9252
J. Phys. Chem. 1994,98, 9252-9259
Dynamics of Zeolite Cage and Its Effect on the Diffusion Properties of Sorbate: Persistence of Diffusion Anomaly in NaA Zeolite? Prakriteswar Santikaryt and Subramanian YashonathfS*§ Solid State and Structural Chemistry Unit and Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore 56001 2, India Received: March 9, 1994; In Final Form: July 1 , 1994”
Recent computer simulations on zeolites Y and A have found that the diffusion coefficient and the rate of intercage diffusion exhibit, apart from a linear dependence on the reciprocal of the square of the sorbate diameter, a n anomalous peak as sorbate diameter approaches the window diameter. Here we report molecular dynamics simulations of zeolite NaA incorporating framework flexibility as a function of sorbate diameter in order to verify the existence of anomalous diffusion. Results suggest persistence of anomalous diffusion or ring effect. This suggests that the anomalous behavior is a general effect characteristic of zeolites Y and A. The barrier for diffusion across the eight-ring window is seen to be negative and is found to decrease with sorbate size. The effect of sorbate on the cage motion has also been investigated. Results suggest that the window expands during intercage migration only if the sorbate size is comparable to the window diameter. Flexible cage simulations yield a higher value for the diffusion coefficient and also the rate of intercage diffusion. This increase has been shown to be due to an increase in the intercage diffusions via the centralized diffusion mode rather than the surface-mediated mode. It is shown that this increase arises from a n increase in the single particle density distribution in the region near the cage center.
1. Introduction
Studies on diffusion of small molecules in the cavities or void spaces of zeolites have been reported by several workers.1-8 Molecular dynamics (MD) calculations on xenon in zeolite NaY and argon in NaCaA have shown that the rate of intercage diffusion in the two systems are 2.9 X lO”J/sorbate/s and 6.05 X lOlO/sorbate/s, respectively, in spite of the fact that the ratio of the window diameter to the sorbate diameter is 1.95 and 1.32 in the two system^.^ This result is surprising since, according to the well-accepted view, the molecular sieve and the diffusion properties are mainly determined by the size and shape of the sorbate and void dimensions. Systematic investigations on the dependence of transport properties on the sorbate diameter have shown the existence of an anomaly in the rate of intercage diffusion, rate of cage visits, and the diffusion coefficient when the size of the sorbate approaches that of the window in zeolites NaY and NaCaA.1OJI The recent work of Demontis et al.I2and Brickmann and co-workersl3 on flexible cages suggests that some of the sorbate properties in flexible cages are significantly different from those obtained from a rigid framework. Titiloye et al.14 found that significant changes in the sorbate properties occur for those sorbates whose sizes are comparable to the dimensions of the void spaces in the zeolites. I n the light of these two findings, it may be justifiably asked whether the anomaly in the diffusion coefficient and related properties observed in zeolites NaY and NaCaA would continue to be observed if the rigid framework approximation is relaxed by carrying out a full-scale molecular dynamics calculation using a flexible framework model. This question is particularly relevant since the reported anomaly in the various properties occurs when the diameter of the sorbate approaches that of the void dimensions. The anomalous diffusion observed by us has been predicted by Derouane and co-workers a few years ago. In their pioneering work based partly on calculations and partly on theoretical treatment,15-I7 they have predicted the possibility of observing t Contribution No. 1032 from the Solid State and Structural Chemistry Unit. f Solid State and Structural Chemistry Unit. Supercomputer Education and Research Centre. @Abstractpublished in Advance ACS Abstracts, August 15, 1994.
0022-365419412098-9252%04.50/0
“floating molecules” in systems with restricted regions. The diffusion anomaly reported by uslOJ1 is essentially the same as the ”floating molecules” of Derouane and co-workers. In the earlier work,I1we termed the effect as “ring” effect or “levitation” effect. These results are also likely to have important ramifications in the separation of molecular mixtures using molecular sieves. Here we report MD calculations on monatomic spherical Lennard-Jones atoms sorbed in zeolite NaA. The cage is treated as flexible, and results of several runs for different sorbate diameters are reported. Results are also reported for argon in NaA, for rigid and flexible framework, and comparisons are made between the two sets of calculations with specific reference to intercage diffusion. 2. Structure of Zeolite NaA The structure of zeolite NaA was taken from the work of Pluth and Smith.18 The space group is Fmsc with a = 24.555 A. The unit cell formula is Na96Si96A1960384. One unit cell of this zeolite consists of eight a-cages also known as supercages (each of diameter 11.4 .&). Each a-cage is connected to six other a-cages in an octahedral fashion, and the interconnection is via the eightring window of approximate diameter 4.5 .&. Extraframework sodium cations are known to occupy three distinct locations. Type I cations, eight in number, are located at the center of six-membered rings and are strongly bound. Type I1 cations, of which there are three per cage, are near the center of the eightring window. Type I11 cations are located well inside the cage in front of the four-membered ring. These are highly mobile, and there is only one such cation per cage. Figure 1 shows the different crystallographic positions occupied by the sodium atoms of types I, 11, and 111. In our present simulations, we have used a model of NaA zeolite where there are no cations of the types I1 and 111. This corresponds to a situation where the unit cell formula becomes Na6&i128A1&84 and the Si/Al ratio changes from 1.0 in the original model of Pluth and Smith to 2.0 in the model used by us.
