J. Phys. Chem. 1988,92, 6783-6789 reservoir-acid concentrations, electrostatic repulsion of the co-ion by the fixed-charge sites does not allow for high co-ion concentrations within the membrane. The length of the diffuse portion of the double layer that extends from the pore surface toward the pore center line (which might be referred to as a Debye length in treatments in which the dielectric constant is invariant) is larger for the low acid concentration results, which tends to inhibit the incorporation of co-ions into the membrane pores. The diffuse portion of the double layer could be reduced, for example, by adding another salt, which would increase the ionic strength of the pore fluid. The model calculations yield a co-ion concentration within the membrane that is lower than the experimental results at higher acid concentrations. This result may be due to specific adsorption of counterions, which is not included in the simulation. Specific adsorption of counterions would effectively reduce the surface charge density and allow for higher co-ion concentrations. The traditional Poisson-Boltzmann treatment overestimates the c H ~variations O ~ ~ C J in C the N Odielectric ~ ~ , ratio c H ~ ~ ~ , ~ ~ ~ ~ ~ ~ ~ ~ / since constant and hydration effects are ignored, which results in a reduced potential difference between the pore wall and the pore center line, reflecting the higher amount of co-ions occupying the pore fluid.
Conclusions The equilibrium properties of the Nafion-l17/sulfuric acid system have been characterized. Experimentally obtained membrane porosities, counterion, co-ion, and water concentrations are reported. A radiotracer technique in which both 3H and 35S concentrations are obtained simultaneously is employed. The mathematical model of Pintauro and Verbrugge is shown to accurately predict the ratio of acid within the membrane to that in the adjacent reservoir. Since the mathematical model represents a substantial improvement over the traditional Poisson-Boltzmann approach, it should be coupled with transport equations to provide a more precise description of the microscopic events that take place
6783
within ion-exchange membranes and influence the rate at which ions can transport across such membranes.
Acknowledgment. We acknowledge helpful discussions concerning the equilibrium properties of ion-exchange membranes with Professor Pintauro of the Department of Chemical Engineering at Tulane University.
Glossary pore radius, m concentration of species i, kg mol/m3 d membrane disk diameter, m charge on the electron, 1.602 X lo-]' C e electric field, V/m E equivalent weight (H' form), kg((kg.equiv) EW F Faraday's constant, 9.6487 X 10 C/(kg.equiv) J/K Boltzmann's constant, 1.381 X kb n index of refraction radial coordinate, m r radius of ion i, m Ti R gas constant, 8314 J/(kg.mol.K) temperature, K T u velocity, m/s V volume, m3 charge number on species i zi dielectric constant parameter of Booth's equation (eq 4), PB m/v t' pore-fluid dielectric constant €0 pore-fluid dielectric constant at the pore center line c permittivity of vacuum, 8.854 X C/(V.m) e porosity solvent dipole moment, J Md stoichiometriccoefficient of species i Vi dry-membrane density (H' form), kg/m3 Pdry charge density of pore fluid, C/m3 Ps U surface charge density, C/m2 @ electrical potential, V Registry No. H2S04,7664-93-9; Nafion 117, 66796-30-3. a
ci
Lattice Vibrational Calculation of A-Type Zeolite Kyoung Tai NO,* Byung Hee Seo, Je Myung Park, Department of Chemistry, Soong Si1 University, Sang Do 1-1 Dong, Dong Jak Gu, Seoul, Korea
and Mu Shik Jhon* Department of Chemistry, Korea Advanced Institute of Science and Technology, Cheong Ryang Ri, Seoul, Korea (Received: December 29, 1987; In Final Form: May 2, 1988) The potential energy functions of Na A-type zeolite are determined subject to the constraints of the crystal structure. Next, the force constants are calculated with these potential functions. Applying the pseudolattice method, the normal mode calculations are carried out, and the IR bands of zeolite A are assigned with the normal modes. The potential functions give reasonable crystal geometries and force constants.
Introduction Vibrational spectra of zeolites are very widely used for the investigations of the properties of zeolite frameworks'" and the cations7-" or molecules b o ~ n d ~ to * -them. ~ ~ In particular, the (1) Milkey, R. G . Am. Mineral. 1960, 45, 990. (2) Kiselev, A. V.; Lygin, V. I. Infrared Specira of Adsorbed Species; Little, L. H.,Ed.; Academic: London, 1967; pp 361-367. ( 3 ) Wright, A. C.; Rupert, J. P.;Granquist, W. T. Am. Mineral. 1968,53, 1293. (4) Flanigen, E. M.; Khatami, J.; Szymanski, M. A. Adu. Chem. Ser 1971, 101, 201. ( 5 ) Joshi, M. S.; Bhoskar, B. T. Indian J. Pure Appl. Phys. 1981,19, 560. (6) Angell, C. L. J . Phys. Chem. 1973, 77, 222. (7) (a) Dutz, V. H. Ber. Drsch. Keram. Ges. 1969, 46, 75. (b) Stubican, V.; Roy, R. Z . Kristallogr. 1961, 115, 200. (8) Ozin, G. A,; Baker, M. D.; Godber, J. J . Phys. Chem. 1984,88,4902.
