J . Phys. Chem. 1994, 98, 9354-9359
9354
Probing Potential Energy Surfaces in Confined Systems: Behavior of Mean-Square Displacement in Zeolitest M. Ghosh,$ G. Ananthakrishna,s and S. Yashonath'tl Solid State and Structural Chemistry Unit, Materials Research Centre, and Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore-56001 2, India
P. Demontis and G. Suffritti Diparmento Chimica, Universitii di Sassari, Via Vienna 2, I-07100, Sassari, Italy Received: May 9, 1994"
Time evolution of mean-squared displacement based on molecular dynamics for a variety of adsorbate-zeolite systems is reported. Transition from ballistic to diffusive behavior is observed for all the systems. The transition times are found to be system dependent and show different types of dependence on temperature. Model calculations on a one-dimensional system are carried out which show that the characteristic length and transition times are dependent on the distance between the barriers, their heights, and temperature. In light of these findings, it is shown that it is possible to obtain valuable information about the average potential energy surface sampled under specific external conditions. parameter a = 24.85 A. One unit cell has a composition Na48SiluA1480384. The structure of Linde 4A zeolite (NaCaA) was obtained from the work of Pluth and Smith.14 The space group is Fm3c with a = 24.555 A. The unit cell composition is Na32Ca32A196Si960384. The structure of cloverite was taken from the high-resolution synchrotron powder diffraction study of Estermenn et a1.15 The space group is Fm3c with a = 52.712 A. One unit cell has 768 gallium, 768 phosphorus, and 2976 oxygen atoms along with 192 terminal hydroxyl groups. Details of the structure of the channels and the nature of interconnection between the channels, etc., are available from the above references. In our calculations on the above zeolites, the positions of the cations have been assumed to be the same as those given by the crystallographic studies except for zeolite NaY where a Si/AI ratio of 3.0 was employed in the present study. For this ratio, we have placed the sodium atoms in all available SI and SI1 sites. The guest-guest and guest-zeolite (or alpos as the case may be) interactions were modeled in terms of simple Lennard-Jones (6-12) atom-atom potentials between the guest on the one hand and the oxygen of the framework and the extraframework cations, if any, on the other. The silicon in the zeolite and Ga or P in cloverite are surrounded by the oxygens and hence the guests do not approach them sufficiently closely. This obviates the need for inclusion of short-range interactions between the guest and the silicon or Ga or P of the framework. The parameters and other details for Xe-NaY,l0 Ar-NaCaA,16 and CH4 in Y and A zeolite," as well as CHI in rigid silicalite and flexible silicalite6 and Xe in cloverite18 have been taken from the references cited and from Kiselev and Du.19 They are listed in Table 1. Calculations are also reported on silicalite for various diameters of theadsorbate. Hereonly the Lennard-Jones parameter between the adsorbate and the oxygen of the zeolite has been varied.
Introduction A number of experimental and, more recently, computational methods have been employed to investigate and understand the nature of diffusion of sorbates in microporous substances.lV2Pulsed field gradient N M R has been successful in measuring accurately the microscopic diffusion ~oefficient.~ Rowlinson and co-workers4 have theoretically predicted the distribution of cage occupancy in the cages of zeolites which have been verified experimentally.s Influence of cage flexibility on the sorption properties have been investigated by several groups."* Recent calculations suggest that the present understanding of the nature of diffusion in confined regions, in particular, in the channels and cavities of zeolites are far from ~ o m p l e t e .In ~ this context, it is worthwhile to mention that the ballistic to diffusive behavior for xenon in Nay10 is reported only recently even though such a crossover is well-known in the solution of the Langevin equation." El Amrani and Kolb have suggested that the transition time in argon-silicalite systems could be used in the choice of molecular dynamics time steps.12 However a systematic study of this problem and the relevance of the study to our understanding of the diffusive motion in zeolites is lacking in the literature. Here, we report a systematic study of the transition times in different sorbate-zeolite systems at various temperatures (2') using molecular dynamics (MD) simulations. The transition times (7) obtained from the MD simulations exhibit different trends with temperature for different adsorbate-zeolite systems. These results are not easily understandable. To get a better insight into these results, we have carried out a series of calculations on model one-dimensional systems in an effort to understand the different observed trends.
