10469
J. Phys. Chem. 1992,96, 10469-10477
A Molecular Dynamics Study of Xenon Sorbed in Sodium Y Zeolite. 1. Temperature and Concentration DependenceA Prakriteswar Santikary,? Subramanian Yashonath,*yt**and G. Ananthakrishod Solid State and Structural Chemistry Unit, Supercomputer Education and Research Centre, and Materials Research Centre, Indian Institute of Science, Bangalore 560 012, India (Received: March 23, 1992)
The sorption properties of xenon sorbed in sodium Y zeolite as a function of adsorbate concentration and temperature have been obtained from molecular dynamics simulations. The properties reported include the various site-site a center of cage-center of mass radial distribution functions, distribution of guest-host energy, guest-guest pair and bon ' g energy, self-diffusion coeffcients, the power spectra, and the distribution of site residence times. The location of the physical adsorption site for xenon is the same as for methane. The guest-host energy distribution function for xenon differs significantly from the bimodal function observed for methane. It is shown that the mean square displacement shows a crossover from ballistic to diffusive behaviour and the activation energy for diffusion is 4.1 kJ/mol. We suggest that the 15-cm-l frequency mode seen in the power spectra of the autocorrelationfunction corresponds to the X e X e dimers. The nonavailability of sorption sites at higher adsorbate concentrations is reflected in the guest-host distribution function, center of cage-center of mass radial distribution function, power spectra, and other properties.
d
study of the sorption of small- and medium-sized molecules in and Demontis et al.16J7 ZSM5. June et al.,I3J4Pickett et have reported sorption of xenon and methane in silicalite. Smit and Ouden18 examined the properties of methane in mordenite. Water in fenierite has been studied by Leherte et a1.I9 Adsorption in zeolite Y have been investigated by several workers recently.20'26 Rowlinson and co-workerszov2'have carried out extensive grand canonical Monte Carlo calculations of thermodynamic and structural properties for xenon in zeolite X and Y. These studies show that there is good agreement of the computer simulation results with the available experimental data?z-24 Yashonath et aL2>%have studied the properties of methane in sodium Y zeolite. Benzene in sodium YZ9has been studied by molecular dynamics, and results have been shown to be in good agreement with the experimental data. Here we present the results on xenon in sodium Y obtained from molecular dynamics calculations. Thermodynamic, structural, and dynamical properties are reported as a function of temperature and concentration. The calculated results are compared with experimental data wherever available. Long simulation runs of 2.6 ns were required to obtain dynamical properties of interest and also for the determination of properties such as site-to-site migration.
Introduction The interest in the study of adsorption properties in restricted regions is growing. Several materials in nature permit the adsorption of small to medium sized guest molecules in voids of the order of several molecular diameters. These are known by the general name micropores. Some of these are simple adducts such as the urea-alkanes with purely physical interaction. The others such as the zeolites not only sorb guest species but often chemically transform the guest giving rise to different product species. Zeolites are well-known for their molecular sieve and catalytic properties which are industrially important. Zeolites are aluminmilicatts built with the basic building blocks of silicon-oxygen and aluminum-oxygen tetrahedra. The interconnected tetrahedra results in a three-dimensional network of cages of varying sizes. The large variety in the structure of zeolites found in nature as well as synthesized in the laboratory makes the study of sorption properties in these systems both interesting and challenging.' The problem is made more difficult by the large number of other factors that influence the adsorption properties such as the extraframework cations, the Si/A1 ratio, etcS2 During the last few decades a large number of experimental studies have been reported in the literature on different species of guest adsorbed in zeolites. However, the experimental limitations and the difficulty in obtaining the microscopic description from experiments have resulted in only a limited understanding of these systems. The developments in the past decade have enabled the use of the Monte Carlo and the molecular dynamics methods for investigating the properties of solids, liquids and sorbed species in These methods provide a powerful and flexible tool in the hands of a physical chemist. Model calculations and simulations of guest species sorbed in small pores of width of the order of a few molecular diameters have been reported by several workers.6-" Subramanian and Davis6 and more recently Powles' et al. have carried out calculations of monatomic particles in pores or slits of various sizes and shapes. MacElroy and Suhe examined the effect of surface roughness on the sorbate diffwivity. Derouane examined the effect of surface curvature on properties of physi~orption.~ Recently, there have been several realistic calculations of adsorption in ze01ites.l~'~Silicalite and its analogue, ZSM5, as well as zeolite Y and its related forms have been most commonly investigated. Vetrivel et al.lz have carried out quantum chemical
Structure of Sodium Y Zeolite The structure of zeolite Y recently obtained from neutron diffraction by Fitch et al.30has been used in this study. The structure is similar to that reported by Olson.3' The lattice parameter of the bare zeolite at room temperature with a = 25.8536 A has been employed. The space group is FdJm. One unit cell has 48 Na, 48 AI, 144 Si, and 384 0 atoms for a Si/Al ratio of 3.0. For this particular ratio the extraframework sodium atoms occupy the SI and SI1 sites completelyzs (see Figure 1). The zeolite Y structure consists of sodalite and a-cages, also known as supercages. Earlier studies show that the guest species are adsorbed in the a-cages. The diameter of xenon (4.6 A) is too large to be accommodated in the sodalite cage. Each a-cage is interconnected tetrahedrally to four other a-cages through 12membered rings. The approximate diameter of the a-cage is about 11.8 A and that of the interconnecting 12-ring window 8 A. There are eight a-cages in one unit cell of zeolite Y. Below we refer to an a-cage simply as cage.
'Solid State and Structural Chemistry Unit. *SupercomputerEducation and Research Centre. 1 Materials Research Centre. Contribution No. 795 from the Solid State and Structural Chemistry Unit.