0022-3654/88/2092-6783$01.50/0
vibrational spectra of absorbed molecules and ions bound to several types of zeolites have been intensively studied during the past decades.'620 However, there are only a few works dealing with (9) Baker, M. D.; Godber,J.; Ozin, G.A. J . Phys. Chem. 1985,89,2299. (10) Ozin, G . A.; Baker, M. D.; Godber, J.; Shihua, W. J. Am. Chem. Soc. 1985,107, 1995. (1 1) Baker, M. D.; Godber, J.; Ozin, G. A. J . Am. Chem. Soc. 1985,107, 3033. (12) De Lara, E. C. Mol. Phys. 1972, 23, 355. (13) De Lara, E. C.; Delaval, Y . J . Phys. Chem. 1974, 78, 2180. (14) De Lara, E. C . ; Tan, T. N. J . Phys. Chem. 1976,80, 1917. (15) De Lara, E. C.; Geisse, J. V. J . Phys. Chem. 1976, 80, 1922. (16) Peuker, C.; Kunath, D. J . Chem. SOC.,Faraday Trans. 1 1981, 77, 2079. (17) Butler, W. M.; Angell, C . L.; McAllister, W.; Risen, Jr., W. M. J . Phys. Chem. 1977, 81, 2061.
0 1988 American Chemical Society
6784
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988
the lattice vibration of zeolite framework^.^',^^.^^ The framework vibrations were investigated by using infrared and Raman spectrometry by Flanigen4 et al. and Angel,6 respectively. In a theoretical study, there are two main difficulties for the calculation of normal modes of a zeolite framework. These are the description of a three-dimensional crystal having a very large unit cell with a finite cluster and the preparation of a physically realistic force constant set that is suitable for the representation of the physical properties of a zeolite framework. Recently N o et a1.22have calculated the normal mode of A-type zeolite by applying the pseudolattice method to the double four ring (D4R) subunit, and these normal modes were used for the assignment of the experimentally obtained vibrational spectra. But the force constants used in that calculation2' do not form a suitable set for the description of the minimum energy structure obtained by X-ray diffraction study. Although several empirical and values2' are available for the force constants of the internal coordinate motions, the force field type crystal potential function may fail in the description of the equilibrium geometry of the zeolite framework. Since the force constants in a covalently bonded crystal are very sensitive to the structure and the couplings between internal coordinates are quite different from one structure to another, it is difficult to constitute a physically realistic force constant set from empirical rules and empirically obtained force constants. The purpose of this work is to obtain the intraframework potential energy functions suitable for the description of Na A-type zeolite and to obtain the force constant for each atom in Cartesian coordinates from the potential functions. Next the normal mode calculations will be carried out using the pseudolattice method, and the intensity of the infrared spectra also calculated from the normal mode vectors.
Methods and Models The geometry of the model used in this study was taken from the X-ray crystallographic study of dehydrated Na A zeolite by Pluth and Smith.25 The space group is Fm3c, the cell constant is 24.555 A, and the experimental conditions are at 350 OC and 10" Torr. Determination of the Potential Energy Functions. In this study, three sets of potential functions were obtained (Sl, S2, and S3). The potential parameters of each set were determined subject to the constraints of the crystal structure. In geometric parameter space, the point corresponding to the crystal structure must be an energy minimum in the potential energy surface. Therefore the potential parameters can be obtained by the minimization of all net forces subject to the constraints of the crystal and the partial first derivative of the potential energy with respect to all the geometric parameters, on the atoms in crystal. The potential parameters are determined by the minimization of the following function f:
No et al. ai is ith potential parameter, and q! and qo! represent the j t h coordinate of Ith atom and that at the equilibrium position, respectively. V and p represent the stabilization energy of the crystal and that at equilibrium position, respectively. There are eight environmentally different atoms in Na A-type zeolite (0(l), 0 ( 2 ) , 0(3), Na(l), Na(2), Na(3), Si, and AI); N is 8, therefore the maximum number of the potential parameters that can be refined must be equal to or less than 24. Potential Energy Functions. The stabilization energy of the crystal (V) is expressed as a sum of several interaction terms:
V = Vel + Vpol+ V,
+ VNB + Vknd
(3)
Electrostatic (Vel) and Polarization ( VPl) Energies. The average net atomic charge of N a atoms in A-type zeolite was obtained as 0.581 by No et a1.26 Since the other parts of the framework, (SiA104),, form strong covalent bonds, the electronegativity of the atoms, except Na, may be equalized. Therefore the net atomic charges are calculated by using Sanderson's electronegativity e q ~ a l i z a t i o nconditions ~ ~ ~ ~ ~ with Huheey's atomic electronegativity set:32
X 6, = - 6 N a + b,6, = an + b,6,
(4)
m#Na
a,
(5)
