4286
J. Phys. Chem. 1995, 99, 4286-4292
Diffusion Anomaly in Silicalite and VPI-5 from Molecular Dynamics Simulations’ Sanjoy Bandyopadhyay$and Subramanian Yashonath*g**s Solid State and Structural Chemistry Unit and Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore-560012, India Received: September 19, 1994; In Final Form: December 12, 1994@
Molecular dynamics (MD) simulations on rigid and flexible framework models of silicalite and a rigid framework model of the aluminophosphate VPI-5 for different sorbate diameters are reported. The sorbatehost interactions are modeled in terms of simple atom-atom Lennard-Jones interactions. The results suggest that the diffusion coefficient exhibits an anomaly as y approaches unity. The MD results c o n f i i the existence of a linear regime for sorbate diameters significantly smaller than the channel diameter and an anomalous regime observed for sorbate diameters comparable to the channel diameter. The power spectra obtained by Fourier transformation of the velocity autocorrelation function indicate that there is an increase in the intensity of the low-frequency component for the velocity component parallel to the direction of motion for the sorbate diameter in the anomalous regime. The present results suggest that the diffusion anomaly is observed irrespective of (1) the geometry and topology of the pore structure and (2) the nature of the host material. The results are compared with the work of Derouane and co-workers, who have suggested the existence of “floating molecules” on the basis of earher theoretical and computational approaches.
1. Introduction Recent investigations in confined regions have suggested that the sorbates exhibit high diffusivities or superdiffusivities under certain conditions.’ Molecular dynamics calculations by us of spherical sorbates in zeolites Y and A as a function of the sorbate size indicated that high diffusivities occur when the sorbate size is close to the dimensions of the narrowest part of the cage, which is the 12-membered window in the case of zeolite Y or eight-membered window in the case of zeolite A. It was shown by us that the peaks in the rate of intercage diffusion, k,, rate of cage visits, k,, and diffusion coefficient, D , are observed when y , defined as 2.2%,,
Y =-
ow
approaches unity.2 Here, a,, is the sorbate-zeolite LennardJones interaction parameter and a, is the window diameter taken to be the distance between the centers of the two oxygen atoms. In the past few years, Derouane and c o - ~ o r k e r s ~have - ~ reported valuable results relating to the effect of surface curvature. In their thorough analysis they look at, among other things, the dependence of energy and force on the sorbate as a function of sorbate size. Even though such an analysis in which only the sorbate diameter is changed without changing the mass and other parameters relating to the sorbate may be considered artificial, it is still worthwhile to look into their results. In particular they have found that the concave surface provides the most favorable adsorption sites for sorbates in the zeolitic cages. They also find among other things that the force on the sorbate is nearly zero when the sorbate is comparable to the size of the void. They further suggest that such an effect should be seen in any type of confined region, irrespective of its geometry, etc. So far, “floating molecules” or “levitation effect” or superdiffusivity t Contribution No. 1054 from the Solid State and Structural Chemistry Unit. Solid State and Structural Chemistry Unit. Supercomputer Education and Research Centre. Abstract published in Advance ACS Absrracfs, February 15, 1995. @
has been observed only in zeolites Y and A. The present work reports MD simulations on a zeolite and an aluminophosphate which have a porous structure completely different from those of zeolites Y and A: silicalite and VPI-5 with a view to examine this crucial question. The sorbate diameter alone is varied, keeping the mass and other parameters related to the sorbate constant. Even though this might be considered artificial, this would be necessary firstly to examine the issue under consideration and to compare with the results of Derouane and coworkers, who also have carried out similar analysis. It should be noted that few techniques have the capability to allow for such variations. Molecular dynamics is one such.
