystalline with a high degree of order. The zeolite samples were put in two flat rectangular aluminum sample cells and dehydrated by evacuating for a period of about 48 h at a temperature 573 K under vacuum (10-5 torr). One of them was loaded with propylene gas (purity >99.99%) to saturation (∼11 molecules/unit cell) at ambient pressure. Quasielastic neutron scattering measurements were performed for both dehydrated (bare) zeolite and zeolite with propylene gas. Two instruments having different energy windows, QENS spectrometer31 and triple axis spectrometer (TAS),32 at Dhruva, Trombay, were used to record the QENS spectra. In the QENS spectrometer, energy analysis is done in multiangle reflecting crystal (MARX) mode in combination with a linear position sensitive detector.31 In the configuration used in the experiments reported here, the spectrometer has an energy resolution of ∼200 µeV with incident energy of 5.1 meV as obtained from standard vanadium sample. The wave vector transfer (Q) range covered is 0.67-1.8 Å-1. The TAS was used in inverted geometry in which incident energy Ei was varied and final energy was kept fixed at Ef ) 20 meV.32 In this configuration, this has energy resolution ∼3 meV in the Q range 0.8-2.5 Å-1 as obtained from standard vanadium sample. Data from the bare zeolite were used to estimate the contribution from bare zeolite to the data obtained from propylene loaded zeolite sample. III. Molecular Dynamics Simulation Molecular dynamics simulation of propylene molecules confined in ZSM5 (an isostructural analog of Na-ZSM5) zeolite have been carried out in the microcanonical ensemble. Atomic positions of ZSM5 zeolite were taken from the ref 33. The simulation cell consisted of a (2 × 2 × 2) unit cell of the ZSM5 zeolite with 64 propylene molecules at a loading of 8 propylene molecules per unit cell. The zeolite atoms were kept fixed throughout the simulation. Propylene molecule was assumed to be rigid and modeled in united atom model. Lennard-Jones potential parameters between guest-guest and guest-zeolite were taken from literature.24,34 Reorientation motion of propylene was studied using quaternion formalism.35 Only few features from MD simulation study relevant to the present experimental results will be discussed here. Details of MD simulation results will soon be reported separately.
J. Phys. Chem. B, Vol. 113, No. 23, 2009 8067 IV. Theoretical Aspects In a neutron scattering experiment, the scattered intensity is analyzed 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 ) p dω into the solid angle dΩ.6 In case of a system containing hydrogen atoms the incoherent scattering from hydrogen dominates the scattering and one can write
d2σ k ∝ [σincS(Q, ω)] ∂ω∂Ω k0
(1)
where S(Q,ω) is the incoherent scattering law, and k and k0 are the final and initial wave vectors. Q ) k - k0 is the wave vector transfer [Q ) 4π sin θ/λ, where 2θ is the scattering angle in case of elastic scattering], and pω ) E - E0 is the energy transfer. For a molecular system different kinds of motion, translational, rotational, and vibrational, can exist and generally it is assumed that these motions are dynamically independent for mathematically tractable solution. In the quasielastic regime ((2 meV), the vibrational contribution will be only through the Debye-Waller factor, e-2W, W ) 1/2 Q2 where is the mean square displacement. The total S(Q,ω) can be expressed as a convolution product of the respective scattering functions under the assumption that the motions are not coupled
Stot(Q, ω) ) exp(- Q2)[Strans(Q, ω) X Srot(Q, ω)] (2) Generally, the rotational scattering law consists of an elastic component and a series of Lorentzians for the quasielastic component. However, in absence of any knowledge of the model, scattering law for the rotational motion can be approximated as6
Srot(Q,ω) ) B(Q)δ(ω) + [1 - B(Q)]LR(ω,ΓR)
(3)
where the first term is the elastic part and the second is the quasielastic one. LR(ω,ΓR) is a Lorentzian function, where ΓR is the half width at half-maximum (HWHM) of the Lorentzian function, inversely proportional to the reorientation time τ. It is convenient to analyze the data in terms of elastic incoherent structure factor (EISF), which provides information about the geometry of the molecular reorientations. If Iel(Q) and Iqe(Q) are the elastic and quasielastic intensities, respectively, then EISF is defined as6
EISF )
Iel(Q) Iel(Q) + Iqe(Q)
(4)
Therefore, B(Q) in eq 3 is nothing but the EISF. There will be no elastic component for the translational motion and the scattering law will have only a Lorentzian function with HWHM of the Lorentzian function as ΓT. Scattering law, Stot(Q,ω) can also be expressed as the Fourier transform of intermediate scattering function Itot(Q,t)
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Stot(Q, ω) )
1 2π
∫-∞∞ Itot(Q, t)exp(-iωt)dt
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(5)
The incoherent intermediate scattering function, Itot(Q,t) is a time correlation function for a single particle and can be expressed in terms of the position vector of the scatterer
Itot(Q, t) ) 〈exp{iQ.(R(t) - R(0))}〉
(6)
where R(0) and R(t) are the positions of scatterer at time t ) 0 and time t, respectively, and the angular brackets indicate ensemble average. Therefore, eq 2 can also be expressed in terms of intermediate scattering function for translational and rotational motion as
Itot(Q, t) ) Itran(Q, t)Irot(Q, t)
(7)
where
Itran(Q, t) ) 〈exp{iQ.(R′(t) - R′(0))}〉
(8)
Irot(Q, t) ) 〈exp{iQ.(D(t) - D(0))}〉
(9)
Figure 1. Fitted QENS spectra from the propylene adsorbed in NaZSM5 zeolite using QENS spectrometer at some typical Q values. Instrument resolution is shown by dashed line in middle panel.