Intermolecular Potential Functions Gwst Zeolite Interaction Potential, All guest zeolite interactions were modeled in terms of short-range interactions. The potential employed to us in the present study is the one proposed by Kiselev et al.32 In this, the xenon-zeolite Y interactions are modeled in
0022-3654/92/2096-10469$03.00/0
0 1992 American Chemical Society
10470 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
Santikary et al. TABLE 11: Equilibrium " ~ ~ Y M Q Properties ~ C for Xtnoa ia sodium Y at Different Temperatures and Adsorbate Concentrations from 2600-psRun C
(K) 188 285 386 414 479 311 304
(Xe/cage) 1 1 1 1 1 2 3
Was (U ) 411 (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol)
-0.64 -0.57 -0.51 -0.50 -0.45 -1.11 -1.62
-15.10 -13.41 -12.33 -12.17 -11.77 -13.76 -14.45
-15.74 -13.98 -12.84 -12.67 -12.22 -14.87 -16.07
17.30 16.35 16.05 16.11 16.20 17.46 18.60
0
0
Figure 1. Structure of a supercage of zeolite Y showing the cation site SI and SII. TABLE I: Intermolecular Potential Parameters for GuestGuest and Cueat-Zeolite atom A (10' kJ/mol, A6) B (lo6 kJ/mol, AI2)
guest-guest Xe-Xe guest-zeolite Xe-0 Xe-Na
( T)
34.913 8.2793 2.9143
165.84 11.1345 7.9079
terms of atom-atom pair potential functions. The interactions between the silicon or aluminum and xenon are taken to be zero as the close approach of xenons to these atoms is prevented by the surrounding oxygens. The interactions are of the LennardJones (6-12) form
employed which yielded good energy conservation. Properties were calculated from codigurations stored at intervals of 0.2 ps. The equilibration was performed over a duration of 250 ps during which velocities were scaled to obtain the desired temperature. Long production runs of 2600 ps have been carried out at each of the temperatures and loadings. The long runs assured us of sufficient statistics for dynamical properties such as mean square displacement. Resulb and Discussion cProperties. The guest-guest interaction energy,
U,, is the sum over all the guest pairs: r N N
The guestzeolite interaction energy, U,, is the sum over all atoms of the zeolite for each of the guest atoms:
Induction interactions have not been taken into account as it The cut off was set at 12 A both for the guest-guest and the is computationally prohibitive due to the many-body nature of guest-zeolite interactions. The total interaction energy (U)is the interactions. The parameter values for A,, and B,,, a = Xe (5) and z = 0, Na were taken from the work of Kiselev and D U . ~ ~ Rowlinson and co-workers2' have used modified values of A and The values of (U ), (U,),(U), and other properties for different B so as to yield better agreement of the calculated thermodynamic temperatures an7 adsorbate concentrations, are listed in Table properties. We have, however, used the values obtained by Kiselev 11. The experimental value of the isosteric heat of adsorption, and D u , in ~ all ~ our calculations reported here. These parameters qst,near room temperature is 18.0 k . T / m ~ l .At ~ ~285 K, we obtain have been found to yield results in reasonable agreement33with ~ ~ a value a value of 16.35 kJ/mol for qBt.Kiselev and D u obtained available experimental data. The potential parameters are listed of 17.25 kJ/mol at 323 K. In view of the difference in the in Table I. More details about the potential parameters are temperature and Si/Al ratio, the agreement between the experavailable from ref 32. The atomic mass of the xenon was taken imental and the calculated value seems to be reasonable. Part to be 131 amu. of the discrepancy between our estimate and that of Kiselev and Guest-Guest Interaction Potential. The xenon-xenon interDu is due to the neglect of the polarization interactions. actions were taken to be of the Lennard-Jones (6-12) form Structure and Ewrgetics. Radial distribution functions (rdfs) A B between the adsorbate and the zeolite atoms 0 and Na at different u(r) = -temperatures are shown in Figure 2. At 188 K, distinct peaks p + p are observed in the Xe-Na and Xe-0 rdfs, indicating that the where the parameter A = 4cd and B = 4td2. The values of tXtXc guest atoms are largely localized in the supercages. These may = 221 K and cXtXe= 4.1 A were taken from l i t e r a t ~ r e . ~ ~ be compared with rdfs for methane in sodium YZ7at 50 and 150 Rowlinson and wworkersZ1have obtained reasonable agreement K when they were found to be localized. At temperatures higher with experimental data over a range of pressures with these pathan 188 K the peaks become less distinct and the rdfs show rameters. characteristics of a fluid phase. The transition from a localized to a delocalized xenon takes place between 188 and 285 K. In Molecular Dynamics Calculations the case of methane such a transition occurred between 150 and 220 K.2527As we shall see, other properties such as the distribution All calculations were carried out in the microcanonical ensemble of guest-host interaction energyf(U,) also confirm this. In Figure at fixed (N,V,E).35Cubic periodic boundary conditions were 3 we show the radial distribution functions Xe-0 and Xe-Na at employed. The zeolite atoms were assumed to be rigid and were different adsorbate concentrations near 300 K. There is a small not included in the integration. Calculations were performed at but discernible decrease in the modulations at higher concena loading of c = 1, 2, and 3 Xe/cage at room temperature. For trations. Similar behavior was observed in earlier studies of c = 1 Xe/cage the calculations are reported at four other nominal methane adsorbed in sodium Y.27 The reason for this will be temperatures of 190,400,425, and 470 K in addition to the room discussed in the later part of this section. temperature. In addition, two runs at 570 and 640 K were carried The Xe-Xe rdf is shown at different temperatures for c = 1 out to get a better estimate of the temperature and time depenXe/cage in Figure 4a. At all temperatures, the rdfs are marked dence of mean square displacement. One unit cell of the zeolite by the absence of the second peak. At 188 K, there is a distinct Y was employed in all calculations. shoulder or another additional peak near 6 A. At all temperatures, All runs were carried out with a starting configuration with the first peak is observed at 4.5 A corresponding to 21/6ax,x,. the xenons placed at the cage centres. A time step of 40 fs was
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10471
Xenon Sorbed in Sodium Y Zeolite
-l @ K
c = 1 Xelcage
2 .o
(b)
285K
[ ......
