J. Phys. Chem. 1996, 100, 11101-11112
11101
Mobility of Aromatic Molecules in Zeolite NaY by Molecular Dynamics Simulation Harald Klein*,† and Hartmut Fuess Fachgebiet Strukturforschung, Fachbereich Materialwissenschaft, Technische Hochschule Darmstadt, Petersenstrasse 20, D-64287 Darmstadt, Germany
Gerhard Schrimpf Institut fu¨ r Physikalische Chemie, Technische Hochschule Darmstadt, Petersenstrasse 20, D-64287 Darmstadt, Germany ReceiVed: February 23, 1996X
Molecular dynamics (MD) simulations of benzene, p- and m-xylene, and m-nitroaniline in zeolite NaY were performed between 20 and 700 K at guest molecule coverages of 1, 16, and 32 molecules per unit cell (uc) to interpret and compare the results of quasi-elastic neutron scattering (QENS) and nuclear magnetic resonance (NMR) studies at the atomic scale. The predominant rotational motion of benzene and xylenes is the rotation about the 6-fold (pseudo-6-fold) axis in good agreement with QENS studies. The rotational motions of m-nitroaniline are predicted to be essentially slower than those of benzene and xylenes. The diffusional mechanism for benzene and xylenes can be described as a jump model with high residential probability at the cation adsorption site. The jumps are tetrahedral rearrangements of the arene molecules, in good agreement with recent NMR studies.
1. Introduction The dynamic properties of the reactants and products of technically important reactions within the voids of the zeolite host, as well as the concentration and nature of acid sites, determine the activity and selectivity of zeolite catalysts.1,2 Information about the dynamics of these molecules, together with knowledge of adsorption sites, forms the basis for understanding the various properties of zeolites, i.e., heterogeneous catalysis, molecular separations, and adsorption. The dynamic processes of interest are translational, rotational, and vibrational motions. The dynamic properties of guest molecules in zeolites can be investigated experimentally at the atomic scale, especially by NMR,3-5 QENS,6-10 and inelastic neutron scattering11 studies. QENS experiments give information about fast motions (correlation times 1 ns). The molecular dynamics of benzene and m- and p-xylene obtained by these two complementary spectroscopic methods are summarized in Table 1. Computer simulation on zeolites to study the sorptive behavior of guest molecules is one of the significant developments in zeolite science in recent years. These simulations back the experimental results because they are able to resolve the structure and dynamics of single molecules. Spectroscopic or diffraction studies, however, mostly yield mean values, from which the individual adsorption sites and molecular dynamics cannot be unambiguously obtained. Calculations of potential energy maps,12 energy minimization techniques,13 and Monte Carlo simulations14 provide information on static and thermodynamic properties. The method of molecular dynamics simulation additionally gives access to time-dependent properties like diffusion coefficients and diffusion mechanisms. This * Corresponding author. † Present address: Davy Faraday Research Laboratory, The Royal Institution of Great Britain, 21 Albemarle Street, London W1X 4BS, United Kingdom. X Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(96)00575-8 CCC: $12.00
technique has been widely applied to the investigation of the diffusional behavior of guest molecules in zeolites.15-27 The present paper analyzes the molecular motions (vibration, rotation) and diffusional mechanisms of benzene and three of its derivatives (m-xylene, p-xylene, and m-nitroaniline) in zeolite Y by using MD simulations. We will present a more detailed picture of the molecular motions of these guest molecules as obtained from QENS10 and NMR3-5 studies alone. 2. The Model: Structure and Force Field The structure of zeolite Y is isomorphous with that of the mineral faujasite, as previously determined by X-ray28,29 and neutron powder diffraction.30 The space group is Fd3hm with lattice parameters in the range 24.60-25.12 Å. The framework is composed of cubooctahedral sodalite cages linked together in a tetrahedral arrangement by 6-membered rings of O(1) atoms to form large cavities, called supercages. The supercages are interconnected by 12-ring windows consisting of 12 Si/Al and 12 O atoms, as shown in Figure 1, where the orientation allows the 6-membered ring in the back to be seen through the 12ring window at the front. Three additional 12-ring windows are at the top, lower left, and lower right. One unit cell contains eight sodalite cages (centered at position 8a: 1/8, 1/8, 1/8) and eight supercages (centered at position 8b: 3/8, 3/8, 3/8). Extra-framework cation sites were located31,32 at the following positions, as shown in Figure 1: (i) at the center of the hexagonal prism (SI site, situated at the origin 3hm (0, 0, 0), where octahedral coordination to O(3) is achieved. (ii) Along the (111) 3-fold axis for lower coordinated cations at position 32e. Site I′ (SI′) (x ) y ) z ≈ 0.06) inside the sodalite cage enables coordination to three O(3) framework oxygens. Further toward the supercage is site II′ (x ) y ) z ≈ 0.21), where coordination to three O(2) oxygens of the sodalite to the supercage aperture is obtained. Slightly inside the supercage is site II (x ) y ) z ≈ 0.23). Site II* (x ) y ) z ≈ 0.25) is nearly analogous to SII, but shifted to the center of the supercage. Cations at these two sites are coordinated with three O(2) oxygens of the 6-ring window. (iii) At sites III and III′ © 1996 American Chemical Society
11102 J. Phys. Chem., Vol. 100, No. 26, 1996
Klein et al.