where a,, b,, and 6, represent the inherent electronegativity, charge coefficient, and net atomic charge of m atom, respectively. Vel =
C C 6mSn/trmn m n>m
(6)
where t and r,, represent the dielectric constant and the interatomic distance between m and n atoms, respectively. In this calculation t is assumed to be 1, because the expression of t as a function of the interatomic distance r,, is a very difficult problem. Since the net atomic forces contributions from Vel are relatively small compared with the contributions of other potential terms, the changes in dielectric constant may not lead to serious differences in determining the other potential parameters. For good convergence of the electric fields Madelung sums, the summation was carried out for all the (SiA104)Na units within the cubic crystal ( R = 2a0),and fast convergency in electric field was obtained? N
Vpol
N
= -1/2Can[(C &mx)2 n
n#m
N
+ ( C zmn')2 + ( C t n z ) 2 1 n#m
n#m
(7)
where a, is the atomic polarizability of n atom and zm* is the x directional electric field at the position of atom n created by atom m. Bonding Potential Functions ( VB).For bonding atomic pairs, Si-0 and AI-0, harmonic and Morse type potential functions are used alternatively:
VBH = YzCC kmn(rmn - rmn? rn n>m
(8)
where
(18) Ward, J. W. J . Phys. Chem. 1968, 72, 4211. (19) (a) Angell, C. L.; Schaffer, P. C. J. Phys. Chem. 1965,69, 3463. (b) Uytterhoeven, J. B.; Schoanbeydt, R.; Liengme, B. V.; Hall, W. K. J . Carol. 1969, 13, 425. (20) (a) Ward, J. W. J . Coral. 1968, 11, 238. (b) Ibid. 1969, 14, 365. (21) (a) Blackwell, C. S. J . Phys. Chem. 1979,83, 3251; (b) Ibid. 3257. (22) No, K. T.; Bae, D. H.; Jhon, M. S. J . Phys. Chem. 1986, 90,1772. (23) Gordy, W. J . Phys. Chem. 1964, 14, 305. (24) Badger, R. M. J . Chem. Phys. 1934, 2, 128. (25) Pluth, J . J.; Smith, J. V. J . Am. Chem. SOC.1980, 102, 4702. (26) (a) No, K. T.; Kim, J. S.; Huh, Y. Y.; Kim, W. K.; Jhon, M. S. J . Phys. Chem. 1987,91,740. (b) No, K. T.; Huh, Y. Y.; Jhon, M. S. J . Phvs. Chem., in press. (27) Song, M. K.; Chon, H. J.; No,K. T.;Jhon, M. S. Zeolires, in press. ( 2 8 ) Choi, K. J.; No, K. T.; Jhon, M. S. Bull. Korean Chem. SOC.1987, R ISR -, _--. (29) No, K. T.; Kim, J . S.; Jhon, M. S. J . Phys. Chem., in press.
where k,, and r,: are the harmonic potential parameters and De,, a, and' P , are the Morse potential parameters for atomic pair m,n. Since the bonds are formed between (Si,AI) and (0(1),0(2),0(3)), for harmonic potential functions, there are 12 potential parameters, 2 parameters for each pair. In Morse type potential function, since the oxygen atoms are in different environments, O( 1),0(2),0(3), are not distinguished; therefore six parameters are used. Nonbonding Potential Functions ( VNB),Lennard-Jones (6-12) type potential function was used for describing the nonbonding interaction of 0-0 and 0-Na atomic pairs:
(30) Sanderson, R. T.J . Chem. Educ. 1945, 31, 2. (31) Sanderson, R. T.Chemical Periodiciry; Reinhold: New York, 1960. (32) Huheey, E. J . Phys. Chem. 1965.69, 3284.
Lattice Vibrational Calculation of A-Type Zeolite
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6785
TABLE I: Refined Potential Parameters at 6 , = 0.581 mdyn" Slb
A1-0, Si-0 (bonding)
0-0,O-Na (non bonding)
6o-o LTSN~ €0-0
e+Na
s2
s3
1.783 1.760 1.794 1.590 1.57 1 1.595 3.036 3.285 2.858 4.938 5.708 4.774
1.782 ( # " M a = 1.770) 1.761 1.794 1.590 (psi-= 1.588) 1.578 1.595 3.062 ( ( ~ ~ 1=- 01.386) 3.306 2.874 4.93 1 ( ( ~ ~ 1=- 01.413) 5.708 4.764 114.8 185.0
2.637 2.485 0.231 0.246
2.642 2.477 0.227 0.247
2.645 2.477 0.227 0.247
109.5' 109.6; 150.6; 0.888; 0.427; 0.085; 0.089; 0.043; 0.009*
111.1 109.1 138.8 0.905 0.432 0.076 0.089 0.043 0.009
115.5 109.3 141.5 0.910 0.432 0.076 0.089 0.043 0.009
;
where E,, and a,, are Lennard-Jones potential parameters. Bending Potential Functions ( Vknd). For bond angle bendings S i U A l , 0-Si-0, and 0-AI-0, the following potential function was used: (1 1)
The second term of right-hand side is introduced for the description of anharmonicity of angle-bending motions. The potential parameters of each potential sets to be refined are summarized in Table I. Normal Mode Calculation. Mass- Weighted Cartesian-Coordinates-Adopted Pseudolattice Method. For the normal mode calculation, the pseudolattice method proposed by No and Jhon22*33-35 is described in a mass-weighted Cartesian coordinate sys I(
[email protected]) -il=0
Bending
t
(12)
where R is the similarity transform matrix (RI-R = I), CP is the force constant real matrix represented by Cartesian coordinates, and M is the diagonal matrix of the masses. In this calculation, the normal modes and frequencies will be obtained by applying the pseudolattice method to the double four ring (D4R)subunit of A-type zeolite. For the description of a crystal with a finite model, each atom in the model must feel the same or similar potential fields as in the crystal. For this purpose, the pseudolattice method utilizes the cyclic boundary conditions and the translational symmetries of atoms. The methods based on the Block theorem (33) No, K. T.; Jhon, M. S . J. Phys. Chem. 1983, 87, 226. (34) Kim, J. S.;No, K. T.; Jhon, M. S. Bull. Korean Chem. SOC.1984, 5, 61.