where R′(t), R′(0) is a radius vector of center of mass of propylene molecules in space-fixed frames at time t and time 0, respectively, and D is a radius vector of a hydrogen atom with respect to center of mass of propylene molecule in body fixed frames. Therefore, by using eqs 8 and 9, one can easily calculate the intermediate scattering function from MD simulation trajectories. In case the motion is unbounded in space, the intermediate scattering function is expected to decay to zero at tf∞. However, in case of localized motion, this function is expected to have a nonzero value even at tf∞. This nonzero value is equivalent to EISF that can be obtained from QENS spectra as the fraction of elastic component in the total spectra. V. Results and Discussion To analyze the QENS data, it is customary to assume a model scattering function, convolute with the instrumental resolution function, and then obtain the dynamical parameters involved in the model scattering function by least-squares fit to the experimental data. The resolution function is determined by measuring standard sample like vanadium or ZrH2. The quasielastic spectra were recorded in the wave vector transfer (Q) range of 0.67-1.8 Å-1 at 300 K for both bare zeolite and propylene-loaded zeolite samples on QENS spectrometer having energy resolution ∼200 µeV at Dhruva, Trombay. Significant quasi elastic broadening was observed in the case of propylene adsorbed in Na-ZSM5 zeolite whereas dehydrated Na-ZSM5 did not show any broadening over the instrument resolution. Thus, this broadening is related to the dynamical motion of propylene molecules adsorbed in Na-ZSM5 zeolite. Data as obtained from bare zeolite were subtracted from propylene-loaded sample. Subtracted data then correspond to propylene molecules only. To analyze the data, first it was attempted to separate the elastic and quasielastic contributions. It was found that elastic part is insignificant and a single Lorentzian function fits the data very well (Figure 1). Therefore, it could be concluded that the observed dynamics at QENS spectrometer is related to pure translation motion. The variation of width of the Lorentzian function, Γ, with Q2 as obtained from
Figure 2. Variation of HWHM with Q2. The lines correspond to the fit assuming different models as described in the text.
the fits, shown in Figure 2, suggests it is not a simple Brownian motion of Fickian type, but a jump diffusion process. In jump diffusion, it is assumed that diffusion occurs through jumps, that is, for a time interval τ, a species remains at a given site, oscillating about a center of equilibrium and after this time moves to another site in a negligible jump time. The time τ, spent at a given site, is called the residence time. Several different models of jump diffusion can be envisaged differing in the distribution of jump lengths, namely, Chudley and Elliott (CE), Singwi and Sjo¨lander (SS), and Hall and Ross (HR) models, which are the plausible ones for the present situation. In the Chudley and Elliott36 model, molecules are assumed to undergo jumps of a fixed length l, that is, all the jumps are identical and hence there is no distribution of jump lengths. However in systems like Na-ZSM5 zeolites, this is not expected to be a correct model because of the complex structure of the channel framework. In their model, Singwi and Sjo¨lander37 implicitly employed a random jump length distribution whereas Hall and Ross38 proposed a situation where the distribution of jump lengths in any direction exhibits a simple Gaussian behavior.