TABLE 1: Molecular Dynamics of Benzene and Xylenes as Obtained by NMR3-5 and QENS10 Studies benzene3,4
T (K) >250 250-150 100-150
NMR QENS NMR QENS NMR QENS
m- and p-xylene5,10
liquid-like diffusion
liquid-like diffusion libration about 6-fold axis tetrahedral jumps, rotation about 6-fold axis libration about 6-fold axis rotation about 6-fold axis rigid molecule
tetrahedral jumps, rotation about 6-fold axis rotation about 6-fold axis
NaY22 with a Si/Al ratio of 3.0. This value enables the complete filling of SI and SII by sodium. The resulting system consists of 192 Si/Al, 384 O, and 48 Na atoms. No distinction is made between Si and Al. In the following these atoms are described as T. The mass of silicon is used for these T atoms. The zeolite framework is flexible, whereas the aromatic molecules are kept rigid by using the SHAKE algorithm.38 The rationalization for employing rigid aromatic molecules is contained in the structural studies of these guest molecules in zeolite NaY,30,33-37 where it could be shown that the bond lengths and angles are not changed by the interaction with the zeolite. The potential energy V of the simulated zeolite-aromatic guest molecule system is calculated by the summation of three terms: the bonded interaction energy of the zeolite, Vzeo; the nonbonded interaction energy between zeolite and guest molecule, Vgm-zeo; and the nonbonded interaction energy between the guest molecules, Vgm-gm.
V ) Vzeo + Vgm-zeo + Vgm-gm
Figure 1. Structure of zeolite Y (faujasite), the vertices of which are occupied by T atoms (T ) Si/Al). The projection is along [111]. The four crystallographic oxygen framework atoms are marked by the numbers 1-4, and the cation positions are characterized by roman numerals (see text).
[position 192i, e.g., (x ) 0.017, y ) 0.418, z ) 0.084)], near the four-membered rings inside the supercage. (iv) Special cation positions, often found in hydrated faujasite samples, are site IV at the center of the supercage (position 8b, x ) y ) z ) 3/8) and site U at the center of the sodalite cage (position 8a, x ) y ) z ) 1/8). The preferred adsorption site of benzene30 is situated in front of the 6-ring window, with a distance from the center of mass to the Na-SII cation of 2.7 Å. We denote this position as cation site A or CA site (see Figure 2a). The aromatic plane intersects the [111] axis at right angles. A second benzene position in the 12-ring window of the supercage was found at high coverages. This position is called window site or W site (see Figure 2b). Similar positions in zeolite Y could be detected for xylenes, mesitylene, and aniline33-36 by neutron powder diffraction. The window site was not occupied, with the exception of aniline, while the cation site is known to exist in two different molecular orientations, namely, cation sites A and B (CA and CB, cf. Figure 2c,d). In position CB, the molecule is rotated by 30˚ relative to position CA. The carbon and hydrogen atoms in position CA are in the general position 192i of space group Fd3hm. The guest molecule atoms in position CB are located in the special position 96g. X-ray diffraction studies37 showed that polar aromatic molecules like m-nitroaniline (Figure 3) and m-dinitrobenzene are adsorbed at other positions. The preferred adsorption sites of these molecules are unique, depending on the positions of cations and the formation of H-bonding to framework oxygen. One unit cell, containing eight supercages, is used to construct the simulation box. We used the Demontis model of zeolite
(1)
All parameters used in this study are listed in Table 2. The bonded interaction energy Vzeo of the zeolite framework was modeled with the anharmonic approximation, as described in one of our recent papers.26 This force field is based on effective T-O, O-O, and Na-O pair potentials between neighboring atoms. Additionally, the repulsive interactions between the sodium ions are taken into account by shifted Coulomb potentials. It was demonstrated in ref 26 that this force field, despite its simplicity, reproduces the main structural and dynamic properties of zeolite NaY and allows for an efficient exchange of energy between the zeolite and the sorbate. The interaction energy Vgm-zeo between the guest molecule and the zeolite is described by the sum of a Lennard-Jones and a Coulomb potential:
(
Vgm-zeo ) ∑∑ BRβrij-12 - CRβrij-6 + i
j
qRqβ 4π0rij
)
(2)
where rij is the interatomic distance. The Lennard-Jones parameter CRβ describes the dispersion interaction between atoms R and β, whereas BRβ characterizes the short-range repulsive energy. Subscript i runs over all atoms of the guest molecule, and j runs over all atoms of the zeolite. The short-range Lennard-Jones interactions with Si and Al are ignored because they are well shielded by the oxygen atoms of the SiO4 and AlO4 tetrahedra.22 The Lennard-Jones parameters used in this study were described in one of our recent papers.37 The interactions with the methyl and amino groups are modeled via the united-atom potential approximation39 to reduce the number of force centers and the degrees of freedom. The nitro group is not described via the united-atom potential approximation. It is kept rigid, so that the nitrogen and oxygen atoms are located in the plane of the aromatic ring. The force field did not include H-bonding. Interaction with the amino group of the guest molecules therefore is only
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J. Phys. Chem., Vol. 100, No. 26, 1996 11103
Figure 2. Adsorption sites of nonpolar aromatic molecules as determined by neutron powder diffraction: (a) the cation site A (CA site) and (b) the window site (W site) of benzene.30 Substituted benzenes, like p-xylene,34 often show two orientations at the cation site, cation site A (CA site) (c) and cation site B (CB site) (d). For the p-xylene molecules (c, d) only one of three symmetry-equivalent configurations are shown. The other two configurations can be obtained by rotation of 120° around the pseudo-6-fold axis.
described by a Lennard-Jones and an electrostatic interaction potential. The interaction between the guest molecules Vgm-gm (eq 3) is modeled by the sum of a Coulomb and a Buckingham potential corresponding to ref 22. Subscripts i and j run over all guest molecule atoms.
Vgm-gm )
1
(
ARβe-r /F ∑ ∑ 2 i j ij
Rβ
- CRβrij-6 +
qRqβ
)
4π0rij
(3)
Lennard-Jones, Buckingham, and Coulomb potentials have been modified to shifted-force potentials Vsf (rij) according to
{
( )
r3 - rc2 dV(rij) Vsf(rij) ) V(rij) - V(rc) drij 3r 2 0
}
for rij g rc
c
rij)rc
for rij < rc
(4)
by using periodic boundary conditions and a cutoff radius of
12 Å. The use of a shifted-force potential instead of the Ewald summation method40 for the calculation of the Coulomb interaction between guest molecule and zeolite is justified for this special case because Dufner41 could demonstrate that this shifted-force potential reproduces the electrostatic field of faujasites with a good approximation compared to the timeconsuming Ewald summation technique. These long-range electrostatic interactions are calculated with all zeolite (including silicon and aluminum) and guest molecule atoms. The partial atomic charges of the guest molecules used in this study (Table 2) were described in our recent paper.37 The charges for the framework atoms (Si/Al ) +1.2, O ) -0.7, Na ) +0.8) were taken from ref 22. 3. Calculations The equations of motion were integrated by using the Verlet algorithm42 with a time step of 1.0 fs. Periodic boundary conditions were used to take into account the periodicity of the crystal. By starting from experimental coordinates for the zeolite as well as for the guest molecules (obtained by diffraction
11104 J. Phys. Chem., Vol. 100, No. 26, 1996
Klein et al. TABLE 2: Potential Parameters (a) Guest Molecule-Zeolite NaY, Short-Range Parameters37 (Lennard-Jones Interaction) Lennard-Jones interaction atom pair (host atom-guest atom)
BRβ (106 kJ mol-1 Å12)
CRβ (103 kJ mol-1 Å6)
O-C O-H O-CH3a O-N O-NH2a O-O Na-C Na-H Na-CH3a Na-N Na-NH2a Na-O
2.258 0.251 5.288 1.734 4.656 1.452 1.293 0.105 3.662 0.909 3.289 0.624
3.107 0.833 5.062 2.728 4.446 3.444 0.952 0.236 1.534 0.849 1.333 1.066
(b) Guest Molecule-Zeolite NaY, Long-Range Parameters37 (Partial Charges for Coulomb Interaction) partial charges of the guest molecule atoms (e) guest molecule Figure 3. Adsorption site of m-nitroaniline as determined by X-ray powder diffraction.37 The nitro group is located in front of the Na SII cation, and the amino group points into the 12-ring window, presumably stabilized by hydrogen bonding with oxygen framework atoms.