Xe-Na
386K ----479K
-
-
Xe Xe c = 1XeIcoge
---
c = 3Xe/coge
c = 1 Xelcoge - 188 K 285K
__-
..... 386K
-188K
c = 1 Xelcase
in)
I,A Figure 4. Radial distribution functions for guest-guest (Xe-Xe) at (a) different temperatures for an adsorbate concentration of c = 1 Xe/cage and at (b) different adsorbate concentrations near 300 K.
til
l i t
1
I
4
r,
B
48
I
0
12 32
Figure 2. Radial distribution functions between the guest xenon and the zeolite atoms (a) 0 and (b) Na shown near 188,285,386. and 479 K for adsorbate concentration of c = 1 xenon/cage.
2*ol
16 .
xe-Na cv 300K
-c= 1 ---
.
-
Xelcage c - 3 Xe/cage c L Y
01
-
1.5 N
3OOK
-c = I
Xe/cage
- - - c - 3 Xe/cage 8 .O
4 .O r,
12
A
Figure 3. Radial distribution functions between the guest xenon and the zeolite atoms (a) 0 and (b) Na shown at different adsorbate concentrations of 1 and 3 xenon/cage near 300 K.
This indicates that even at the low adsorbate concentration of 1 Xe/cage pairs of xenon exist. The absence of second peak in the Xe-Xe rdf suggests that no clusters involving more than the first-shell neighbors are observed at these temperatures (between 188 and 479 K) and adsorbate concentration investigated here (c = 1.2, and 3 Xe/cage). We attribute the shoulder at 6 A and 188 K to xenons in neighboring adsorption sites in zeolite Y. We note that two neighboring adsorption sites in a cage in zeolite Y are separated by a distance of about 6 A. The X t X e rdf is shown in Figure 4b as a function of different adsorbate concentrations
8
0
188 K I
2
.
4
r, A Figure 5. Center of cagecenter of mass radial distribution functions g-(r) at different temperatures plotted as a function of the distance from the cage center for an adsorbate concentration of 1 Xe/cage.
near 285 K. Little change is observed except for a small decrease in the height of the first peak and a small shift of the first peak toward lower distance with increase in xenon concentration. The distribution of xenon atoms at different temperatures as a function of the radial distance from the cage centre is shown in Figure 5 . The radial distribution function, g-,,,(r), was calculated from
where ( n ( r ) ) is the average number of guests between r and r + dr, p is the density and equals N / V,where N is the total number of guests in volume V, the volume of the simulation cell. The curve was obtained by averaging Over all the eight cages. A few features of this curve are noteworthy. Firstly, a prominent peak is observed in the rdf near 4.2 A. This peak is due to the preference of the adsorbate for the region near the inner surface of the cage. A similar behavior has been observed in methane and benzene sorbed With increase in temperature it is found that in sodium Y.26*2729
10472 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
c
A 32.0 m
Santikary et al.
i I-
0
2.0
LO
6 3
r (8,
Figure 6. Center of cage-center of mass radial distribution functions at different adsorbate concentrations of 1,2, and 3 Xe/cage and near 300 K plotted as a function of the distance from the cage center. g-&)
,r\
c:l Xelcago
Figure 8. A display of the trajectory of the xenon when (a) -12.5 < Ugh C -1 1.5 Id/mol where a peak is observed inAU,) and (b) ush < -18.0 kJ/mol. The points show the position of the xenon every fifth MD step. The picture indicates that the guest is not localized to any particular region in the cage in the case of (a). For (b) the guest is localized and corresponds to the adsorption site of xenon in sodium Y. Ugh, kJlmoi
Figure 7.
Distribution of energy of interaction between the guest and the hOSt,AUgh) at different temperatures and an adsorbate concentration of c = 1 Xe/cage.