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3. Intermolecular Potential Functions 3.1. Sorbatesorbate Interactions. Sorbate-sorbate interactions have been modeled in terms of (6-12) Lennard-Jones pair 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9253
Persistence of Diffusion Anomaly in NaA Zeolite
model reproduces satisfactorily the main features of aluminosilicate frameworks. Further, this is the only model that has been specifically proposed for zeolite A. The values of ro are consistent to a good approximation with the values of LennardJones parameters of Kiselev and Du. TYPQ I Type 11 Type 111 Figure 1. Location of different types of cations inside the a-cage of
zeolite NaA. Plus, open circles and the filled circle indicate respectively type I, 11, and I11 positions.
TABLE 1: Interatomic Potential Parameters for Sorbate-Sorbate and Sorbate-Zeolite Interactions
Ar-Ar Ar-0 Ar-Na
Sorbate-Sorbate 120.0 Sorbate-Zeolite 159.2 35.8
3.41 3.10 3.33
TABLE 2 Interatomic Potential Parameters for Zeolite Framework Interactions. atom Si-0 A1-0 0-(Si)-0 O-(Al)-O Na-0 rex, is the Na-0
k (kJ mol-l A-2) 2092.0 2092.0 431.0 430.0 125.5
TO
(4
1.605 1.760 2.617 86 2.870 70 2.385 (if re,, < 2.5 A) 2.614 (if rex, < 2.5 A)
distance.
potential
where ushas been varied to study the dependence of the properties on the sorbate diameter or size. In all calculations reported in this paper, es was taken to be 120 K. 3.2. Sorbatezeolite Interactions. These have been modeled again in terms of short-range (6-1 2) Lennard-Jones potential
where s stands for sorbate and z = Si, 0,and Na. The potential depth e,, and the diameter us, were obtained from the LorentzBerthelot combination ruleslg
The values of c, and u, (z = Si, 0,and Na) were obtained from the work of Kiselev and Du.20 The interaction parameters for sorbatesorbate and sorbatezeolite atoms are listed in Table 1. 3.3. Zeolitezeolite Interactions. The intrazeolite potential for interaction amongst the zeolite atoms (Si, Al, 0, Na) was chosen to be of the harmonic type between the bonded atoms. This model has been earlier proposed and studied by Demontis et a1.12 The potential energy is given by (4)
where ro is the equilibrium distance and k is the force constant. The values of k have been derived from the spectroscopic data of natrolite. Thevibrations between bonded pairs of atoms Si+, A1-0, and Na-0 as well as 0-(Si)-0 and O-(Al)-O have been included in the present calculation. The values of ro and k are listed in Table 2. Even though many sophisticated models have been proposed in recent times,l3.*1 the harmonic
4. Computational Details All calculations were carried out in the microcanonical ensemble on one unit cell of zeolite NaA, consisting of eight supercages. Periodic boundary conditions were employed. A cutoff of 10 8, was used for calculating the sorbate-sorbate and sorbate-zeolite interaction energies and forces. Two sets of calculations have been carried out. In the first set, termed as set a, thedependence of properties as a function of the sorbate diameter has been investigated. Simulations employing the flexible framework are reported over the range 2.21-4.09 8, of sorbate diameter us.A time step of 1 fs was found to be adequate for the integration of both sorbate and zeolite atom coordinates. The calculations have been performed at 140 K. The system was equilibrated for 200 ps, and properties were accumulated over 1600 ps at an interval of 0.10 ps. Integration was carried out using the velocity form of the Verlet algorithm first proposed by Swope et a1.22 In a second set of calculations, termed set b, simulations were carried out on argon (us = 3.41 8,) at 340 K, one employing flexible zeolite framework and another employing rigid framework. These calculations have been carried out to elucidate the effect of framework motion on intercage diffusion and related properties. For rigid framework calculations, a time step of 20 fs and the Verlet scheme19 have been employed. This time step has been found to be adequate, yielding good energy conservation. Both set a and b calculations have been carried out at a sorbate concentration of 1 sorbate/a-cage. The method employed by us for the calculation of the rate of cage-to-cage or intercage diffusion (k,) and rate of cage visits (k,) is straightforward. The cage in which a particular sorbate atom is resident at a given time step is found by calculating the distance between all the cage centers and the sorbate. The resident cage is that cage whose sorbate-cage center distance is the smallest. The instant at which the sorbate has diffused to a neighboring cage, that is, the instant at which an intercage diffusion has occurred, is taken to be the time step when the resident cage is different from that of the previous time step. At this instant the perpendicular distance between the sorbate and the plane of the eight-ring window is nearly 0. For the calculation of the rate of intercage diffusion, the number of diffusions from one cage to another is noted during the course of the simulation. The rate of cage visits is obtained from the rate of intercage diffusion by eliminating all the diffusion events from cage i to j which are either followed or preceded by diffusion from cage j to i. More details are discussed in earlier work on zeolite NaY.23