experiments of the host-guest systems),30,33-37 and a MaxwellBoltzmann distribution of the atomic velocities, the systems were equilibrated for 50 ps. Subsequently, we performed MD runs in the NVE ensemble over an additional period of 100 ps and extended MD runs over an additional period of 1.5 ns. The 100 ps runs carried out with the guest molecules benzene, m-xylene, p-xylene, and m-dinitrobenzene at five different temperatures (20, 100, 300, 500, 700 K) have been used for comparison with the QENS data.10 The results at 20 K must be seen as more inaccurate than the higher temperatures due to the neglect of quantum effects, which also contribute to the total potential energy at such low temperatures. The guest molecule coverage corresponds to the zero coverage limit (1 molecule/uc). However, we have performed these MD runs using 16 molecules/uc without any interaction between the guests to improve the statistics for the subsequent evaluations. To study the effect of guest molecule loading, we performed two additional MD runs with benzene at loadings of 16 and 32 molecules/uc, respectively. The extended MD simulations of 1.5 ns were performed with the guest molecules benzene and p-xylene at the zero coverage limit (1 molecule/uc, 16 molecules without guest-guest interaction) for comparison with the results of the NMR experiments.3-5 4. Results 4.1. Internal Adsorption Energy and Energy Histograms. The internal adsorption energy ∆Uads is defined as the time average of the potential energy and can be calculated by summation of the mean guest-guest 〈Vgm-gm〉, the mean guesthost 〈Vgm-zeo〉, and the destabilization interaction energy ∆Vzeo:
∆Uads ) 〈Vgm-gm〉 + 〈Vgm-zeo〉 + ∆Vzeo
(5)
The destabilization interaction energy ∆Vzeo is the difference between the mean potential energies of the bare and the loaded zeolites. This interaction energy is small compared to 〈Vgm-gm〉 and 〈Vgm-zeo〉, and ∆Vzeo was less than 1 kJ/mol for all simulations.
benzene m-xylene
p-xylene m-nitroaniline
CH3a, NH2a, N, O
C
H
C ) -0.153 C1 ) -0.018 C2 ) -0.168 C4 ) -0.166 C5 ) -0.142 C1 ) -0.029 C2 ) -0.155 C1 ) +0.244
H ) +0.153
C2 ) -0.079 C3 ) +0.183 C4 ) -0.108 C5 ) -0.148 C6 ) -0.146
H2 ) +0.147 H4 ) +0.150 H5 ) +0.151 H2 ) +0.150 H2)+0.163 H4 ) +0.205 H5 ) +0.169 H6 ) +0.171
CH3 ) +0.040
CH3 ) +0.039 N ) +0.158, O ) -0.316 NH2 ) -0.179
(c) Guest Molecule-Guest Molecule, Short-Range Parameters22 (Buckingham Potential)b Buckingham potential parameters atom pair (guest atom-guest atom)
ARβ (kJ mol-1)
FRβ (Å)
CRβ (kJ mol-1 Å6)
C-C H-H C-H
369743.0 11971.0 65485.0
0.2777 0.2673 0.2724
2439.8 136.4 573.0
a
United-atom potential approximation. b Buckingham potentials have been used for benzene-benzene interactions only.
