3849
J . Phys. Chem. 1993,97, 3849-3857
Xenon in Sodium Y Zeolite. 2, Arrhenius Relation, Mechanism, and Barrier Height Distribution for Cage-to-Cage Diffusion? Subramanian Yashonath'.**s and Prakriteswar Santikaryt Solid State & Structural Chemistry Unit and Supercomputer Education & Research Center, Indian Institute of Science, Bangalore 560 01 2, India
Received: March 23, 1992
Various geometrical and energetic distribution functions and other properties connected with the cage-to-cage diffusion of xenon in sodium Y zeolite have been obtained from long molecular dynamics calculations. Analysis of diffusion pathways reveals two interesting mechanisms-surface-mediated and centralized modes for cageto-cage diffusion. The surface-mediated mode of diffusion exhibits a small positive barrier, while the centralized diffusion exhibits a negative barrier for the sorbate to diffuse across the 12-ring window. In both modes, however, the sorbate has to be activated from the adsorption site to enable it to gain mobility. The centralized diffusion additionally requires the sorbate to be free of the influence of the surface of the cage as well. The overall rate for cage-to-cage diffusion shows an Arrhenius temperature dependence with Ea = 3 kJ/mol. It is found that the decay in the dynamical correction factor occurs on a time scale comparable to the cage residence time. The distributions of barrier heights have been calculated. Functions reflecting the distribution of the sorbate-zeolite interaction a t the window and the variations of the distance between the sorbate and the centers of the parent and daughter cages are presented.
1. Introduction
In the last few years, there have been several computer simulation investigations of molecules adsorbed in zeolites. Demontis et al.' have carried out a molecular dynamicscalculation of methane in silicalite and water in Linde zeolite 4A to understand the structural and dynamical properties. Theodorou and cow o r k e r ~have ~ , ~ reported detailed investigations of xenon, methane, and other alkanes in silicalite. Recently, Rowlinson and cow o r k e r ~ ~reported ,~ extensive grand canonical Monte Carlo calculations of xenon in sodium Y zeolite to understand the thermodynamic and structural characteristics. Their results are in reasonable agreement with the molecular dynamics calculations of xenon in sodium Y zeolite obtained by usa6 There have also been other simulation studies in recent times. The calculations of Pickett et al.' of sorption in silicalite are interesting. Titiloye et alas recently investigated the effect of zeolite framework relaxation on the heats of sorption of small- and medium-sized hydrocarbons by energy minimization. The above papers are concerned mainly with the overall equilibrium and dynamical properties of small molecules in zeolites. In the present work, we haveinvestigated the microscopicdetails of xenon in sodium Y zeolite relevant to cage-to-cage migration. The overall adsorption properties of xenon sorbed in sodium Y were reported in part 1.6 Here, properties such as the mechanism, activation energy, and dynamics for xenon diffusing from one cage to another have been obtained from the molecular dynamics method. The next section gives brief details of the structure and intermolecular potential functions. Section 3 introduces the relevant theoretical methods. Details of the molecular dynamics calculations and trajectory analysis are also given in section 4. Results and discussion are presented in section 5.
2. Structure and Intermolecular Potential Functions The structure of sodium Y zeolite is taken from the neutron diffraction study of Fitch et aL9 The space group is Fdjm with a = 24.85 A. A Si/A1 ratio of 3.0 was assumed. For this Si/Al ratio the extraframework sodiums occupy site I and site I1 ' Contribution No. 802 from the Solid State &Structural Chemistry Unit. I
Solid State & Structural Chemistry Unit. Supercomputer Education & Research Center.
