4188
J. Phys. Chem. 1987, 91, 4188-4193
styrene), 24936-53-6; poly@-methylstyrene), 24936-41-2; poly(cwmethylstyrene), 25014-3 1-7; poly(oxymethylene), 9002-8 1-7; poly(oxyethylene), 25322-68-3; poly(oxymethy1ene-oxyethylene), 28726-47-8; poly(oxytrimethylene), 26636-41-9; poly(oxytetramethylene), 25 190-061; poly(oxyoctamethylene), 51033-60-4; poly(oxymethy1ene-oxytetramethylene), 7 1325-33-2; poly(oxypropylene), 34465-52-6; poly(methy1acrylate), 9003-2 1-8; poly(ethylacrylate), 9003-32- 1; poly(butylacrylate), 9003-49-0; poly(isobutylacrylate), 26335-74-0; poly(octadecylacrylate), 25986-77-0; poly(methacry1ic acid), 25087-26-7; poly(methy1 methacrylate), 901 1-14-7; poly(:thyl methacrylate), 9003-42-3; poly(buty1 methacrylate), 9003-63-8; poly(isobuty1 methacrylate), 901 1-15-8; poly(acrylonitrile), 25014-41-9; polyglycolide (homopolymer), 2620208-4; polyglycolide (SRU), 26009-03-0; poly(ethy1ene oxalate)(copolymer), 2761 5-22-1; poly(ethy1ene oxalate) (SRU), 2761 3-96-3; poly(propiolactone) (homopolymer), 25037-58-5; poly(propio1actone) (SRU), 24938-43-0; poly(butyro1actone) (homopolymer), 3 12 13-03-3; poly(butyrolactone) (SRU), 28728-97-4; poly(valero1actone) (homopolymer), 26354-94-9; poly(valero1actone) (SRU), 26499-05-8; poly(capro1actone)
(homopolymer), 24980-41-4; poly(capro1actone) (SRU), 25248-42-4; poly(undecano1actone) (homopolymer), 91628-63-6; poly(undecano1actone) (SRU), 25735-90-4; poly(tridecano1actone) (homopolymer), 91628-64-7; poly(tridecano1actone) (SRU), 91629-33-3; poly(pentadecanolactone) (homopolymer), 36486-90-5; poly(pentadecano1actone (SRU), 73207-55-3; poly(buty1ene adipate) (copolymer), 25103-87-1; poly(buty1ene adipate) (SRU), 24936-97-8; poly(ethy1ene sebacate) (copolymer), 25037-32-5; poly(ethy1ene sebacate) (SRU) Nylon 6, 25038-54-4; Nylon 6,6, 32131-17-2; Nylon 6,12,26098-55-5; Nylon 6,12 (SRU), 24936-74- 1; poly(methacry1amide) (homopolymer), 25014-12-4; poly(parapheny1ene) (SRU), 25 190-62-9; poly(ethy1ene terephthalate), 25038-59-9; poly(oxy-l,4-phenylene) (SRU), 25667-40-7; poly(thio1.4-phenylene) (SRU), 2521 2-74-2; poly(paraxyly1ene) (SRU), 2572233-2; poly(oxy-2,6-dimethyl-l,4-phenylene)(homopolymer), 25 134-01-4; poly(oxy-2,6-dimethyl-l,4-phenylene) (SRU), 24938-67-8; poly(4,4’isopropylidene diphenylene) (copolymer), 25037-45-0; poly(4,4’-isopropylidenediphenylene) (SRU), 24936-68-3; poly(oxy- 1,4-phenyleneoxy-1.4-phenylene-carbonyl- 1,4-phenylene) (SRU). 3 1694-16-3
Thermal Expansion of Structure I Ethylene Oxide Hydratet John S. Tse,* W. Ross McKinnon, Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K1 A OR9
and Massimo Marchi Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4 M l (Received: October 1, 1986; In Final Form: March 30, 1987)
The cubic unit cell parameter of a structure I clathrate of ethylene oxide was measured from 18 to 260 K by X-ray powder diffraction. I n agreement with previous studies, the thermal expansion of the hydrate is found t o be considerably larger than those for hexagonal ice a t the same temperatures. T h e larger thermal expansitivity in the clathrate hydrate indicates greater anharmonicity in the force field governing the dynamics of the water lattice. This observation is consistent with the smaller thermal conductivity detected in the clathrate hydrates. Constant pressure molecular dynamics calculations employing simple point charge intermolecular potentials for water and ethylene oxide were used to simulate the thermal expansion in hexagonal ice, hypothetical empty structure I hydrate, and the clathrate hydrate oT ethylene oxide. The theoretical results show that the larger anharmonic potential in the hydrate arises primarily from the interactions between the water and the guest molecules.
