Levitation Effect: Role of Symmetry and Dependence of Diffusivity on

Mar 14, 2011 - seen at the value predicted by the levitation effect. ... maximum as a function of diameter of the diffusant often referred to as the l...
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Levitation Effect: Role of Symmetry and Dependence of Diffusivity on the Bond Length of Homonuclear and Heteronuclear Diatomic Species Manju Sharma and S. Yashonath* Solid Sate and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India - 560 012 ABSTRACT: Molecular dynamics investigation of model diatomic species confined to the R-cages of zeolite NaY is reported. The dependence of self-diffusivity on the bond length of the diatomic species has been investigated. Three different sets of runs have been carried out. In the first set, the two atoms of the diatomic molecule interact with the zeolite atoms with equal strength (example, O2, the symmetric case). In the second and third sets which correspond to asymmetric cases, the two atoms of the diatomic molecule interact with unequal strengths (example, CO). The result for the symmetric case exhibits a well-defined maximum in self-diffusivity for an intermediate bond length. In contrast to this, the intermediate asymmetry leads to a less pronounced maximum. For the large asymmetric case, the maximum is completely absent. These findings are analyzed by computing a number of related properties. These results provide a direct confirmation at the microscopic level of the suggestion by Derouane that the supermobility observed experimentally by Kemball has its origin in the mutual cancellation of forces. The maximum in diffusivity from molecular dynamics is seen at the value predicted by the levitation effect. Further, these findings suggest a role for symmetry in the existence of a diffusivity maximum as a function of diameter of the diffusant often referred to as the levitation effect. The nature of the required symmetry for the existence of anomalous diffusivity is interaction symmetry which is different from that normally encountered in crystallography.

1. INTRODUCTION There have been a large number of investigations into the diffusion of linear molecules in zeolites. A comprehensive review of these studies is available in the literature.1-4 Kolokolov et al. have studied the mobility of a linear alkane, n-butane, adsorbed in zeolite ZSM-5 by deuterium solid-state NMR (H-2 NMR). The study suggests that n-butane molecules are essentially located in both straight and zigzag channels. The molecule performs fast intramolecular as well as reorientational motion as a whole.5 Krishna and co-workers have investigated the Maxwell-Stefan diffusivity of small molecules such as He, Ne, Ar, Kr, H2, N2, CO2, and CH4 in all silica zeolites MFI, AFI, FAU, CHA, DDR, and LTA as a function of loading.6 They find that the diffusivities are strongly dependent on loading. For a given molecule, this variation with loading depends on the zeolite structure and can exhibit either a decreasing or increasing trend. Further, they find that this is determined by the molecular dimensions. Laloue et al. have investigated the diffusion of a binary mixture consisting of n-hexane and 2,2-dimethylbutane in zeolite ZSM-5 through kinetic Monte Carlo.7 They report self-diffusivities and corrected diffusivities and acceleration/deceleration effects due to correlation between the adsorbed components. Smit and co-workers have investigated the entropy effects during sorption of mixtures of alkanes in different zeolites with the help of configurational-bias Monte Carlo.8,9 Equilibrium adsorption isotherms of linear and branched alkanes with six carbon atoms have been experimentally studied with the help of a tapered element oscillating microbalance by Zhu et al.10 A small kink in the isotherm is observed at about four molecules for n-hexane, while 2-methylpentane and r 2011 American Chemical Society

3-methylpentane show a second step in the adsorption isotherm at loadings over four molecules at 303 K. Hansenne et al. have investigated the role of entropy in the properties of benzene in zeolite L.11 Tsekov and Ruckenstein reported stochastic dynamics of a hydrocarbon molecule with a view to understanding resonant diffusion.12 Unusual behavior such as resonant diffusion is not uncommon in the field of confined fluids. Another example of a surprising behavior is the diffusivity of small molecules adsorbed in zeolites and on other surfaces studied by Kemball.13 These experimental measurements by Kemball showed that for certain adsorbate-adsorbent systems, such as xenon on mercury, the loss of entropy on adsorption is small.13 He suggested that in these systems the adsorbate can perform essentially unhindered translational motion on the surface (internal or external) of the adsorbent.13 Derouane and co-workers made a detailed analysis of some of these systems and found that those adsorbates whose diameter is similar to the void diameter within the adsorbent suffered minimum loss of entropy.14-16 Adsorbates whose diameter is smaller than the void diameter present within the adsorbent suffered a larger loss of entropy. Derouane and coworkers suggested that the force exerted on the adsorbates by the adsorbent is a minimum when the diameter of the adsorbate is comparable to the diameter of the void present within the adsorbent (see Figure 1). In a related but independent study, it Received: October 17, 2010 Revised: January 8, 2011 Published: March 14, 2011 3514

