J . Phys. Chem. 1987, 91, 5789-5791
5789
Dynamical Properties of the Structure I I Clathrate Hydrate of Kryptont John S. Tse* and Michael L. Klein Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K I A OR9 (Received: February 25, 1987)
The phonon density of states and the infrared spectrum of the structure I1 clathrate hydrate of krypton in the translatiosal mode region were obtained from molecular dynamics simulations. We found that the host lattice vibrational density of states is broadly similar to the type I hydrates. However, there are some subtle differences in the region from 100 to 190 cm-I. The frequencies of translational motion of the krypton atoms in the clathrate cages were calculated to be 9 and 34 cm-I. These theoretical results are in good agreement with the values derived from a recent heat capacity analysis of the krypton hydrate. The simulated infrared spectrum only show gross resemblance with experiment. However, frequenciesof the predicted absorptions correlate reasonably well with the observed spectral features.
Introduction It was shown in the mid-1950s by von Stackelberglsz that water clathrates can be classified into two distinct crystallographic structures. Both structures have a cubic unit cell. The type I structure3 consists of 46 water molecules per unit cell and forms 2 cages of pentagonal dodecahedra (12-hedra) and 6 cages of tetrakaidecahedra (1Chedra). The type I1 structure4 has a larger unit cell and contains 136 water molecules, forming 16 pentagonal dodecahedra and 8 hexakaidecahedra (16-hedra). It was long believed that small guests (atoms or molecules) form the type I structure and large molecules prefer the type I1 s t r ~ c t u r e . ~In 1982, Holder challenged this conjecture and suggested that very small guests should favor the type I1 structure.6 This prediction was later confirmed by the experimental observation that the hydrates of argon,7 k r y p t ~ n ,nitrogen,* ~ and oxygen9 indeed adopted the type I1 structure. Even though the local ordering of the water molecules in the clathrate hydrates is very similar to ice, some of their physical properties, in particular, the thermal expansitivitylO*land thermal conductivity,1z-16show substantial differences. To rationalize these unusual observations, it is essential to understand the fundamental differences in the dynamical properties of the water lattice in both ice and the hydrates. In previous work, we have reported the vibrational density of states of several polymorphic forms of ice17 and type I hydrates enclathrated with a variety of guests.18-20 The results showed that the vibrational frequency distribution functions for the water lattice are broadly the same in all cases. However, coupling between the host lattice and the guest's vibrations can induce frequency shifts and enhancement in the phonon density for selected vibrational modes. In this investigation, we extend our study to the vibrational properties of a type I1 clathrate hydrate and compare with the findings for the type I hydrates. The dynamical properties of krypton hydrate were studied through the calculation of its vibrational density of states and infrared spectrum employing the technique of molecular dynamics simulations. Computational Details The technical details of the molecular dynamics simulation can be found in our earlier paper^.'^-^^ A brief account of the theoretical procedure is given below. The simulation was performed at 33 K on a fully occupied type I1 hydrate with 136 water molecules and 16 krypton in the 12-hedra cages and 8 krypton in the 16-hedra cages. The length of the cubic simulation box is taken to be 17.012 A, which is the experimental cell constant determined from neutron diffraction at 5 K.7 The initial coordinates for the oxygen atoms of the water molecules were taken from the X-ray crystal structure of the type I1 tetrahydrofuran h ~ d r a t e .Since ~ the,proton arrangement in the clathrate hydrate is orientationally disordered, their positions were first determined +Published as NRCC 27608. * To whom correspondence should be addressed.