Introduction Clathrate hydrates are a class of inclusion compounds consisting of water molecules which form a hydrogen-bonded skeleton with voids where small atoms or molecules can be encaged.’ The short-range ordering of the host lattice is similar to t h a t in hexagonal ice, a n d so are the properties such as the phonon density of states: ir~frared~-~ and Raman spectra: and the bulk and shear These properties largely depend on t h e vibrational frequencies of t h e solid,I2 which in turn are determined by t h e q u a d r a t i c t e r m s in t h e expansion of the solid’s potential energy as a function of atomic d i ~ p l a c e m e n t . ’ ~ Properties J~ like thermal conductivity and t h e r m a l expansion, however, which are more sensitive to the higher order terms in this e x p a n s i ~ n , ’differ ~~’~ between clathrates and ice. Thermal conductivities of t h e hydrates are monotonic between 150 and 270 K,17-22which is quite unusual for a crystalline solid. Furthermore, over this temperature range, the magnitudes of t h e thermal conductivites are found to be five to twenty times lower t h a n t h a t of ice.23 Recently, it has also been observed t h a t there is a marked iccrease of the shear velocity with rising t e m p e r a t u r e in t h e s t r u c t u r e I1 hydrate of tetra-
h~drofuran.~~ As a further study of these differences between ice and clathrates we measured the thermal expansion of the unit cell parameter ‘Published as NRCC 27818. *To whom correspondence should be addressed.
0022-3654/87/209l-4188$01.50/0
of a type I clathrate hydrate of ethylene oxide2s and compare it t o constant pressure molecule dynamics calculation^^^^^^ for (1) (a) von Stackelberg, M.; Meuthen, B. Z . Elekfrochem. 1958, 58.40. (b) Davidson, D. W. In Water-A Comprehensive Treaties, Franks, F., Ed., Vol. 2, Plenum: New York, 1973; p 115. (2) Naumann, A. W.; Safford, G. J. J . Chem. Phys. 1967, 47, 867. (3) Bertie, J. E.; Bates, F. E.; Hendricksen, D. K. Can. J. Chem. 1975,53, 71. (4) Bertie, J. E.; Jacobs, S. M. Can. J . Chem. 1977, 55, 1777. (5) Bertie, J. E.; Jacobs, S. M. J . Chem. Phys. 1982, 77, 3230. (6) Bertie, J. E.; Jacobs, S. M. J . Chem. Phys. 1978, 69, 4105. (7) Klug, D. D.; Whalley, E. Can. J . Chem. 1973, 51, 4062. (8) Gerbaux, M . M. X.; Berthel, C.; Hadni, A. Spectrochim. Acta, Part A 1973, A31, 1901. (9) Johari, G. P.; Chew, H. A. W. Phil. Mag. 1984,49, 281. (10) Kiefte, H.; Clouter, M. J.; Gagnon, R. E. J . Phys. Chem. 1985, 89, 3103. (1 1) Gammon, P. H.; Kiefte, H.; Clouter, M. J. J . Phys. Chem. 1983, 87, 4025. (12) Born, M.; Huang, K. Dynamical Theory of Crystal Lattices, Oxford University: London, 1954. (13) Maradudin, A. A,; Montrell, E. W.; Weiss, G. H.; Ipatova, I. P. Solid State Phys. 1971, 5 , 1. (14) Califano, S.; Schettino, V.; Neto, N. Lattice Dynamics of Molecular Crystals, Springer-Verlag: Berlin, 198 1. (15) Barron, T. H. K. In Lattice Dynamics: Proceedings of the 1963 Internal Conference, Wallis, R. F., Ed.; Pergamon: New York, 1965; p 247. (16) For a recent review see, Barron, T. H. K.; Collins, J. g.; White, G. K. Adu. Phys. 1980, 29, 609. (17) Ross, R. G.; Anderssen, P.; Backstrom, G.Nature (London) 1981, 290. 322.