5. Results and Discussions 5.1. Ring Effect or DiffusionAnomaly. Here we report results of calculations carried out under conditions referred to as set a earlier. Recently, we have reported the existence of anomalous diffusion, termed ring effect in zeolites Y and A.IO Below, we describe briefly this effect. Rate of intercage diffusions, k,, rate of cage visits, k, and the diffusion coefficient D when plotted against 1/us2 shows linear behavior for us 3 A) and centralized ($ < 3 A) diffusion modes for Ar in Nab;: continuous line, rigid framework model; dashed line, flexible framework model via surface-mediated route; and dotted line, flexible framework via centralized diffusion mode.
-12
'
-1 4
I
-3
-6
I
1
I
0
3
6
d.A
-10 3
0
E \ 7
i
-12
3
-1 4
-3
-6
0
3
6
d, A
-10
-
-12
-
2
0
E \
i Y
4
Rigid
3
Flexible
T -3LOK
-14 -8
I
I
I
I
-4
0
4
8
d,A Figure 12. (a) Variation of U vs d during intercage diffusion for rigid as well as flexible frameworks. The inset shows a plot of U,,vs time just before, during, and after the crossover in the interval -4 to +4 ps. The timer = 0 corresponds to the time when the sorbate is close to the window plane. The curves are shown for intercage diffusion events via (b) the surface-mediatedroute > 3 A) and (c) the centralized diffusion route
(C < 3 A).
(c
In Figure 12a we show the variation of the total interaction energy U against d averaged over all intercage diffusion events. Also shown in Figure 12b,c are the U vs d curves averaged over all intercage diffusion events taking place via s-m ($ > 3 A) and sd ($< 3 A) modes. The inset of Figure 12a shows thevariation of the sorbate-zeolite interaction energy, (Us,), with time t . It
TABLE 5: Rate of Intercage Diffusions (k,)and Fractional Contribution from Centralized ($ < 3 di) and Surface-Mediated ($ > 3 A) Modes for Rigid as Well as Flexible Zeolite Frameworks at around 340 K fractional contribution from s-m mode k, X lVIO cd mode model (/sorbate/s) (t'33) (t'33) rigid 4.50 0.46 0.54 flexible 5.48 0.49 0.5 1 TABLE 6 Rate of Cage Visits (kv)and Fractional Contribution from Centralized ($< 3 A) and Surface-Mediated ($ > 3 di) Modes for Rigid as Well as Flexible Zeolite Frameworks at around 340 K fractional contribution from kv X 1V'O cd mode s-m mode model (/sorbate/s) (tf"33~) (tf'3A) rigid 2.55 0.52 0.48 flexible 3.00 0.60 0.40 is seen that the magnitude of the barrier for diffusion across the eight-ring window for the overall intercage diffusion as well as for the s-m and cd modes is lowered on going from the rigid to the flexible framework model. The barriers are negative in all cases, in agreement with earlier studies.24 Another change that is observed is the uniform shift of the Uvs d curve toward higher energy for the flexible model in the case of s-m diffusion but not the cd mode (see Figure 12b,c). The shift in the case of Figure 12a is in between the shifts observed for s-m and cd modes, which is reasonable since the overall curve shown in Figure 12a is an average over s-m and cd modes. This result suggests that there is an increase of the potential energy near the inner surface of the cage and the window but not near the cage center when the rigid cage assumption is relaxed. Earlier studies have found that the potential energy surface in zeolite A when the rigid cage model is employed is as depicted in Figure 13.24 The changes in the underlying potential energy surface when the rigid framework approximation is relaxed, suggested by the present study, are depicted in Figure 13. The observed increase of U near the inner surface of the cage in fact explains the increase in the sorbate population near the cage center. These changes lead to significant change in the rate as well as the mechanism of intercage diffusion. Rate of intercage diffusions, k,, and rate of cage visits, k,, and their respective fractional contributions from s-m and cd modes are listed in Tables 5 and 6, respectively. These results suggest that rate of intercage diffusion, k,, and rate of cage visits, k,, increase by about 15-20% on going from rigid to flexible cage. This increase may be attributed largely to the increase in the diffusion rate via the cd mode. Figure 14 shows the evolution of mean square displacement with time. The inset shows a In-ln plot of the same; the transition from ballistic to diffusive motion is clearly evident from this figure.