Structure and Intermolecular Potential The structure of various zeolites were taken from X-ray crystallographic and neutron diffraction studies. Structure of zeolite Y was taken from the neutron measurements of Fitch et al.13 The structure is cubic (space group F d h ) with a lattice
Details of Molecular Dynamics All calculations werecarried out in the microcanonical ensemble by the molecular dynamics method. The guest species consisted of monatomic adsorbates in all cases; for methane a single-site model was used. Integration was carried out using the Verlet scheme. For integration of the guest coordinates, a time step varying between 1 and 40 fs was used depending on the mass of the adsorbate. Flexible cage simulations in the case of silicalite were carried out using a time step of 1 fs. A cut-off radius varying
t Contribution No. 900 from the Solid State and Structural Chemistry Unit. 3 Solid State and Structural Chemistry Unit. 8 Materials Research Centre. 1 Supercomputer Education and Research Centre. * To whom correspondence should be addressed. a Abstract published in Advance ACS Abstracts, August 15, 1994.
0022-3654 f 94 f 2098-9354SO4.50f 0
1
1994 American Chemical Society
Potential Energy Surfaces in Confined Systems
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9355 I
-20'
2 .o
I
I
I
1.0
4 .O
7.0
-
In t , p s
Ar in NaCaA
CHI in zeolite A
Xe in cloverite
~
10,
A (103
B(106
kJ/moI A6)
kJ/moI A1ZI
~~
8
~~
34.913
11.1345 7.9079
6.264
9.849
4.701 1.641 18.390
4.173 2.261 27.421
4.843
5.339
5.919 1.776 16.187
3.991 0.413 20.230
8.2793 9.921
I
165.84
8.2793 2.9143
34.913
6
argon/a-cage.
Guest-Guest and Guest-Zeolite atom guest-guest Xe-Xe guest-zeolite XS-0 Xe-Na guest-guest Ar-Ar guest-zeolite Ar-0 Ar-Na Ar-Ca guest-guest CH4-CH4 guest-zeolite CH4-0 CH4-Na CH4-Ca guest-guest Xe-Xe guest-zeolite Xe-0 Xe-OH
3
Figure 2. log-log plot of mean-squared displacement against time of argon sorbed in zeolite A at an adsorbate concentration of 1
TABLE 1: Intermolecular Potential Parameters for
~
0 In t , p s
Figure 1. log-log plot of mean-squared displacement against time of xenon sorbed in zeolite Y at an adsorbate concentration of 1 xenon/acage.
svstem Xe in zeolite Y
3
-71
-3
I
-1
I
I
I
I
1
3
5
7
10
165.84
326 K
11.1345 21.288
between 10 and 14 A was used in evaluating the guest-guest and guest-host interactions. More details of the simulation are available from the references cited: Xe-NaY,lO Ar-NaCaA,16 and CH4 in Y and A ~ e o l i t e , as ' ~ well as CH4 in rigid silicalite and flexible silicalite6 and Xe in cloverite.'* All calculations in zeolites Y and A were performed at an adsorbate concentration of one sorbate per cage. In the case of silicalite, the adsorbate concentration was 12 CH4/crystallographic unit cell. In the case of xenon in cloverite there were 25 xenons/unit cell of cloverite. The mass of the adsorbate with variable adsorbate diameter simulations in silicalite was taken to be 40 amu.
Results and Discussion Previous work on the behavior of the mean-squared displacement of adsorbates in zeolites suggest that there is a distinct transition from the ballistic to the diffusive behavior."JJ2 In Figures 1 - 6 we show several log-log plots of the mean-square displacement against time. These have been obtained from M D simulation data of a variety of adsorbates in different zeolites. From the figures it is clear that all of them exhibit ballistic to diffusive transition. During the initial period, a log-log or In-ln (used interchangeably below) plot of the mean-squared displacement with time shows a slope of nearly 2. After the transition
9
In t , p s
6N
.a
6
4-
Y
N
9 -
20-2'
-3
-1
I
I
1
1
I
1
3
5
7
9
In t,ps Figure 3. log-log plot of mean-squared displacement against time of CH4 in (a, top) zeolite Y and (b, bottom) in zeolite A at an adsorbate concentration of one methane/a-cage.
to the diffusive behavior the slope is closer to 1 (see below for related comments). The temperature at which the simulation was carried out and the time a t which the transition is observed (as seen from Figures 1-5) along with the slope of straight lines fitted to the data before and after the transition are listed in Table 2. The time of transition was taken to be the point
Ghosh et al.