Published 1987 by t h e American Chemical Society
Structure I Ethylene Oxide Hydrate hexagonal ice and structure I ethylene oxide hydrate. A knowledge of the thermal expansitivity is crucial for the conversion of the heat capacity data,Z8g29which is often measured under isobaric condition, into constant volume heat capacity. The constant volume heat capacity can then be used for the analysis of volume dependent crystal properties on the basis of quasi-harmonic the~ry.~~,~' The lattice parameters of ethylene oxide hydrate have been measured many times in the past. The type I structure was first established by von Stackelberg by X-ray powder diffraction.la McIntyre and Peterson measured the cell parameter from 130 to 250 K as a function of the concentration of ethylene oxide in the specimens.32 It was shown that the relative thermal expansion of the hydrate is greater than hexagonal ice at the same temperature range. Subsequently, Bertie and co-workers arrived at the same conclusion by comparing their unit cell parameter obtained at 110 K3 to that of McMullan and Jeffreyz5 measured at 248 K. Except in the work of McIntyre and Peterson, thermal expansitivities were often estimated by uing the cell parameters determined by different laboratories and it is difficult to ascertain the absolute errors. In this paper, we extended the measurement of the unit cell parameter of ethylene oxide hydrate to low temperature in order to provide a set of thermal expansion data of uniform accuracy for future reference. Experimental Section The ethylene oxide (EO) hydrate was prepared by slow cooling a solution of ethylene oxide in distilled water mixed in the stoichiometric ratio E0.7.0Hz0. The solid hydrate was ground and loaded into a vacuum-sealed aluminum sample holder equipped with a beryllium window at liquid nitrogen temperature. The sample holder was then mounted on the cool tip of a closed-loop liquid helium cryostat (Displex Model CS-202). The temperatures were measured with a gold-iron thermometer inserted at the back of the sample holder. A tight vacuum seal was found to be very important as ethylene oxide started to vaporize at 200 K in an unsealed sample. The powder diffraction pattern was measured with a STOE powder diffractometer using Cu Ka (1.542 A) radiation. Since ethylene oxide hydrate has a cubic structurez5 (Pm3n), the cell parameter a is the only dependent lattice parameter. The thermal expansion of the unit cell was measured by monitoring four strong Bragg reflections, [222], [321], [530], and [531], as a function of the sample temperature. The reproducibility of the data was ensured by cycling the temperature of the measurements. Because the position of the sample shifts with temperature, both the cell parameter a and the off-axis parameter were calculated from a least-squares fit to the position of the four Bragg peaks.33 The displacements were usually about 0.3 mm, small enough that the focusing condition of the goniometer was not significantly affected. We checked this procedure for Si using eight Bragg peaks, and found results within 0.001
(18) Ross, R. G.; Anderssen, P. Con. J . Chem. 1982, 60, 881. (19) Cook, J. G.; Laubitz, M. J. Proceedings of the 17th International Thermal Conductiuity Conference; Gaitherburgs, MD, 1981. (20) Cook, J. G.;Leiast, D. G. Geophys. Res. Lett. 1983, 10, 397. . (21) Anderssen, P.; Ross, R.G.; Bachtrom, G. J. Phys. C 1983,16, 1423. (22) Ashworth, T.; Johnson, L. R.; Lai, L. High Temp. Press. 1985, 7,413. (23) (a) Slack, G. A. Solid State Phys. 1979, 34, 1. (b) White, M. A,, private communication. (24) Bathe, M.; Vagle, S.; Saunders, G. A.; Lambson, E. F. J . Muter. Sci. Lett. 1984, 3, 904. (25) McMullan, R. C . ; Jeffrey, G.A. J . Chem. Phys. 1966, 42, 2725. (26) Andersen, H. C . J . Chem. Phys. 1980, 72, 2384. (27) Parrinello, M.; Rahman, A. Phys. Reu. Lett. 1980, 45, 1196. (28) Leiast, D. G.; Murray, J. J.; Post, M. L.; Davidson, D. W. J . Phys. Chem. 1982.86, 4175. (29) White, M. A,; MacLaren, M. T. J . Phys. Chem. 1985, 89, 1380. (30) Barron, T. H. K.; Berg, W. T.; Morrison, J. A. Proc. R. Soc. London, Ser. A 1957, 242, 478. (31) Leadbetter, A. J. Proc. R. SOC.London, Ser. A 1965, 287, 403. (32) McIntyre, .I. A,; Petersen, D. R. J . Chem. Phys. 1967, 47, 3850. (33) Lyon, K. G.; Salinger, G. L.; Swenson, C. A,; White, G.K. J . Appl. Phys. 1977, 48, 865.
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4189 d
XX X
:1
t
~ i
J
0
50
l
100
l
150
l
l
200
250
Temperature (K) Figure 1. Unit cell parameter for the structure I clathrate of ethylene oxide: ( 0 ) experimental data point; (0) measurements at which the sample temperature had not been equilibrated; (+) NMD results for the hypothetical empty structure I hydrate; ( X ) NMD results for ethylene oxide hydrate.
A or 0.02% of published values,34with only four peaks, the accuracy of the lattice parameter should be about 0.05% or 0.005 A in 11 A. Constant Pressure Molecular Dynamics Calculations
A change of temperature for a crystalline solid under equilibrium generates an elastic stress; the crystal unit cell has to adjust in order to alleviate this stress. The spontaneous response leads to thermal expansion or phase transformation. In principle, these phenomena can be studied by a constant pressure molecular dynamics technique. In a constant pressure molecular dynamics (NMD) calculation, the shape as well as the volume of the molecular dynamics (MD) cell is allowed to vary in response to any imbalance between thermally generated internal stress and an external p r e s s ~ r e . ~ ~The , ~ ' theoretical background of this newly developed technique34 and its potential applications can be found in a recent review article.36 So far this method has been applied to studies of the mechanism of phase transitions in molecular solid^^',^^ and to calculation of elastic constants of Lennard-Jones solids,39 with very promising results. N M D calculations of structure I clathrate of ethylene oxide were carried out for a M D cell containing 92 water molecules.40 The positions of the oxygen atoms were taken from the X-rays crystal structure.z5 Hexagonal ice was transformed to an orthorhombic M D cell containing 128 water molecules.41 Since the proton arrangement of the water molecules in both ice and clathrate hydrates are orientationally disordered, their initial positions were assigned in an ad hoc manner following BernalFowler rules.4z Then the proton configurations were randomized until a zero dipole moment cell was obtained.43 In the case of the hydrate, the ethylene oxide molecules were introduced into the large 14-hedral cages according to the position given in the X-ray s t r u c t ~ r e . ~We ~ *also ~ performed calculations for the empty (34) Dahn, J. R. Ph.D. Dissertation, Department of Physics, University British Columbia, Vancouver, Canada, 1980. (35) Nbse, S. Mol. Phys. 1984, 52, 255. J . Chem. Phvs. 1984, 81, 511. (36) Klein, M. L. Annu. Reu. Phys. Chem. 1985, 36, 525. (37) Nbe, S.; Klein, M. L. J . Chem. Phys. 1983, 78, 6928. (38) Nbse, S.; Klein, M. L. Phys. Reu. Lett. 1983, 50, 1207. (39) Sprik, M.; Impey, R. W.; Klein, M. L. Phys. Reu E 1984, 29, 4368. (40) Tse, J. S.; Klein, M. L.; McDonald, I. R. J . Chem. Phys. 1984, 81, 6146. (41) Tse, J. S . ; Klein, M. L.; McDonald, I. R. J . Chem. Phys. 1984, 81, 6124. (42) Bernal, J. D.; Fowler, R.H. J . Chem. Phys. 1935, I , 515. (43) Rahman, A,; Stillinger, F. J . Chem. Phys. 1972, 57, 4009.