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Table 1. Lennard-Jones Interaction Parameters for the Symmetric Guest and Host (A and B Sites Are Identical for Symmetric Interaction) type of interaction

Figure 1. (a) Variation of self-diffusivity as a function of the diameter of a spherical sorbate in zeolite-Y at 190 K.17 (b) Schematic representation of force acting when sorbate is small relative to the bottleneck and when it is comparable to the bottleneck. In the latter case, the forces from diagonally opposite directions are equal and opposite, leading to negligible net force on the sorbate.

was found that self-diffusivity of a guest within a zeolite exhibits a nonmonotonic dependence on the diameter of the guest.17 For small sizes of guest or diffusant, the diffusivity decreases with an increase in the diffusant diameter (σdd), exhibiting a 1/σ2dd dependence. This is termed the linear regime (LR, see Figure 1a). With further increase in the diffusant diameter, diffusivity increases with an increase in the diameter of the diffusant before decreasing at still larger diameters. Thus, we see a maximum in diffusivity. This is the anomalous regime (AR). This diameter at which selfdiffusivity is maximum is comparable to the bottleneck diameter of the pore network provided by the zeolite within which the diffusant diffuses. The effect observed here is referred to as the levitation effect (LE). Ghorai et al. have studied linear molecules in zeolite NaY and showed that the linear molecule exhibits a maximum in selfdiffusivity when the bond length of the linear molecule is varied.18 They further showed that the bond length at which diffusivity is maximum is determined by the diameter of the bottleneck. We report here an investigation that extends this study of linear diatomic molecules in zeolite NaY to understand underlying factors responsible for the existence of the anomalous dependence of self-diffusivity on bond length in the case of diatomic species. The bond length d of the diatomic molecule (AA) is varied to see if the self-diffusivity exhibits a maximum as a function of the bond length. More significantly, simulations of the heteronuclear diatomic molecule AB have been carried out with different degrees of asymmetry in the interaction of sites A and B with the zeolite atoms. These have been carried out to test the suggestion of Derouane and co-workers that the force cancellation is essential. Further, the role of symmetry in leading to the diffusivity maximum of the levitation effect has been explored. The results provide interesting insights into the nature of symmetry required to observe the levitation effect.

2. METHODS 2.1. Model and Intermolecular Potential. Molecular dynamics simulations have been carried out on model homonuclear diatomic species AA (e.g., O2, N2, etc.) in zeolite NaY as a function of the bond length d. We refer to this (AA) as the symmetric diatomic species, since both sites of the diatomic species interact with equal strength with the zeolite. This is referred to as sym. We have also carried out simulations on heteronuclear diatomic species AB (e.g., CO, HF, HCl, etc.) diffusing within zeolite NaY. Here, the interaction strength (given by the Lennard-Jones parameter ε) between sites A and B on the one hand and the zeolite atoms on the other hand have

