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A molecular dynamics study of methane in zeolite NaY
Volume 153, number
6
CHEMICAL
A MOLECULAR Subramanian Deparrmmt
DYNAMICS
YASHONATH,
STUDY OF METHANE Pierfranco
DEMONTIS
of Chemistr_v, lJniversit_v of Pennqbania,
Received I5 July
PHYSICS LETTERS
1988; in final form 19 October
Philadelphia,
30 December
IN ZEOLITE
USA
1988
1. Introduction Faujasite is a mineralogical rarity but typical of zeolites widely used industrially as a catalyst for a variety of reactions and separation processes [ l-31. The structure is based on interconnected cubooctahedra with (hexagonal) double-6 rings [4]. Faujasite may be synthesized with different values for the Si/Al ratio. It is then referred to as NaX or NaY depending on whether the ratio is low or high [ 41. A wide range of experiments exist on the adsorption characteristics of methane in zeolites [ 5,6 ] and in particular in faujasite: NMR studies [7,8], uptake [9-l 1 ] and neutron scattering [ 121 measurements have been carried out. Calorimetric studies on the adsorption of methane in faujasite are also available in the literature [ 13-161. Theoretical studies treating the general problem of adsorption in zeolites and specific calculations by Monte Carlo and energy minimization methods on the thermodynamics of adsorption of methane in faujasite already exist in the literature [ 17-201. However, as yet no dynamical investigations have been reported_ Accordingly, in the present article we have used molecular dynamics (MD) to probe the dynamical aspects of methane adsorption in faujasite in order to make contact with the NMR and neutron data. In this preliminary investigation we have studied the temperature dependence of methane mobility at a constant di Chimica, sitB di Sassari. Via Vienna 2, 07100 Sassari, Italy.
0 009-2614/88/$03.50 (North-Holland
NaY
’ and Michael L. KLEIN PA 19104-6323.
Molecular dynamics calculations have been used to investigate theeffect of temperature At room temperature, methane molecules reside mostly in the vicinity of the supercage the zeolite structure occurs predominantly via surface diffusion.
’ On leave of absence from Dipartimento
1988
Univer-
0 Elsevier Science Publishers Physics Publishing Division)
on the mobility of methane in faujasite. walls. Migration between supercages of
loading of six molecules per supercage. In a subsequent publication we will report on the effect of varying Che cage loading and effects on spectroscopic data.
2. The model The basic approach we adopt is to use a realistic methane-methane potential [2 1 ] for the guest molecules but treat the host faujasite as an idealized rigid crystal. Neglect of an explicit guest-host dynamical coupling is justified because, at the present time, little is known about the time scales for intercage diffusion or the mechanism involved. Moreover, it is known that the zeolite structure is little changed when it is loaded with guest molecules [ 22 1. 2.1. The host lattice The host faujasite, whose crystal structure is Fd3m with a=24.85 8, [22], is modelled by assigning effective charges to the atoms of the zeolite. Since in our model the zeolite lattice is rigid we do not need to specify any interatomic potential for host atoms. We work with a fixed Si/Al ratio of 3.0 which, in turn, implies that we are studying Nay. The Si/Al ratio uniquely determines the Na content; for the chosen ratio the so-called extra framework cation sites (SI, SII) are fully occupied. Following previous workers [ 201, we do not distinguish between Si and Al atoms and assign to them a uniform charge. The values B.V.
551
Volume 153. number 6 Table 1 Intermolecular
CHEMICAL
potential
parameters
A,, (lO’kJ/mol)
awp (A)
cc
845.3870 3.5146 0
3.23 3.77 0
CH HH Cr, (kJ/mol
C, (kJ/mol
A”)
7740
Table 2 Potential
for methane
aP
A*)
C,O (kJ/mol
PHYSICS LETTERS
parameters
13 ( lo6 kJ/molA’*)
Si
0
Na
s1
0
Na
C
0
H
0
1.7159 0.5802
0.5547 0.1506
0 0
1.0169 0.1118
0.6985 0.0758
A’“)
A$+ g ’ ,a
chosen for the effective charges, based in part on MO calculations [20], are 1.151el for Si, -0.701el for 0 and l.Olel for Na. interaction potential
We have used the modified RMK methane potential of Meinender and co-workers [ 23,241 which has been fitted to a variety of solid and gas phase properties. The potential consists of a short-range repulsive part give by exp( -aupr)
and a dispersion atoms
,
ffy,P=C, H
term that acts only between carbon
where ilr)=exp(
-
(5’1~2ur)*) forr<5.18
and f(r)=1
for r>5.18
The electrostatic interactions signing a charge of +O. 143 1el oms and -0.572 lel to the potential parameters are listed 2.3. Guest-host The interaction host is modelled potential 552
zeolite
A ( IO3 kJ/molA6)
Atom
g,, = -
v,,p(r) =A,,
for methane-Nay
1988
1058552
89956
2.2. Guest-guest
30 December
were included by asto the hydrogen atcarbon atom. The in table 1.