l
4190
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
hydrate without the present of the ethylene oxide molecules. Periodic boundaries were imposed to mimic an infinite system. A conventional constant volume M D calculation was first carried out to equilibrate the solid to the desired temperature. Then the constant pressure (NMD) calculation was initiated with the pressure set at 1 bar. The last two steps were repeated for several temperatures. In the N M D calculations, the water-water interactions were described by the pairwise additive simple point charge model The SPC model was used in our (SPC) of Berendsen et previous constant volume M D calculations of clathrate hydrat e ~ ~and, ice4' ~ *with ~ satisfactory ~ results. It was also found to be suitable in reproducing the structures and dynamics of the proton-ordered ice VI1 and IX.48 The ethylene oxide molecule was represented by three interaction sites;40two methylene groups (centered on the carbon atoms) and the etheric oxygen atom. Each site-site interaction consisted of a Lennard-Jones (12-6) potential plus a coulomb term. The Lennard-Jones interaction parameters were those in the TIPS model of J~rgensen.~'The partial charges on the respective atoms (groups) were evaluated from the known dipole moment.50 The guest-water interaction potential is also represented in the Lennard-Jones form with the relevant parameters estimated from the usual combination rules.51 The longrange electrostatic interactions was handled by the Ewald met h ~ d . ~ ~
Experimental Results The cubic cell parameter for structure I clathrate hydrate of ethylene oxide was measured from 18 to 260 K. Figure 1 shows the experimental data fitted to a quadratic polynomial of the form 4T)
(4= + 2.2173 X
11.835
10W5T(K-l)
+ 2.2415 X
10-6T2 (K-2) (1)
In agreement with previous findings, the thermal expansion of the hydrate unit cell is considerablely larger than that in normal ice. Over the temperature range from 20 to 250 K, the cell parameter of the ethylene oxide hydrate increased by 0.13 8,or about 1.1%. In contrast, the increment in the crystallographic a and c axis of hexagonal ice in the same temperature range is only 0.4% and 0.6%,539Mrespectively. Within the limits of accuracy of our measurements, we found no evidence of negative thermal expansion over any part of this temperature region. The present results can be compared with several previous measurements at selected temperatures. The unit cell parameter of ethylene oxide hydrate was found to be 12.03 (1) 8,at -25 O C from single-crystal X-ray d i f f r a ~ t i o n ,11 ~ 3~ 9 (3) A at 110 K from X-ray powder d i f f r a ~ t i o n and , ~ 11.87 (1) A at 80 K from single-crystal neutron diffraction on a deuteriated sample.44 Our results at the corresponding temperatures are 11.992 (5), 11.865 ( S ) , and 11.850 (5) A respectively. Even though our values seem to be consistently lower,, the agreements are satisfactory considering the limits of experimental accuracy and the uncertainty in the temperature measurement in the previous studies. There is (44) Hollander, F.;Jeffrey, G. A. J. Chem. Phys. 1977, 66, 4699. (45) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces, Pullman, B., Ed.; Reidel: Dordrecht, Holland, 1981. (46) Tse. J. S.; Klein, M. L.; McDonald, I . R. J . Chem. Phys. 1983, 78, 2096. (47) Tse, J. S.; Klein, M. L.; McDonald, I. R. J . Phys. Chem. 1984, 87, 4198. (48) Impey, R. W.; Klein, M. L.; Tse, J. S . J. Chem. Phys. 1984, 81, 6406. (49) Jorgensen, W. L. J . Am. Chem. Soc. 1981, 103, 335. (50) Reid, R. C.; Prausnitz, J. H.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. (51) Hirschfelder, J. 0.;Curtiss, C. F.;Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (52) Cowley, E. R.; Jacucci, G.; Klein, M. L.; McDonald, I. R. Phys. Rec. B 1976, 14, 1758. (53) (a) La Placa, S.; Post, B. Acta Crystallogr. 1960, 13. 503. (h) Brill, R.; Tippe, A. Acta Crystallogr. 1967, 23, 343. (54) For a recent review on the thermal expansion of ice see, Kuhs, W. F.; Lehmann, M. S . In Water Science Review, Vol. 2, Franks, F.,Ed.; Cambridge University: London, 1986.