2. Host Structure Two different host materials with cavity structures completely different from each other and from those of zeolites Y and A have been chosen in the present study. As discussed in the previous section, this is important since our aim here is to verify the existence of diffusion anomaly in host materials which provide cavities or void structures completely different from those on which the earlier calculations have been reported. The structure of silicalite obtained from the single-crystal X-ray diffractin study of Price et aL7 has been adopted in this study. Silicalite crystallizes in the orthohombic space group Pnma, where a = 20.07 A, b = 19.92 A, and c = 13.42 A. The framework structure of silicalite comprises straight and sinusoidal channel systems consisting of 10-membered rings. Straight channels are slightly elliptical with a cross section of approximately 5.7 x 5.2 A* and are parallel to the [OlO] direction (see Figure la), whereas the zigzag sinusoidal channels with a nearly circular cross section of 5.4 8, run along the [loo] direction. Cavities of about 9 8, in diameter are formed at the intersections of the two channel system^.^ Note that the dimensions as well as the geometry and topology of these channel systems are entirely different from those of zeolite Y and A (see Figure IC). The structure of VPI-5 obtained from the X-ray powder diffraction study of Crowder et a1.8 has been adopted in this study. VPI-5 belongs to the hexagonal space group P63cm with
0022-365419512099-4286$09.00/0 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 12, I995 4287
Diffusion Anomaly in Silicalite and V P I J
Wmol or 181.6 K. The interactions among the sorbates were also of the Lennard-Jones form. The self-interaction parameter css for the sorbate was 1.837 kJ/mol (221 K). The sorbate diameter a,, was varied between 1.5 and 4.5 8, for silicalite, whereas for VPI-5, it was varied between 2.8 and 6.4 8,. The cross-interaction parameters between the sorbates and the oxygens were obtained from the combination rules.12 For modeling the framework, the harmonic potential model proposed by Demontis et al.13 has been adopted here. Interactions of Si-0 as well as 0-(Si)-0 have been taken into account using the potential form
Here, k is the force constant and ro is the equilibrium distance. The values for k were derived from spectroscopic data while ro values were deduced from structural data. The values of the force constants k and e uilibrium distances ro are 2.0921 x lo3 kJ/mol.8,-2 and 1.605 for Si-0 and 4.309 x lo2kJ/mol.8,-2 and 2.618 8, for O-(Si)-0.13 Only the first neighbors are taken into account in obtaining the force and the interaction energy.
R
4. Computational Details
(d)
(e)
Figure 1. (a) Schematic diagram illustrating the channel structure in silicalite. (b) Cross section of the straight and sinusoidal channels. (c) Supercage in zeolite Y surrounded by sodalite cages. (d, e) Crosssectional views of the channel structure in VPI-5. Note the elliptical nature of the channel cross section. The minor and the major axes of the ellipse are respectively 13.2 and 21 A.
a = 19.0 8, and c = 8.11 8,. It is an aluminophosphate-based molecular sieve containing 18 AlP04 formula units per unit cell with an alternating arrangement of AlP04 and PO4 tetrahedra. The channel system comprises of straight channels parallel to the c axis (see Figure Id). This channel has an 18-membered ring pore opening?JO and the free diameter of the channel is about 13 8,. The channel system in VPI-5 is different from that of silicate in that sorbate placed in one channel is unable to enter another channel due to the absence of intersections. These channels also have a much larger dimension and are more elliptical than the channels in silicalite.
3. Potential Functions The sorbate-zeolite interactions were modeled in terms of short-range interactions using atom-atom pair potentials:
where s and z stand for sorbate and zeolite atoms. The dispersion (A,,) and repulsion (Bs,) constants are related to the well depth (E,,) and the diameter (asz)of the Lennard-Jones potential: A,, = 4 ~ , ~ a ,and , ~ B,, = 4 ~ , , a , , ~ ~Interactions . between the framework atoms Si or A1 or P and the sorbate atoms were neglected since the close approach of the sorbates to these atoms is prevented by the surrounding oxygens. Hence, only the interactions between the sorbate and the oxygen atoms of the host framework were considered. The self-interaction parameters for the zeolite atoms were taken from the well-known work of Kiselev and Du:" ao-o = 2.529 8, and 60-0 = 1.51
All simulations have been carried out in the microcanonical ensemble (N, V, E) with cubic periodic boundary conditions. Calculations on silicalite were carried out on 32 unit cells (4 x 4 x 2) consisting of 6144 0 atoms and eight unit cells (2 x 2 x 2) consisting of 1536 0 atoms for rigid framework (rf)and flexible framework (ff) simulations, respectively. Calculations on VPI-5 were carried out on 72 unit cells (6 x 6 x 2) which consist of 5184 0 atoms. In the case of the rigid silicalite model, there were 26 sorbates corresponding to a sorbate concentration of 0.81 per unit cell and approximately one sorbate per channel, whereas the calculations on flexible framework model of silicalite were performed with 16 sorbates at a concentration of 2.0 sorbate atoms per unit cell and one sorbate per channel. Calculations on VPI-5 were performed on 35 sorbates corresponding to an approximate concentration of one sorbate per channel. The mass of the sorbate atom was assumed to be 40 amu. Integration was carried out using simple Verlet14 and modified VerleP algorithms for rigid framework and flexible framework simulations, respectively. For rigid framework simulations time steps of 5 and 10 fs were found to be adequate for simulations in VPI-5 and silicalite, respectively. A time step of 1 fs was used in the case of the flexible framework model. The temperature of the runs for the rigid framework model of silicalite was about 310 K, and that of the runs for the flexible framework model calculations was close to 300 K. For VPI-5, the temperature of the runs was around 620 K. A spherical cutoff of 12 8, for both sorbate-sorbate and sorbatezeolite interactions was employed. Equilibration was performed over a duration of 100 ps for the rigid framework model of both silicalite and VPI-5. This was followed by runs of 400 ps duration, during which the properties reported here have been calculated. An equilibration period of 150 ps was allowed in case of the flexible model of silicalite, which was followed by a production run of 550 ps.