0022-3654/93/2091-3849104.00/0
completely.i0 One unit cell of the zeolite Y has the formula Na4&i144A1480~g4.There are eight a-cages or supercages in one unit cell of zeolite Y. The adsorbate chosen in the present study is xenon. Since little is known about the nature of cage-to-cage migration in any of the zeolites, we preferred to select a simple monatomic adsorbate. The short range xenon-zeolite interaction potential parameters are those given by Kiselev and Du." The shortrange guest-host potential is of the 6-12 Lennard-Jones form,
'az
'az
The interaction parameters A and B for the xenon-zeolite Y atoms 0 and Na are the same as those used in part 1 of the work.6 The induction energy interactions are neglected in the present calculations since they are computationally expensive due to the many-body nature of the interactions. The guest-guest interaction was assumed to be of the formi2
3. Theory By cage-to-cage migration we mean the migration of an adsorbed atom (xenon) from one supercageto another neighboring supercage through the 12-ring window. Below we refer to a supercage or an a-cage simply as a cage. Two cages are connected through a 12-ring (Figure 1a). The 12-ring is made up of 12 0 and 12 Si/Al atoms. Of the 12 oxygens, 6 are in one plane, which we define as the plane of the window, and three other oxygens lie on one side of this plane toward one cage while the remaining three oxygens lie on the other side of the plane closer to the center of the other cage. Each cage is connected to four other cages in a tetrahedral manner. Each of these four cages are similarly connected to four cages in a tetrahedral fashion and so on, leading to a three-dimensional networkof interconnectedcages(see Figure lb). It is known that xenon can only enter the supercages that have a radius of 5.9 A. The sodalite or the &cages are too small compared to the size of xenon (-2.2-A radius). The 12-ring interconnecting two a-cages has a radius of 4 A. Hence, xenon can easily pass from one a-cage to another. 0 1993 American Chemical Society
3850 The Journal of Physical Chemistry, Vol. 97,No. 15, 1993
Yashonath and Santikary ZwanzigI6 has recently discussed the various aspects of this in detail. There are many processes in nature with a rate constant k(B) where k(B) is a function of some control variable, B, which may fluctuate according to the Langevin equation or assume discrete values. The most common control variable is the barrier height. In a situation where the barrier height takes on more than one value and does so continuously the average time dependence of the concentration of the ?eactant” is given by
where
k(B) = k, exp(-B/RT) (5) is the well-known Arrhenius form for the rate constant. Here AB) is the distribution of B. If B is a random function of time, i.e., B(t) then the process is said to display dynamical disorder, otherwise it is referred to as static disorder. For example, Austin et al.17 found a continuous distribution of barrier height with dynamic disorder for ligand binding to myoglobin. In the present problem the barrier height, Ua(S), is a function of the path or trajectory s of the diffusing particle. In a given molecular dynamics run, let us suppose that there are m cageto-cage diffusion events yielding a set of trajectories indicated by the set (si) over which to calculate the average properties, then the average barrier height is
where
u,‘(s’=)V(si;c)- u‘(si;w)
Figure 1. (a) Two neighboring cages of sodium Y zeolite and 12-ring window interconnecting the cages. (b) Topology of the eight cages of zeolite Y through which the xenon diffuses. Each cage is connected to four other cages tetrahedrally.
In the present study, there are in all eight cage centers, which is the number of cages in one unit cell of zeolite Y. The minimum distance, ccmin, is the minimum value of the eight distances between the xenon and the different cage centers, c,. Here i refers to the particle and c to the cage. It was found that ccmin was always less than 5.9 A showing that the xenon particle cannot enter the sodalite cage. The various properties such as the distribution of cage residence times and other properties discussed below are based on the above criteria of assigning a particle to a given cage. At all temperatures, the properties presented in section 5 have been obtained by averaging over the corresponding number of trajectories or cage-to-cage crossovers. Even though this number is smaller at lower temperatures, the error is not large due to the smaller fluctuations in the average properties. A brief review of the terms used by us in section 5 is helpful as there is some ambiguity in the literature about the exact meaning of some of the terms. The Arrhenius activation energy Ea is a phenomenological quantity, and it is defined in terms of the slope of an Arrhenius plot of the rate constant, k(7) versus 1/T, where T is the temperatureI3
i = 1, m
(7)
Here, U(si;c) and U(sf;w) are the potential energies for the ith trajectory, si, at the cage and the window, respectively. &‘is the barrier height for the s’th trajectory. We note that there are, in fact, an infinite number of trajectories leading from one cage to another which may be indicated by the set (Si).The set of actually traversed trajectories for a given set of external conditions such as temperature, etc., is given by (si), which is a subset of (Si).The distribution of barrier heights, AUa) may be directly obtained from thevarious U,i values. We report the distribution of barrier heights for cage-to-cage migration at different temperatures in section 5. Chandler, Berne, and co-workers have studied trans-gauche isomerization dynamics of n-butane and other higher alkanes such as n-pentane and n-decane by stochastic and molecular dynamics methods.18J9 They have devised a method to calculate the rate constants for the isomerization in these systems. The method has been extended by Voter and Do1120 to include multistate transitions and also to study rare events. In this method, the fact that every crossing of the potential barrier-between the reactant and the product-need not necessarily lead to a successful completion of the reaction is taken into account. The rate, kj for exiting from the ith cage has to be corrected as the recrossed sorbates do not contribute to the successful completion of reaction. More precisely, the rate, kj is related to the rate constant for the transition from state or cage i t o j
where
Ea = -R d In k ( T ) / d ( l / T )
(3) The Arrhenius activation energy is often interpreted approximately as the energetic threshold or barrier height, Ua, for the reaction, but as has been pointed out by Truhlar13and others,14J5 it may be either larger or smaller than Ua.