The internal adsorption energy ∆Uads increases nearly linearly with temperature for all guest molecules. At 300 K, at the zero coverage limit, the following internal adsorption energies were determined: -89 kJ/mol, benzene; -76.1 kJ/mol, m-xylene; -76.1 kJ/mol, p-xylene; -116 kJ/mol, m-nitroaniline. A higher benzene coverage causes a decrease in ∆Uads (16 molecules/ uc, ∆Uads ) -90 kJ/mol; 32 molecules/uc, ∆Uads ) -98 kJ/ mol), which can be explained by additional guest-guest interactions. Reasonable agreement with the experimentally determined adsorption heat, Qads (Qads ) -∆Uads + RT), of benzene in zeolite NaX43 (Si/Al ) 1.18) at 323 K (83.5 kJ/mol-1) has been obtained (calculated heat of adsorption for benzene, 1 molecule/ uc at 300 K, is 91.4 kJ/mol-1). For p-xylene we obtained, at the zero coverage limit at 300 K, a heat of adsorption of 78.5 kJ/mol-1. This compares well with the calculated heat of adsorption of Schrimpf et al.,44 who obtained Qads ) 83 kJ/ mol-1 at 300 K from MD simulations at the zero coverage limit. Good agreement with experimental results (Ruthven and God-
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J. Phys. Chem., Vol. 100, No. 26, 1996 11105
Figure 4. Radial distribution functions between the center of mass (COM) of benzene and the position of the experimentally determined adsorption site (ADS) between 20 and 700 K at zero coverage limit (1 molecule/uc).
dard,45 77 kJ/mol-1 between 403 and 443 K; Simonot-Grange et al.,46 85 kJ/mol-1 at 300 K) is also observed. The differences between experimental and calculated heats of adsorption presumably are due to the inaccuracy of both the theoretical model and the experimental method. 4.2. Radial Distribution Functions. Analysis of the radial distribution functions gAB(r) yields information about the structural characteristics of sorbates in the pores of zeolites. The radial distribution function is defined as the probability that two centers A and B are separated by a distance r with regard to a statistical distribution of both centers A and B:
gAB(r) )
〈
〉
dnAB 1 dr 4πr2ρ N B A
(6)
NA describes the number of A centers, FB is the macroscopic density of the B centers, and nAB is the number of AB pairs. The integral IAB(r) of the radial distribution function specifying the number of B centers surrounding an A center within a distance range from 0 to r often completes the interpretation of gAB(r):
IAB (r) ) 4πρB∫0 gAB(r ′)r ′2 dr ′ r
(7)
The radial distribution functions yield scalar information only. To obtain spatial information, we have calculated three radial distribution functions between the center of mass (COM) of the guest molecule and the following positions of the zeolite Y structure: center of supercage (COC), center of the 12-ring window (WIN), and experimental adsorption site of benzene in zeolite NaY30 (ADS). A selection of these radial distribution functions is shown in Figures 4 and 5. Figure 4 shows that benzene (zero coverage limit) has the highest g value above 100 K at the experimentally determined adsorption site. At 100 and 20 K, the maximum of the radial distribution function is shifted about 0.4 Å from the experimental adsorption site. The shift is mainly lateral, away from the [111] axis toward the zeolite wall. This is in good agreement with recent results.37 We have indeed shown by static molecular mechanics calculations with a rigid zeolite lattice that the minimum of the potential energy of benzene and the xylenes is laterally slightly shifted off the [111] axis. At higher temperatures (T > 100 K), sites with higher potential energies such as the experimental adsorption site are also occupied, resulting in the calculated maximum
Figure 5. Radial distribution functions between the center of mass (COM) of different aromatic guest molecules and the center of the 12ring window (WIN) (a) and the center of the supercage (COC) (b) at 300 K at zero coverage limit (1 molecule/uc).