5. Results and Discussion 1. Rigid and Flexible Silicalite. Figure 2 shows the evolution of the mean squared displacement curves for several values of a,,. By fitting straight lines to the points in the 5-25 ps region to the mean squared displacement curves shown in Figure 2, we have calculated the diffusion coefficient D using the well-known Einstein's relation.12 Figure 3a,b shows a plot of the diffusion coefficient D against the square of the reciprocal
4288 J. Phys. Chem., Vol. 99, No. 12, 1995
Bandyopadhyay and Yashonath 1.5
I
Silicalite ~~
1.2
-04
n A-
&
0 9)
N
3
0.9
NE a
9 X
0.6
n 0
5
15
10
20
d
25 0.3
l !
0
0 ’I
I
0
5
I
I
I
1
10
15
20
25
0.0
0.1
0.0
0.1
0.2
0.3
0.4
0.5
t, ps Figure 2. Time evolution of mean-squared displacement for the (a) rigid framework and (b) flexible framework of silicalite for a range of sorbate diameters us,. sorbate diameter, l/a,,*, for rigid (rf) and flexible framework (ff) simulations of silicalite. For small a,,, the diffusion coefficient varies linearly with l/a,?. Further increase in a,, results in an increase in the diffusion coefficient instead of a decrease. This is the diffusion anomaly which occurs as y approaches 1 leading to a high diffusion coefficient or superdiffusivity. In Figure 4, the diffusion coefficient is plotted against y . These results suggest that the diffusion anomaly is observed in silicalite at around the same value of y as is observed in zeolites Y and A. Note that the diffusion anomaly is more pronounced in rigid framework simulations. This suggests that the diffusion anomaly does not arise from a strong coupling of the sorbate motion with the framework motion. In fact, thermal motion of the framework decreases the intensity of the peak observed as y approaches unity. The peak is observed when y 0.95. In the cases of zeolites Y and A also, the peak was observed near y 0.95. Table 1 lists the values of D for various values of a,, and y along with the sorbate-zeolite potential energy and average temperature. Figure 5 shows the power spectra for four different values of a,,, two from the linear regime and two from the region exhibiting anomalous diffusion. The spectra for both rigid and flexible framework simulations are shown. It is observed that there is an increase of the intensity of the low-frequency motion associated with the sorbate for diameters in the anomalous regime. In particular, the intensity is higher for a,, = 2.25 and 3.00 8, below 20 cm-’ for rigid framework simulations. For flexible framework simulations, the intensity is higher for a,, = 2.15 8, below 10 cm-’. Figure 6a,b shows the power spectra obtained from the Fourier transform of the x , y . and z components of the velocity-velocity autocorrelation function for a,, = 1.7 and 2.15 A, respectively. These results have been obtained from flexible framework simulations. Note that there is an increase in the intensity of the power spectra associated
-
-
I
0.3
I
I
0.2
0.3
1
I
0.4
0.5
/d
Figure 3. Plot of the diffusion coefficient D against l/u,,Z for the (a) rigid framework and (b) flexible framework of silicalite. In the rigid framework model for sorbate diameters uss< 1.75 A, D varies linearly with l/u,,2. For us,> 1.75 A, a maximum in the value of D is observed at a, = 2.25 A. In the flexible framework model, the linear dependence of D on l/u,,Z is observed for a,, < 1.7 A. For a,, > 1.7 A, a maximum in the value of D is observed at a,, = 2.15 A. with the x and y components below about 5 cm-’ on going from the linear regime to the anomalous regime. The channels in silicalite run along x and y directions. Thus, there is an increase in the low-frequency component along the direction of motion. 2. VPI-5. The mean squared displacement for different values of a,, is shown in Figure 7. The diffusion coefficients obtained from the well-known Einstein’s relation and from the slopes of the curves shown in Figure 7 are plotted against the reciprocal of the square of the sorbate diameter l/a,? and y in Figure 8, parts a and b, respectively, for VPI-5. The figure clearly shows the existence of a diffusion anomaly for y 1. Due to the elliptical nature of the channel cross section, the peak occurs at a value of y lower than 0.95. The channels in V P I J run along the c axis. The channels deviate from the perfectly cylindrical shape; the cross secton perpendicular to the long axis reveals the noncircular, elliptical shape of the channel (see Figure 1). For the purpose of calculating y , we have used a value of 13.2 8,for a, which is the minor diameter