σ, Å

εsym , kJ/mol εi-asym , kJ/mol εe-asym , kJ/mol

AA

3.78

0.867

0.867

0.867

AO

3.16235

1.5858

0.835805

0.0858050

ANa

3.57465

0.2766

0.126636

0.0266360

BB

3.78

0.867

0.867

0.867

BO BNa

3.16235 3.57465

1.5858 0.2766

2.33580 0.426636

3.08580 0.526636

been kept unequal. That is, εAz 6¼ εBz for z = O, Na. Two cases have been chosen: (i) with only a slight difference in the interaction strength between site A and zeolite atoms (εAz) and site B and zeolite atoms (εBz) (indicated by i-asym for intermediate asymmetry); (ii) with a large difference in the interaction strength between site A and zeolite atoms and site B and zeolite atoms (e-asym for extreme asymmetry). Thus, in all, there are three sets: sym, i-asym, and e-asym. The unit cell coordinates of zeolite Y employed in the present study have been taken from Fitch et al.19 Interactions between the diatomic species and the atoms of the zeolite are accounted for by the site-site interaction between the two sites on the diatomic molecule (A, B) and zeolite atoms O and Na. The interactions between the diatomic and the zeolite are accounted for in terms of the Lennard-Jones potential. There are no longrange forces. The united atom parameters of ethane given by Jorgensen have been employed in the present work with suitable modification.20 The zeolite-zeolite parameters have been taken from the work of Kiselev and co-workers:21 (σOO = 2.5447 Å, εOO = 1.2891 kJ/mol, σNaNa = 3.3694 Å, and εNaNa = 0.0392 kJ/ mol). The ε parameter for intermediate asymmetry is related to the parameters for the symmetric case through εi-asym xNa (x = A, B) i-asym (x = A, B) = εsym = εsym xNa ( 0.15 kJ/mol and εxO xO ( 0.75 kJ/mol. Similarly, the ε parameter for the extreme asymmetry is related e-asym (x = A, B) = εsym = through εe-asym xNa xNa ( 0.25 kJ/mol and εxO sym εxO ( 1.5 kJ/mol. The cross parameters for guest-zeolite interaction have been obtained from the Lorentz-Berthelot combination rules, and these (along with the self-interaction parameters) are listed in Table 1. The total interaction energy of the guest-zeolite system is given by Utot ¼ Ugh þUgg

ð1Þ

2.2. Computational Details. Simulations have been carried out in the microcanonical ensemble on 2  2  2 unit cells of zeolite NaY. The initial configuration has three diatomic (AB) guest molecules per supercage. The DLPOLY package and Verlet leapfrog integration scheme have been employed for the present study.22 A time step of 1 fs was used, as it yielded conservation in energy of the order of 10-5. A somewhat large cutoff radius of 20 Å has been used to calculate guest-guest and guest-host interactions. Most of the computations have been made at a temperature of 200 K, and the bond length of a diatomic molecule is varied in the range 1.54-4.0 Å in increments of 0.2 Å. Simulations were carried out for an equilibration period of 500 ps with a production duration of 1 ns. The position coordinates, velocities, and forces on the guests were accumulated every 25 fs. 3515

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Figure 2. Radial distribution functions between the centers of mass of two guest molecules for a few guest bond lengths d and for sym, i-asym, and e-asym interactions.

3. RESULTS AND DISCUSSION In Figure 2, we show the center of mass (com)-com radial distribution function (rdf) between pairs of diatomic species. This is shown for different guest bond lengths d as well as for different degrees of asymmetry in interaction. For d = 1.6 Å, the rdf can attain nearly 3.0 as well as exhibit structure and welldefined peaks for all three sets (sym, i-asym, and e-asym). Note that the second and third shells near 12 and 16 Å are also prominent. The rdf's are akin to that observed for well ordered solids. In contrast to this, the larger diatomic species with d = 3.2 and 4.0 Å exhibit a more fluid-like rdf with less structure and reduced height of peaks. We shall see in the next section that the structure seen is a consequence of the associated dynamics of these species. In Figure 3, the rdf's between the center of mass of diatomic species and oxygen of the zeolite are plotted for three different bond lengths and three different degrees of asymmetry in interaction. Once again, the rdf for d = 1.6 Å exhibits structure and well-defined peaks, while for the longer diatomics the rdf's are more fluid-like and less structured. The radial distribution of the molecular center of mass from the center of the supercage is also of interest. These are shown in Figure 4. Considerable differences in the distribution within the supercage can be seen. d = 1.6 Å for sym exhibits a sharper distribution at larger distance

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Figure 3. Center of mass of the guest-host atom radial distribution function, gcom-h(r) for a few guest sizes, d, for sym, i-asym, and e-asym.