Tse et al. also the possibility that nonuniformity in the ethylene oxide concentration in different hydrate samples lead to small differences in the cell parameters. Our observations, however, differ significantly from the measurements of McIntyre and Peterson.32 In this earlier work,32the hydrate specimens were prepared by quenching aqueous solutions of ethylene oxide of different concentrations. From the measurment of the X-ray diffraction from only one Bragg peak ([732/651]) as a function of temperature, it was reported that the thermal expansion of the hydrate was dependent on the amount of ethylene oxide in the sample. For instance, the cell parameters reported for the hydrates at 130 K varied from 11.94 to 11.95 8, depending on the stoichiometry of the sample. Although a large thermal expansitivity of the hydrate was observed, the magnitudes of the unit cell parameters disagree with other independent studies. Extrapolation of the equation given for the 12.8 mol % ethylene oxide hydrate32 to 110 K gave a lattice constant of 11.944 A. This value is much greater than the present result (11.865 (5) 8,) and the 11.89 (3) A reported in ref 3. Moreover, from a very careful analysis of the compoiti ion,^^ it was found that the stoichiometry of the ethylene oxide hydrate can only be EO(6.76-7.21)*0.07H20 regardless of the concentration of the solution where the hydrates were prepared. Therefore, it is unlikely that the concentration of ethylene oxide in the hydrate can be varied to the extent mentioned in the previous s f ~ d y Owing . ~ ~ to the uncertainty in the integrity of the samples and that only one Bragg reflection was used in the determination of the unit cell parameter, it is our opinion that their results may be at fault. The thermal expansion results presented here for the structure I clathrate of ethylene oxide can be compared with a similar report on the structure I1 clathrate of t e t r a h y d r o f ~ r a n . ~The ~ experimental thermal expansitivity at 100, 150, and 200 K are 40 X and 77 X loW6K-' for the structure I hydrate of lo", 58 X 42 X and 52 X K-' ethylene oxide, and 28 X for the structure I1 hydrate of tetrahydrofuran, respectively. The corresponding values for hexagonal ice are 7 X 28 X and 56 X K-' for the a axis53and 8 X 25 X and 57 X 10" K-I for the c axis.53 Despite the structural differences between the two types of clathrate hydrate^,^' both show a relatively larger expansitivity than ice. It is suggestive this property may be common to all hydrates5* It is interesting to note that the deviation of the hydrate's thermal expansitivity from the ice value becomes smaller with rising temperature. It appears to be quite natural to attribute the large crystal anharmonicity in ethylene oxide hydrate to the intrinsic structural differences from ice. This argument was invoked to explain the large frequency shift of the translational vibrations with the increase external pressure observed in the Raman spectra of the structure I1 hydrate of tetrahydrofuran.' The translational mode Griineisen constant for this hydrate is about 30% higher than in hexagonal ice. The larger Griineisen constantm implies a greater anharmonicity and, therefore, a larger thermal expansitivity. Similar experiments have not been performed for the structure I hydrate. In hexagonal ice, the local arrangement of the water molecules is nearly tetrahedral. The O-O-O angles are 109.33 (2)' and 109.61 ( 2 ) O and the 0-0 distances are 2.750 (1) and 2.752 (1) For ethylene oxide hydrate at 80 K,44there are eight 0Q-o valence angles in the range 106.7 (3)" to 124.7 (1)" and four 0-0 distances, 2.724 (2), 2.750 (9), 2.768 (3), and 2.784 (4) A. Therefore, in comparison with ice, the deviations of the valence O-O-O angles in the hydrate structure are considerable. The arguments presented above assume the guest-host interactions to be weak and only relate the structural differences in the host lattices to their thermal expansitivities. Thus, if an empty hydrate lattice ever existed, the thermal expansitivity would also (55) Glew, D. B.; Rath, N. S . J . Chem. Phys. 1966, 44, 1710. (56) Roberts, R. B.; Andrikdis, C.: Tainsh, R. J.; White, G. K. Proceedings of ICEC-IO, Helsinki, 1984. (57) Mak, T. C. W.; McMullan, R. K. J . Chem. Phys. 1965, 42, 2732. (58) Sargent, D. F.; Calvert, L. D. J . Phys. Chem. 1966, 70, 2689. (59) Kuhs, W .F.; Lehmann, M. S . J . Phys. Chem. 1983, 87, 4312.
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
Structure I Ethylene Oxide Hydrate
h
5,
4191
X X
I 0
X
X
I
0
,
50
A
A
A
A
I
100
I
150
200
250
300
Temperature (K) Figure 3. Theoreticalcell parameters for a hypothetical empty structure I hydrate. 0, A, and V represent the three independent MD vectors defining the simulation box.