the intensity of this peak decreases and simultaneouslythere is an increase in the intensity of the rdf near smaller values of r. This means that the xenons increasingly occupy regions near the center of the cage. As a result, the main peak near 4.2 A also shows a small shift toward lower values of r with increase in temperature. These changes crucially determine the mechanism and properties relating to cage to cage migration of xenons through the 12-ringwindow.M Evidence from experiments for the behavior observed in Figure 5 is available in the literature. Cohen de Lara and Khan3’ have reported neutron scattering results of methane in zeolite A. Around 200 K, the region near the periphery of the cage is found to be populated predominantly. At room temperature, the central region of the cage is also found to be populated although sparsely. The distribution of xenon atoms, g-&) as a function of the distance from the cage center is shown in Figure 6 for different adsorbate concentrations. The guests approach slightly closer to the cage center at higher xenon loadings. The distribution of the energy of interaction,flU@), between the guest and the host zeolite is shown in Figure 7 at different temperatures. At low temperatures, especially at 188 K,most xenons exhibit a strong interaction in the range -19 < U, < -17 kJ/mol with the surface. As we shall see at low temperatures, the particles are close to the adsorption site. There is an intense peak near -12 kJ/mol at all ternperatm. In the case of methane sorbed in sodium Y,a bimodal distribution function was observed forflU,,,), with a shoulder near -12 kJ/mo1.25-26In the present
case, the peak near -12 kJ/mol is rather sharp. In order to get an insight into the origin of the peaks, we have examined the trajectories of particles having energies in the range of the two peaks inAU,,). Figure 8a displays a stroboscopic snapshot of a particle with U,,between -12.5 and -1 1.5 kJ/mol, for a fairly long length of time, 60 ps. It is clear from the figure that there is no particular region in the supercage where the particle is localized for any reasonable fraction of time. Thus, the particles with energies in this range are essentially delocalized. In contrast, particles with ener ies Ugh < -18.0 kJ/mol remain more or less as can be seen from Figure 8b where the localized within 1 trajectory is shown for 25 ps, although the localization in the same region was seen for the entire 60 ps. This position where the particle is localized should be interpreted as the adsorption site of xenon in the zeolite. The location of the adsorption site is clear from Figure 8b; there are three four-membered rings which connect two six-membered rings. The adsorption site is located in the middle of the central four-membered ring. This location of the sorption site in the supercage is almost identical to the site obtained for methane.21*25 This suggests that for small molecules the geometry of the zeolite structure predominantly determines the position of the adsorption site. This is true for the physisorption site and is not likely to be true for the chemisorption site. The Xe-Na rdf (see Figure 2) shows an intense second peak which suggests fairly strong interaction with the sodium cation. Hence, apart from the geometry of the cage, the nature and the position of the cations and their positions also appear to play a role in the determination of the physisorption site. Earlier molecular dynamics calculations on benzene on sodium Y zeolite correctly predicted the sorption sites obtained from neutron diffraction and IR st~dies.~OJ~ Therefore, we believe that the physisorption site
i3,
The Journal of Physical Chemistty, Vol. 96,No. 25, I992 10473
Xenon Sorbed in Sodium Y Zeolite -300 K
- c = lXelcoge
--.....
c c:
2Xe/coge 3Xelcoge
Ugh,KJ/mol
F i 9. Distribution of energy of interaction between the guest and the host, flu,,) at different concentrations and temperature near 00 K. I
I
-2
-1
0
U d , kJlmol
Figure 11. Distribution of the energy of interaction between any two xenon atoms, flu,),shown at different adsorbate concentrations.
r 1. -300H
(b)
Xe/cqe -2 Xe/coge - - - 3 Xelcoge -1
,,60
I
I
I
-2
-1
--
0
U d , kJ /mol
Figure 10. Distribution of the energy of interaction between any two xenon atoms, flu,),shown for different temperatures for an adsorbate concentration of 1 Xe/cage.
for xenon obtained here is correct and it should be possible to verify this by X-ray and neutron diffraction studies. We are not aware of any such investigations of xenon in faujasite at low temperatures. The energy distribution,f(U&), is shown in Figure 9 at different adsorbate concentrations near 300 K. There is a small but noticeable decrease in the intensity between -18 and -13 kJ/mol on going to c = 3 Xe/cage. The absence of decrease in the intensity for c = 2 Xe/cage is attributed to the higher temperature (311 K) of this run. This means that the additional xenons available at higher concentrations are unable to occupy regions close to the adsorption site and instead occupy regions near the cage center in agreement with Figure 6. The energy distribution function,f(Ud),is shown in Figure 10. Here u d is the interaction energy between any two guest particles. The functionflu,,) yields information regarding the pairs, that is, dimers, pairs in trimers, and higher n-mers. The main peak near zero is due to the guests at large distanca (monomers) where the interaction is negligible. The peak near -2 kJ/mol is due to pairs. These pairs are expected to be at a distance of 2'/6uxxtxc = 4.6 A from each other where the interaction is sXtXc = 221 K or -2 kJ/mol. This interpretation is supported by the peak near 4.5 A in the g(r) for guest-guest (Figure 4). The intensity of the peak near -2 kJ/mol is less than one-fifth of the peak near zero. Thus, the fraction of guests existing as-pairs at any given instant is le@ than 15%. There is a decrease in the dimer population with increase in the temperature in agreement with earlier observation (Figure 4). Figure 11 shows A&) for various adsorbate concentrations near 285 K. The increase in intensity of the peak near
I
c:l
Xekage
- 188 K
160
--- 285 H ......479 K
Ub,kJlmol
Figure 12. Distribution of the energy of interaction of the guest atom with all the other guest species,flUb), shown (a) at different temperatures and (b) adsorbate concentrations. There is an increase in the higher sized clusters with increase in the adsorbate concentration.
-2 kJ/mol is attributed to increase in the number of pairs with increase in adsorbate concentration. The distribution function,AUJ, where U, refers to Ubfor ith particle and N u b i = CVkj) j=i
(7)
is the energy required to remove ith guest atom to infinity in the absence of the host is shown in Figure 12 at three different tem-
10474 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
Santikary et al. (a)
N
- 1.0
aC
N
-
*a
2.0
6.0-
-0.0
C
-2.01 -2.0
I
I
1.o
4 .O
0
1 7.0
o
0
o
In 1 , p s Figure 13. Log-log plot of mean square displacement as a function of time is shown for different temperatures.