from the cage center. This species therefore stays closer to the internal surface of the supercage. For the longer diatomics, a shift of the maximum in the distribution toward smaller values of r is seen (for all three sets). Thus, nonzero population is seen for the diatomic of bond length 4.0 Å even at a distance of 2.0 Å from the cage center for all three sets. Note the skewed distribution and appearance of a shoulder is seen for the asymmetric species (both i-asym and e-asym). These may be compared with the symmetric nature of the curve around the maximum in the distribution for the symmetric case for all bond lengths (sym). Also, the symmetric interaction species exhibits a tail toward the cage center as compared to the other two models (i-asym and e-asym). It may be noted that previous studies have shown that molecules which remain close to the internal zeolite surface of the R-cage of zeolite Y generally have lower potential energies while those closer to the cage center have higher guest-zeolite potential energy of interaction.23,24 Further, it is also known that those close to the internal surface encounter a positive energetic barrier at the 12membered ring window while those nearer to the cage center encounter a negative barrier at the 12-membered ring window. In fact, this is related to the levitation effect as we shall see and helps it to have higher self-diffusivity. 3516

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Figure 4. Radial distribution function between the center of cage to center of mass of the diatom. With increase in d, the distribution maximum shifts toward the cage center for all three degrees of asymmetry in interaction (sym, i-asym, and e-asym).

The mean square displacement (MSD) of the center of mass of the diatomic species with different bond lengths and degrees of asymmetry obtained from molecular dynamics simulation are plotted in Figure 5. The MSD curves are straight up to 300 ps, which suggests good statistics, and the resulting diffusivities estimated from the slope of the MSDs are likely to be reliable. First, we note that the MSD values reach as high values as 600 Å2 for some bond lengths for the symmetric case. For the intermediate asymmetric case (i-asym), this value is at best 300 Å2, while, for the extreme asymmetric or e-asym case, the values of MSD are about one tenth of the values seen for the symmetric case. Thus, we see that for a given guest bond length, d, the magnitude of the slope is least for guests of the e-asym or extreme asymmetric potential. We show in Figure 6 the variation in self-diffusivity, D, as a function of the bond length of the diatomic molecule for the three different cases of (a) sym (b) i-asym, and (c) e-asym . D increases with d in the range 1.54-3.2 Å for the symmetric case but decreases for still larger bond lengths. Also shown are the error bars which are significantly smaller than the changes in

Figure 5. Time evolution of the mean square displacement of guests for different values of d. They are shown for the three sets involving different degrees of asymmetry in interaction: sym, i-asym, and e-asym.

diffusivity on change of bond length. Thus, the trend seen here is reliable and beyond uncertainities seen in the values themselves. Another feature that is worth noting from the figure is the somewhat flat regions followed by a sudden increase in diffusivity as bond length is varied. We see that D increases significantly on going from 1.6 to 1.8 Å or 2.0 to 2.2 Å or 2.4 to 2.6 Å or 3.0 to 3.2 Å. With the introduction of asymmetry in interaction for the i-asym case, the maximum in D decreases significantly. For the extreme asymmetry case or e-asym, there is no increase in D at all with an increase in d. In fact, a slight decrease is seen up to 2.0 Å before leveling off. Previous studies by Derouane and co-workers as well as molecular dynamics studies from this laboratory have shown that the average force on the guest is a minimum for the size of the guest for which self-diffusivity is maximum.14-17 Derouane suggested that the supermobility observed experimentally by 3517

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Figure 6. Diffusivity of the guest molecule as a function of the bond length d in zeolite-Y at 200 K in the presence of different types of guest-host interactions: sym, i-asym, and e-asym cases.

Figure 7. The distribution of the center of mass of the guest from the center of the window when the center of mass of the diatomic species is in the plane of the window for guest lengths of 1.6 and 3.2 Å (sym) and 3.2 Å (i-asym and e-asym).