= [ ( 2 ~ m / k T ) ” 2 ( u i ( o ) e , ()It ) (9) is the dynamic correction factor. U j ( 0 ) is the velocity of the sorbate particle at the window between the two cages at time 0. The time when the particle is located at the dividing surface (12-ring fij(r)
Xenon in Sodium Y Zeolite window) is taken as zero. The dividing surface is the surface separating the reactant from the product and is common to both. The col is located on the dividing surface. In the present problem the plane of the 12-ring window may be conveniently taken as the dividing surface. q t ) is a switching function which takes on a value of + 1 when the particle is in thejth cage and zero otherwise. From now onward, for purposes of clarity, we will refer to the initial cage as the parent cage (the ith cage above) and the destination cage (the jth cage) as the daughter cage. k,,refers to the rate for the particular transition from the ith to thejth cage. k, is the rate for exiting from the ith cage. The brackets (( )) indicate that averaging is to be carried out over the phase space distribution given by Ui(d=O;r=O)exp(-H/RT) where H is the Hamiltonian of the system. The function V'(d=O;r=O) simply indicates that averaging is to be carried out for all sorbate particles which are located on the dividing surface at r = 0. For more details the reader is referred to the references mentioned above. Typically, the dynamic correction factor, fdl,(t), decays during a time r,,,, that soon reaches a plateau. The above treatment is valid under the condition that rcorr 3 ~
rp
285 K L79K
60
O
‘p -=3A
°
F
100
oo
____
285K
L-
-56
00
56
d,A
Figure 9. Variation of the distance between the diffusing particle and the window center, r , , * , with d, the distance between the particle and the window plane through which the particle is assing during the cage-to>3 cage crossover, for rpm< 3 A and rpnl
1.
and centralized cage-to-cage diffusion. These often show contrasting characteristics. In Figure 9 the variation of rawwith d is shown. The second derivativeor thecurvaturedecreases slightly with temperature for both the modes. For a given temperature the curvature is marginally smaller for the barrierless mode of migration than that for the surface-mediated migration. The behavior of rpand r d during the cage-to-cage diffusion is shown in Figure 10. It is seen that rpand rd are significantly closer to the cage center for rpm< 3 A as is to be expected. The change in the nature of the rp(t) - rd(r) curve with temperature is more significant for the surface-mediated migration (Figure 1Ob) than for the barrierless mode of migration (Figure loa). The distributionflr,,) a t d = 0 is shown in Figure 11 for the two modes of migration. The width of the distribution is higher for rpm< 3 A. More particles pass through the region near the center and the outer extreme of the window for the barrierless mode in comparison with the surface-mediated mode. Increase in temperature results in similar changes in both the modes, viz., a small increase in the intensity for extreme values of rdW. The residence time of xenon in the cage is an important property that is essential in modeling the diffusion of xenon in zeolites. We show the distribution of cage residence times at different temperatures in Figure 12. At temperatures below 300 K, the distribution peaks near 7 ps. At higher temperatures, the position of the peak shifts toward lower values. It is seen that at all temperatures investigated in this work, residence times as high as 80 ps are observed. The average cage residence times at different temperatures are listed in Table 111. As can be seen from Figure 12, it is likely that the distribution has not decayed completely and hence the average residence times listed in Table 111 may present a lower bound for the actual values. Average cage residence times at room temperature are about 17 ps, which is significantly larger than the site residence times that is of the order of 1 ps.6 Several workers have obtained the cage residence times from neutron scattering data and nuclear magnetic relaxation measurements, for methane in zeolite A and X.27.28 Neutron and IR investigations of Cohen de Lara and co-workers3’ as well as Stockmeyer2’ suggest cage residence times to be
u
20 20
60
‘p,
A
100
Figure 10. Behavior of r p- r d during cage-to-cage migration at different temperatures for (a) the centralized mode, rpm< 3 A, and (b) the surfacemediated mode, rpm> 3 A.