0022-3654/87/2091-5789$01.50/0
in an ad hoc manner according to the Bernal-Fowler rulesz1and then randomized until a zero dipole moment cell was obtained.zz The water-water interactions were described by the pairwise additive simple point charge potential model of Berendsen et aLZ3 The krypton-krypton interaction potential used was the Lennard-Jones potential given in ref 24. The krypton-water interaction potential is also represented in the Lennard-Jones form with the relevant parameters estimated from the usual combination rules. Electrostatic interactions were handled by the Ewald method,25and the equations of motion of the water and krypton were solved numerically.26 In each simulation, the trajectories
(1) Von Stackelberg, M. Naturwissenschaften 1949, 36, 327. (2) Von Stackelberg, M.; Mliller, H. R. Z. Elektrochem. 1954, 58, 25. (3) McMullan, R. K.; Jeffrey, G. A. J . Chem. Phys. 1965, 42, 2725. (4) Mak, T. C. W.; McMullan, R. K. J . Chem. Phys. 1965, 42, 2732. (5) Davidson, D. W. In Wutet-A Comprehensive Treatise; Frank, F., Ed.; Plenum: New York, 1973; Vol. 5. (6) Holder, G. D.; Manganiello, D. J. Chem. Eng. Sci. 1982, 37, 9. (7) Davidson, D. W.; Handa, Y. P.; Ratcliffe, C. I.; Tse, J. S.; Powell, B. M. Nature (London) 1984, 311, 142. (8) Davidson, D. W.; Handa, Y. P.; Ratcliffe, C. I.; ILipmeester, J. A.; Tse, J. S.; Dahn, J. R.; Lee, F. L.; Calvert, L. D. Mol. Cryst. Liq. Cryst. 1986, 138, 1 . (9) Tse, J. S.; Ratcliffe, C. I.; Handa, Y . P.; Powell, B. M. J. Inclusion Phenom. 1986, 4, 235. (10) Roberts, R. B.; Andrikdis, C.; Tainsh, R. J.; White, G. K. Proceedings of ICEC-10, Helsinki, 1984. (11) Tse, J. S.; McKinnon, R. E.; Marchi, M. J . Phys. Chem. 1987, 91, 4188. (12) Ross, R. G.; Anderssen, P.; Backstrom, G. Nature (London) 1981, 290, 322. (13) Ross, R. G.; Anderssen, P. Can. J. Chem. 1982, 60, 881. (14) Cook, J. G.; Laubitz, M. J. Proceedings of the 17th International Thermal Conductivity Conference, Gaithersburg, MD, 198 1. (15) Cook, J. G.; Leaist, D. G. Geophys. Res. Lett. 1983, 10, 397. (16) Anderssen, P.; Ross, R. G.; Backstrom, G. J. Phys. C 1983, 16, 1423. (17) Tse, J. S.; Klein, M. L.; McDonald, I. R. J . Chem. Phys. 1984, 81, 6124. (18) Tse, J. S.; Klein, M. L.; McDonald, I. R. J. Chem. Phys. 1983, 78, 2096. (19) Tse, J. S.; Klein, M. L.; McDonald, I. R. J. Phys. Chem. 1983, 87, 4198. (20) Tse, J. S.; K1ein;M. L.; McDonald, I. R. J . Chem. Phys. 1984, 81, 6146. (21) Bernal, J. D.; Fowler, R. H. J . Chem. Phys. 1933, 1 , 515. (22) Rahman, A,; Stillinger, F. H. J. Chem. Phys. 1972, 57, 4009. (23) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, Holland, 198 1. (24) Horton, G. K. In Rare Gas Solids; Klein, M. L., Venables, J. A., Eds.; Academic: London, 1976; Vol. 1 . (25) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J . Comput. Phys. 1977, 23, 321. (26) Cowley, E. R.; Jaccuci, G.; Klein, M. L.; McDonald, I. R. Phys. Rev. B: Solid State 1976, 14, 1758.
Published 1987 by the American Chemical Society
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The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
9
V
Frequency in wavenumber Figure 1. Phonon density of states for the (a) water lattice and (b) krypton vibrations in type I1 hydrate of krypton. were followed for a total of 30 ps, about one-half of which is allowed for equilibration. The structure was found to be stable within the period of the simulation. The power spectra (phonon density of states), Z(o),of the atomic motions were obtained as the Fourier transforms of the corresponding velocity autocorrelation functions (acf). Z ( w ) = xmexp(-iwt)(V(t)-V(0)) dt
(1)
In principle, the infrared absorption coefficient a ( w ) is rigorously related to F(w), the acf of M(t), the crystal dipole moment. However, in a molecular dynamics calculation we can only obtain the classical acf, F,(w), and not the required quantum mechanical counterpart. If we ignore the question of detailed balance, one can show thatz7 a(@) 0: w2Fc(w)
(2)
F J w ) = l+mexp(-iwt)(M(t)-M(0)) dt
(3)
where
The infrared spectrum was calculated assuming the molecular field interaction is through interaction-induced dipoles2*without the consideration of short-range collision-induced effect. Technical details on the computations are given elsewhere.29 Results and Discussion
The vibrational frequency distribution functions for the water lattice and the krypton guests are displayed in Figure 1, a and b, respectively. The “ripple” of nearly equal energy spacing in the powder spectra is due to termination error with the Fourier transforms. The features of the translation vibrational density of state for the water lattice are very similar to that in ice” and the type I clathrate hydrates.’8-20 The vibrational distribution can be divided arbitrarily into two regions separated by a gap at ca. 150 cm-I. These two regions loosely correspond to the acoustic (27) Berens, P. H.; Wilson, K. R. J . Chem. Phys. 1981, 74, 4872. (28) Califano, S.; Schettino, V.; Neto, N . Lartice Dynamics of Molecular Crystals; Springer-Verlag: New York, 1981. (29) Marchi, M.; Tse, J. S.; Klein, M. L. Trans. Faraday Soc., in press.