(35) No, K. T.; Jhon, M. S . Bull. Korean Chem. SOC.1985,6, 183. (36) Gwinn, W. D. J. Chem. Phys. 1971, 55, 477. (37) Tyson, J.; Claassen, H. H.; Kim, H. J . Chem. Phys. 1971, 54, 3142. (38) Kim, H. Biopolymers 1982, 21, 2083.
+
Bending
+ Non-Bonding t Bending Figure 1. Top: since atoms 21 and 25 (22-26,23-27,24-28) have the same chemical environments, located at the same potential fields, the cyclic boundary condition is applied along the line connecting atoms 21-1-13-25 (22-5-9-26,23-3-15-27,247-11-28). Bottom: from the cyclic boundary conditions described above, the force constants results from the interaction of atom 25 with 13 (stretching, bending), 12, 14, 19 (bending) are stored to the submatrices of a, which represent the interactions between 21 and (13, 12, 14, 19). It is well described in ref ;
'The potential sets S1 and S2 consist of the electrostatic, harmonic bonding, nonbonding and bending potential functions. And in S3, the harmonic bonding potential function is replaced by the Morse type bonding potential function. k and k' are in mdyn/A, k" is in mdyn/ A2, ro is in A, @ is in degrees (Y is in l / A , De is in kcal/mol, e is in kcal/mol, and u is in A. bValues with an asterisk are taken from the ref 21 and use force field type harmonic bending potentials; these parameters are not included in our refinement.
vknd= yzz(k,,ye, - e,,y - k/(en - e,0)3) n
Bonding
: Non-Bonding Bonding
22.
have utilized the translational symmetries of the unit cell of crystal. Figure 1 shows how the force constant matrix, CP, is prepared by applying the pseudolattice method to the D4R model. Detailed descriptions of the model are written in our previous paper.22 In setting up CP, the cyclic boundary condition is made along the line connecting two O(2) atoms having an inversion center. Calculation of Force Constants from the Potential Functions. Force constants are described in Cartesian coordinates system as follows:
= (d2V/d#'dqy)
= C(d2V//d@"d@) = C q ( l ) I
/
(13)
where m, n and i, j represent the atoms m, n and the Cartesian coordinates, respectively, and I represents all types of the potentials in V. Since the potential functions are described by internal coordinates, the calculation of force constants includes the coordinate transformations. The force and force constants contributed from each potential terms are written in Appendix I. The calculating method of the forces acting on the atoms in a bending coordinate due to its bending motion represented in Cartesian coordinates is described in Appendix 11. Intensity Calculation of IR-Active Modes. Since the changes in dipole moment during the vibrational motion of zeolite framework are difficult to calculate, in this study, the dipole moment change was calculated from the displacement vector of the normal mode of the cluster shown in Figure 2. The displacements of atoms A', B', c', and D' in a normal mode can be obtained by symmetric operations of the displacements of atoms A, B, C, and D. These operators correspond to the symmetric elements of the normal mode. For a harmonic oscillator, the infrared absorption intensity for a fundamental transition of ith mode is simply given39 as 1,= f / 4 a € 0 ( N ~ a / 3 C Z ) ( ~ i / w i ) C ( a ~ / a ~ i )(1' 4) g
6186
No et al.