interaction potential between methane and the NaY by means of a short-range L-J
) i=Si, 0, Na , a=C,
H
’ ,a
plus the guest-host Coulomb contribution. The potential parameters which are listed in table 2 are the same as those of ref. [ 201. The charges on the atoms of the guest methane give rise to a more reasonable estimate for the guest-host Coulombic interaction energy than in the earlier MC study which underestimated this contribution [ 20 1. The Coulombic contribution is now about 20% of the guest-host potential which yields a static single molecule adsorption energy of 21 lcJ/mol in reasonable agreement with a previous estimate of 23 kJ/mol [ 19 1. The difference between these two values can be accounted for by the polarization energy of the guest species by the host lattice. It is known that for a single guest molecule this contribution is lOoh of the adsorption energy [25]. However, we do not include the induction energy term in our calculations because this many-body term would significantly increase the computational time.
3. Molecular
dynamics calculations
One unit cell of NaY contains eight supercages: 192 Si/Al, 384 0 and 48 Na in sites SI and SIX. The simulation system consists of a single periodically replicated unit cell of faujasite and 48 CH, molecules, initially disposed uniformly in the 8 cages. One such cage is shown schematically in fig. 1. Since the host lattice is rigid it provides an external field that acts on each methane molecule. The contribution from this field is included in the dynamic equations for the methane molecules. The simulations are carried out in the microcanonical ensemble with constant volume and constant total energy. The equations of motion were integrated using the predictor-correc-
CHEMICAL
Volume 153, number 6
30 December
PHYSICS LETTERS
1988
4. Results and discussion 4. I, Thermodynamics
Table 3 lists the the guest-host and the energy, and the tential energy (U}
MD results for the temperature, the guest-guest contributions to heat of adsorption. The total poof the system is 7
(u>=(~#l>+(~i,> where
is the guest-host
Fig. I. A schematic view of a single faujasite supercage taken from the MD run at 50 K showing some methane molecules in favoured adsorbed sites. tor method and quaternions for orientational degrees of freedom [26] with a time step of 1.0 fs. Typically, the system was allowed to thermalize for lo-20 ps. The calculated properties were then obtained from averages over 25 ps. Six runs were carried out at nominal temperatures of 50, 150,220, and 300 K. The lowest two temperatures were studied by heating the minimum energy configuration and by cooling the room temperature run.
interaction
energy and
is the guest-guest interaction energy. We note that the lowest temperature studied heating and cooling runs yield slightly different values for ( L!,, >. Careful examination of the MD trajectories revealed energetically competitive contigurations close to the minimum one and indeed some residual mobility even at 50 K which is the origin of the small hysteresis in (U,,). This point is further amplified in the following sections. 4.2 Energy and spatial distributions In order to characterize the behaviour of CH, in the host zeolite we have calculated the distribution of total energy averaged over the MD trajectories.
Table 3 Molecular dynamics Run
cooling cooling cooling heating heating 0, -qs=
results for methane in zeolite NaY at a loading of six molecules per super cage
(W (kJ/mof)
(U,h> (k.J/mol)
<%i,> (kl/mol)
4srD’ (kJ/mol)
306.3 226. I 154.3 53.8 147.7 54.2
-5.71(0.05) -8.81(0.09) - 12.32(0.08) - 17.85(0.03) - 12.71(0.04) - 18.38 (0.03)
- 13.35 - 14.45 - 16.16 -19.19 - 16.40 - 19.74
-11.68 - 12.72 - 14.43 -17.14 - 14.68 - 18.68
- 1.67 _ 1.73 - 1.73 -2.05 - 1.72 - 1.06
15.90 16.33 17.44 19.64 17.62 20.19
( U) -RT.
