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J. Phys. Chem. B 1997, 101, 6290-6292
Dynamic Behavior of Diatomic Guest Molecules in Clathrate Hydrate Structure II Shinichiro Horikawa* Institute of Low Temperature Science, Hokkaido UniVersity, N19W8 Sapporo 060, Japan
Hidenosuke Itoh and Jiro Tabata Department of Applied Physics, Hokkaido UniVersity, N13W8 Sapporo 060, Japan
Katsuyuki Kawamura Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Tokyo 152, Japan
Takeo Hondoh Institute of Low Temperature Science, Hokkaido UniVersity, N19W8 Sapporo 060, Japan ReceiVed: October 11, 1996; In Final Form: May 7, 1997X
Molecular dynamics simulation studies were carried out on N2- and O2-hydrates to reveal distributions and intramolecular vibrational spectra of diatomic guest molecules in two different sized cages. It was found that the guest molecules in the large cages distribute apart from the centers and move around in the cages while those in the small cages are located at the centers and that the guest molecules in the small cages have a preferred orientation which lie on a crystallographic {111} plane. These two results are well explained in terms of the size difference of the two cages and the distortion of the small cages, respectively. Vibrational spectra calculated for the stretching modes of the guest molecules showed double maxima for those in the large cages and a single maximum for those in the small cages. No significant difference between nitrogen and oxygen was found in all behaviors examined, except double maxima in the vibrational spectra of oxygen even in small cages.
1. Introduction
2. Computational Details
It was shown in recent studies on air-hydrates included in polar ice cores that the density distribution of air molecules encaged in hexakaidecahedra (16-hedra) of type II structure has four maxima displaced from the center of the cage while that for pentagonal dodecahedra (12-hedra) has a single maximum at the center1 and that intensities of stretching vibrational spectra of nitrogen and oxygen varied by the rotation of a single crystal under a polarized incident laser beam.2,3 For a better understanding of the behavior of the guest molecules, molecular dynamics (MD) simulation studies on the same hydrate structures are required. MD simulation studies on dynamic behavior of small guest molecules were first carried out on Kr-hydrate by Tse et al.4 They found significant displacement of the guest atom positions from the symmetry center of the large cages. Similar behavior was also found on Ar-hydrate by Tabata et al.5 Tanaka carried out MD simulations on ethane- and propane-hydrates to observe the anisotropic nature of the guest molecules.6 He found that rotational motion was restricted by the interaction with host molecules. Our aim in the present study is to observe the dynamic behavior of diatomic guest molecules by MD simulations. Since an atom-atom potential model was used, stretching vibrational spectra of the diatomic molecules were also calculated as well as motion and orientation of the whole molecules. The results obtained are compared with those from experiments.
Potential Model. In the present simulations, we use the KKY potential model developed by Kumagai, Kawamura, and Yokokawa which allows intramolecular degrees of freedom.7 Pair potentials due to van der Waals, repulsive, and Coulomb interactions are taken into account for all atom pairs. The pair potential for atom pairs that have covalent bonds are given by Morse potential function. The total pair potentials are given by
* Corresponding author. E-mail:
[email protected]; Fax: 81-11-706-7142. X Abstract published in AdVance ACS Abstracts, June 15, 1997.