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988
TABLE 11: Net Forces of the Atoms Calculated at the X-ray Positions with the Potential Functions Obtained at = 0.581 mdyn
5
Cz
ex
5, c2 Ex 5
cz ex 5 Cz Ex 5
Ex 5 cz tX
5 F,
s1 -0.019
0.000 0.015 -0.013
0.000 0.012
-0.008 0.000 0.011
-0.001 0.000 0.028 0.006
-0.003 0.009
0.005 0.005 0.005 0.008 0.000 -0.008 0.002 331
Figure 2. In the calculation of dipole moment changes of the normal modes, the atoms A‘, B’, C’, and D’ are attached to the model, and the net atomic charges of the boundary atoms take the one-half of 60. From
application of the symmetric operations, the symmetric elements of the mode to the displacements of A, B, C, and D and the displacements of A’, B’, C‘, and D’ are obtained, respectively. where Q,, 6, and g represent the normal coordinates of the ith mode, dipole moment vector, and degeneracy of the given mode, respectively. This equation can be reduced as li = 974.8644x(ail/aQi)2
(15)
g
where Ii is in km/mol. In the calculation of aji/a&, two approximations are used. The atoms at the surface of the model in Figure 2, A, B, C, D, A’, B’, C’, and D’, get half of the net atomic charge of O(2) for neutrality, and the net atomic charges do not charge during the vibration. Since the mass-weight coordinate system was used in this calculation, the displacement matrix R can be obtained as
R = M’12.X
(16)
where X represents the atomic displacement matrix in mass-weight coordinates. The dipole moment change matrix is
D = R-6
(17)
where 6 represents the diagonal matrix of the net atomic charges. Therefore, the intensity of ith normal mode is given by N
I , = 974.8644[(Cd,(ij))2 j-1
N
N
+ (/=1c E y ( i j ) ) 2+ (Cd,(ij))’] j= I (18)
where d,(ij) and N represent the dipole moment change due to thejth atom in the x direction of ith normal mode and the number of atoms in model. Results and Discussions Potential Parameters. The refined potential parameters and the net atomic force still remaining on each atom after the optimization of the potential parameters are listed in Tables I and 11, respectively. Since in SI and S2 for each T-0 bonding pair (Si-O( l ) , Si-0(2), Si-0(3), A1-0( I ) , A1-0(2), and A1-O(3)) has one set of the harmonic potential parameters ( k , @), the net forces remaining on the atoms are relatively small compared with those on atoms calculated with potential S3. But in S3, all the Si-0 bonds, or all the A1-0 bonds, are described by one potential (39) Swanton, D. J.; Bacskay, G. B.; Hush,N. S. J . Chem. Phys. 1986, 84, 5 7 1 5 .
s2 -0.017
s3 -0.032
0.000
0.000
0.013 -0.013 0.000 0.013 -0.005 0.000 0.01 1 -0.004 0.000
0.091 0.068
0.023
0.005 -0.003 0.008 0.006 0.006 0.006
0.009 0.000 -0.007 0.001 898
0.000 -0.013 0.007
0.000 -0.001 -0.025
0.000 0.030 -0.016 -0.03 1 0.010 0.006
0.006 0.006 0.009 0.000 -0.007 0.018 681
curve. Therefore S3 is more realistic than SI and S2, although it gives larger net atomic forces. S3 can be used for the theoretical studies of the other aluminosilicates or even silicates and alumina. Since the 0-T-0 bending and nonbonding 0-0 coordinates are not independent, the coupling between these coordinates is large; therefore, the potential parameter related to 0-0 and 0-T-0 coordinates of S1 are quite different from those of S2 and S3, because the bending potential parameters of SI are excluded in optimization. These parameters are 0, k’, k”, and €. The packing effect of oxygen atoms also plays an important role for the determination of potential parameters of these bonds stretching and the Si-0-A1 angles bending. The potential paare very sensitive to the rameters ro, uw, OT+T, and structure of the framework. Force Constants. The force constants of each atom are calculated in the Cartesian coordinate system with the exceptionally obtained crystal geometry. Only the diagonal terms contributed from each potential energy function (electrostatic, harmonic or Morse, Lennard-Jones, bending, and torsional potential functions) are listed in Table 111. The force constants contributed from the polarization interaction are excluded in the calculation because their contribution is negligibly small and their calculation is very time consuming. Since the electrostatic potential is a long-range interaction, its contribution to the force constant is very small compared with the other potential functions, although most of the stabilization energy, more than 90%, is contributed from the electrostatic interaction. Since the Si atom is located at a sharp and deep potential minimum,26 it has a large stability in the framework. But A1 atoms can be removed in an acid-catalyzed condition. The force constants k , and kyyof O(1) atom, the vector 2 y’ directed toward the center of eight-ring, are relatively small, and