553
Volume 153, number 6
CHEMICAL
The results for four trajectories are shown in fig. 2. Even at the highest temperature, the energy distribution is bimodal. The relative intensities of the two peaks shift systematically as the temperature is lowered. At 50 K the energy distribution is determined by peaks around - 20 and - 12 kJ/mol. Monte Carlo results [20] on the same system but with cage occupancy of only a single CH, molecule indicated that at 50 K the guest is localized. The present results suggest that at a loading of six molecules per supercage there exists some small residual mobility which will be discussed further below. Fig. 3 shows the temperature dependence of the spatial distribution functions for the methane center-of-mass with respect to the center of the supercage. The main point to note is that at all temperatures studied molecules are confined to the proximity of the cage surface. Only around 300 K is there any indication of molecules near the center of the cage. The lowest temperature curve mirrors the corresponding energy distribution shown in fig. 2. The bimodal energy distribution suggests two energetically distinct regions of the supercage surface. The main energy peak around - 20 kJ/ mol corresponds to guest molecules at or near the most favoured site, shown in fig. 1. The higher energy peak arises from molecules that are delocalized
ENEMY
DISTRIBUTION
30 December
PHYSICS LETTERS
1988
on the supercage surface. The latter are highly mobile, in contrast to the molecules contributing to the main peak. 4.3. Mobility of methane In this initial study, we have characterized the diffusive motion of CH, molecules in terms of the two quantities z, and z, that are, respectively, the cage and site residence times. The individual molecular trajectories have been analyzed to determine the length of time a given molecule spends in an environment with potential energy less than some specified value E*. In the present case E* was chosen to be the value of E at the minimum separating the two peaks shown in fig. 2, viz. - 14 kJ/mol. The ensemble average of this time is b. Values of T, determined in this way are plotted in fig. 4. We have verified that modest changes in the choice of site binding energy, E*, have little effect on these results. The low value of 7s at high temperatures arises from the high mobility of CH4, even though the molecules are confined to the vicinity of the surface. At low temperature ( 50 K), a small fraction of the molecules are still mobile, and this is why qsl in table 3 has not reached its minimum value. CAQE RDF
FUNCTION. N(E)
5.0
sw 4.0
5
-20
-15
-10
-5
0
E (kJ/mol) Fig. 2. Energy distribution function for methane in zeolite NaY at a loading of six molecules per supercage and nominal temperatures 300 K (continuous line), 220 K (dotted line), 150 K (dashed line) and 50 K (dash-dotted line).
554
Fig. 3. Radial distribution functions for methane centers-of-mass with respect to the supercage center at nominal temperatures 300 K (continuous line), 220 K (dotted line), 150 K (dashed line) and 50 K (dash-dotted line) taken from the heating run. The bimodal distribution at the lowest temperature should be noted.
Volume 153,
number
6
CHEMICAL
PHYSICS LETTERS
30 December 1988
5. Conclusion
. g
0.5
We have described some preliminary MD results on the mobility of CH4 in the zeolite Nay. The calculations suggest that the zeolite holding potential is such that even at 300 K the motion of CH, molecules is largely confined to the cage walls [20]. The calculated site binding energy and diffusion coefftcient agree suffkiently well with calorimetric and NMR studies [ 8,lS ] that it is of interest to explore the dynamical behaviour of the CH, molecules in more detail and to examine the effect of changing the loading of the guest in the supercage.
To
‘5
I10 .
7
s-
. .
OI
I
0
50
/
100
.,
150
.
I
I
I
200
250
300
I
T (K)
Fig. 4. Temperature dependence of the site (5,. dots) and cage (r_ squares) residence times for methane in NaY at a loading of six molecules per supercage. Statistical uncertainties are of order 20% for the three highest temperature points and at least twice as
Acknowledgement This research was supported in part by the US National Science Foundation. MLK thanks the Donors of the Petroleum Research Fund administered by the American Chemical Society for partial support of this research.
large for the lowest temperature.