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TABLE 1: Jump Length (l), Residence Time (τ), and Diffusion Coefficient (DT) As Obtained from QENS Data and MD Simulation for Different Models Considered QENS experiment
MD simulation
model
l(Å)
τ (ps)
DT (10-5 cm2/sec)
l (Å)
τ (ps)
DT (10-5 cm2/sec)
CE SS HR
1.9 1.3 1.6
12 5 8
0.50 0.56 0.53
1.6 1.2 1.4
15 7 10
0.28 0.33 0.32
All three models mentioned above were tried to fit the experimentally obtained HWHM as shown by the lines in Figure 2. As evident from the figure all the three models fit the QENS data in a very similar way. Obtaining the parameters, root-meansquare jump length l, and residence time τ, from the fit, diffusion coefficient DT, was determined using Einstein relation DT ) /(6τ). Obtained values of diffusion coefficient, jump length, and residence time are given in table 1. Different types of motion are driven by intermolecular interactions and the sites at which the species spend most of their time are the sites of potential energy minima. The distribution of the jump lengths in this type of motion depends upon the topology of the confining medium, which determines the potential energy landscape. We have calculated the potential energy landscape experienced by the CH3 site of a propylene molecule due to the host matrix of ZSM5 zeolite. The position of zeolite atoms are held fixed and the same set of potential parameters are employed to calculate the potential energy as used in MD simulation. In order to focus on the interesting features in regions of low potential energy, the potential energy landscapes were constructed with setting all positive energies to zero (white background). Two such potential landscapes in X-Z plane at Y )15 Å and in X-Y plane at Z )14 Å are shown in Figure 3a,b, respectively. A clear effect of sinusoidal channels of ZSM5 zeolite in X-Z plane as mentioned in introduction is evident in Figure 3a. It may be noted that within the channels, there are several regularly arranged favorable positions for a guest molecule where the potential energy is minimum as shown by dark-blue region. Figure 3b provides a pictorial view of straight channels in X-Y plane, the sites of minimum energy (dark-blue region) in the potential energy landscape looks very well ordered. This implies that in zeolites, although the jump lengths are not expected to be constant, the spread in the jump length distribution is very small. Because of sharp distributions of jump lengths, it does not matter whether the jump lengths are distributed randomly, Gaussian, or constant. Therefore, all the three models fitting the experimental data in a similar way (Figure 2) is justified. Intersections of sinusoidal and straight channels are also seen in Figure 3a,b. Exact location of intersection was obtained by comparing few potential landscapes at different planes. It was found that at the channel intersections, potential of a propylene molecule is higher than that when it is nearest to sinusoidal channels or straight channels. This indicates that propylene molecule does not prefer the intersections of channels even though the free space at the intersection is large compared to the diameter of the channel. Therefore, it can be inferred that during the motion, propylene molecules spent more time in channels rather than at intersections. Similar behavior was also observed for methane molecules adsorbed in zeolite A.20 In the case of benzene confined in H-ZSM5, it was also shown that although the molecules were trapped at intersection because of their large size, translational diffusivity was very small.10 Trajectories of the center of mass as obtained from MD simulation provide a demonstration of how molecules move inside the channels of ZSM5 zeolite in different planes. The
same is shown in Figure 4. In X-Y plane, it gives a nice picture of straight channels whereas in X-Z plane, it reflects the geometry of sinusoidal channels of ZSM5 zeolites reflecting the effect of potential landscape. It may also be noted that one single propylene molecule diffuses to other channels through channel intersections and within the whole production run (1.3 ns), it covers several such channels. To obtain the translational diffusivity from MD simulation, intermediate scattering functions corresponding to center of mass of propylene molecule Itran(Q,t) within the channels of Na-ZSM5 zeolite are calculated using eq 8. Various possible combinations of Gaussians and exponentials was used to fit Itran(Q,t) versus t. It was found that the sum of a Gaussian and two exponentials as given below describes the Itran(Q,t) in entire Q and t ranges. 2 2
Itran(Q, t) ) A1(Q)e(-t Γ1(Q)/2) + A2(Q)e(-tΓ2(Q)) + A3(Q)e(-tΓ3(Q)) (10) Itran(Q,t) is expected to behave differently in two different regimes: (i) short time and small length scales and (ii) long time and large length scales. At short time and small length
Figure 3. Potential energy landscape of a CH3 site due to ZSM-5 zeolite framework (a) through the sinusoidal channels in the X-Z plane at Y ) 15 Å and (b) through straight channels in the X-Y plane at Z ) 14 Å.