X
x
X
X
I
0
50
100
150
200
250
300
Temperature - (K) Figure 2. Cell parameters for the a and c axis of hexagonal ice: ( 0 ) experimental data from ref 53a; ( X ) and (+) theoretical NMD results.
be greater than for hexagonal ice. To test this hypothesis, we have carried out constant pressure molecular dynamics calculations on hexagonal ice, the empty structure I hydrate, and the ethylene oxide hydrate.
Molecular Dynamics Results The calculated thermal expansion of hexagonal ice along the crystallographic a and c axes are displayed in Figure 2, a and b, respectively. Since we have transformed the trigonal unit cell into an orthorhombic MD two of the MD cell vectors, in principle, are related to each other by a constant factor of d3. However, in the NMD simulation, both the MD vectors and the shape of the box are allowed to vary.36 Therefore, the accuracy of the water potential in predicting the true structure of hexagonal ice can be evaluated by noting the changes to the axes and to the box angles. As shown in Figure 2a, the two formally equivalent axes deviates by no more than 0.05 8, (at 196 K). This error is well within the statistical limits of accuracy of the MD calculations. The variation in the box angles are also very small with a maximum deviation about 2.0’. Compare to experiment, the slopes (expansitivity) of the thermal expansion curves are slightly exaggerated in both the a and c axes. Since we have only computed the cell parameters at a few temperatures, it is inappropriate to fit the computed data points to a curve. A better representation of the thermal expansitivity is to calculate the relative increase of the crystal volume a a function of temperature. In Figure 5 we present the results with reference to 70 K. The overall agreement with experiments is remarkable in view of the simplicity of the SPC potential model and the limited number of water molecules employed in the calculations. These results show that a NMD calculation employing a realistic intermolecular potential model for water interactions is capable to yield, a t least, semiquantitative results on the thermal expansion of molecular crystals consisting of water molecules. In passing, it is important to mention that the equivalence of the two MD vectors for the orthorhombic cell is lost if a nonzero dipole moment MD cell was
used. Preliminary calculations employing a MD cell consisting of 96 water and having a residual total dipole moment yield far inferior results. The results of the NMD simulations on a hypothetical empty structure I hydrate are given in Figure 3. Even though empty hydrate does not exist in the nature, it is mechanically stable within the time span of the simulation under an external pressure of 1 bar. Such structural (“meta”) stability was also observed in previous constant volume simulations.40 In a cubic structure, the three vectors defining the system should be equivalent and the average MD box angles should remain 90’ during the course of the NMD calculation. In this case, the maximum deviation in the box angles is about 1.0’ and the three MD vectors differ from one another by at most 3% (Figure 3). Once again, these discrepancies can be attributed to the small number of water molecules (2 X l X l unit cells comprising of 92 water molecules) employed in the simulations. Due to limitations in computational facilities, it is presently impracticable for us to perform calculations on a system conists of a larger number of water molecules. Nonetheless, the relative differences of the cell parameters are small and nearly constant with temperature (see Figure 3). Therefore, the temperature dependence of the expansion of the unit cell size should not be adversely affected by the deficiency in the calculations. As shown in Figure 5 , the calculations predict the expansion of the empty hydrate to be only slightly greater than that of ice. Even though these differences are much smaller than that detected e~perimentally?~ it demonstrates that the structure of the water lattice do play a minor role in determining the thermal expansion of the crystal. It is probable that the more opened water framework in the hydrate enhances the anharmonicity of the crystal potential and leads to a higher thermal expansitivity. In the N M D calculations of the structure I hydrate of ethylene oxide, the three MD vectors defining the cubic box were slightly different from each other (Figure 4). These differences, however, are appreciably smaller than that in the case of the empty hydrate. The largest discrepancy in the length of the MD vectors was 0.2 8, and the maximum deviation of the box angles was only 0.6’. It is important to note that the temperature dependency in the expansion of all the MD vectors is almost identical. Figure 1 compares the calculated average and experimental cell parameters for the ethylene oxide hydrate. The inclusion of ethylene oxide molecules in the large cages of the structure I hydrate has a profound effect on the thermal expansion. The predicted cell parameters are now much larger than the empty structure and increase more rapidly with rising temperature. Although the calculations overestimate the expansion, the ratios of the relative volume change of the hydrate to that in ice are in better agreement with experiment (Figure 5 ) . For instance, at 230 K, the relative volume change of ethylene oxide hydrate if almost three times
4192 The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
Tse et al.
5
= I f
I
3'
h
'F
,'
-
3
I
0
50
I
100
150
1
200
I
250
300
Temperature (K) Figure 4. Theoretical cell parameters for a structure I clathrate of ethylene oxide. 0 , A, and v represent the three independent vectors defining the MD simulation box.
that of hexagonal ice. The corresponding theoretical ratio is 2.8. Therefore, the constant pressure M D calculations employing a SPC potential model for water interactions were successful in predicting the qualitative trend in the thermal expansitivity of the ethylene oxide hydrate. Furthermore, the computed relative volume expansion are also in reasonable agreement with experiment.