Kemball in some systems arises due to lowered force on the diffusant or guest exerted by the zeolite or porous solid in which it is diffusing.13 He also showed that such a reduction in force arises from the mutual cancellation of forces from diagonally opposite directions. The force exerted on the diffusant from a given direction is equal and opposite to the force exerted on it by the atoms of the porous solid in the diagonally opposite direction. This happens only when the diameter of the diffusant is comparable to the void diameter in which it is located. Previous studies were carried out on monatomic guests within zeolites. In the present study, the diatomic molecule has two sites through which the molecule interacts with the zeolite. For the symmetric case, when the sum of l, half the bond length (l = d/2), and 21/6  the Lennard-Jones diameter (σAO) between the guest site and the oxygen of the zeolite are comparable to the radius of the 12-ring window, then a maximum is expected in D. This radius is the distance from the center of the 12-ring window to the center of the oxygen atom (σW/2) that defines the 12-ring window. In other words, a maximum is seen when the levitation parameter17,25 defined as lþ21 = 6 σAO ð2Þ γ¼ σw =2 is close to unity. The values of γ for different half-bond lengths, l, for the symmetric case, computed from this equation are (where

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the value of γ is shown in parentheses) l (γ) = 0.8 Å (0.4985), 1.6 Å (0.997), and 2.0 Å (1.246). Thus, the maximum appears at the exact place that previous simulation studies on monatomic diffusants have suggested, viz., when γ is close to unity.17 From the above expression, it is evident that the diffusant bond length at which the maximum in self-diffusivity is seen is related to the bottleneck diameter. The absence of symmetry in interaction between site A and oxygen of the zeolite on the one hand and site B and oxygen of the zeolite on the other hand destroys the diffusivity maximum completely. This is evident from the absence of a maximum for the e-asym case. When symmetry is broken by a weaker perturbation (as is the case with the intermediate asymmetry or i-asym), this leads to a weak maximum. The present study demonstrates unambiguosly that the diffusivity maximum owes its existence to the mutual cancellation of forces suggested by Derouane and co-workers.14-16 It is worthwhile to recollect that the forces have their origin in the dispersion term which is attractive in nature and not the repulsive interaction.17 We now examine other related properties which give deeper insight into the nature of the diffusivity maximum or levitation effect. Figure 7 shows a plot of the distribution, f(cow-com), of the distance from the center of the window to the center of mass of the diatomic, dcow-com, when the center of mass of the diatomic molecule is in the plane of the window. That is, when the distance between the molecular center of mass from the window plane, dwg, is zero. When dwg is zero, the molecule is exiting from one of the R-cages and entering a neighboring R-cage. For the symmetric case, the distribution exhibits a maximum near dcow-com = 1.5 Å for the diatomic molecule away from the diffusivity maximum (here, we have chosen the molecule with d = 1.6 Å). In contrast to this, the maximum in the distribution is near dcow-com = 0 Å for the diatomic molecule at the diffusivity maximum (d = 3.2 Å). The coincidence of the center of mass for the diatomic species with the window center (dcow-com = 0 Å) for d = 3.2 Å is interesting. Note that the window centers are located at 3m, which are centers of inversion symmetry (the space group of zeolite Y is Fd3m). Thus, the diffusivity maximum is related to the passage of the diffusant through the crystallographic inversion center of symmetry. This is a significant observation. The presence of inversion symmetry is required for mutual cancellation of force that is responsible for the diffusivity maximum. Here, we must emphasize that the coincidence of the center of mass of the diatomic species with the window center is meant only in the statistical sense: the average position of the molecular center of mass passes through the center of inversion symmetry. For the i-asym as well as e-asym cases, we see that the maximum is again close to dcow-com = 0 Å. Here, although a similar coincidence of the molecular center of mass with the center of the window does occur, there is an asymmetry in the interaction between the two sites of the diatomic molecule with the zeolite. Therefore, such a cancellation of forces does not take place. As a result, no maximum in self-diffusivity is seen even when the bond length d is similar to the window diameter. The asymmetry arises because εAh 6¼ εBh for h = O, Na. These results can be summarized by the statement that passage of the center of mass of the diffusant through the center of crystallographic inversion symmetry is essential for the observance of the levitation effect. This provides the necessary condition. The passage through the inversion center, however, is not a sufficient condition for the existence of size-dependent diffusivity maximum or levitation effect. 3518

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Figure 8. Distribution of angle between the unit vector perpendicular to the window plane and the molecular axis of the diatomic molecule: 1.6 and 3.2 Å (sym), 3.2 Å (i-asym and e-asym).