9356 The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 8
5,
6
A h
s v
4
N
3
V
r
2
3
I
-101
-4
-7
I
I
-1 I n t,ps
2
0
5
-2 -4
0
-2
2
In
.U
6
Figure 6. log-log plot of mean-squared displacement against time for adsorbates of different sizes in silicalite at room temperature. The values for u, one of the parameters in the adsorbateadsorbate Lennard-Jones interaction potential, were 1.65,1.75, 2.25,and 3.0 A.
C
N
4
t
6 c
+-
N
TABLE 2 Results Obtained from the Mean Squared Displacement Curves of Different Zeolite Systems system T(K) 7 (ps) h (1Vcm) slope I slope I1
C
-E
Xe in zeolite Y argon in zeolite A -1
c
-4
2
-1 In t,ps
CH4 in zeolite Y
I
CH4 in zeolite A
Figure 4. log-log plot of mean-squared displacement against time of CH4 in silicalite for (a, top) fixed and (b, bottom) mobile framework at an adsorbate concentration of 12 methane/silicalite unit cell.
8
716K
CHI in silicalite mobile
6
xenon in cloverite
N
.a -
4
A c
+-
I
N
r
2
0
-2
-3
CHI in silicalite fixed
I
-1
I
I
I
1
3
5
7
Int,ps
Figure 5. log-log plot of mean-squared displacement against time of xenon in cloverite at an adsorbate concentration of 25 xenon/crystallographic unit cell.
of intersection of the two straight lines fitted to the data before and after the transition. The following trends may be observed from the Figures 1-6 and from Table 2: (a) The transition time, 7,is different for different sorbatezeolite (or gallophosphate, as the case may be) systems.
168 637 139 261 341 226 322 171 234 326 167 220 298 167 220 298 397 494 716
0.68 1.17 0.68 0.83 0.96 0.52 0.64 0.45 0.64 0.75 0.13 0.13 0.13 0.20 0.20 0.20 0.92 1.05 1.11
1.21 4.07 2.01 3.36 4.44 3.08 4.53 2.36 3.86 5.34 0.69 0.79 0.92 1.03 1.18 1.37 2.52 3.22 4.09
2.00 1.89 1.92 1.82 1.86 1.66 1.72 1.76 1.76 1.76 1.96 1.93 1.93 1.92 1.92 1.90 1.85 1.90 1.98
0.89 0.85 0.86 0.86 0.86 0.96 0.92 0.92 0.94 0.88 0.73 0.78 0.78 0.80 0.81 0.82 0.69 0.69 0.74
(b) The temperature dependence of the transition time 7 could be of several types: (i) The transition time changes with temperature significantly. This is the case, for example, in XeN a y , Ar-NaCaA, Xecloverite, etc. (ii) The transition time is independent of the temperature. Both rigid and flexible simulation of CH4-silicalite system show this trend. (c) The transition time is nearly independent of the size of the adsorbate in silicalite in the range of diameters 1.65-3.00 A. Note that the channel diameter is approximately 5.4 A. These are the main observations from the mean-squared displacement curves obtained from the MD calculations. It may be worthwhile to note that often the slope in the “diffusive regime” is much less than 1, meaning that it could be subdiffusive. In the last section, we shall comment on this. One-Dimensional Model. To gain insight into the results presented above, we have carried out calculations on a onedimensional model. As mentioned earlier, the linear Langevin equation predicts ballistic behavior (u2 t2) for short times and diffusive behavior (u2 t ) for long times.10911 However, this analysis does not contain any information about the distances between the barriers or about the barrier heights. To investigate the possible relationship between these and the transition, we
-
-
Potential Energy Surfaces in Confined Systems
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9357 I
-8
Figure7. Schematicdiagramof the one-dimensionalmodel termed model 4.