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Figure 4. Trajectory of the center of mass of a propylene molecule in ZSM5. The two panels show the projections of the trajectory in (a) X-Y plane showing the straight channels and in (b) X-Z plane showing the sinusoidal channels of ZSM5 zeolite.
scales, a molecule does not interact with other molecules and therefore acts as a free particle (ideal gas). In this regime Itran(Q,t) should be a Gaussian.39,40 At longer time and at large length scales molecules experience several interactions, and hence no longer remain free. Itran(Q,t) in this case behaves as an exponential function similar to bulk liquid. However, in the case of diffusion in zeolite, dynamics is determined by its pore topology and it is found that two exponentials are required to fit the Itran(Q,t) at long times. Thus, in all, for a wide range of Q and t, the intermediate scattering function is approximated by a sum of a Gaussian and two exponentials. The three different components in Itran (Q,t) are shown in Figure 5a. The fastest component dominates at short time and short length scale indicating free particle behavior. The slowest component represents very slow motion of molecules similar to the one observed by Jobic et al. in the case of xylene adsorbed in X-type zeolites from their NSE experiment (D ) 10-8-10-9 cm2/s).17 These two components are either two fast or too slow to be observed in the present spectrometer. Variations of Γ values corresponding to three components at different Q values are shown in Figure 6. Half width at half maxima of the quasielastic component as observed from QENS experiment is also shown in Figure 6. It is clearly seen that the intermediate component falls in the range of QENS spectrometer. A variation of the width of this component with Q2 was fitted by jump diffusion models as was done for the experimental one. Parameters (jump length, residence time, and diffusion constant) so obtained are also shown in Table 1. The parameters obtained from experiment and that from MD simulation are found to be in agreement. Intermediate scattering functions corresponding to rotational motion, Irot(Q,t) are also calculated from positions of CH3 site of propylene molecules in center of mass frame (as described
Sharma et al.
Figure 5. Variation of (a) translational and (b) rotational intermediate scattering functions calculated from MD simulation trajectories with time t at Q ) 0.4452 Å-1. Existence of three different components in the translational intermediate scattering function is also shown in (a).
Figure 6. The behavior of Γ values in eq 10 corresponding to the three components of the intermediate scattering function Itran(Q,t) with Q. Filled circle symbols show HWHM of quasielastic width as obtained from QENS experiment.
in eq 9). Very different features can be noted from the behavior of the calculated intermediate functions corresponding to translational and rotational motions as shown in Figure 5 for a typical value of Q ) 0.4452 Å-1. The intermediate scattering functions corresponds to the rotational motion reaches to a nonzero minimum value within t ∼ 100 ps whereas that of the translational goes to zero at t ∼ 900 ps. The nonzero value of the minimum, due to the presence of local motion, provides information toward EISF as explained earlier. A much faster decay of the rotational component indicates an order of magnitude faster motion than translation. This is corroborative to the fact that rotational motion was not observed in the QENS measurement described above.
Adsorbed Hydrocarbon in Nanoporous Zeolite Framework
Figure 7. Variation of EISF as obtained from QENS data using TAS with respect to wave vector Q. The solid line correspond to the isotropic rotational diffusion model.
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Figure 9. Variation of rotational diffusion coefficient with Q.
Figure 8. Typical fitted experimental data observed with TAS assuming isotropic rotational diffusion model for propylene adsorbed in Na-ZSM5 zeolite at some typical Q values.
To investigate the fast rotational dynamics of propylene molecules adsorbed in Na-ZSM5 zeolite, as predicted by the MD simulation results, we have carried out an experiment using TAS at Dhruva, Trombay having wider energy resolution (∆E ∼ 3 meV).32 Indeed, significant broadening was observed for the propylene-loaded Na-ZSM5 zeolite whereas no broadening was observed for the bare Na-ZSM5 zeolite. Data for the bare zeolite was subtracted from propyleneloaded Na-ZSM5 data to get the contribution from the propylene molecules alone. Here both the fast and slow components are expected to contribute to the data and therefore the scattering law would have the contribution from both translation and rotation as a convolution product as given in eq 2. The solace in the translational contribution is already available from the measurement of the QENS spectrometer, and using those as known parameters the contribution of the rotational part are to be evaluated. Here we used the total scattering law as given in eq 2. Convoluting the total scattering function with the instrumental resolution function, the parameters of the rotational part (B(Q) and ΓR) were determined by least-squares fit with the measured data. The EISF extracted from the fit is shown in Figure 7. Out of the various plausible models that can be envisaged for describing the rotational motion of the propylene adsorbed in Na-ZSM5, isotropic rotational diffusion described the experimental EISF very well as evident in Figure 7. Radius of gyration
Figure 10. Trajectory of the CH3 site with respect to the center of mass of the molecule for (a) 1311 ps and (b) first 50 ps of the simulation production run in X-Y plane.