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994
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Santikary and Yashonath 12
0 -
1800 vl C
“-4
A
2 1200
r.4
5
0
3
15
10
20
t , ps
600
- Rigid ----
Flexible
”
0
50
100
150
200
t.ps
Figure 14. Evolution of the mean squaredisplacement with time for rigid and flexible framework models. The inset shows In-ln plot of the same, showing the transition from ballistic to diffusive motion.
TABLE 7: Values of Diffusion Coefficients of Argon in Zeolite NaA for Rigid and Flexible Zeolite Models at around 340 K model D x lo8 (m*/s) rigid 0.84 0.86 flexible The transition point seems to show no significant shift on going from rigid to flexible framework. As can be seen from this figure, the diffusion coefficient obtained using Einstein’s relationI9 is higher for flexible cage as compared to rigid cage. The D values are listed in Table 7. The trend obtained here on going from rigid cage to flexible cage is also the same as obtained recently by Demontis et aLzl The increase in D is consistent with increase in qs, observed earlier. The increase in D is also consistent with the increase in k , and k,, on going from the rigid to the flexible framework model. As pointed out by us earlier, diffusion of sorbate in zeolite A (as well as Y ) seems to consist of two subprocesses. First, the sorbate has to free itself from the inner surface of the a-cage. The sorbate is then free to move within the a-cage. The second important subprocess consists of the diffusion of the sorbate across the window, that is, the rate of intercage diffusion. The results presented here suggest that the increase in D on going from rigid to flexible cage is likely to be due to the increase in the rate of intercagediffusion. The increase in the rate of intercage diffusion itself appears to be mainly due to the increase in the intercage diffusions via the cd mode. In Figure 15 the velocity autocorrelation function (see inset) and the power spectra obtained by Fourier transformation are depicted. It is seen that the maximum in the power spectra undergoes a shift toward lower frequency. This is accompanied by a decrease in the intensity in the high-frequency regime. This trend is in agreement with the observed changes in the potential energy surface: As noted above, the potential energy associated with the region near the inner surface of the a-cage shows a shift toward higher values. Since the site of adsorption or the overall minimum in the potential energy is located in the vicinity of the inner surface of the a-cage, it is to be expected that there will be a decrease in the sorbate frequency of the sorbates adsorbed in these potential wells, due to a decrease in the well depth and consequently a decrease in the second derivative at the minimum. Therefore, the observed decrease of intensity on the high-frequency side and the accompanying increase in the intensity near the low-frequency side are consistent with the changes observed in other properties.
0
30
60
90
120
t,PS
Figure 15. Power spectra of the sorbate obtained by Fourier transformation of the velocity autocorrelation function for both rigid and flexible cage models. The inset shows the decay of the velocity autocorrelation function with time.