-
Diffusion Anomaly in Silicalite and V P I J
J. Phys. Chem., Vol. 99, No. 12, 1995 4289
TABLE 1: Diffusion Coefficients for Different Sorbate Diameters ( u ~Obtained ) from Molecular Dynamics Simulation for Both Rigid and Flexible Framework Models of Silicalite
1.50 1.55 1.60 1.65 1.70 1.75 1.85 2.00 2.15 2.25 2.50 3.00 3.50 4.00 4.20 4.50
0.822 0.832 0.842 0.853 0.863 0.873 0.894 0.924 0.955 0.975 1.026 1.128 1.230 1.332 1.373 1.435 1.5
328 324
-8.45 -8.22
1.221 0.986
307 301 296 303 307 311 310 304 305 322 319 306 298
-9.53 -9.36 - 10.25 -10.58 - 10.66 -9.97 -10.15 -9.73 -13.28 -18.18 -25.01 -28.60 -30.60
0.771 0.830 0.63 1 0.838 0.956 1.107 1.269 1.044 0.787 0.549 0.449 0.239 0.109
Silicalite
(4 I
rf
1.2
Q)
0.9
E
03
P X
0.6
n
'9 0.3
0
0.8
1
1.4
1.2
1.6
Y 1.5
8
1.1
"?, N
E
co
0, X
0
0.7
D
0.3
0.75
1.25
1
1.50
Y Figure 4. Plot of the diffusion coefficient D against y for (a) rigid framework and (b) flexible framework models of silicalite. In both cases, the peak in D occurs as y 1, which seems to be valid irrespective of the geometry and topology of the confined region.
-
of the ellipse. Table 2 lists the values of temperature T, the sorbate-zeolite potential energy (Ua), and the diffusion coefficient D for various values of us,. Note that the diffusion
-7.37 -7.83 -7.35 -8.29
1.276 1.219 1.184 1.031
301 296 303 300 300
-8.94 -9.16 -9.77 -9.81 -13.33
1.118 1.191 1.120 1.043 0.875
299 299
-25.11 -28.27
0.586 0.489
TABLE 2: Diffusion Coefficients for Different Sorbate Diameters (us)Obtained from Molecular Dynamics Simulation for Host VPI-5 2.80 3.00 3.15 3.30 3.45 3.60 4.00 5.00 5.50 6.00 6.20 6.40
0
"?, hl
298 302 297 301
0.458 0.475 0.487 0.500 0.513 0.526 0.560 0.645 0.687 0.730 0.747 0.764
624 607 615 635 629 622 634 638 610 615 60 1 618
-7.36 -8.46 -9.50 -9.65 -10.45 -10.84 -12.95 -20.02 -24.10 -28.66 -30.75 -32.45
4.096 3.173 3.276 3.100 2.508 2.343 2.698 3.426 3.635 5.643 4.815 4.676
coefficient shows an increase in spite of the fact that the physisorption energy, (U,), is decreasing. In the case of silicalite, the physisorption energy at 2.25 A, the value of ass corresponding to the peak in the diffusion coefficient (see Figure 3), was about the same as the value of the physisorption energy for a,,in the linear region. Figure 9 shows the power spectra for two different values of a,,, one from the linear regime (as,= 3.6 A) and one from the region exhibiting anomalous diffusion (ass = 6.0 A). The power spectra corresponding to the x , y, and z components of the velocity autocorrelation function are shown in Figure 10, parts a and b, respectively, for the linear (ass= 3.6 A) and anomalous regimes (ass = 6.0 A). Note that here the channels are parallel to the z axis. For the value of Y , % 8 cm-', there is a significant increase in the intensity of the z component power spectra for values of a,, in the anomalous region as compared to the linear region. This observation is consistent with the results obtained by us in the previous section on silicalite for rigid and flexible framework simulations where it was found that the intensity was higher at low frequencies along x and y directions-the directions along which the channels are found in silicalite. For diffusion in zeolites Y and A which have large a cages of nearly 11.8 in diameter with narrower windows of 8 and 4.5 8, in diameter separating these cages, no such behavior is expected. The diffusion anomaly in these zeolites was observed when the sorbate diameter approached the value of the window diameter.* An explanation to the diffusion anomaly is proposed here which relates the force on the sorbate. When the sorbate is significantly smaller than the channel or cage diameter, as the case may be, the sorbate is strongly attracted by the surface nearest to it. This binds the particle strongly to one of the surfaces, leading to a significantly low mobility of the sorbate. With increase in the sorbate diameter, there arises a situation in which