was obtained as 1.91 ( 0.01 Å, which is equivalent to the average distance of all hydrogen atoms from the center of mass of propylene molecules. The scattering law for isotropic rotational diffusion41 is given by
Srot(Q, ω) ) j20(Qr)δ(ω) +
∞ l(l + 1)DR 1 A (Q) π l)1 l (l(l + 1)D )2 + ω2 R (11)
∑
where Al(Q) ) (2l + 1)j2l (Qr) is the weight factor of the quasielastic part, jl’s are the spherical Bessel function of order l, and DR is rotational diffusion coefficient. With the maximum allowed wave vector transfer, Q, as 2.5 Å-1 in TAS, it was found that contribution only up to l ) 6 toward Al(Q)in eq 11 are sufficient.22 Using eqs 11 and 2, the total scattering function can be written as
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Stot(Q, ω) ) j02(Qr)
Sharma et al.
∞ ΓT(Q) 1 1 (2l + 1)j2l (Qr) × + 2 π (Γ (Q))2 + ω2 π l)1 T [ΓT(Q) + l(l + 1)DR]
∑
[ΓT(Q) + l(l + 1)DR]2 + ω2
(12)
To compare with the experimental data, eq 12 was convoluted with instrumental resolution and the parameters were determined by least-squares fit. Here only DR is to be determined. Typical data with fits obtained are shown in Figure 8. As evident from the figure, the model has provided very good description of the experimental data. Variation of rotational diffusion coefficients obtained from fits is shown in Figure 9. It is found that rotational diffusion coefficient is more or less constant with respect to Q with an average rotational diffusion coefficient of DR ) 0.37 meV. For isotropic rotational motion, the trajectory, as obtained from MD simulation, of a CH3 site of the propylene molecule with respect to the center of mass should trace a sphere in three dimensions. Figure 10a,b) shows the same for up to 1.3 ns and 50 ps in the X-Y plane. It is clearly seen from Figure 10a that the path traced by the CH3 site with respect to the center of mass of the molecule encompasses the surface of a sphere at long times, thereby showing that the molecule has had all the possible orientations during the time interval of the simulation. This shows that the rotational motion is indeed isotropic on the average over large time. But at short time, the motion is restricted by channel framework and not truly isotropic which is evident in Figure 10b. This is probably as the channel size of the ZSM5 zeolite is comparable with the size of the propylene molecule, the molecule does not have the full freedom to rotate in all the directions in a particular channel at short times. However, at long times the molecule can migrate to other channels through channel intersection thereby covering larger orientational space. Therefore, at sufficiently long time on the average a molecule can cover all the different orientations. This is in contrast to Na-Y zeolite where the large size of the host cage leaves the isotropic rotation of propylene.15 VI. Conclusion Dynamics of propylene confined in channel framework of NaZSM5 have been carried out using QENS measurements and MD simulations. Slower translational diffusion was studied using QENS spectrometer having energy resolution of ∼200 µeV and the data were found to be describable by jump diffusion models. Calculated potential energy landscape corroborated the experimental finding that the distribution of jump lengths is very sharp. MD simulations showed the translational motion of the propylene molecules in the channels of the ZSM5 zeolite consists of three components with different time scales and only the intermediate one corresponds to the experimentally observed one. It also showed the presence of faster rotational motion, which would be beyond the dynamical range of the QENS spectrometer. This was confirmed by measurements carried out using a much wider energy window spectrometer (∼3 meV) and it was found that propylene undergoes isotropic rotational diffusion in the channels of ZSM5 zeolite. MD simulation also indicated isotropic rotational diffusion consistent with the experimental findings; however, it showed that at shorter times the rotational motion deviates from isotropic behavior, which is probably expected from the topology of the ZSM5 zeolite framework. Here experiments and simulation were carried out simultaneously and independently. The results as obtained from these studies not only gave insight of the total dynamical process but also were consistent within each other.
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