i
A
,--. c v- 1 2
have undertaken to simulate the effect of distribution of distances between the barriers and the barrier heights in the following manner. Consider the conventional Langevin equation:
+
dv(t)/dt -flu(?) ~ ( t ) (1) where ~ ( t is) the Gaussian white noise with zero mean and p is the damping coefficient. Since this equation does not contain any information about distances between the barriers or barrier heights, we include these dependences explicitly in the following way. Consider the simplest situation wherein only one type of barrier is used with a distance d between them. Between the barriers the time development is that of free flight. The effect of damping as well as the random force is accommodated only when the particle attempts to surmount the barrier. We consider the particle to surmount the barrier of height b with a probability
1
I / -24 -8
- 1 0
-2
- 4
-6
In
0
t
Figure 8. log-log plot of mean-squared displacement against time for models 1,2, and 3 obtained from model 4 by putting bl = b2 for different values of d. -8
1
-13
-12
where e ( t ) is the kinetic energy of the particle a t the timet. Once the barrier is surmounted, the velocity of the particle is chosen randomly consistent with the velocity distribution a t that temperature. If it does not succeed in surmounting the barrier, the particle velocity is reversed. We also use the consistency condition inherent in the fluctuation4issipation theorem, namely, therelationship between@and the width of thevelocity distribution and also the temperature. Several simulations were carried out for distinct values of barrier heights bi, distances d between the barriers, and temperature T . First, we start with a series of calculations by varying the distance between the barriers to bed, 2 4 and 4d, for a fixed barrier height b and temperature T . The barriers are all assumed to have a width w. Note that e and b are measured in units of kT. d is expressed in arbitrary units. These cases are referred to as model 1 , 2, and 3 respectively. In another series of calculations, simulations were carried out at two different temperatures namely (model 4 ) , kT = 1 and kT = 0.76, on a system consisting of two barriers of heights, bl = 2.2kT and b2 = 3.64kT separated by a distance d. The periodicity in this case is 2d 2w (see Figure 7). A log-log plot of the mean-squared displacement against time is shown in Figure 8 for the three models 1,2, and 3 forb = 2.2kT and kT = 1 . All the runs have been carried out at the same value of ( v 2 ) the velocity of the adsorbate. The slopes of the ballistic and diffusive regions, the transition times along with the corresponding characteristic length A, obtained from the 7 ’ s using the relation X = (u2)1/2 T are listed in Table 3 . It is seen that when the distance between the barriers is changed from d to 2d, the transition time approximately doubles. Increasing thedistance to 4d (model 3) results in further increase in T,to a value nearly 4 times that obtainedfor model 1 . Themagnitudeofthe transition time is therefore a reliable indicator of the distance between the barriers since all the calculations have been carried out a t the same temperature. This is consistent with the intuitive picture that time of free flight between the barriers is a good indicator of the distance between them. Figure 9 shows a log-log plot of the mean-squared displacement curves for two different temperatures for the model shown in Figure 7 (model 4 ) . The lower of the two temperatures was chosen such that the probability of surmounting both the large and the small barriers was small. The effective distance between the barriers is, therefore, d. Note the large-amplitude oscillatory behavior exhibited by the mean-
+
A
-16
n
&
3
-16
V
c
i
-20 /’
2. kT
=
0.76
--10
-8
-4
-6
-2
0
In t Figure 9. log-log plot of mean-squared displacement against time for model 4 at two different temperatures. Inset shows an expanded plot of
the mean-squareddisplacement after the transition average over a single run and 20 runs with different initial configurations. Note the decrease
in the amplitude of the oscillations.
TABLE 3: Results Obtained from the Mean-Squared Displacement Curves of the Model System system r(10-3) ~(10-3) slope1 s model 1 2.26 0.35 2.00 0.78 model 2 4.80 0.76 2.00 0.90 model 3 8.65 1.37 2.00 0.8 1 model 4 ~
kT= 1 kT = 0.76
6.82 3.92
1.07 0.54
2.00 1.94
E
0.82 0.62
squared displacement curve immediately following the ballistic part. This behavior is typical of any inhomogeneous random walk.20 This feature arises from the relaxation effects due to confinement of the particle between the barriers. The effect is more dominant a t low temperature. This oscillatory behavior is entirely due to the fact that we have not averaged over the initial distribution of velocities. The averaged behavior corresponding to an average over 20 different initial values of the velocities and positions is shown in the inset of Figure 9 along with the result for a single initial condition. It is clear that the amplitude of the oscillations have decreased. If a completely smooth behavior is desired, an average over a much larger set of initial conditions would be required. However, for the purposes of our discussion, it is sufficient to have a smoothness adequate for obtaining a proper extrapolation. Since sampling over 20 initial values gives
Ghosh et al.