at 0.0 Å. The g functions of the MD runs with higher benzene loadings (not shown) yield approximately the same result. Figure 5, which shows the g functions between COM and WIN (panel a) and COM and COC (panel b) for different aromatic guest molecules, indicates that m-nitroaniline has a sorption behavior different from that of benzene or the xylenes. m-Nitroaniline is located closer to the 12-ring window (Figure 5a) and farther away from the center of supercage (Figure 5b) than benzene or the xylenes. These results are consistent with the guest molecule positions obtained by neutron30,33-36 or X-ray powder diffraction;37 nonpolar aromatic guest molecules like benzene (Figure 2a) and the xylenes (Figure 2c,d) are adsorbed at the cation adsorption site, whereas polar aromatic molecules like m-nitroaniline (Figure 3) have completely different adsorption sites. Furthermore, nearly identical g functions were found for m- and p-xylenes (not shown), and therefore they differ only slightly in their sorptive behavior. 4.3. Velocity Autocorrelation Functions and Spectral Densities. Time correlation functions and their Fourier transforms are useful tools to examine the dynamic properties of host-guest systems. In this section, we analyze the velocity autocorrelation functions (VACF) and the corresponding Fourier transforms (spectral densities) of the zeolite framework and the center-of-mass motion of the sorbates. The velocity autocorrelation function (VACF) and its Fourier transform are calculated as
Cvv(t) ) and
〈v(t) ‚ v(0)〉 〈v(0) ‚ v(0)〉
(8)
11106 J. Phys. Chem., Vol. 100, No. 26, 1996
C ˆ vv(ω) ) x2/π∫0 Cvv(t) cos(ωt) dt ∞
Klein et al.
(9)
respectively. The averaging was performed over time origins, t0, spaced by 0.1 ps and over all guest molecules. The Fourier transform of VACF has been calculated by using a weighting function (Blackman’s window)47 to suppress oscillations near the maxima caused by the finite summation. The spectral densities of the bare zeolite framework have been discussed in detail in a recent paper.26 The spectral densities of the zeolite framework show slight changes upon loading with guest molecules (results not shown). The influence of temperature on the frequencies of the maximum amplitudes was found to be negligible. However, the intensities slightly depend on temperature. To distinguish between different vibration modes of the guest molecules, we calculated the VACF for two directions by using a vector rCOC-COM defined as the connection of the center of supercage (COC) and the center of mass (COM) of the guest molecule. The VACF Cvv,par(t) and Cvv,nor(t) give information about the vibration frequencies parallel and perpendicular to the vector rCOM-COC, respectively. They are defined as
Cvv,par(t) )
〈[v(0) ‚ rCOM-COC(0)][v(t) ‚ rCOM-COC(0)]〉 〈[v(0) ‚ rCOM-COC(0)]2〉
(10)
〈[v(0) × rCOM-COC(0)][v(t) × rCOM-COC(0)]〉 〈[v(0) × rCOM-COC(0)]2〉
Cvv,par(t)
(11)
The locations of the peaks of the spectral densities of Cvv,par(t) as well as those of Cvv,nor(t) are shown in Table 3. The parallel vibrational motion of benzene (wavenumber ≈ 55-65 cm-1) is considerably faster than the normal vibrational motion (wavenumber ≈ 25 cm-1) for all temperatures. An increase in the temperature results in a slight shift of the maximum amplitudes toward smaller wavenumbers for both vibrational motions and also causes a broadening of the bands. At higher guest molecule coverage (benzene at 300 K, 32 molecules/uc) only a broadening of the normal mode is observed. The normal vibration frequencies of m-xylene and mnitroaniline are quite similar to those of benzene. However, a distinct change is observed for the parallel vibration motion. The high-frequency band at 55-65 cm-1 is lacking. A new band at lower wavenumbers, with values similar to those of the parallel vibrational motion (wavenumber ≈ 25 cm-1), occurs. Demontis et al.22 obtained in the power spectrum of their MD simulation of benzene in zeolite NaY at 300 K peaks at 20 and 110 cm-1. The difference from our results is mainly explained by the fact that we modeled a dynamic zeolite lattice, whereas Demontis et al.22 only used a rigid zeolite framework. The validity of our results is backed by inelastic neutron scattering experiments of benzene in zeolite NaY carried out by Jobic et al.48 Several peaks are found in ref 48 between 20 and 60 cm-1, with two maxima situated at 20 and 45 cm-1, respectively. That means that the peaks around 20 cm-1 can be assigned to motions perpendicular to the vector rCOC-COM, and the peaks with the maximum at 45 cm-1 can be assigned to motions parallel to the vector rCOC-COM. 4.4. Orientational Autocorrelation Functions. In this section, we analyze the rotational motion of the guest molecules by orientational autocorrelation functions of unit vectors e(t),