O( 1) has a small stabilization energy compared with the other oxygen atoms. Therefore, the first step of dealumination may be the dissociation of A1-0( 1) bond. Assignments of Calculated Normal Modes and Frequencies to the IR Absorption Bands. The vibrational frequencies, symmetric species, and relative infrared absorption intensities of the normal modes calcuated with each potential set SI, S2, and S3 are summarized and are compared with the Raman and IR spectra of Na A-zeolite in Table IV. In the D,d symmetry point group, B2 and E modes are both IR and Raman active, A1 and BI modes are Raman active only, and A2 modes are both Raman and IR inactive. In the stretching region, the frequencies of S2 are smaller than those of S3. But in the low-frequency region, the vibrational frequencies of S2 and S3 belong to the same mode, have similar magnitudes. Since the vibrational motions of the pseudolattice D4R model are invariant under the translational operation dif-
+
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6787
Lattice Vibrational Calculation of A-Type Zeolite
TABLE 111: Diagonal Terms of the Force Constants Contributed from Each Potential Energy Function (in mdyn/A)
bonding Kx Ky K, K, Ky
A1
Si
K, O(1)
O(3) 00)
K, Ky K, K, Ky K,
nonbonding
elctrst"
S1
s2
S3
0.034 -0.010 -0.024 -0.004 0.006 -0.002 0.139 0.123 -0.347 -0.131 -0.017 0.147
3.671 3.676 3.968 6.840 6.541 7.012 -0.088 0.748 7.139 3.280 3.946 0.284 -0.052 5.668 3.273
3.689 3.698 3.997 6.832 6.524 7.002 -0.087 0.750 7.157 3.287 3.942 0.283 -0.057 5.679 3.278
4.431 4.378 4.788 6.571 6.584 6.908 -0.061 0.844 7.735 3.681 4.159 0.372 -0.103 5.604 3.218
Kx
KY
SI
0.045 0.065 0.286 0.164 0.198 0.055 0.086 0.283 0.173
S2
0.045 0.065 0.289 0.165 0.199 0.055 0.087 0.286 0.174
bending
total
S3
Slb
S2
S3
torsionOsb
SI
S2
S3
0.046 0.066 0.294 0.168 0.202 0.560 0.088 0.290 0.177
1.630 1.577 1.615 0.904 0.955 0.958 0.773 0.706 0.071 0.338 0.416 0.736 0.689 0.261 0.447
1.726 1.656 1.692 0.925 0.914 0.923 0.740 0.678 0.086 0.339 0.436 0.708 0.735 0.293 0.457
1.731 1.661 1.697 0.925 0.914 0.923 0.741 0.679 0.086 0.339 0.437 0.709 0.736 0.293 0.457
0.355 0.218 0.295 0.249 0.417 0.348 0.821 0.058 0.007 0.252 0.250 0.498 0.053 0.019 0.032
5.691 5.460 5.854 7.989 7.919 8.316 1.775 1.699 7.156 3.903 4.792 1.720 0.776 6.230 3.925
5.805 5.562 5.961 8.002 7.862 8.271 1.658 1.674 7.192 3.912 4.810 1.691 0.817 6.277 3.941
6.551 6.247 6.756 7.741 7.921 8.177 1.686 1.769 7.774 4.309 5.031 1.781 0.772 6.206 3.884
"The contribution of electrostatic and torsional potential functions to the force constants of the S1, S2, and S 3 are the same. bTaken from the ref TABLE IV: Vibrational Frequencies, Symmetric Species, and Relative Infrared Absorption Intensities of the Normal Modes Calculated with Each Potential Function Set" s1 s2 s3
IR 1090 (w) (1091 (E)) 1050 (w) (1068 (E))
995 (s) (981 (E))
740-750 (VW,sh) (733 (E))
660 (w) (679 (B2))
550 (ms)(589 (B2))
464 (m)(450 (B2))
378 (ms)(381 (E))
260 (W) (260 (B2))
freq
int
freq
int
freq
1071 B2 1066 A1 1057 E 1031 E 1025 A1 1007 B2 971 A2 969 E 966 B1 857 B2 853 E 788 B1 787 A1 775 E 774 A1 774 B2 772 A2 743 E 723 A1 718 B2 707 E 688 A1 668 E 646 B2 621 A1 612 E 553 B2 552 B1 550 A2 509 E 474 E 473 B1 450 B2 449 A2 448 A1 435 E 355 A2 306 B1 390 E 262 E 254 B2 252 B1 243 A2 229 E
1511
1072 B2 1067 A1 1058 E 1032 E 1027 A1 1008 B2 970 A2 969 E 965 B1 861 B2 857 E 791 AI 790 B1 781 E 778 A1 775 A2 774 B2 745 E 725 A1 723 B2 708 E 689 A1 668 E 649 B2 621 A1 615 E 553 B1 552 B2 551 A2 508 E 469 E 468 B1 448 A2 447 B2 443 A1 434 E 352 A2 316 B1 294 E 260 E 252 B1 251 B2 247 A2 230 E
1525
1095 B2 1091 A1 1086 E 1028 E 1027 A1 1000 B2 991 A2 985 B1 980 E 903 E 899 B2 828 B1 812 A1 811 A2 197 E 790 B2 789 A1 759 E 737 B2 732 E 727 B2 712 A1 688 E 680 8 2 650 A1 641 E 573 B1 571 A2 567 B2 520 E 476 B1 475 E 456 B2 455 A1 452 A2 445 E 360 A2 307 B1 288 E 258 B1 258 E 245 B2 244 A2 229 E
20 646 68 785 185 749 26
10 275 88 14 28 1204 962 496 12 14 99 122 4 53 20 2
16 666 68 782 188 754 19 16 287 109 11 29 1165 942 496 12 14 101 125 0.4 50 17
3
int
Raman
1779
1100 (w) (1088 (Bl))
0.02 1121
1037 (w)
0.01 977 (w) (977 (B2)) 536 773 129
39 4 209 0.0009 17 225
700 (w)
40 875 719 540 10 490 (s) (474 (E)) 9 77 410 (w) (412 ( A l ) ) 146 0.07 44 7
340 (w) 280 (w) (259 (A2))
3
"Vibrational frequencies and intensities are in cm-' and km/mol, respectively. The values in parentheses are the assignments of ref 22. ferent from the vibrational motions of D4R in zeolite framework, only 3N-3(69) vibrational modes may be obtained. In the cluster model, 3N-6(66) vibrational modes will be obtained. To classify the vibrational modes into the mainly contributing internal coordinate motions, we carried out the normal mode