S1089-5647(96)03210-5 CCC: $14.00
Uij ) f0(bi + bj) exp[(ai + aj - rij)/(bi + bj)] - cicj/rij6 + zizje2/rij + f0Dij {exp[-2βij(rij - rij*)] 2 exp[- βij(rij - rij*)]} (1) where f0 is constant, rij is the interatomic distance, a, b, c, and z are parameters of atom species, and Dij, βij, and rij* are intramolecular potential parameters. In addition, three-body interaction is also introduced to confine the H-O-H angle. Details of the model are given in the literature.7,8 To apply the model to N2- and O2-hydrates, in the present study, intermolecular potential parameters for nitrogen and oxygen molecule were determined so that the densities of the liquid phase were correctly reproduced by the simulations using those parameters. The intramolecular potential parameters for N-N and O-O interactions were determined by using experimental data on the basis of the Morse potential function. The potential parameters used for nitrogen and oxygen are tabulated in Table 1. All other parameters are given in the previous paper.8 Method of Calculation. The molecular dynamics simulations were carried out by the computer program MXDORTO9 © 1997 American Chemical Society
Diatomic Guest Molecules in Clathrate Hydrate
J. Phys. Chem. B, Vol. 101, No. 32, 1997 6291
TABLE 1: Potential Parameters for N2 and O2 intermolecular a (Å) b (Å) c (kJ1/2 Å3 mol-1/2) z intramolecular D (Å) β (Å-1) r* (Å)
N (N2)
O (O2)
1.73 0.122 21.4 0.00
1.597 0.122 19.4 0.00
228.4 2.46 1.0976
119.4 2.66 1.208
developed by Kawamura et al. Since Stackelberg’s structure II was confirmed for the crystallographic structure of airhydrate,1 the unit cell of this structure was used as a fundamental cell with a three-dimensional periodic boundary condition in the present simulations. Then, the cell contains 136 water molecules forming 16 small cages (12-hedra) and eight large cages (16-hedra), and full occupation of both types of the cages by guest molecules is assumed. For vibrational spectra of guest molecules, however, the fundamental cell was extended to twice the unit cell length in all three crystallographic axes to obtain better statistics in the calculation of power spectra. Therefore, 128 small cages and 64 large cages were included in the extended fundamental cell. The simulation was started at the initial positions of atoms determined by X-ray diffraction of air-hydrate crystals.1 Although the distribution with four maxima was obtained for large cages, we started the simulation from the position of guest molecules sitting at the center of the cages. Both the van der Waals and the repulsive interactions were cut off at the interatomic distance of half the cell length in the case of the single unit cell and 15 Å in the case of the extended cell. For calculating Coulomb long-range interaction, the Ewald method was used. The equations of motions were solved using the Verlet algorithm with a time step of 0.4 fs, and each simulation run was performed for 8 ps (2 × 104 steps) in the case of the single unit cell and for 4 ps (1 × 104 steps) in the case of the extended cell. The position data and the velocity data were recorded at every 100 steps (40 ps) and at every step, respectively. Power spectra of motion of guest molecules or intramolecular vibrational spectra were calculated by the Fourier transform of the velocity autocorrelation function. To observe the temperature dependence of the behavior of guest molecules, the system was cooled to different temperatures of 10, 50, and 150 K after keeping the system at 250 K for sufficient time to attain the equilibrium. The pressure was fixed at 10 MPa in all simulations runs. 3. Results and Discussion Trajectories of Guest Molecules. Trajectories of nitrogen molecules (center of mass) in both types of cages calculated for 8 ps at 250 K were projected on the X-Y plane or (001) plane as shown in Figure 1. The small and large circles in the projection correspond to 12-hedra cages and 16-hedra cages, respectively. The radius of the circle expresses a radius of the cavity space which is equal to 0.5 Å for a small cage and 1.3 Å for a large cage. These radii were calculated by subtracting van der Waals radius of both the water molecule and the nitrogen (oxygen) molecule from the average radius of the cage. No significant difference was found in the trajectories between nitrogen and oxygen molecules. Even at a high temperature of 250 K, guest molecules move around very close to the center of the cavity, as shown in Figure 1. The distributions of guest molecules become sharper at lower temperatures. In contrast, guest molecules in large cages move apart from the center, and
Figure 1. Trajectories of nitrogen molecules (center of mass) at 250 K, projected on the (001) plane. The radius of the circle expresses a radius of cavity space (see text).
those molecules occasionally sit in certain sites displaced from the center at the lowest temperature of 10 K. The same behavior was reported in our previous paper on argon-hydrates.5 To evaluate a width of the distribution, the mean-square displacement (MSD) of guest molecules was calculated as a function of time. The MSD of the guest molecule in the large cage after equilibration is, for example at 250 K, 2.3 Å2 for nitrogen and 2.9 Å2 for oxygen, with a long-term oscillation. A larger MSD value for oxygen is due to a smaller parameter of the atom radius for oxygen than that for nitrogen. The oscillation of MSD in the large cage corresponds to translational motion of the molecules in the cage. The MSD in the small cage is as small as 1 Å2 at 250 K with no oscillation. The results obtained on the molecules agree well with density distributions deduced by X-ray diffraction of air-hydrate crystals.1 The present simulation, however, was not able to reproduce the four maxima in the large cage, which was probably due to a limited time of simulation period. Orientational Distribution of Guest Molecules. To examine orientational distributions of the diatomic guest molecules, axes of the molecules were plotted in a stereographic projection. The method of projection used was Schmidt’s equal area projection. We found preferred orientations only for small cages. The molecular axes distribute approximately parallel to (111) plane as shown in Figure 2, and this preferential tendency is enhanced at lower temperatures. This preferred orientation can be well explained in terms of a distortion of the small cage in structure II. The small cage is formed by 20 oxygen atoms, which are classified into three types, as shown in Figure 2a.