References The cage residence time rc was determined from the MD trajectories by using a distance criterion to decide whether or not a given molecule belonged to a specific cage. A molecule initially within a distance of 5.9 A of a cage center is assigned to the cage in question. The ensemble average of individual residence times yields the values of 7, shown in fig. 4. At higher temperature, the molecules are sufficiently mobile that rc is well defined. As the temperature is lowered, r, increases rapidly and due to the limited length of our MD runs the values of TV are only a crude estimate, particularly at 50 K. Nevertheless, the values oft, indicate that mobility increases significantly around 200 K. We have used the MD trajectory at 300 K to estimate a value for the diffusion coeffkient based on the Einstein relation. Clearly, a diffusion coefficient estimated in this fashion need not necessarily be relevant to mass transport through the zeolite. However, the calculated value 2.0~ 1O-’ m2 s- ’agrees well with the NMR value of 1.5~ 1O-8 mZ s-’ [8].
[ I] R.M. Barrer: Zeolites and clay minerals as sarbents and molecular sieves (Academic Press, New York, 1978); The hydrothermal chemistry of zeolites (Academic Press, New York, 1982). [ 21 F. Schwochow and L. Puppe, Angew. Chem. Intern. Ed. Engl. 14 (1975) 620. [ 31 W. Holderich, M. Hesse and F. Naumann, Angew. Chem. Intern. Ed. Engl. 27 (1988) 226. [4] C.A. Fyfe, J.M. Thomas, J. Klinowski and G.C. Gobbi, Angew. Chem. Intern. Ed. Engl. 22 (1983) 259. [5] E. Cohen De Lara, R. Kahn and R. Seloudoux, J. Chem. Phys. 83 (1985) 2646. [ 61R. Kahn, E. Cohen De Lam and K.D. Moller, J. Chem. Phys. 83 (1985) 2653. [ 7 ] J. Kruger, H. Pfeifer, J. Cam, M. Berlow, J. Richta-Mendau, B. Fahlke and L.V.C. Rees, .I. Appl. Catal. 24 ( 1986) 187. [ 8 ] J. Karger and H. Pfeifer, Zeolites 7 ( 1987 ) 90. [9] D.M. Ruthven, K.F. Lougblin and R.I. Derrah, Advan. Chem. Ser. I2 1 ( 1973) 330. [lo] D.M. Ruthven, R.I. Derrah and K.F. Lougblin, Can. J. Chem. 51 (1973) 3514. [ I I] D.M. Ruthven, Am. Chem. Sot. Symp. Ser. 40 ( 1977) 320. [ 121 R. Stockmeyer, Zeolites 4 (1984) 81. [ 131 R.M. Barrer and J.W. Sutherland, Proc. Roy. Sot. A 237 (1956) 439. [ 141 H.W. Habgood, Can. J. Chem. 12 (1964) 2340.
555
Volume 153, number 6 [IS] R.J. Neddenriep.
CHEMICAL
J. Colloid Interface Sci. 28 ( 1968) 293.
[ 161 H. Stach, U. Lohse, H. Thamm and W. Schirmer, Zeolites 6 (1986) 74. [ 171 P.A. Politowicz and J.J. Kozak, Mol. Phys. 62 (1987) 939. [ 181 G.B. Woods, A.Z. Panagiotopoulos and J.S. Rowhnson, Mol. Phys. 63 (1988) 49. [ 191 A.G. Bezus, A.V. Kiselev, A.A. Lopatkin and P.Q. Du, I. Chem. Sot. Faraday Trans II 74 (1978) 367. [ 201 S. Yashonath, J.M. Thomas, A.K. Nowak and A.K. Cheetham, Nature331 (1988) 601. [ 211 R. Righini, K. Maki and M.L. Klein, Chem. Phys. Letters 80 (1981) 301.
556
30 December
PHYSICS LETTERS [22] A.N. Fitch, H. Jobic and A. Renouprez, (1986) 1311. [23] N. Meinender
1988
J. Phys. Chem. 90
and G.C. Tabisz, J. Chem. Phys. 79 (1983)
416. [24] A.R. Penner. N. Meinender
and G.C. Tabisz, Mol. Phys. 54 (1985) 479. [ZS] E. Cohen De Lam and T. Nguyen Tan, J. Phys. Chem. 80 (1976)
1917.