Tse and Klein mode and the optic mode vibrations of the water lattice, respectively. The low-frequency region shows three strong peaks at 57, 77, and 94 cm-l with the first one being the most intense. Three peaks at approximately the same positions were also observed in the theoretical vibrational density of states for the type I clathrate hydrates. However, in that case, the first peak had the lowest intensity. In addition to these prominent features, we also found a small but distinct peak at 31 cm-l. This peak was very weak and, in most cases, either not observed or unresolved in the type I hydrates. Other vibrational structures can be identified at 112, 124, and 139 cm-I. At the high-frequency region, the most intense peak is located at 277 cm-I. In comparison with the type I hydrate, the relative distribution of the density of vibrational states between the low frequency and high frequency is more even in the type I1 hydrate. Other prominent features were observed at 176, 195, 208, 237, 267, 293, and 312 cm-’. There is no experimental vibrational density of states available for type I1 clathrate hydrate for direct comparison. Such spectra, in principle, can be measured by neutron incoherent inelastic scattering experiments. However, infrared spectra had been reported for several type I1 hydrate^.^"-^^ In general, the gross spectral features are very similar. In the type I1 hydrate, more pronounced infrared absorption features were observed in the frequency region from 100 to 190 cm-’. Thus, it has been suggested that the distribution of the water lattice vibrational modes is similar to, but should show more feaures than, that of type I hydrates in this region. This o b ~ e r v a t i o nagrees ~~ with an u n p ~ b l i s h e dtheoretical ~~ density of vibrational states of the hypothetical empty type I and I1 hydrates obtained from the force field calculations. A careful comparison of the phonon density of states for krypton hydrate with other type I hydrates indeed reveals that the vibrational features in the frequency region from 100 to 200 cm-’ are more distinct and resolved.18-20 It has been conjecturedZothat the anomalous behavior of the thermal conductivity of the clathrate hydrates may be attributed to the presence of low-frequency guest motions that can coupled with the vibrations of the host lattice. Therefore, it is relevant to calculate the vibrations of the guest atoms in the clathrate cages. The vibrational density of states for the krypton in the two type of cages is presented in Figure 1b. Two peaks at 9 and 34 cm-’ are evident from the figure and can be assigned to the vibrations in the 16-hedra cages and the 12-hedra cages, respectively. The vibrational frequency of krypton in the 12-hedral cages can be compared with that in the 0-quinol clathrate in which the size of the cage is c o m ~ a r a b l e . In ~ ~the latter compound, the rattling motion of the trapped atom was found to be 36 cm-I from its far-infrared spectrum.34 The theoretical vibrational frequencies for the krypton motions are also in good agreement with the experimental values derived from an analysis of the heat capacity data36employing the Posch-Teller potential The 12-hedra cages are present in both type I and type I1 hydrates with almost identical dimensions. In type I methane hydrate, the corresponding methane rattling motion was calculated at 47 cm-’.18 Since the intermolecular potentials for methane and krypton are very similar,lg the shift to a lower frequency in krypton hydrate is largely due to its heavier mass. In methane hydrate the translational vibrations of methane in the 14-hedra cages occur at 35 and 54 cm-I. Since the mean volume of the 16-hedra cage in krypton hydrate is significantly larger, it is not surprising that the guest’s translational vibrations occur at a much lower frequency. We have also calculated the displacement of the krypton positions from the symmetry center of the cages. The results are summarized in Figure 2. We found that at no time the krypton (30) Klug, D. D.; Whalley, E. Can. J. Chem. 1973, 51, 4062. (31) Gerbaux, M. M. X.; Barthel, C.; Hadni, A. Spectrochim. Acta, Part A 1973, 31, 1901. (32) Bertie, J. E.; Jacobs, S. M. J . Chem. Phys. 1978, 69, 4105. (33) Birss, F. W., unpublished results cited in ref 32. (34) Burgiel, J. C.; Meyer, H.; Richards, P. L.J. Chem. Phys. 1965, 43, 4291. (35) Neece, G. A.; Pairier, J. C. J . Chem. Phys. 1965, 43, 4291. (36) Handa, Y. P.; Tse, J. S.J . Phys. Chem. 1986, 90, 5917.
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
Structure I1 Krypton Clathrate Hydrate
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0
L O
a 2a
9-
50
0
300
250
I
200
v.
150
100
I
50
Frequency in wavenumber Figure 3. Theoretical infrared spectrum for type I1 hydrate of krypton.