Bandyopadhyay and Yashonath
4290 J. Phys. Chem., Vol. 99, No. 12, 1995 4
us: -1.658
...... 1.758
---- 2.258
5 Silicalite
-3 1OK
rf
ass: 1.708
a,
[a1
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C a,
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0 0
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20
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-1
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0 0 1 ‘
0
I
I
10
20
u,cm
20
10
v,cm
I
-1
30
40
Figure 5. Plot of the power spectra for four different values of uss. Two of these (1.65 and 1.75 A) lie in the linear regime and the remaining two (2.25 and 3.0 A) lie in the region exhibiting anomalous diffusion for the rigid framework model (a). For the flexible framework model of silicalite, two values of uss(1.60 and 1.70 A) from the linear regime and one (2.15 A) from the anomalous region have been chosen (b). the force on the sorbate from the surface on one side is exactly balanced by the force on the sorbate from the opposite side. This happens when y is slightly lower than unity. A consequence of this is that the force is a minimum near the value when D is a maximum. Figure 11 shows the variation with y of the y component of the force due to silicalite. It is seen that the force indeed is lower near the anomalous region as compared to the force in the linear region. In WI-5, the diffusion coefficient along the z direction has been obtained from the slope of the z component of the mean squared displacement in W I - 5 (see Figure 12), which is plotted in Figure 8c. The z component of the force on the sorbate is also plotted as a function of y . It is again found that the force is lower in the anomalous region as compared to the linear region. It is worthwhile to note that Derouane and co-workers have suggested that the force goes to zero as the sorbate diameter becomes comparable to the void dia~neter.~.’~ They termed this as “floating molecules”. We
-1
30
40
Figure 6. Plot of the x , y , and z components of the power spectra for the flexible framework model of silicalite for sorbate diameter from the (a linear region (ass = 1.70 A) and (b) anomalous region (uss= 2.15 ). Note the increase in intensity at low frequencies along the direction of motion.
1
600 VPI-5 Us:
N
400
-
200
-
/’
o 3.OH
04:
= A-
N
a
0 0
5
15
10
20
25
t, ps Figure 7. Time evolution of the mean-squared displacement for sorbates in VPI-5 for different values of uss.
J. Phys. Chem., Vol. 99, No. 12, I995 4291
Diffusion Anomaly in Silicalite and VPI-5 c
VPI-5
-3.68
uss:
..... 6.0%
0.15
-620K
I
I
7
14
I
I
I
21 28 35 -1 v,cm Figure 9. Plot of the power spectra for a,, = 3.6 8, (linear regime) and a,, = 6.0 8, (anomalous regime) in VPIJ.
0
-620K
12
VPI-5
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9
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1
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-
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0.5
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0.8
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I
'
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-
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I
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I
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I
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T
0
.J 0)
C a, c, C
6
U
3
0.25
0.4
0.5
0.6
0.7
0.8
Y
Figure 8. Plot of the diffusion coefficient D against (a) l/a,? and (b) y for VPI-5. For a,, < 3.6 A, there is a linear dependence of D on
l/a,?. The anomalous regime is observed for a, > 3.6 A, with a maximum in the value of D at a,, = 6.0 8,. In the plot of D against y. the peak occurs as y 0.8. In c, the diffusion coefficient along the channel direction (z axis) and the z component of the force due to the zeolite are shown as a function of y . See the text for discussion.
-
0
I
1
I
I
I
7
14
21
28
35
Y , cm-l
Figure 10. Plot of the x, y, and z components of the power spectra for VPIJ for (a) a, = 3.6 8, ( h e a r regime) and (b)a, = 6.0 8, (anomalous regime). Note the increase in intensity at low frequencies for the z component, which is parallel to the channel axis.