[26] M.P. Alien and D. Tildesley, Computer
(Clarendon
Press, Oxford,
1987).
simulation of liqwds
6
CHEMICAL
A MOLECULAR Subramanian Deparrmmt
DYNAMICS
YASHONATH,
STUDY OF METHANE Pierfranco
DEMONTIS
of Chemistr_v, lJniversit_v of Pennqbania,
Received I5 July
PHYSICS LETTERS
1988; in final form 19 October
Philadelphia,
30 December
IN ZEOLITE
USA
1988
1. Introduction Faujasite is a mineralogical rarity but typical of zeolites widely used industrially as a catalyst for a variety of reactions and separation processes [ l-31. The structure is based on interconnected cubooctahedra with (hexagonal) double-6 rings [4]. Faujasite may be synthesized with different values for the Si/Al ratio. It is then referred to as NaX or NaY depending on whether the ratio is low or high [ 41. A wide range of experiments exist on the adsorption characteristics of methane in zeolites [ 5,6 ] and in particular in faujasite: NMR studies [7,8], uptake [9-l 1 ] and neutron scattering [ 121 measurements have been carried out. Calorimetric studies on the adsorption of methane in faujasite are also available in the literature [ 13-161. Theoretical studies treating the general problem of adsorption in zeolites and specific calculations by Monte Carlo and energy minimization methods on the thermodynamics of adsorption of methane in faujasite already exist in the literature [ 17-201. However, as yet no dynamical investigations have been reported_ Accordingly, in the present article we have used molecular dynamics (MD) to probe the dynamical aspects of methane adsorption in faujasite in order to make contact with the NMR and neutron data. In this preliminary investigation we have studied the temperature dependence of methane mobility at a constant di Chimica, sitB di Sassari. Via Vienna 2, 07100 Sassari, Italy.
0 009-2614/88/$03.50 (North-Holland
NaY
’ and Michael L. KLEIN PA 19104-6323.
Molecular dynamics calculations have been used to investigate theeffect of temperature At room temperature, methane molecules reside mostly in the vicinity of the supercage the zeolite structure occurs predominantly via surface diffusion.
’ On leave of absence from Dipartimento
1988
Univer-
0 Elsevier Science Publishers Physics Publishing Division)
on the mobility of methane in faujasite. walls. Migration between supercages of
loading of six molecules per supercage. In a subsequent publication we will report on the effect of varying Che cage loading and effects on spectroscopic data.
2. The model The basic approach we adopt is to use a realistic methane-methane potential [2 1 ] for the guest molecules but treat the host faujasite as an idealized rigid crystal. Neglect of an explicit guest-host dynamical coupling is justified because, at the present time, little is known about the time scales for intercage diffusion or the mechanism involved. Moreover, it is known that the zeolite structure is little changed when it is loaded with guest molecules [ 22 1. 2.1. The host lattice The host faujasite, whose crystal structure is Fd3m with a=24.85 8, [22], is modelled by assigning effective charges to the atoms of the zeolite. Since in our model the zeolite lattice is rigid we do not need to specify any interatomic potential for host atoms. We work with a fixed Si/Al ratio of 3.0 which, in turn, implies that we are studying Nay. The Si/Al ratio uniquely determines the Na content; for the chosen ratio the so-called extra framework cation sites (SI, SII) are fully occupied. Following previous workers [ 201, we do not distinguish between Si and Al atoms and assign to them a uniform charge. The values B.V.
551
Volume 153. number 6 Table 1 Intermolecular
CHEMICAL
potential
parameters
A,, (lO’kJ/mol)
awp (A)
cc
845.3870 3.5146 0
3.23 3.77 0
CH HH Cr, (kJ/mol
C, (kJ/mol
A”)
7740
Table 2 Potential
for methane
aP
A*)
C,O (kJ/mol
PHYSICS LETTERS
parameters
13 ( lo6 kJ/molA’*)
Si
0
Na
s1
0
Na
C
0
H
0
1.7159 0.5802
0.5547 0.1506
0 0
1.0169 0.1118
0.6985 0.0758
A’“)
A$+ g ’ ,a
chosen for the effective charges, based in part on MO calculations [20], are 1.151el for Si, -0.701el for 0 and l.Olel for Na. interaction potential
We have used the modified RMK methane potential of Meinender and co-workers [ 23,241 which has been fitted to a variety of solid and gas phase properties. The potential consists of a short-range repulsive part give by exp( -aupr)
and a dispersion atoms
,
ffy,P=C, H
term that acts only between carbon
where ilr)=exp(
-
(5’1~2ur)*) forr<5.18
and f(r)=1
for r>5.18
The electrostatic interactions signing a charge of +O. 143 1el oms and -0.572 lel to the potential parameters are listed 2.3. Guest-host The interaction host is modelled potential 552
zeolite
A ( IO3 kJ/molA6)
Atom
g,, = -
v,,p(r) =A,,
for methane-Nay
1988
1058552
89956
2.2. Guest-guest
30 December
were included by asto the hydrogen atcarbon atom. The in table 1.