Discussion The dramatic increase in the cell size after the inclusion of the ethylene oxide molecules into the empty hydrate structure clearly shows that the guest molecule plays a subtle role in governing the thermal expansion of the hydrate. In the following, .we explore several possibilities that the motions of the guest may contribute to the dynamics of the water lattice. The collisions of the guest molecules with the cage wall exert an internal pressure to the water lattice and weaken the interactions between the water linkages. The volume expansion (AVe/Ve)from the empty structure is related to the internal pressure (Pin,) generated by the kinetic motions of the guest molecules and the bulk modulus (B) of the lattice by the following relationship:
The minus sign preceding Pi,, indicates the pressure is originated within the crystal. The internal pressure is simply the partial pressure created by the guest molecule in the cavity and can be approximated with the ideal gas law (3) where k is the Boltzmann constant, T i s the temperature of the system, and V fis the free volume of the cavity in which the guest molecule can move freely. Substituting eq 3 into eq 2 we get
-AVe = - - 1 kT (4) Ve B Vf Under this assumption, the volume of the enclathrated structure, V,,is simply the sum of the empty structure and the expanded volume, V, + AVe. For the convenience of comparison with M D results, we evaluated the ratio given by (5) From the radical distribution function presented earlier,40the average onset of the ethylene oxide molecule distribution was found at 3.0 A. Since the average radius of the large cavity in the
50
100
150
250
200
Temperature
-
300
(K)
Figure 5. Relative volume change normalized to 70 K: ( 0 and A) experimental data for hexagonal ice from ref 53a and 53b, respectively; (0) experimental data for ethylene oxide hydrate; ( X ) theoretical results for hexagonal ice; (+) theoretical results for empty structure I hydrate; (*) theoretical results for ethylene oxide hydrate.
structure I hydrate is 4.33 A,' the free volume V, is the space enclosed by the spheres of radius 1.33 %, (= 4.33 - 3.0). Taking the bulk modulus ( B ) of the hydrate to be 0.85 X 1O'O N / m 2 as reported for tetrahydrofuran hydrate'Oqllthe ratios of the volume of ethylene oxide hdyrate to that of the hypothetical empty hydrate evaluated from eq 4 and 5 at 100, 150, 200, and 250 K are 1.016, 1.025, 1.033, and 1.041, respectively. The corresponding "exact" Vh/Vevalues calculated by using the M D cell parameters of the filled and empty hydrates62interpolated from the data depicted in Figure 1 are 1.021, 1.024, 1.035, and 1.047. The overall agreements are exceedingly good showing that the simplistic model gave a reasonable account on the effect of the guest molecules to the volume expansion of the empty hydrate. As described earlier the experimental thermal expansitivity of the hydrate is much greater than ice at low temperature and gets smaller as temperature rises. It is possible that, at low temperature, the motions of the ethylene oxide molecule are localized and the interactions with the water are stronger and creating a greater anharmonicity. At higher temperature, the motions of the guest are less restricted and the interactions with water diminish. In this regime, the thermal expansion of the hydrate cell is largely determined by the vibrations of the water molecules. Consequently, the thermal expansitivity is not very difference from ice. The trapping of guest molecules in local sites have been established in the MD calculation of CF4 hydratem and suggested indirectly from the analysis of the oxygen motions in the low temperature crystal structure of O2 hydrates6' We have demonstrated that the motions of the guests can interact with the lattice modes and induced frequency shifts and enhancement of phonon density for selective vibrational states.40 Mofeover, since the frequencies of the localized rotational and translational vibrations of the guest molecules are very close to that of the lattice acoustic modes, coupling between these vibrations can increase the crystal anharmonicity and provide a mechanism for the scattering of thermal phonons resulting in larger thermal expansitivity and lower thermal conductivity. Another probable factor contributing to the large anharmonicity in the crystal force field is the long-range dipolar interactions between the water and the ethylene oxide. In a previous report:' we showed that the electrostatic interactions are responsible for shifting the lattice optical translational vibrations to higher frequencies. It is noteworthy that the inter(60)Barron, T. H.K. Ann. Phys. 1957,1 , 77. (61)Tse, J. S.;Ratcliffe, C. I.; Handa, Y. P.; Powell, B. M. J . Inclusion Phenom. 1986,4, 235. (62) The unit cell sizes obtained from MD calculations at 100, 150, 200 and 250 K are 11.904,11.949,12.035,and 12.131 8, for ethylene oxide hydrate and 11.820,11.854,11.897,and 11.9468, for the empty structure.