Bandyopadhyay and Yashonath
4292 J. Phys. Chem., Vol. 99, No. 12, 1995 2
-310K
Silicalite
-620K
N
rf
,-
0.15
N
400 200
--/-et---
__--
__---/a-
*-e---
0
2
0.10
2
5
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N
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0 (u
200
/--'
0.5 0
0 ' 0.8
I
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I
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1
1.4
I
The anomalous behavior appears to be universal, and the following conclusions can be drawn from the present work: (1) The diffusion anomaly seems to be independent of the host substance: zeolites (Y, A, or silicalite) and an aluminophosphate (WI-5) both exhibit anomaly. These results suggest that diffusion anomaly should be prevalent in any substance in which confined regions are present. (2) The existence of diffusion anomaly seems to be independent of the geometrical and topological details of the restricted or confined region. (3) There is a significant increase in the intensity of the power spectra toward lower frequencies for the component parallel to the direction of diffusion. These results are of great importance to all areas where diffusion in confined or restricted regions occurs: fast ion conduction, diffusion across membranes, molecular sieve properties of zeolites and other materials, and guest-host chemistry among others. Work is under progress in this laboratory to understand and develop new applications arising out of the above results.
Acknowledgment. S.B. gratefully acknowledges the Council of Scientific and Industrial Research (C.S.I.R), New Delhi, India, for the award of a fellowship.
20
25
15
20
25
15
20
25
ass=5.08
Y
6. Conclusions
15
t, ps
1.6
believe this is a better explanation for the observed diffusion anomaly since it is uniformly applicable to all types of zeolites irrespective of the specific details of the structure. We would also like to point out that in real zeolites, unlike in models, the channels are rarely perfectly cylindrical and hence the force never really goes to zero. This, however, does not decrease the importance of the diffusion anomaly.
10
600
0.05
Figure 11. Diffusion coefficients along the direction of straight channels obtained from the mean squared displacement curve for y component shown as a function of y for the rigid framework model of silicalite. The variation of the y component of force as a function of y is also shown.
5
'*., N
200 0
5
10
t. DS
v
-
0
5
10
t, ps Figure 12. Variation of the x, y , z components of the mean squared displacement for VPIJ as a function of various values of the sorbate diameter.
References and Notes (1) Yashonath, S.;Santikary, P. J . Chem. Phys. 1994, 100, 4013. (2) Yashonath, S.;Santikary, P. J . Phys. Chem. 1994,98,6338,9252. (3) Derycke, I.; Vigneron, J. P.; Lambin, Ph.; Lucas, A. A,; Derouane, E. G. J . Chem. Phys. 1991, 94, 4620. (4) Derouane, E. G.; Andre, J. M.; Lucas, A. A. Chem. Phys. Lett.
1987, 137, 336. ( 5 ) Derouane, E. G.; Andre, J. M.; Lucas, A. A. J . Catal. 1988, 110, 58. (6) Lambin, Ph.; Lucas, A. A.; Derycke, I.; Vigneron, J. P.; Derouane, E. G. J. Chem. Phys. 1989, 90, 3814. (7) Price, G. D.; Pluth, J. J.; Smith, J. V.; Bennett, J. M.; Patton, R. L. J. Am. Chem. SOC. 1982, 104, 5971. (8) Crowder, C. E.; Garces, J. M.; Davis, M. E. A&. X-ray Crysrallogr. 1989, 32, 507. (9) Davis, M. E.; Saldaniaga, C.; Montes, C.; Garces, J.; Crowder, C. Nature 1988, 331, 698. (10) Davis, M. E.; Saldaniaga, C.; Montes, C.; Garces, J.; Crowder, C. Zeolites 1988, 8, 362. (11) Kiselev, A. V.; Du, P. Q.J. Chem. SOC., Faraday Trans. 2 1981, 77, 1. (12) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U.K., 1987. (13) Demontis, P.; Suffriti, G. B.; Quartieri, S.; Fois, E. S.;Gamba, A. J. Phys. Chem. 1988, 92, 867. (14) Verlet, L. Phys. Rev. 1967, 159, 98. (15) Swope, W. C.; Andersen, H. C.; Berens, P. H.; Wilson, K. R. J. Chem. Phys. 1982, 76, 637. (16) Derouane, E. G.; Leherte, L.; Vercauteren, A. A,; Lucas, A. A,; Andre, J. M. J . Catal. 1989, 119, 266. JF'9425070