interaction potential between methane and the NaY by means of a short-range L-J
) i=Si, 0, Na , a=C,
H
’ ,a
plus the guest-host Coulomb contribution. The potential parameters which are listed in table 2 are the same as those of ref. [ 201. The charges on the atoms of the guest methane give rise to a more reasonable estimate for the guest-host Coulombic interaction energy than in the earlier MC study which underestimated this contribution [ 20 1. The Coulombic contribution is now about 20% of the guest-host potential which yields a static single molecule adsorption energy of 21 lcJ/mol in reasonable agreement with a previous estimate of 23 kJ/mol [ 19 1. The difference between these two values can be accounted for by the polarization energy of the guest species by the host lattice. It is known that for a single guest molecule this contribution is lOoh of the adsorption energy [25]. However, we do not include the induction energy term in our calculations because this many-body term would significantly increase the computational time.
3. Molecular
dynamics calculations
One unit cell of NaY contains eight supercages: 192 Si/Al, 384 0 and 48 Na in sites SI and SIX. The simulation system consists of a single periodically replicated unit cell of faujasite and 48 CH, molecules, initially disposed uniformly in the 8 cages. One such cage is shown schematically in fig. 1. Since the host lattice is rigid it provides an external field that acts on each methane molecule. The contribution from this field is included in the dynamic equations for the methane molecules. The simulations are carried out in the microcanonical ensemble with constant volume and constant total energy. The equations of motion were integrated using the predictor-correc-
CHEMICAL
Volume 153, number 6
30 December
PHYSICS LETTERS
1988
4. Results and discussion 4. I, Thermodynamics
Table 3 lists the the guest-host and the energy, and the tential energy (U}
MD results for the temperature, the guest-guest contributions to heat of adsorption. The total poof the system is 7
(u>=(~#l>+(~i,> where
is the guest-host
Fig. I. A schematic view of a single faujasite supercage taken from the MD run at 50 K showing some methane molecules in favoured adsorbed sites. tor method and quaternions for orientational degrees of freedom [26] with a time step of 1.0 fs. Typically, the system was allowed to thermalize for lo-20 ps. The calculated properties were then obtained from averages over 25 ps. Six runs were carried out at nominal temperatures of 50, 150,220, and 300 K. The lowest two temperatures were studied by heating the minimum energy configuration and by cooling the room temperature run.
interaction
energy and
is the guest-guest interaction energy. We note that the lowest temperature studied heating and cooling runs yield slightly different values for ( L!,, >. Careful examination of the MD trajectories revealed energetically competitive contigurations close to the minimum one and indeed some residual mobility even at 50 K which is the origin of the small hysteresis in (U,,). This point is further amplified in the following sections. 4.2 Energy and spatial distributions In order to characterize the behaviour of CH, in the host zeolite we have calculated the distribution of total energy averaged over the MD trajectories.
Table 3 Molecular dynamics Run
cooling cooling cooling heating heating 0, -qs=
results for methane in zeolite NaY at a loading of six molecules per super cage
(W (kJ/mof)
(U,h> (k.J/mol)
<%i,> (kl/mol)
4srD’ (kJ/mol)
306.3 226. I 154.3 53.8 147.7 54.2
-5.71(0.05) -8.81(0.09) - 12.32(0.08) - 17.85(0.03) - 12.71(0.04) - 18.38 (0.03)
- 13.35 - 14.45 - 16.16 -19.19 - 16.40 - 19.74
-11.68 - 12.72 - 14.43 -17.14 - 14.68 - 18.68
- 1.67 _ 1.73 - 1.73 -2.05 - 1.72 - 1.06
15.90 16.33 17.44 19.64 17.62 20.19
( U) -RT.