9358 The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 ‘Potential energy surface
dl.1.23A
.d,=4.11A
Figure 10. Schematic plot of the potential energy surface as suggested by the probe xenon in zeolite Nay. The characteristic lengths are respectively 1.23 and 4.11 A at 168 and 637 K.
us the desired accuracy to obtain reasonable estimates for 7,we have not carried out a better averaging. By extrapolating, we have obtained the transition time, 7. The high-temperature run was chosen such that the probability of surmounting the higher barrier was small, and the probability of surmounting the lower barrier was high. The mean-squared displacement for this run is shown in Figure 9. Note that the amplitude of the oscillations are much smaller as compared to the lower temperature. The values of transition times 7 and the characteristic length, A, are listed in Table 3. It is seen that the characteristic length A has nearly doubled when we go from the lower temperature to the higher temperature. This result is easy to understand considering the fact that a t higher temperatures the effectivedistance between the barriers is that corresponding to the larger barriers. Consequently, the effective distance should nearly double. Note the distance between the two taller barriers is approximately 2d. Thus, it is clear that the change in the characteristic length X with temperature reflects the distance between the effective barriers at that temperature. We now make use of these results for a proper interpretation of the molecular dynamics results at various temperatures and on different sorbate-zeolite systems. The characteristic length, X are also listed in Table 2. These have been obtained from the relation A = (3kT/m)’I27 (3) We start with the molecular dynamics resultson xenon incloverite for which the distances between the barriers are 2.55, 3.26, and 4.15 A at 397, 494, and 716 K. Our room-temperature calculations on methane in zeolite NaY leads to a X of 3.3 8, which is significantly larger than 0.92 8, obtained for methane in silicalite at 298 K. The lower characteristic length for the latter may be attributed partly to the smaller sized voids and partly to the higher adsorbate concentration of methane (12 methane/unit cell) in silicalite. In the case of xenon in zeolite N a y , the characteristic length increases from 1.23 to 4.11 A when the temperature is increased from 168 to 637 K. This suggests a picture of the potential energy surface which is similar to that of model 4. The barrier height, bl = 168k = 1.4 kJ/mol are separated by a distance of 1.2.k Thedistance between barriers of height bz = 5.3 kJ/mol are about 4.1 8, (see Figure 10). In contrast, the results for methane in silicalite listed in Table 2 suggest that with increase in temperature there is little change in the characteristic length. This indicates a rather flat potential energy surface at least as experienced by methane in the temperature range 167-298 K. Thus, it is seen that in the temperature ranges investigated the potential energy surface in zeolite NaY (as probed by Xe) is rather more structured with “hills and valleys” of greater depth and heights as compared to zeolite NaCaA (as probed by methane). Thus, if one were to calculate from a molecular dynamics simulation, the behavior of mean-squared displacement as a function of temperature, then one could derive from such
information the potential energy landscape inside the zeolite. Since both equilibrium and dynamical properties of adsorbates in zeolite in the high dilutionn limit depend principally on this potential energy landscape,21such information would be valuable for understanding the behavior of adsorbates in zeolites. It must be pointed out that such a landscape gives exactly what an adsorbate would “see” or “sense” in a zeolite as against the potential energy surface obtained by direct evaluation of the interaction energy without carrying out a simulation. Finally we consider the independence of the transition time with thevariation of the adsorbate diameter. Earlier calculations have shown that the diffusion coefficient D varies linearly with 1 /u2over a wide range of the diameters of the adsorbate u. There is, however, a peak in the diffusion coefficient as the diameter of the adsorbate approaches that of the channel diameter. This is referred to as the diffusion anomaly.23 In light of this, it would be worthwhile to see if there is any change in r . We have chosen two values of the diameters to fall in the linear regime and two in region of the peak. A log-log plot of the mean-squared displacement with time is shown in Figure 6 for four different sizes of the adsorbate diameters, viz., 1.65, 1.75, 2.25, and 3.0 A. The points with 0 equal to 1.65 and 1.75 8, fall in the linear regime, whereas the diameters 2.25 and 3.0 8, fall in the regime where the diffusion anomaly has been found. The latter have a value of y close to unity. These calculations were carried out at room temperature. Interestingly we do not find any change in the value of the transition time 7 . This possibly implies that the distances between the barriers are not affected. At the temperature a t which these simulations have been carried out, it appears that the changes in the barrier heights, if any are not felt by the sorbate. Any such changes can only be verified by performing simulations at lower temperatures.