Cvv,nor(t)
main minor main minor peak peaks peak peaks
system benzene (1 molecule/uc), 20 K benzene (1 molecule/uc), 300 K benzene (1 molecule/uc), 700 K benzene (32 molecule/uc), 300 K m-xylene (1 molecule/uc), 300 K m-nitroaniline (1 molecule/uc), 300 K
65 60 55 55 30 37
55 51 92
26 25 24 25 25 30
37 37
which describe the orientation of the guest molecules. The orientation of a molecule is unambiguously defined by the three Eulerian angles. However, the determination of these three angles by comparison of the initial and final configurations is not easy because the sequence of rotations cannot be changed (commutativity is not valid for rotation). Therefore, to analyze the rotational motion of the planar aromatic guest molecules, we have used a vector of the aromatic plane (plane vector, epl) and a vector normal to the aromatic plane (enorm). The orientational autocorrelation function is defined as49
Cee,n(t) )
and
Cvv,nor(t) )
TABLE 3: Location of Main and Minor Peaks (cm-1) in the Spectral Densities of the Center-of-Mass Motion of Different Guest Molecules Parallel, Cvv,par(t) and Perpendicular, Cvv,nor(t) to the Vector rCOM-COC (The Influence of Temperature and Guest Molecule Coverage Is Also Shown)
〈Pn[e(0) ‚ e(t)]〉 〈Pn[e(0) ‚ e(0)]〉
(12)
Since different spectroscopic methods exhibit a decorrelation depending on the order n of the Legendre polynomial Pn (IR, n ) 1; NMR, n ) 2), different rotational autocorrelation functions Cee,n(t) have been calculated, mainly Cee,1(t) and Cee,2(t). The decorrelation times τn are determined by fitting an e-t/τn function to Cee,n(t). Furthermore, we have calculated the rotational diffusion coefficient Drot49 as
Cee,n(t) ) e-n(n+1)Drott
(13)
This formula is due to a Fickian phenomenology of the rotational diffusion only. If this assumption is valid, the proportion between τ1 and τ2 corresponds to a value of 3. We have calculated Drot from Cee,1(t). Table 4 shows a selected survey of the decorrelation times τ1 and τ2 and the rotational diffusion coefficient Drot for both rotational motions (plane, normal), resulting from the MD simulations. The orientational correlation functions Cee,1(t) are shown in Figures 6-8. The nonpolar aromatic guest molecules (benzene, m-xylene, and p-xylene) exhibit a considerably faster decorrelation of the plane vector than of the normal vector (Figures 6 and 7). The decay of the orientational correlation function Cee,1(t) for the normal vector to zero could be observed for benzene and p-xylene only at 700 K (Figure 7). For all other temperatures, a partial decorrelation of the normal vector has been obtained. The decorrelation times for the orientation of the normal vector are on the order of 100 times greater than those of the plane vector. Decorrelation of the normal vector by rotation about one of the 2-fold axes is sterically not possible at the cation adsorption site and is always connected with a jump to another cation adsorption site (four symmetry equivalent cation adsorption sites per supercage). The residence times at the cation adsorption site (cf. next section) are considerably longer than the decorrelation times of the plane vector. Hence, rotation about the 6-fold (pseudo-6-fold) axis at the cation adsorption site is the predominant rotational motion of benzene and xylenes in zeolite NaY.
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Figure 7. Orientational correlation function Cee,1(t) of the normal vector for benzene (a) and p-xylene (b) at zero coverage limit (1 molecule/ uc) between 100 and 700 K.
Figure 6. Orientational correlation function Cee,1(t) of the plane vector for benzene (a), m-xylene (b), and p-xylene (c) at zero coverage limit (1 molecule/uc) between 20 and 700 K.