peratures and adsorbate concentrations. Here, flub) has been averaged over all particles, and 0.01
5.0
This function yields information about the formation of larger clusters of guest species. We note that the above function does not yield any information about the lifetime of the clusters. The tail near low energy below -2 kJ/mol indicates the presence of trimers and larger guest clusters. The presence of clusters of size three and higher sizts have been reported earlier by Demontis et al.17 There is a small decrease in the population of larger clusters at higher temperatures as can be seen from Figure 12a. The decrease is accompanied by an increase in the number of dimers and monomers. The change influb) with adsorbate concentration is more significant (see Figure 12b). The distribution function flub) shown in Figure 12b and flud)shown in Figure 11 are roughly similar for c = 1 Xe/cage. For higher adsorbate concentrations, the nature offlub) is markedly different fromf(Ud). This shows that at the low adsorbate concentration of 1 Xe/cage the guests may be treated as isolated particles or dimers moving in the field of the zeolite. The disappearance of the peak near zero in u b (see Figure 12b) shows that at 2 Xe/cage the number of monomers is small. At c = 3 Xe/cage, there are no monomers and even the fraction of pairs responsible for the peak near -2 kJ/mol is small at this concentration. The increase in the number of large-sized clusters is reflected in the average guest-guest interaction energy, (U ) (see Table 11). This suggests that, at c = 2 Xe/cage and 3 %/cage, the guest particles can no longer be treated as single particles moving in the external field though thii is to a good approximation true at c = 1 Xe/cage except for a small number of pair population. One expects that at c = 2 and 3 Xe/cage the mean free path of the sorbate would be less than at c = 1 Xe/cage, closer to that of the liquid than for c = 1 Xe/cage. The distribution functionf(UJ displays a shape which is close to the characteristic unimodal distribution observed for the bonding energy distributions at liquid d e n ~ i t i e s . ~ ~ , ~ Dyn8mid Properties. A study of the variation of the mean square displacement, ( u2(t)) as a function time and temperature yields important insight into the nature of the motion of the sorbate in the cages of zeolites. In order to obtain more accurate time and temperature depemhx of mean square displacement we have carried out two more runs at 567 and 637 K. In Figure 13, In ( u2(t)) has bem plotted against In t, for fne different temperama for c = 1 Xe/cage. It is sem that, for all temperatures, In (u2(r)) exhibits a slope of nearly 2 for short times changing over to a slope of unity for t > 1, which is about 1.3 ps. Thus, while for short time the motion is ballistic (free), for long times the motion of the particle is diffusive. In Figure 14 open and filled circles
5.5
6.0 inT. K
6.5
Figure 14. (a) Logarithm of mean square displacement (I&)), plotted against reciprocal temperature for c = 1 Xe/cage. The filled circles show the short-time mean square displacement before the crossover (seeFigure 13). The open circles show the long-time mean square displacement after the crossover. The best least-squares line for the long-time mean square displacement is also shown. (b) Logarithm of mean square displacement before the crossover, ( u 2 ( t ) ) is plotted against the logarithm of temperature. The best least-squares line for the short-time mean square displacement is also displayed. The points correspond to 188,285,386, 414,479,567, and 637 K.
TABLE Ill: Diffusion Coefficients Obtaiwd from the Slope of Mean Quare %placement after the Crossover T (K) 108D (m2/s) T (K) 108D(m2/s) 188 0.29 414 1.30 285 0.77 479 1.43 386 1.29
represent the long and short time behaviors, respectively. Shown in Figure 14a is a plot of In (u2) versus 1/T. The long- and short-time behaviors are represented by open and filled circles, respectively. Clearly, the points lie on a straight line. From the slope of In ( uz) against 1 / T for long time, the activation energy was found to be 4.1 M/mol. Using Einstein's relation and the intercept from Figure 14a, the value of Do in D = Do wrp(-E,/RT) was found to be 2.54 X m2 s-l. The diffusion coefficients at different temperatures are liited in Table 111. The diffusion coefficients (m2s-l) at different adsorbate concentrations are 0.613 X 10-8 (c = 1 Xe/cage and 285 K); 0.727 X 10-8 (c = 2 Xe/cage and 311 K), and 0.555 X 10-8 (c = 3 Xe/cage and 304 K). Due to the temperature being different for different adsorbate concentrations and the small differences in D at different wncentrations, the trend in the variation of D with c cannot be deduced without ambiguity. We have also calculated the D values from the velocity autocorrelation function using the expression D = l l f m ( o ( r ) a(0)) dt 3
0
(9)
where t,, was taken to be 600 ps. These values were about 15-20'36 higher than those from Figure 13. There are lBXe NMR investigations of Xe adsorbed in X- and Y-type zeolites by Fraissard and co-~orkers."~~~ However, we could not find any reports on diffusion coefficients of Xe in zeolite Y. Interestingly, for short times (fdled circles), we find that the behavior of In (uz)
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10475
Xenon Sorbed in Sodium Y Zeolite is linear with In T as is clear from Figure 14b. Thus, in the short-time limit,
(u2(t)) - T whereas in the long-time limit
-
(u2(r))
-
-
t2
(10)
t
(11)
(a)
exp(-E,/RT)
The microscopic origin of the observed dependence of the mean square displacement on temperature is not clear. Below we show that the inverse exponential temperature dependence of mean square displacement arises from the dependence of the residence time at the sorption sites on the temperature. Here T is the mean residence time before the particle surmounts a classical barrier and hops to a neighboring site, and therefore T is expected to have the form T
= r0 exp(-E,/RT)
(12)
For the sake of completeness we briefly summarize results of the Langevin equation and point out the origin of the temperature dependence, dv/dt = -flu + v(r) (13) where ~ ( t is) taken as a Gaussian white noise, with ( ~ ( t ) =) 0 and ( ~ ( t~)( 2 ' ) ) = 2Ds(t-t'). Here D is the velocity and /3 the conventional 'frictional force". The above equation predicts the crossover behavior in time of the mean square displacement from t2to t and also the linear temperature dependence at short times. Since the molecular dynamics simulation has been carried out in the microcanonicalensemble, the temperature and, therefore, the velocity fluctuates. Hence, the velocity of the sorbate xenon is the random variable in eq 13. The equivalent Fokker-Planck equation for the Langevin equation (13) is43
ap(v,t) -I at
a(p(u,r)v) av
+
WP(U,~) a02
(14)
where u = nAv and r = mT; n, m = -w, ..., -1, 0, +1, ,.. +m are discrete random variables. R is an arbitrary number which determines the extent of unequal left-right jump probabilities and equals (1 r u/R)/2. Quation 15 goes over to the continuous form given by (14) in the limit lim
a2 --cQ
27
-
1 lim-+/3 4 RT
I
(16)
R--
From (16) it is clear that T . Thus,the inverse exponential temperature dependence of the microscopic residence time given by (12) is essentially responsible for the observed behavior of the mean square displacement given by (1 1). The expression for mean square displacement is"*'*
where ( ) refers to average over the statistical properties of ~ ( t ) and the bar refers to the initial velocity distribution of the diffiing particles. We note that only the initial velocity distribution, 9
1
I
30.0
I
60.0
-
0.0
where P is the probability that the sorbate atom has a velocity u at time 1. Here, the temperature dependence of /3 is entirely controlled by the fluctuation-dissipation theorem which cannot explain the observed Arrhenius temperature dependence. The physical meaning of /3 is, however, more transparent and the relationship between diffusion in zeolites and the Langevin approach becomes clearer if we consider the discrete onedimensional random walk version of eq 14
a . 4
I 0.0
30.0
6 .O
J , cm-'
Figure 15. Power spectra for xenon sorbed in sodium Y obtained by Fourier transformation of the velocity autocorrelation function. The power spectra (a) for an adsorbate concentration of 1 Xc/cage at dif-
ferent temperatures; the inset shows the power spectra correspondmg to the velocity components parallel and perpendicular to the cage surface; (b) near 300 K for different adsorbate concentrations. See text for discussion.