Figure 9. Variation of the interaction energy of the guest with the host as a function of the distance of the center of mass of the guest from the window plane, dwp. The energy profiles are shown for 2.6 and 3.2 Å guest bond lengths for symmetric, intermediate asymmetric, and extreme asymmetric cases.

The latter follows from the fact that for i-asym and e-asym we did not observe the diffusivity maximum. The angle between the molecular axis of the diatomic molecule and the unit vector perpendicular to the window plane when the molecular center of mass coincides with the window plane is shown in Figure 8. We see that the angle is nearly 90° and the plane of the molecule coincides with the plane of the window, suggesting that the molecule prefers to go with its long axis parallel to the window plane rather than perpendicular. This is because the molecule optimizes the interaction between the diatomic molecule and the zeolite by being in-plane, better than when the molecular axis is perpendicular to the window plane. As a result, the barrier at the window is significanly lower for the inplane orientation. In the presence of the symmetric interaction, a slight tilt of around 1° is seen around the perpendicular orientation. We do not understand the reason for this. The energy barrier at the window center is a characteristic of the diffusant’s size. As previous studies have shown,17 this barrier is positive and noticeably larger when the diffusant diameter is significantly lower than the bottleneck diameter. The diatomic near the diffusivity maximum encounters either a lower barrier more often or a negative barrier. Figure 9 shows the variation of the guest-zeolite interaction energy as a function of the distance from the window plane for the diatomic molecule. We see that for

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Figure 10. Velocity autocorrelation function of diatomic guests of different bond lengths d diffusing in zeolite Y at 200 K in the presence of different degrees of asymmetry in the guest-zeolite interaction: sym, i-asym, and e-asym cases.

the symmetric interaction there is a significant barrier at dwg = 0 for the 2.6 Å sized guest. However, for the larger size of 3.2 or 3.4 Å, the barrier is lower. This is in agreement with the previous findings on monatomic species within zeolites.17 For i-asym as well as e-asym simulations, we see that the barrier is larger for 2.6 as well as 3.2 Å. Thus, the absence of a diffusivity maximum could arise from the presence of a barrier for all sizes when interaction is asymmetric. Further, note the strong association between the energy barrier at the bottleneck and the absence of mutual cancellation of forces arising from asymmetric interaction. Also, note the higher population seen earlier in Figure 4 toward the cage center for the longer diatomics. As the potential energy of interaction is higher between the guest and the zeolite toward the cage center as compared to the periphery of the cage, the energy barrier across the 12-ring window bottleneck is now lowered so much as to lead to a negative barrier. This is the centralized mode of transport and is different from the surface-mediated diffusion seen for diatomics of smaller bond lengths.24 The velocity autocorrelation function (VACF) for a few of the guest sizes for the three different cases—sym, i-asym, and e-asym—are shown in Figure 10. The guest with a bond length of 1.6 Å shows oscillatory behavior irrespective of the nature of the interaction potential, that is, for sym, i-asym, as well as e-asym . The VACF of the guest with a diffusivity maximum in symmetric interaction (3.2 Å) has less backscattering as compared to the i-asym and e-asym cases of the same bond length. Larger backscattering is also seen for the diatomic molecule of 1.6 Å bond length with symmetric interaction. As we have seen, the guest with maximum diffusivity in the case of symmetric interaction has a lower energy barrier at the window leading to its facile passage past the window. The guest with a bond length of 1.6 Å has a higher energy barrier at the window which is responsible for the negative correlations in the VACF. With an increase in asymmetry in the guest-host potential, the diffusivity maximum at intermediate guest sizes disappears. The guests from i-asym and e-asym are characterized by an oscillatory behavior of the VACF even for the intermediate guest sizes. Figure 11 shows the Arrhenius plots for a few of the guest bond lengths 1.6 and 3.2 Å: sym, i-asym, and e-asym between log D and inverse of average temperatures obtained from the 3519

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Figure 11. Arrhenius plot of the different guest lengths, d, for cases sym, i-asym, and e-asym obtained from the diffusivities at average temperatures (obtained from the simulation runs performed at nominal temperatures of 200, 225, 250, 275, and 300 K).