significantly larger than a few picoseconds. Freude28reported N M R results suggesting an estimate of 500 cts at 300 K for methane in Na-A. We attribute thediscrepancy to possible grain boundary effects and crystal defects in the samples. Activation energies for diffusion in the presence of grain boundaries would be large resulting in a higher value for the residence times. In contrast, the simulation is for a single crystal without grain boundaries or defects. 6. Conclusions
We state a few points pertinent to the above discussion. The 3-A value for rpmused to demarcate the centralized or barrierless mode of diffusion from the surface-mediated diffusion was chosen since it is one-half the radius of the cage. A more appropriate choice, however, would have to be based on the underlying potential energy surface. It was pointed out by us in section 5.3, during the discussion ofthedistribution of barrier heights, that there was some similarity between our distribution and that reported by Frauenfelder and c o - ~ o r k e r s . ’Here ~ we would like to note that while the present results are based on a rigid zeolite framework and therefore correspond to a situation of static disorder, the work of Frauenfelder and co-workers actually corresponds to what is commonly referred to as dynamic disorder, E = E ( t ) . If the framework atoms were included in the molecular dynamics integration then the zeolite atom coordinates would be time dependent and hence would result in dynamic disorder of the barrier heights. The activation energy obtained from the Arrhenius plot of the rate of cage-to-cage diffusion is 3 kJ/mol. The calculation of the dynamic correction factor has shown that rcorr values associated with the time for recrossings are comparable to the cage residence times, T~ at all temperatures. This is not surprising since the activation energy for cage-to-cage diffusion is of the order of kT. The process of cage-to-cage diffusion may be said to comprise
3856 The Journal of Physical Chemistry, Vol. 97, No. IS, 1993
Yashonath and Santikary TABLE I V Rate of Cage-to-Cage Diffusion at Different 3 A and rpm k, x IO I o (s ' ) T (K) (rpm > 3 A) (rpm c 3 A) 188 285 386 414 479
row,
Figure 11. Distribution functionf(r:,,),
for the particle-window center distance a t d = Ocorresponding to the particle in the plane of the window. The curves are shown for (a) the centralized mode, rpm< 3 A and (b) the surface-mediated cage-to-cage mode, rpm> 3 A. 012
I
I" I(
- 188K _ _ _ _ 285K - - 479K
I
1
2.37 3.74 4.26 4.87 4.40
0.52 I .85 3.47 3.83 4.66
mode, the first subprocess results in the activation of the sorbate from the sorption site. The particle is mobile after activation and is capable of diffusing to a neighboring cage. The barrier height encountered by this sorbate while passing through the 12-ring window is positive but small. In the centralized diffusion mode the sorbate is first activated from the sorption site to the region near the center of the resident cage. Thus, in centralized diffusion the particle is not only free from the adsorption site but is also free of the immediate influence 3f the inner surface of the cage. The second subprocess of diffusion across the 12-ring window now is found to be facile since the barrier height for this subprocess is negative. It must be noted that even though a sorbate approaches the window from the central region of the cage in the centralized mechanism, it often is in close proximity of the surface in the daughter cage. In fact (see Figure 6a) this is the rule rather than the exception especially at lower temperatures. Hence, it is not possible to speak of the two modes of diffusion as completely separate. Therefore, one has to view Figure 6c as an idealization of cage-to-cage diffusion process. The predominant mechanism for cage-to-cage diffusion at low temperatures is the surface-mediated route since the sorbates do not have sufficient kinetic energy to excite to the center of the cage. At higher temperatures, the centralized mode becomes increasingly preferred since there already exist a significant number of activated sorbate species near the cage center and since the barrier is negative for diffusion across the 12-ring window (see Table IV). Preliminary investigations on cage-to-cage diffusion in zeolite A suggest that the overall activation energy itself may be negative.jO Acknowledgment. Theauthors wish to thankoneof the referees for useful comments which resulted in estimating the dynamical correction factors. We also thank Dr. J.-P.Ryckeart and Professors N. Sathyamurthy and G. Ananthakrishna for helpful discussions. We also thank Professor C. N. R. Rao for kind encouragement. References and Notes
Tcv
PS
Figure 12. Distribution of cage residence times, f(r,), a t three different temperatures.