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Horikawa et al.
Figure 3. Stretching vibrational spectra of (a) nitrogen and (b) oxygen. The solid and dashed lines indicate spectra at 250 and 50 K, respectively.
Figure 2. Stereographic projection of directions of nitrogen molecules encaged in a small cage at 50 K. Oxygen atoms of host water molecules, designated O1, O2, and O3, form the small cage which is slightly compressed in the direction O1-O1 or [111].
The oxygen atoms designated O1, O2, and O3 are different in elements of symmetry. The distance between a pair of the oxygen atoms such as O1-O1 is a measure of the cage diameter in the direction. Using crystallographic data,1 those distances for all pairs were calculated as 3.73, 3.82, and 3.93 Å for O1-O1, O2-O2, and O3-O3 pairs, respectively. The cage, therefore, is slightly compressed in the direction parallel to O1-O1. This distortion of the cage was also confirmed by the present simulation. Those distances calculated from the simulated structure were equal to 3.8, 3.9, and 4.0 Å, respectively. Since the direction of O1O1 is parallel to the crystallographic 〈111〉 axis, axes of the molecules were projected on (111) plane in Figure 2b. This projection indicates that the diatomic guest molecules preferably lie on (111) plane, the normal of which corresponds to the direction of O1-O1. This preferred orientation of the guest molecules found by the present simulation provides an adequate interpretation for the anisotropy observed in Raman spectra of air-hydrates.2,3 Vibrational Spectra for Guest Molecules. Taking an advantage of the KKY potential model, we calculated stretching vibrational spectra of the guest molecules encaged in either type of the cages, as shown in Figure 3. The most interesting feature is double maxima in the spectra for the guest molecules in the large cages. The frequency difference between two maxima decreases as the temperature decreases, and the spectra have a single maximum at 50 K. The difference between nitrogen and oxygen was also found. The spectrum for oxygen in the small cage has double maxima at 250 K, while only a single maximum was found for nitrogen in the small cage at all temperatures examined. It is also shown by the spectra at 50 K that the peak
frequency of the guest molecules in the large cage is lower than that in the small cage. The vibrational frequency of the guest molecules must be changed by a location and an orientation of the guest molecule in a cavity space. Since, as described in the preceding subsections, the guest molecule in the small cage has a localized distribution at the center of the cage and a preferred orientation in the molecular axes, the spectra should have a single maximum. The guest molecule in the large cage, however, has a wide variety in both locations and orientations and then should have a large width of the spectrum peak, although we do not have an adequate explanation for the double maxima. In addition, recently Kuhs et al.10 reported the results that the large cages are doubly occupied by nitrogen molecules. Further study is necessary to clarify the behavior of guest molecules in the large cages as well as that in the small cages. References and Notes (1) Hondoh, T.; Anzai, H.; Goto, A.; Mae, S.; Higashi, A.; Langway, Jr., C. C. J. Inclusion Phenom. 1990, 8, 17. (2) Ikeda, T.; Fukazawa, H.; Hondoh, T.; Lipenkov, V. Y.; Duval, P.; Mae, S. In Proceedings of the 2nd International Conference on Natural Gas Hydrates; Toulouse, France, 1996; p 117. (3) Ikeda, T.; Fukazawa, H.; Hondoh, T.; Mae, S. J. Phys. Chem. B, in press. (4) Tse, J. S.; Klein, M. L. J. Phys. Chem. 1987, 91, 5789. (5) Tabata, J.; Kawamura, K.; Goto, A.; Hondoh, T.; Mae, S. In Computer Aided InnoVation of New Materials II; Doyama, M., Kihara, J., Tanaka, M., Yamamoto, R., Eds.; Elsevier Science: Dordrecht, 1993; p 329. (6) Tanaka, H. J. Chem. Phys. 1994, 101, 10833. (7) Kumagai, N.; Kawamura, K.; Yokokawa, T. Mol. Simul. 1994, 12, 177. (8) Itoh, H.; Kawamura, K.; Hondoh, T.; Mae, S. J. Chem. Phys. 1996, 105, 2408. (9) MXDORTO, Japan Chemistry Program Exchange No. 029. (10) Kuhs, W. F.; Chazallon, B.; Radaelli, P.; Pauer, F.; Kipfstuhl, J. Proceedings of the 2nd International Conference on Natural Gas Hydrates; Toulouse, France, 1996; p 9.