Table 2. Activation Energies of Guests of Different d Obtained from the Arrhenius Plot for Different Cases: sym, i-asym, and e-asym l, Å

Ea, kJ/mol

sym

0.8

7.4358

sym

1.6

3.8927

i-asym i-asym

0.8 1.6

7.8917 7.5054

e-asym

0.8

11.7714

e-asym

1.6

10.8939

case

simulation runs. The diffusivities for the Arrhenius plot have been obtained by simulating the systems at five different temperatures (200, 225, 250, 275, and 300 K). The activation energies for diffusion obtained from the slope of the Arrhenius plot are shown in Table 2. The activation energy (Ea) is smallest for the guest length 3.2 Å with symmetric interaction with the zeolite. Even the shorter diatomic species with d = 1.6 Å (sym) has nearly twice this activation energy. When the interaction is asymmetric, the activation energies of both 1.6 and 3.2 Å are comparable and higher. Thus, only when the interaction is symmetric, the activation energy is lower for the diatomic whose bond length is comparable to the 12-ring window diameter of zeolite Y. This is in agreement with the predictions of the levitation effect.

4. CONCLUSIONS The present work reports results for diatomic species AB in zeolite NaY. The results show that, when A = B, there is a maximum in the self-diffusivity of the diatomic molecule at large bond lengths dAB. This suggests that the levitation effect exists for diatomic species AA. Results when A 6¼ B with small asymmetry in the interaction between A and B on the one hand and atoms of the zeolite on the other shows a weak maximum in the selfdiffusivity. When the interaction strength between A and host atoms is made very different from that of B with host atoms, it is seen that such a maximum in D disappears completely. This suggests the absence of levitation effect for such a system. It is important to be able to design experiments to verify the results reported here. It is difficult to find either homonuclear or heteronuclear diatomic species of varying bond length. However, the discussion given above is completely valid even if the guest molecule has more than two atoms, provided it is geometrically linear. Such species indeed exists in real life. Consider, for example, species such as CH2(CHdCH)nCH2 with n = 1, 2, 3 ... can be

Figure 12. Schematic diagram illustrating the interaction inversion symmetry. The diagram shows the single atom guest species (open circle) and the neighboring zeolite atoms (shaded circles) along a single diagonal direction. There are several zeolite atoms along the chosen direction and the sum of the forces along a given radial direction is equal to the force along the diagonally opposite direction, leading to negligible net force along this chosen direction. If this happens along all the radial directions, then there is an interaction inversion symmetry.

a series of symmetric species with varying bond lengths, although these bond lengths are discrete and not continuous. If at one end of the molecule, the H atoms are replaced by halogens, then the resulting species is asymmetric. Similar series of species with different bond lengths can also be obtained from CH(CtC)nCH. A few remarks on the symmetry necessary for the existence of diffusivity maximum or levitation effect is pertinent. The inversion symmetry which is essential is not the crystallographically defined symmetry which is based on structure alone. Thus, the necessary inversion symmetry we require for ensuring that the levitation effect exists is the inversion symmetry in interaction. Interaction inversion symmetry requires that the force on the diffusant from a given direction is equal and opposite to the force from the diagonally opposite direction. This is less stringent a requirement than the crystallographic inversion symmetry. The latter, however, provides a sufficient condition for the existence of interaction inversion symmetry. Situations where there is no crystallographic inversion symmetry but there is interaction inversion symmetry are when the forces arising from atoms at different distances add up to along a given direction to say, Fp. Now, although the atomic arrangement along the diagonally opposite direction is completely different, if it adds up to also -Fp, then there is interaction inversion symmetry, since the two forces mutually cancel each other (see Figure 12). These results suggest the role of interaction symmetry is intimately related to the levitation effect. The present work demonstrates the need for symmetry in the interaction to observe the diffusivity maximum and levitation effect. Absence of symmetry in the interaction leads to obliteration of the diffusivity maximum or levitation effect.

’ ACKNOWLEDGMENT The authors wish to thank Department of Science and Technology, New Delhi, and CSIR, New Delhi, for financial support in carrying out this work. The authors also acknowledge CSIR, New Delhi, for a research fellowship to M.S. We also wish to thank Ms. Pragna and Mr. Venkatesh for help. 3520

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