6. Conclusions The results suggest the persistence of the novel ring effect observed earlier in simulations where the framework was treated as rigid. It is found that the peak heights in theintercagediffusion rate and diffusion coefficients are somewhat lower than that observed for the rigid cage model. The barriers for diffusion across the the eight-ring window is negative and is found to decrease with increase in the size of the sorbate. The eight-ring window is found to expand slightly when the sorbate is passing through the window only when the sorbatediameter is comparable to that of the window diameter or, more precisely, when y is nearly 1.O. This suggests that not only are the sorbate properties modified due to the flexibility of the framework but there is also the influence of the sorbate on the properties of the zeolite framework. It is found that the single-particle density distribution inside the a-cage undergoes significant change on going from rigid cage to flexible cage. No significant change is found in the atomatom rdfs between the sorbate-sorbate and sorbate-zeolite. The rate of intercage diffusion and the diffusion coefficient are found to increase on the introduction of flexibility. The increase is partly attributed to the increase in the rate of intercage diffusion via the centralized diffusion mode. The potential energy of the region near the inner surface of the a-cage is found to increase, accounting for the higher heat of sorption for the flexible cage model. In contrast, no such change is observed for the region near the cage center. The presence of ring effect can be of vital importance in many areas of condensed matter physics and chemistry. Preliminary investigations suggest that the anomalous diffusion may be independent of the geometry and topology of the confined region. The results presented here would beuseful in many areas including separation of mixtures using zeolites, diffusion in porous glasses, and diffusion of ions in superionic conductors to name a few. It would be relevant in any situation where diffusion in a confined region is being considered. The work on diffusion of monatomic spherical sorbates in zeolites was started in the Solid State and Structural Chemistry Unit in 1988. The founder of this department is Professor CNR Rao, who has encouraged us and shown a continuing interest in this work. His enthusiasm and interest has nurtured the department to grow. It is our pleasant duty to thank him and contribute to this issue. We would like to dedicate this work in his honor.
Persistence of Diffusion Anomaly in NaA Zeolite
References and Notes (1) Thomas, J. M. Philos. Trans. R. Soc. London 1990, A333, 173. (2) Hensen, N. J.; Chectham, A. K.;Peterson, B. K.;Pickett, S. D.; Thomas, J. M. J. Comput.-Aided Mater. Des. 1993, I , 41. (3) Rowlinson, J. S.;Woods, G. B. Physica A 1990, 164, 765. (4) Heink, W.; Karger, J.; Pfeifer, H.; Datema, P. K.;Nowak, A. K. J . Chem. Soc., Faraday Trans. 1992,88, 3505. ( 5 ) Heink, W.;Karger, J.;Pfeifer,H.;Salverla,P.;Datema,K. P.;Nowak, A. K. J. Chem. SOC.,Faraday Trans. 1992, 88, 5 15. (6) Yashonath, S.J . Phys. Chem. 1991, 95, 5877. (7) June, R. L.; Bell, T. A.; Theodorou, D. N. J . Phys. Chem. 1990,94, 1508. (8) Kono, H.; Takasaka, A. J . Phys. Chem. 1987, 91, 4044. (9) Yashonath, S.;Santikary, P. Mol. Phys. 1993, 78, 1. (10) Yashonath, S.;Santikary, P. J . Chem. Phys. 1994, 100, 4013. (1 1) Yashonath, S.;Santikary, P. J . Phys. Chem. 1994, 98, 6368. (12) Demontis, P.; Suffritti, G.B.; Quartieri, s.; Fois, E. S.;Gamba, A. J . Phys. Chem. 1988, 92, 867. (13) Schrimpf,G.;Schlenkrich,M.; Brickmann,J.; Boff, P. J.Phys. Chem. 1992, 96, 7404.
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9259 (14) Titiloye, J. 0.;Parker, S.C.; Stone, F. S.;Catlow, C. R. A. J . Phys. Chem. 1991.95, 4038. (15) Derouane, E. G. Chem. Phys. Lett. 1987, 142, 200. (16) Derycke, I.; Vigneron, J. P.; Lambin, Ph.; Lucas, A. A.; Derouane, E. G. J . Chem. Phys. 1991, 94,4620. (17) Derouane, E. G.; Andrew, J.-M.; Lucas, A. A. J . Catal. 1988,110, 58. (18) Pluth, J. J.; Smith, J. V. J. Am. Chem. SOC.1980, 102, 4704. (19) Allen, M. P.;Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U. K., 1987. (20) Kiselev, A. V.; Du, P. Q. J. Chem. SOC.,Faraday Trans. 2 1981.77, 1. (21) Demontis, P.; Suffritti, G. B.; Fois, E. S.;Quartieri, S . J . Phys. Chem. 1992, 96, 1482. (22) Swope, W. C.; Anderson, H. C.; Berens, P. H.; Wilson, K. R. J. Chem, phys. 1982, 76, 637. (23) Yashonath, S.;Santikary, P. J . Phys. Chem. 1993, 97, 3849. (24) Yashonath, S.; Santikary, P. J . Phys. Chem. 1993, 97, 13778.