TABLE 111: Average Cage Residence Times for Xenon in Zeolite Y at Different Temwratures ~
188 285 386
16.4 14.9 12.5
414 479
~~~
~-
11.6 9.3
twosubprocesses. First, the sorbate has to be activated from the sorption site. Without this activation the sorbate is incapable of motion even within the resident cagee6.I0The second subprocess consists of diffusion of the particle across the 12-ring window. Approximately, cage-to-cage diffusion may be said to occur via two different mechanisms-the surface-mediated and the centralized or barrierless modes. In the surface-mediated
( I ) Demontis, P.; Suffriti, G . 8.; Quartieri, S.;Fois, E. S.; Gamba, A. Zeolites 1987, 7. 552; J. Phys. Chem. 1988, 92, 867. (2) June, R. L.; Bell, T. A.; Theodorou, D. N. J . Phys. Chem. 1990,94, 1508. (3) June, R. L.; Bell, T. A.; Theodorou, D. N. J . Phys. Chem. 1990,94, 8232. (4) Woods, G. B.; Panagiotopoulos, A. Z.; Rowlinson, J. S . Mol. Phys. 1988, 63, 49. (5) Woods, G. B.; Rowlinson, J. S. J . Chem.Soc., Faraday Trans 2 1989, 85, 765. (6) Santikary, P.; Yashonath, S.; Ananthakrishna, G. J. Phys. Chem. 1992, 96, 10469. (7) 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. (8) Titiloye, J. 0.;Parker, S . C.; Stone, F. S.; Catlow, C. R. A. J . Phys. Chem. 1991, 95, 4038. (9) Fitch, A. N.; Jobic. H.; Renouprez, A. J . Phys. Chem. 1986, 90, 1311. (IO) Yashonath,S.;Thomas,J. M.;Nowak,A. K.;Cheetham,A. K. Nature 1988, 331. 601. ( I 1 ) Kiselev, A. V.; Du, P. Q.J . Chem. Soc., Faraday Trans 2 1981, 77, 1.
( I 2) Hirschfelder, 0. J.; Curtiss, F. C.; Bird. B. R. Molecular Theory of Gases and Liquids; John Wiley: Chichester. 1954.
Xenon in Sodium Y Zeolite Truhlar, D. E. J . Chem. Educ. 1978, 55. 309. Sathyamurthy. N . Indian J . Chem. Educ. 1979, 6 , I . Redpath, A. E.; Menzinger, M. Can. J . Chem. 1971, 49, 3063. Zwanzig. R. Acc. Chem. Res. 1990, 23, 148. Austin, R. H.; Beeson, K. W.;Eisenstein, L.; Frauenfelder, H.; Gunsalas, I. C. Biochemistry 1975, 14, 5355. (18) Chandler, D.J . Chem. Phys. 1978, 68, 2959. Berne, B. J . J . Chem. Phys. 1979, (19) Montgomery, J. A.;Chandler, D.; 70, 4056. (20) Voter, A. F.; Doll, J. D. J . Chem. Phys. 1985, 82, 80. (21) June, L. R.; Bell, A. T.; Theodorou, D.N. J. Phys. Chem. 1991,95, (13) (14) (15) (16) (17)
8866.
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3857 (22) Menon. A,; Sathyamurthy, N. J. Phys. Chem. 1981,85, 1021. (23) Laidler. K. J. Chemical Kinetics, 3rd ed.; Harper and Row: New York, 1987. (24) Clary, D. C., Ed. The Theory of Chemical Reaction Dynamics; Dordrecht: Reidel, 1986. (25) Clary, D.C. Annu. Reo. Phys. Chem. 1990.41, 61. (26) Yashonath, S.J. Phys. Chem. 1991, 95, 5877. (27) Stockmeyer. R. Zeolites 1984, 4 , 81. (28) Freude, D. Zeolites 1986, 6, 12. (29) Cohen de Lara, E.; Kahn, R.; Mezei, F. J . Chem. Soc., Faraday Trans. I 1983, 79, 1911. (30) Yashonath, S.;Santikary, P. J . Phys. Chem., submitted.