= k T / m , is used and the equipartition theorem is not used for subsequent times. Expression 17 clearly gives the proper time and temperature dependence. At large times the dominant temperature dependence is controlled by the exponential factor with the linear temperature contributing in an insignificant way. The activation energy, E,, for the diffusion of xenon in the zeolites is related to the baniem encountered by xenon as it diffuses inside the cages of zeolites. This picture is in agreement with the picture suggested by the potential energy surface for Xe in Xand Y-mlites reported by Rowlinson and ceworkers2' where they found several local minima inside the cage. The overall picture one obtains is a zeolite having several sites with bamer heights for migration from one site to another varying for different pairs of sites. The situation here is similar to diffusion on a surface with barriers of differing heights which has been reviewed by Haus and K e l ~ r . ~ ~ The power spectrum obtained by the Fourier transformation of the velocity autocorrelation function is shown in Figure 15a for different temperatures. At 188 K,the power spbctrum shows a low-frequency band extending from 8 to 16 cm-' and a small shoulder near 28 cm-I. The low-frequency band appears to be compmed of two peaks, at 8 and 16 an-'.It should be expected that a motion along a direction parallel to the surface involving small changes in the potential energy should be seen in the lowfrequency part of the power spectrum while comparatively large changes in the perpendicular component should be observed at
10476 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
1'
I
Santikary et al. ~ : i a e i ~ ~ p ~
- t88K
c: 1 X e l c a g e
--- 285K - - 386K '
0 2&
c i2 Xe lcoge
0.24
s.3Xekoqe 0.24
479K
0
LoL o L
ooo
10
7.0
PS
Figure 17. Distribution of site residence timcs, T#, shown for different adsorbate concentrations near 300 K. There is no marked change in the distribution as a function of the adsorbate concentration.
rs,
PS
Figure 16. Distribution of site residence times, T ~ shown , for different temperatures. The site residence time was calculated by taking a particle to be in an adsorption site when the Ughwas less than -14 kJ/mol.
relatively higher frequencies. Therefore, a decomposition of the power spectrum into parallel and perpendicular components has been carried out. We have calculated the parallel and perpendicular component of the velocity from D l ( t ) = a(t).i(t)
q ( t ) = a(?) - a,(t) where i(r)is the unit vector from the particle to the cage center. We indeed find that both the low-frequency modes are present in the power spectrum of the parallel component of the autocorrelation function. The power spectrum of the parallel and perpendicular components are shown in the inset of Figure 15a. The high-frequency component is dominant in the perpendicular component whereas the low-frequency component is relatively more intense in the parallel component. At higher temperatures, there is an increase in the intensity of the lower frequencies which is accompanied by a decrease in the intensity of the 16-cm-' band and 28-cm-l shoulder. The trend with temperature is similar for methane in sodium Y ze0lite.2~The power spectrum for methane, however, showed bands extending upto 150 cm-I. The band at 150 cm-' was assigned to librations associated with rotational motion. The effect of increase in the adsorbate concentration on the power spectrum is shown in Figure 15b. A shoulder at 16 cm-l can be seen to appear with the increase of xenon loading. From Figure 11, we know that the dimer concentrationis considerably more at higher loadings even at the relatively high temperature of 300 K. This suggests that the 16-cm-' peak might be due to dimer frequency. The stretching frequency of xenon dimer can be easily calculated from the relation Y = 1 / 2 n ~ ( k / p ) ' / ~where , c, k, and p are the velocity of light, force constant between the dimers, and the reduced mass of the dimer. We found Y to be 16 cm-I. The force constant k was calculated from the second derivative of the Lennard- Jones interaction potential for Xe-Xe at r = 21/6uxtxe. The population of dimers is large at high loadings and even at c = 1 Xe/cage at low temperatures (see Figure lo), resulting in significant contributions to the 16-cm-' peak of the power spectrum at 188 K from xenon dimers. An increase in the intensity below 7 cm-' is observed with increase in temperature. A decrease in the low-frequency part of the power spectra is observed with increase in the adsorbate concentration. We note that there is a relative increase in the population of delocalized monomers with increase in temperature as is borne out by the increase in the intensity above -12 kJ/mol and other properties such as AU,) and flu,). This in flugh) suggests that the intensity below 7 cm-l is possibly due to the presence of monomers which are not in the proximity of the
adsorption site. This also suggests that the decrease in the intensity of the low-frequency region of the power spectra with increase in xenon loading is due to the decrease in the monomers population which is consistent with the observed changes influ,) (see Figure 12b). The distribution of site residence times is shown in Figure 16 for different temperatures and adsorbate concentrations. The site residence time was obtained by taking a cutoff of E, = -14 kJ/mol. If Ugh < E, when the particle is said to reside in the adsorption site. The increase in temperature results in a considerable increase in the intensity of the site residence times, rs,of particles in the adsorption site (Figure 16). This is accompanied by a decrease in the intensity in the longtime region. The criterion employed by us here for defining the site residence time may not be the best but is adequate. It may be worthwhile to see how different criteria change the distribution of site residence times. The change in adsorbate concentration has little influence on the distribution of site residence times (Figure 17) except for a small increase in T~ = 0.2 ps at higher concentrations. Experimental measurements of residence times are available from NMR. Site residence time ~ ~ large has been estimated to be around 60 ps by F r e ~ d e .The discrepancy between this value and that reported in this work could be attributed to differences in the zeolite structure and grain boundary effects, etc.