calculations with the parts of the potential energy functions in S2 or S 3 . In Tables V and VI, the vibrational frequencies calculated with S 2 and S 3 are listed, respectively. Both potential sets show similar tendencies in the contribution of each potential term to the vibrational frequencies, although the frequencies
6788 The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 TABLE V Vibrational Frequencies Obtained with Several Combinations of the Potential Functions in S2 ( ~ r n - ' ) ~ bonding bonding bonding + bonding + bonding electrostatic nonbonding bending torsion total
+
1049 B2 1041 E 1005 E 986 ~2 924 E ai6
1041 B2 1031 E 1004 E 982 ~2 923 E a12 E a06 ~2 664 B2 661 E 651 E 637 E 617 B2 586 ~2 557 E 514 E 339 E 330 B2 309 E 298 ~2 124 E 120 E
E
ai0 ~2 664 B2 662 E 655 E 642 E 624 B2 585 ~2 558 E 515 E 344 E 331 B2 295 E 292 B2 66 E 63 B2
+
1057 B2 1050 E 1014 E
996 B2 928 E 828 E a22 ~2 679 B2 677 E 666 E 654 E 643 B2 587 ~2 566 E 519 E 357 E 352 82 308 ~2 307 E 101 E ai 82
1066 B2 1054 E 1023 E 999 B2 951 E a48 ~2 a39 E 760 E 730 B2 711 E 691 B2 670 E 642 B2 626 E 592 E 475 B2 456 E 418 ~2 408 E 333 E 257 E
1054 B2 1045 E
iooa E
989 ~2 936 E a21 E a12 82 685 E 669 B2 667 E 663 E 661 B2 611 E 590 B2 543 E 450 B2 386 E 328 E 305 B2 279 E 142 E
1072 B2 1058 E 1032 E iooa 82 969 E 861 B2 a57 E 781 E 774 B2 745 E 723 B2 708 E 668 E 649 B2 615 E 551 82 508 E
469 E 447 82 434 E 294 E 260 E
Here the bonding potential is the harmonic potential. TABLE VI: Vibrational Frequencies Obtained with Several Combinations of the Potential Functions in S3 bonding + bonding + bonding bonding + bonding electrostatic nonbonding bending torsion total
+
1073 B2 1069 E
999 E 977 B2 939 E a59 E a49 ~2 693 E 676 E 665 B2 655 E 645 B2 621 82 584 E 546 E 360 E 351 B2 316 E 306 B2 53 E
(I
1063 B2 1058 E 999 E 976 B2 937 E a55
E
a46 ~2 690 E 671 E 664 B2 656 E 640 B2 622 82 583 E 545 E 356 E 351 B2 329 E 313 B2 133 E
1081 B2 1079 E 1036 E 988 ~2 943 E a69 E a60 ~2 703 E 686 E 674 B2 672 E 669 B2 623 B2 592 E 550 E 373 E 371 B2 327 E 323 B2 95 E 56 B2 47 E
1090 B2 1082 E 1019 E 990 B2 963 E a87 ~2 a84 E 777 E 739 B2 730 E 703 B2 696 E 674 B2 647 E 616 E 487 ~2 468 E 427 82 419 E 336 E 252 E 203 E 72 E
1078 ~2 1073 E 1001 E 980 B2 950 E 865 E 852 ~2 713 E 698 E 678 ~2 664 B2 660 E 639 E 625 B2 573 E 466 B2 404 E 344 E 319 B2 287 E 139 E 125 E a7 ~2
1095 B2 1086E 1028 E 1000 B2 980 E 902 E a98 ~2 797 E 790 B2 759 E 737 B2 732 E 727 B2 688 E 680 ~2 641 E 567 B2 520 E 475 E 456 B2 445 E 288 E 258 E 245 B2 229 E
Here the bonding potential is the Morse type potential.
calculated with S3 are a little higher than those calculated with S2 throughout the whole frequency range. The electrostatic interactions, Madelung potential, lowering the frequencies correspond to stretching motions and raising the low frequencies. Because the forces due to electrostatic and bonding interactions in a T-0 bond have opposite directions. But in low-frequency
No et al. motions, the forces of nonbonding and electrostatic interaction of the 0-0 pair have the same direction. The bending potential raises vibrational frequencies throughout the whole frequency region, especially in the region lower than 800 cm-I, and its contribution to normal mode is dominant. The influence of nonbonding interactions on the T-0 stretching motion is large because the oxygen atom of a T-0 bond feels the repulsive force from the other three oxygen atoms in the same tetrahedron, TO.,, and the sum of these repulsive force vectors almost coincides with the T-0 bond axis. The nonbonding interaction raises the vibrational frequencies over the whole range. The contribution of torsional coordinates to normal modes distinctly appeared lower than 550 cm-l. In this region, the degree of contribution of internal coordinates to the normal modes are different from one mode to another, because the vibrational modes in this range are coupled modes of several internal coordinates. From above analysis, the absorption bands of A-type zeolite may be classified into three regions as listed in Table VII. The modes in region I1 may be a coupled mode of symmetric stretching and bending. In region 111, the coupling of several internal coordinates may produce several characteristic motions of a four-ring or four-rings. Since the D4R is used as a model in this study, the characteristic six-ring motion of zeolite cannot appear here. The band assignments were made with the calculated normal modes based on Flanigen's4 and our previous assignments.22 The assignments of the IR absorption band with the calculated normal modes are summarized in Table VII. The calculated intensities are quite different from the intensities of the observed bands corresponding to those modes. This disagreement may be caused by the