Conclusions The limitations of the above interpretation are obvious. In a two- or three-dimensional system, the particle can often find alternative pathways which obviates the need for it to overcome the barrier. Thus, it should be noted that the mapping of the potential energy surface of a realistic system such as zeolites into one dimension is an oversimplification. However, such a simplification is what may be desirable at times. Such an analysis would indicate the qualitative trends quite correctly. A more accurate analysis would have been to carry out the model calculations in two dimensions. The interest of the present exercise, however, is not so much to obtain accurate estimates of the characteristic length but rather to demonstrate the utility of the approach. There would be differences in the results between one and two dimensions. For example, one would not expect to find oscillations a t low temperatures in two dimensions, since the particle could find alternative pathways. It should also be noted that the picture obtained from the behavior of the mean-squared displacement, in particular, the transition time is an averaged picture. The results presented here does not preclude the possibility of having certain regions in silicalite which are highly structured. Similarly, the results presented here do not exclude the possibility of having rather flat regions in some parts of the zeolite N a y . Rather, the results obtained here suggest that on the average one can term the potential energy surface in zeolite NaY as more structured as compared to that of zeolite NaCaA at the temperatures investigated. In general, an increase in X with temperature corresponds to a more structured potential energy surface of varying barrier heights as compared to a situation where there is little change in A. It appears that the characteristic length X generally increases with temperature suggesting that barriers of greater heights are also usually separated by greater distances. Finally, we comment upon the significant departure of the slope from unity for CH4 in silicalite (-0.75) and Xe in cloverite
Potential Energy Surfaces in Confined Systems
(-0.70). This departure is much too large to be considered as diffusive. The situation here possibly refers to a subdiffusive behavior. There are theoretical models which mimic subdiffusive behavior.22 The particle has a tendency to be moving through the low-energy configurations most of the time, thereby not sampling the whole phase space. This essentially leads to the confinement of the particle to some regions of the porous medium. One other possibility for the departure of the slope from unity is that the time averages are too restrictive to represent the phase space averages. This could imply that in such situations much longer simulations are desirable to confirm the existence of subdiffusive behavior. Acknowledgment. We are extremely happy to dedicate this work in honor of Professor C. N. R. Rao. References and Notes (1) Thomas, J. M. Philos. Trans. R. SOC.London, A 1990, 333, 173. (2) Williams, C.; Yashonath, S.; Thomas, J. M. Int. Reu. Phys. Chem. 1988, 7, 81. (3) Karger, J.; Pfeifer. H. Zeolites 1987, 7, 90. (4) Rowlinson, J. S.; Woods, G. B. Physica A 1993, 164, 117.
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9359 ( 5 ) Chmelka, B. F.; Raftry, D.; McCormick, A. V.; de Menorval, L. C.; Levine, R. D.; Pines, A. Phys. Reu. Lett. 1991, 66, 580. ( 6 ) Demontis, P.; Suffritti, G. B.;Fois, E. S.;Quartieri, S.J. Phys. Chem.
1992, 96, 1482. (7) Schrimpf,G.;Schlenkrich,M.; Brickmann, J.; Boff, P. J . Phys. Chem. 1992, 96, 7404. (8) Titiloye, J. 0.;Parker, S.C.; Stone, F. S.;Catlow, R. R. A. J . Phys. Chem. 1991, 95, 4038. (9) Yashonath, S.; Santikary, P. Mol. Phys. 1993, 78, 1. (10) Santikary, P.; Yashonath, S.;Ananthakrishna, G. J . Phys. Chem. 1992, 96, 1049. (11) Uhlenbeck, G.;Ornstein, L. S. Phys. Reu. 1930, 36, 823. (12) El Amrani, S.; Kolb, M. J . Chem. Phys. 1993, 98, 1509. (13) Fitch, A. N.; Jobic, H.; Renouprez, A. J . Phys. Chem. 1986, 98, 1311. (14) Pluth, J. J.; Smith, J. V. J. Am. Chem. SOC.1980, 102,4704. (15) Estermenn, M.;McCusker, L. B.; Baerlocher, C.; Merrouche, A.; Kessler, H.Nature 1991, 352, 320. (16) Yashonath, S.; Santikary, P. J. Phys. Chem. 1993, 97, 13474. (17) Bandyopadhyay, S.; Yashonath, S. Chem. Phys. Lett., in press. (18) Bandyopadhyay, S.; Yashonath, S. J . Solid State Chem., in press. (19) Kiselev, A. V.; Du, P. Q.J. Chem. Soc., Faraday Trans. 2 1981,77, 1. (20) Ananthakrishna, G.; Balasubramanian, T. Bull. Mater. Sci. 1988, 10, 77. (21) Yashonath, S.; Santikary, P. J. Phys. Chem., in press. (22) John, T. M.; Ananthakrishna, G. Phys. Lett. 1985, 11OA, 41 1. (23) Yashonath, S.; Santikary, P. J. Chem. Phys. 1994, 100, 4013.