0
x
0.0
0.2
0.4
0.6
0.8
1.0
1.2
:
Shift A Figure 2. Displacement of the krypton atoms from the symmetry center of the (a) 12-hedra cages and (b) 16-hedra cages.
x
atom is situated at the s mmetry center. On the average, it is displaced by about 0.18 from the center of the 12-hedral cage. The distribution of the krypton atom is even much wider in the 16-hedral cage. In this case, the maximum shift from the center of symmetry is almost 1.2 A with a mean displacement of about 0.7 A. The large displacement is undoubtly due to the larger volume of the 16-hedra cage. The interactions between the krypton and the host lattice bring the krypton in closer contact with the “wall” of the cage. A similar conclusion was reached in the ~ analysis of the structure of the type I1 oxygen h ~ d r a t e . This observation indirectly suggests that there exist “pockets” of local potential minima in the 16-hedra cage. At very low temperature, it is likely that the krypton atoms will be localized (trapped) inside the potential minima. Low-temperature heat capacity measurements will be very useful to detect this phenomenon. Since the infrared spectra of several type I1 hydrates have been reported in the literature, it is of interest to simulate the infrared spectrum for the krypton hydrate. The water molecules in the clathrate hydrate are orientationally disordered; therefore, in principle, all the lattice vibrations should be active in the infrared spectrum. The theoretical infrared spectrum for the translational mode region of krypton hydrate is displayed in Figure 3. The profile of the simulated spectrum is only in fair agreement with experiment.*32 The low-frequency acoustic translational modes are predicted to be much weaker than the higher frequency optic modes. This is consistent with the observed spectra. However, there are major differences in details. In particular, the most intense infrared-active peak in the experimental spectra is generally observed at ca. 230 cm-I, while the calculation puts the most intense peak at 273 cm-I. In the low-frequency acoustic translation region, infrared-active peaks are predicted at 53, 72, 87, and 97 cm-’. In the intermediate-energy region, weak absorptions are found at 116, 127, 141, and 158 cm-l. The strong features from 170 to 300 cm-l dominate the calculated spectrum. Infrared absorptions are calculated at 177, 196, 209, 239, 256, 273, and 295 cm-‘. A direct comparison with the experimental spectra is also hindered by the lack of a complete assignment on the symmetry of the normal vibrations. Therefore, we can only correlate the calculated peaks to features observed in the actual spectra. To this end, we choose to compare
our theoretical spectrum with that of type I1 hydrate of cyclopropane.3z Since the cyclopropane molecule does not possess a permanent dipole moment, the host lattice vibrations induced by dipole-dipole interaction are anticipated to be small and closely resemble that in krypton hydrate. Below 100 cm-I, vibrational features are found a t 58, 65, 69, 72, 82.5, 93.5, and 99 cm-’ for the cyclopropane hydrate. Above 100 cm-’, apart from numerous weak structures, absorption peaks are registered at 120.8, 173, 193, and 229 cm-’ for a deuteriated hydrate. Allowing for the small shift to lower frequencies due to deuteriation, it is gratifying to note that the overall correlation with the theoretical results is very reasonable.
Conclusion We have presented the results of a the molecular dynamics simulation on a type I1 hydrate of krypton. The calculated translational vibration frequencies of the host lattice are similar to that in the type I hydrates. However, the distribution of the phonon density shows appreciable differences. We found that the phonon density of states is more evenly distributed among the acoustic and optic region than that of the type I hydrate. In addition, the low-frequency mode at 31 cm-’ is strongly enhanced in the type I1 hydrate, and the vibration modes between 100 and 200 cm-’ region are well-resolved. The calculated rattling frequencies of the krypton are in good agreement with the values derived from heat capacity measurement. These localized modes should be observable in the far-infrared spectrum. The agreement of the simulated infrared spectrum of the host water lattice in the translational mode region with experiment is less satisfactory. While the positions of calculated and observed spectral features are in good accord, the relative intensities of the infrared absorptions are poorly described by the simple point charge model for water. The reason for this may be related to the difficulties of obtaining reliable crystal dipole derivatives required for the evaluation of the infrared intensities. In fact, the simple point charge model is rather simplistic and is not expected a prior to give quantitative results.29 Furthermore, due to the limited time evolution of the molecular dynamics simulation, it is possible that some of the vibrational modes are not fully relaxed and hence have enhanced contributions to the density of states. This effect is likely most important when simulation is performed at low temperature. Acknowledgment. The authors thank M. Marchi for providing the starting configuration for the empty type I1 hydrate. Registry No. Kr24-(H20)136,110096-53-2.