assumptions used for the calculation of dipole moment changes described above. In the assignment of 1090-cm-' IR band, the B2 mode with strong intensity (1072 and 1095 cm-I in S2 and S3, respectively) looks to be the most probable candidate for this band. But it was reported in a previous calculation that the frequency of this B2 mode is insensitive to the compositions of Si and Al, contrary to the dependency of the 1090-cm-' band on Si/Al. The band is obtained at 1131 and 1151@cm-I when Si/Al is 1.79 and 3.00, respectively. Therefore the 1090-cm-1band may be assigned to the first E mode. The motions corresponding to 1090- and 995-cm-I bands contain large parts of Si-0 and A1-0 internal coordinates, respectively. In the range of 740-750 cm-', the band corresponding to the characteristic doubling motion of A-type zeolites is observed to be very weak and sharp; this can be assigned to 745 cm-' (S2) (759 cm-I (S3)) of the E mode. The 660-cm-' band is sensitive to %/AI, and the frequency of B2 mode has the same tendency;22therefore this band may be assigned to 649 cm-I (S2) (680 cm-I (S3)) B2 mode. The modes corresponding to 550- and 464-cm-I bands are the characteristic motions of a double four ring and are assigned to B2 modes, 552 cm-I (S2) (567 cm-' (S3)) and 447 cm-' (S2) (456 cm-I (S3)), respectively. The 378-cm-' band corresponds to the characteristic motion of a six-ring, but there is no such six-ring mode near 380 cm-I in our calculation because the D4R model is too small for the description of six-ring or eight-ring motions. The 260-cm-l band is the pore-opening motion of double four-ring and is assigned to 260 cm-' (S2) (258 cm-' (S3)) of the E mode. As a whole the vibrational frequencies in region I1 are higher than the absorption bands. The assumption of linearity of Si-0(2)-Al bonds and the introducing of the torsional potential without optimization may
TABLE VII: Assignment of the Calculated Modes to IR Absorption Bands (crn-')
IR 1090 (w) 1050 (w) 955 (s) 740-750 (VW, sh) 660 (w) 550 (ms)
464 (m) 378 (ms) 260 (w)
s1 1057 (w, E) 1031 (m, E) 969 (s, E) 743 (m, E) 646 (vs, B2) 553 (m,B2) 450 (w, B2)
s2 1058 (w. E) 1032 (m, E) 969 (s, E) 745 (m, E) 649 (vs, B2) 552 (m.B2) 447 (m,B2)
262 (w, E)
260 (w, E)
s3 1086 (vw, E) 1028 (m,E) 980 ( s , E) 759 (m,E) 680 ( s , ~ 2 ) 567 (m.B2) 456 (m, B2)
dominant internal coord' I (str (bonding) bending NB)* I1 (str (bonding) bending)< 111 (str (bonding) bending torsion
258 (w, E)
'NB = nonbonding. *In the region 850-1090 cm-'.c I n the region 550-850 cm-I.
the region 258-550 cm-'
NB)d
Lattice Vibrational Calculation of A-Type Zeolite
Center o f m a s s Figure 3.
be the reasons for the overestimation of the force constants corresponding to the deformational motions of the framework. Since the model size is small, the force constants of the motions including several internal coordinates may be overestimated. Although our potential sets cannot give better vibrational frequencies compared with those obtained from BLSF-BR force constant sets, the framework structures are well described with those potential sets. And all the force constants can be obtained from potential functions as an analytical form different from other empirical force constant sets.21q22Since these potential sets give the equilibrium geometry of zeolite, a reasonably small net force on each atom, and a reasonable first derivative of force on each atom, these potential sets may be used for the dynamic studies of zeolite framework including bound ions.41 Therefore it is possible to calculate the dependency of vibrational frequencies upon not only Si/AI but also the structural changes of the framework. These structural changes may occur due to the binding of the molecule to the framework, especially waters. Since the force constants described by the orthogonal internal coordinates set are very useful and chemists are well acquainted with them, the coordinates transformations between force constant spaces, from a Cartesian coordinate system to an orthogonal internal coordinate system, are being prepared by the mode to be projected to the internal coordinates for more accurate assignments of the vibrational bands.
Appendix I. Forces and Force Constants Represented in Cartesian Coordinates Electrostatic. N
@ = C iim6,,(8 - $)/r,3 n#m N
flm= C 6,6,(3(qp - q;")2/rmn5 - 1/rm,,9 n#m
w = 6,6,(-3(q; = -36,6,(qY
- qf)2/rmn5+ 1/rmn3) - qF)(qy - q y ) / r m 2
Polarization. N
N
n#m
n#m
@ = an(C d n ( 8 - p ) / r m n 3 ( - C an/rm: + N
C 36n(@ - 47")2/rm,5)) n#m Nonbonding (Lennard-Jones Potential). N
@ = nC 4cmn(6(~mn/rmn)~ - 12(umn/rmn)")(8 - P ) / r m ; #m (40) Breck, D. W. Zeolite Molecular Sieves; Wiley: New York, 1974. (41) Shin, J. M.; No, K. T.; Jhon, M. S . J . Phys. Chem. 1988, 92,4533.