Conclusions The guestzeolite radial distribution functions and guestzeolite energy distribution functions suggest that the xenons are largely localized in the supercages at the lowest temperature (188 K) studied. At higher temperatures, xenons are largely delocalized. xenons are mostly confined to the periphery At lower tempera-, of the cage populating the region, r > 3 A. At higher temperatures, xenons populate the central region of the cage as well. The guest-host zeolite energy distribution function shows a peak near -12 kJ/mol which has been shown to correspond to unbound and mobile xenons. Most xenons exist as monomers and the remaining exist as dimers at c = 1 Xe/cage. At c = 3 Xe/cage, the monomer population is found to be negligible; clusters of xenons of size greater than 2 are common at this sorbate concentration. The variation of the mean square displacement with time shows a crossover from a free particle to an Arrhenius behavior. In the short-time limit, the mean square displacement shows a t2 dependence on time and linear temperature dependence. In the long-time limit the mean square displacement shows a linear time dependence and an exponential temperature dependence corresponding to the well-known Arrhenius behavior. The power spectrum exhibits a peak near 16 cm-' which has been shown to correspond to the stretching frequency of X e X e dimers. It is shown that the motions parallel to the inner surface of the cage are reflected in the low-frequency part of the power spectrum while the motions perpendicular to the inner surface of the zeolite appear at relatively higher frequencies (28 cm-I). At higher adsorbate concentrationsa shoulder at 16 cm-I observed is assigned to the Xe-Xe dimers. References and Notes (1) Thomas, J. M.Philos. Trans. R. SOC.London 1990, A333, 173. (2) Barrer, R.M.Zeoliies and Clay Minerals as Sorbenis and Molecular Sieues; Academic Press: New York, 1978. ( 3 ) Allen, M. P.; Tildesley, D. J. Computer Simulation 01Llquids; Clarendon Press: Oxford, U.K., 1987.
J. Phys. Chem. 1992,96, 10477-10483 (4) Klein, M. L.; Lewis, L. J. Chem. Rev. 1990, 90, 459. (5) Rao, C. N. R.; Yashonath, S. J . Solid Stare Chem. 1987, 68, 193. (6) Subramanian, G.; Davis, H. T. Mol. Phys. 1979,38, 1061. (7) Powles, J. G.; Dornforth-Smith, A.; Evans, W. A. B. Phys. Rev. Lett. 1991, 66, 1177. (8) MacElroy, J. M. D.; Suh, S.-H. Mol. Simular. 1989, 2, 313. (9) Derouane, E. G. Chem. Phys. Lert. 1987, 142, 200. (IO) Politowicz, P. A.; Kozak, J. J. Mol. Phys. 1987, 62, 939. (1 1) Woods, G. B.; Panagiotopoulos, A. Z.; Rowlinson, J. S. Mol. Phys. 1988,63,49.
(12) Vetrivel, R.; Catlow, C. R. A.; Colbourn, E. A. J. Chem. Soc. Far-
aday Trans. 2 1989,85, 497-503.
(13) June, R. L.; Bell, T. A,; Theodorou, D. N. J . Phys. Chem. 1990,94, 1508. (14) June, R. L.; Bell, T. A,; Theodorou, D. N. J . Phys. Chem. 1990,94, 8232. (IS) Pickett, S. D.; Nowak, A. K.; Thomas, J. M.; Peterson, B. K.; Swift, J. F. P.; Cheetham, A. K.; den Ouden, C. J. J.; Smit, B.; Post, M. F. M.J . Phys. Chem. 1990,94, 1233. (16) Demontis, P.; Suffriti, G. B.; Quartieri, S.; Fois, E. S.; Gamba, A. Zeolites 1987, 7, 552. (17) Demontis, P.; Fois, E. S.; Suffriti, G. B.; Quartieri, S. J. Phys. Chem. 1990,94,4329. (18) Smit, B.; den Ouden, C. J. J. J . Phys. Chem. 1988,92, 7169. (19) Leherte, L.; Lie, G. C.; Swamy, K. N.; Clementi, E.; Derouane, E. G.; Andre, J. M. Chem. Phys. Lett. 1988, 145, 237. (20) Rowlinson, J. S.; Woods, G. B. Physica A 1990, 164, 117. (21) Woods,G. B.; Rowlinson, J. S . J. Chem. Soc., Faraday Trans. 2 1989, 85, 765. (22) Barrer, ,R. M.; Sutherland, J. W. Proc. R . Soc. London, Ser A 1956, 237, 439. (23) Fomkin, A. A.; Serpinskii, V. V.; Bering, B. P. Bull. Acad. Sci. USSR, Dlv. Chem. Sci. 1975, 24, 1114. (24) Chkhaidze, V. E.; Fomkin, A. A.; Serpinskii, V. V.; Tsitsishili, V. G. Izv. Akad. Nauk. SSSR, Ser. Khim. 1986, 276.