553
Volume 153, number 6
CHEMICAL
The results for four trajectories are shown in fig. 2. Even at the highest temperature, the energy distribution is bimodal. The relative intensities of the two peaks shift systematically as the temperature is lowered. At 50 K the energy distribution is determined by peaks around - 20 and - 12 kJ/mol. Monte Carlo results [20] on the same system but with cage occupancy of only a single CH, molecule indicated that at 50 K the guest is localized. The present results suggest that at a loading of six molecules per supercage there exists some small residual mobility which will be discussed further below. Fig. 3 shows the temperature dependence of the spatial distribution functions for the methane center-of-mass with respect to the center of the supercage. The main point to note is that at all temperatures studied molecules are confined to the proximity of the cage surface. Only around 300 K is there any indication of molecules near the center of the cage. The lowest temperature curve mirrors the corresponding energy distribution shown in fig. 2. The bimodal energy distribution suggests two energetically distinct regions of the supercage surface. The main energy peak around - 20 kJ/ mol corresponds to guest molecules at or near the most favoured site, shown in fig. 1. The higher energy peak arises from molecules that are delocalized
ENEMY
DISTRIBUTION
30 December
PHYSICS LETTERS
1988
on the supercage surface. The latter are highly mobile, in contrast to the molecules contributing to the main peak. 4.3. Mobility of methane In this initial study, we have characterized the diffusive motion of CH, molecules in terms of the two quantities z, and z, that are, respectively, the cage and site residence times. The individual molecular trajectories have been analyzed to determine the length of time a given molecule spends in an environment with potential energy less than some specified value E*. In the present case E* was chosen to be the value of E at the minimum separating the two peaks shown in fig. 2, viz. - 14 kJ/mol. The ensemble average of this time is b. Values of T, determined in this way are plotted in fig. 4. We have verified that modest changes in the choice of site binding energy, E*, have little effect on these results. The low value of 7s at high temperatures arises from the high mobility of CH4, even though the molecules are confined to the vicinity of the surface. At low temperature ( 50 K), a small fraction of the molecules are still mobile, and this is why qsl in table 3 has not reached its minimum value. CAQE RDF
FUNCTION. N(E)
5.0
sw 4.0
5
-20
-15
-10
-5
0
E (kJ/mol) Fig. 2. Energy distribution function for methane in zeolite NaY at a loading of six molecules per supercage and nominal temperatures 300 K (continuous line), 220 K (dotted line), 150 K (dashed line) and 50 K (dash-dotted line).
554
Fig. 3. Radial distribution functions for methane centers-of-mass with respect to the supercage center at nominal temperatures 300 K (continuous line), 220 K (dotted line), 150 K (dashed line) and 50 K (dash-dotted line) taken from the heating run. The bimodal distribution at the lowest temperature should be noted.
Volume 153,
number
6
CHEMICAL
PHYSICS LETTERS
30 December 1988
5. Conclusion
. g
0.5
We have described some preliminary MD results on the mobility of CH4 in the zeolite Nay. The calculations suggest that the zeolite holding potential is such that even at 300 K the motion of CH, molecules is largely confined to the cage walls [20]. The calculated site binding energy and diffusion coefftcient agree suffkiently well with calorimetric and NMR studies [ 8,lS ] that it is of interest to explore the dynamical behaviour of the CH, molecules in more detail and to examine the effect of changing the loading of the guest in the supercage.
To
‘5
I10 .
7
s-
. .
OI
I
0
50
/
100
.,
150
.
I
I
I
200
250
300
I
T (K)
Fig. 4. Temperature dependence of the site (5,. dots) and cage (r_ squares) residence times for methane in NaY at a loading of six molecules per supercage. Statistical uncertainties are of order 20% for the three highest temperature points and at least twice as
Acknowledgement This research was supported in part by the US National Science Foundation. MLK thanks the Donors of the Petroleum Research Fund administered by the American Chemical Society for partial support of this research.
large for the lowest temperature.
References The cage residence time rc was determined from the MD trajectories by using a distance criterion to decide whether or not a given molecule belonged to a specific cage. A molecule initially within a distance of 5.9 A of a cage center is assigned to the cage in question. The ensemble average of individual residence times yields the values of 7, shown in fig. 4. At higher temperature, the molecules are sufficiently mobile that rc is well defined. As the temperature is lowered, r, increases rapidly and due to the limited length of our MD runs the values of TV are only a crude estimate, particularly at 50 K. Nevertheless, the values oft, indicate that mobility increases significantly around 200 K. We have used the MD trajectory at 300 K to estimate a value for the diffusion coeffkient based on the Einstein relation. Clearly, a diffusion coefficient estimated in this fashion need not necessarily be relevant to mass transport through the zeolite. However, the calculated value 2.0~ 1O-’ m2 s- ’agrees well with the NMR value of 1.5~ 1O-8 mZ s-’ [8].
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