Figure 6 reveals a free rotational motion about the 6-fold (pseudo-6-fold) axis for both benzene and xylenes. For benzene, only at 20 and 100 K a sinusoidal function typical of a libration motion is observed in addition to the exponential decay due to the free rotational motion. An increase in the guest molecule coverage [Cee,1(t) not shown, for τ1 and τ2 see Table 4] scarcely influences the orientational correlation functions. An increase in the temperature results in a faster decay of all rotational correlation functions. Benzene shows a faster rotational motion about the 6-fold axis [steeper decay of Cee,1(t)] than do the xylenes at the same temperature about the pseudo6-fold axis. m- and p-xylene have nearly identical rotational motions above 100 K. At 20 and 100 K, a steeper decay and therefore a faster rotational motion have been observed for p-xylene.
The rotational autocorrelation functions of m-nitroaniline (Figure 8) are completely different from those of the nonpolar arenes (Figures 6 and 7). The decorrelation of the normal vector is faster than the decorrelation of the plane vector. However, the decorrelation times of the normal vector are quite large (cf. Table 4) especially at temperatures below 500 K, indicating that m-nitroaniline exhibits slower rotational motions compared to nonpolar sorbates like benzene and xylenes. This different rotational motion is explained by the changed adsorption site37 (cf. section 4.2, especially Figure 5) at which the nitro group interacts the with the sodium SII cation and the amino group forms hydrogen bonds to the oxygens of the 12-ring window. Estimates of the activation energies for the rotational diffusivities have been obtained by an Arrhenius plot. The activation energies for the rotation about the 6-fold (pseudo-6-fold) axis are on the order of 1 kJ/mol-1 for benzene and the xylenes. The activation energy for the rotation about the 2-fold axis is considerably higher for these nonpolar arenes, with values on the order of 15 kJ/mol. The activation energies for the rotation about the pseudo-6-fold and 2-fold axes of m-nitroaniline are 14 and 8 kJ/mol-1, respectively. 4.5. Mechanism of Self-Diffusion. Sorbate self-diffusion coefficients can be calculated from their mean square centerof-mass displacements ∆R(t)2 using the Einstein relation:47
D)
1 〈∆R(t)2〉 6t
(14)
The averaging takes place over time origins t0 spaced by 5 ps and over all guest molecules. The mean square displacements of benzene and p-xylene as a function of temperature are shown in Figure 9. We have determined the slopes of the regression
11108 J. Phys. Chem., Vol. 100, No. 26, 1996
Klein et al.
TABLE 4: Decorrelation Times τ1 and τ2 and Rotational Diffusion Coefficients Drot of the Rotational Motions (Plane, Normal) for Benzene, p-Xylene, and m-Nitroaniline at Different Temperatures T (K) 20 100 300 500 700 300 300 100 300 500 300 700 300 700 300 500 700 300 700 a
guest molecule a
benzene (1 molecule/uc) (p) benzene (1 molecule/uc) (p) benzene (1 molecule/uc) (p) benzene (1 molecule/uc) (p) benzene (1 molecule/uc) (p) benzene (16 molecule/uc) (p) benzene 32 molecule/uc) (p) p-xylene (1 molecule/uc) (p) p-xylene (1 molecule/uc) (p) p-xylene (1 molecule/uc) (p) m-nitroaniline (1 molecule/uc) (p) m-nitroaniline (1 molecule/uc) (p) m-nitroaniline 1 molecule/uc) (n)b m-nitroaniline (1 molecule/uc) (n) benzene (1 molecule/uc) (n) benzene (1 molecule/uc) (n) benzene (1 molecule/uc) (n) p-xylene (1 molecule/uc) (n) p-xylene (1 molecule/uc) (n)
τ1 (ps)
τ2 (ps)
τ1/τ2
Drot (1011 s-1)
312.50 12.50 2.00 1.00 0.83 2.00 2.50 50.00 6.25 2.22 2.8 × 104 77.00 1000.00 8.30 >106 2500.00 111.00 >106 40.00
175.00 5.00 0.77 0.37 0.33 0.83 0.83 29.00 1.25 0.83 2.0 × 104 10.00 500.00 2.00 >106 1100.00 58.00 >106 15.00
1.78 2.50 2.59 2.70 2.51 2.41 3.00 1.72 5.00 2.67 1.40 7.70 2.00 4.15
0.01 0.40 2.50 5.00 6.00 2.50 2.00 0.10 0.80 2.25 1.8 × 10-4 6.5 × 10-2 5.0 × 10-3 0.60