10477
(25) Yashonath, S.; Thomas, J. M.; Nowak, A. K.; Cheetham, A. K. Nature 1988, 331, 601. (26) Yashonath. S.: Demontis. P.: Klein, M. L. Chem. Phvs. h t . 1988. 153, 551. (27) Yashonath, S.; Demontis, P.; Klein, M. L. J. Phys. Chem. 1991,95, 588 1. (28) Yashonath, S. J. Phys. Chem. 1991, 95, 5877. (29) Demontis, P.; Yashonath, S.; Klein, M. L. J. Phys. Chem. 1989, 93, 5016. (30) Fitch, A. N.; Jobic, H.; Renouprez, A. J. Phys. Chem. 1986.90, 1311. (31) Olson,D. H. J. Phys. Chem. 1968, 72, 4366. (32) Kiselev, A. V.; Du, P. Q. J. Chem. Soc., Faraday Trans. 2 1981,77, 1.
(33) Yashonath, S. Chem. Phys. Lett. 1991, 177, 54. (34) Hirschfelder, 0. J.; Curtiss, F. C.; Bird, B. R. Molecular Theory of Gases and Liquids; John Wiley: Chichester, U.K., 1954. (35) Kushick, J.; Berne, J. B. In Statistical Mechanics, Part E: Timedependent processes; Beme, J. B., Ed.; 1977; Chapter 2. (36) Yashonath, S.; Santikary, P. Submitted for publication in J . Phys. Chem.
(37) Cohen de Lara, E.; Kahn, R. J . Phys. 1981, 42, 1029. (38) de Mallmann, A.; Barthomeuf, D. Zeolites 1988, 8, 292. (39) Yashonath, S.; Rao, C. N. R. Proc. R . SOC.(London) 1985, A400, 61. (40) Jorgensen, W. L.; Madura, J. D.; Swenson, C. J. J. Phys. Chem. 1984, 106, 6638. (41) Friassard. J.: Ito. T. Zeolites 1988. 8. 350. (42j Grosse, R.;Burmeister, R.; Boddent&, B.; Gedeon, A.; Fraissard, J. J. Phys. Chem. 1991, 95, 2443. (43) van Kampen, N. G. Stochastic Processes in Phvsics and Chemistrv: .. North-Holland: Amsterdam, 1981. (44) Chandrashekar, S. Rev. Mod. Phys. 1943, 15, 1. (45) Uhlenbeck, G. E.; Ornstein, L. S. Phys. Rev. 1930, 36, 3. (46) Haus, J. W.; Kehr, K. W. Phys. Rep. 1987, 150, 263. (47) Fruede, D. Zeolites 1986, 6, 12.
Measurements of the Cotton-Mouton Effect of Water and of Several Aqueous Solutions Jeffrey H. Williams* and Jim Torbett Institut hue-Langevin 156X, 38042 Grenoble Cedex, France, and Max Planck Institute for High Magnetic Fields 166X, 38042 Grenoble Cedex, France (Received: June 22, 1992; In Final Form: September 8, 1992)
We report a new measurement of the Cotton-Mouton constant, at 632.8 nm of liquid water, together with measurements, at the same wavelength of the Cotton-Mouton effect in several aqueous ionic solutions of varying concentration. We show how these results may be combined with literature values of optical Kerr measurements in the same solutions to separate the optical and magnetic susceptibility contributions to the Cotton-Mouton effect. These results will be of interest to those seeking to measure electrooptic effects in aqeuous solutions as the combination of the two experiments will give new information on the nature of the aqueous state.
Even though the aqueous state is ubiquitous, a detailed understanding of its microscopic structure remains elusive. We are nearly completely ignorant of the electric and magnetic properties of the ions which constitute the solution. This state of affairs is mainly the result of the lack of suitable techniques for the study of the electmoptic properties of solvated ions. We have, therefore, undertaken a series of measurements of the Cotton-Mouton effect in aqueous solutions. These experiments have demonstrated the suitability of this technique for such studies. We were also motivated by recent epidemiological evidence’ which suggests that electromagnetic fields produced, for example, by power cables and household electrical appliances could be harmful to health. As these electromagneticfields oscillate slowly (-60 Hz),it is conceivable that their magnetic field component induces small orientational fluctuationsin ionic distributionswithin tissue thus influencing in vivo interactions. Application of an intense electromagneticfield to a fluid induces an anisotropy in its physical properties. The Cotton-Mouton +Presentaddress: Department of Biochemistry,University of Edinburgh, George Square, Edinburgh EH89XD, U.K.
effect, for example, is the response of a fluid to a strong uniform magnetic field. The field gives rise to differences between the components of the refractive index parallel and perpendicular to the field; thus a beam of linearly polarized light propagating perpendicular to the field direction becomes elliptically polarized on passage through the fluid. Physically, the molecules are being oriented by the magnetic field and the fluid is thus behaving as a uniaxial crystal, with its optic axis parallel to the applied field direction. The Cotton-Mouton constant, C, is defined as C = (n,, - n , ) v / B 2 (1) where n,, - n, is the difference between the components of the refractive index, n, for electromagnetic waves of wavenumber Y with electric vectors parallel and perpedicular to the field direction, the magnetic field, B, being uniform and normal to the optical path. For a pure material, a molar Cotton-Mouton constant, ,C, may then be defined
where V, is the molar volume and po the vacuum permeability.
0022-3654/92/2096-10477%03.00/00 1992 American Chemical Society