J . Phys. Chem. 1987, 91, 4193-4198 actions between the guest and water vibrations have also been vindicated from the substantially smaller ratio in the frequency shift of the translational vibration at 230 cm-' in the infrared absorption spectra between the deuteriated and nondeuteriated ethylene oxide hydrate from the value deduced from simple harmonic oscillator
Conclusion The unit cell parameters for the structure I clathrate of ethylene oxide have been measured by the X-ray powder diffraction method from 18 to 260 K. We confirmed previous findings that the linear expansion coefficient of the hydrate is greater than that in hexagonal ice and so is the anharmonicity. The essence of the experimental and theoretical results for ethylene oxide hydrate are summarized in Figure 5 . The relative volume change over the temperature range 70 to 270 K is about 2% for hexagonal ice and 3% for the structure I hydrate of ethylene oxide. The corresponding cakculated volume change is 4% and 7%, respectively. Even though the absolute magnitudes are overestimated by the calculations, the trend is duly reproduced. The small calculated relative volume expansion between the hypothetical empty structure I hydrate and hexagonal ice suggest that the structural (63) Bertie, J. E.; Othen, D. A. Can. J . Chem. 1972,50,3443. Discussed
in ref 3.
4193
difference between the two compounds only contribute a minor part in the thermal expansion. The interactions between the host lattice and the guest molecules enhance the anharmonic terms in the total crystal potential of the hydrate and is responsible for the larger thermal expansitivity. These interactions also cause the large thermal expansion reported for a number of structure I1 hydrate.56,58 We have demonstrated the usefulness of constant pressure molecular dynamics simulations (NMD) in helping to rationalize the mechanisms of thermal expansion in molecular solids. Even though the simple pairwise additive intermolecular potential models employed in this study are very crude, the major difference between the thermal expansion of hexagonal ice and hydrate is correctly reproduced. Moreover, our experience41has shown that the transverse acoustic and optic modes of hexagonal ice are predicted to be too soft by the SPC model. This failure is due to a basic deficiency inherent with the model which treats the water molecule as rigid and nonpolari~able.~~ This effect is likely more important when the simulation is performed at high temperature. Acknowledgment. The authors express their gratitude to M. L. Klein for his encouragement and to R. W. Impey for technical advice in using his constant pressure M D program. Thanks are extended to J. R. Dahn for his assistance with the X-ray experiments at the early stage of this study. We also thank the referees for many helpful suggestions.
Measurements of Critical Supersaturation for Homogeneous Nucleation of Bubbled Mordecai B. Rubin*$ and Richard M. Noyes* Department of Chemistry, University of Oregon, Eugene, Oregon 97403 (Received: December 9, 1986)
If a solution of a gas in a liquid becomes sufficiently supersaturated, it attains a threshold for homogeneous nucleation at which bubbles form spontaneously. Most previous efforts to measure this threshold have required cumbersome saturation at high pressures. We have worked instead at ambient temperature and pressure with use of chemical reactions to produce supersaturated solutions. The degree of supersaturation is then measured as the amount of gas released when rapid stirring or sonication is suddenly initiated. The threshold for nucleation is the limit beyond which it is impossible to push the level of supersaturation. Thresholds for the diatomic gases Hz, N2, Oz, CO, and NO in aqueous solutions all lie between 0.012 and 0.07 M; that for C02is at least 0.4 M. Continued sonication appeared to accelerate the release of hydrogen from sodium borohydride but not the release of carbon dioxide from acetonedicarboxylic acid. Measured thresholds vary surprisingly little with temperature, indicating that the barrier to nucleation is primarily entropic rather than energetic. The threshold seems to decrease with increasing ionic strength. The effects of changing various parameters such as external hydrostatic pressure have not yet been examined.
Introduction Chemists understand well the equilibrium partition of a volatile solute X between liquid and gas phases. A unique function relates the partial pressure Px in the gas phase to the concentration Cx in the liquid. If the solution is sufficiently dilute, Px will be directly proportional to C x . Even if the proportionality breaks down at greater concentrations, Px will increase monotonically with increasing Cx. Because transport processes in gases are much faster than in liquids, an excess of X in the gas phase will be reduced to the equilibrium value much more rapidly than will an excess of X in a comparable volume of liquid. In other words, if both gas and liquid phases are present, then supersaturated solutions will persist much longer than will supersaturated vapors. *Permanent address: Department of Chemistry, Technion-Israel Institute of Technology, Haifa, Israel. 'This paper is No. 77 in the Series 'Chemical Oscillations and Instabilities". No. 76 is: Ruoff, P.; Hansen, E. W.; Noyes, R. M.J . Phys. Chem., in press.
Because of this persistence of supersaturation, it is possible in principle to observe another function relating Px and Cx which we shall call the threshoZd for homogeneous nucleation. If Cx is increased to this threshold, many bubble nuclei will be created virtually discontinuously. Those bubbles will then grow rapidly and deplete the supersaturation while the equilibrium state is approached. The situation can be illustrated in Figure 1 representing a closed isothermal system containing an inert solvent and an inert gas. If no X is present, the total pressure including vapor pressure of solvent is Po. Now let molecules of X be generated in the liquid phase by some means. If sufficient time is allowed for the system to attain equilibrium, the state can be described by a point on the curve designated "equilibrium" in Figure 1. That point is a unique function of the amount of X in the system. Now let agitation be sufficiently rapid that the compositions of the gas and liquid phases are each virtually uniform. However, let the time scale of observations be short enough that the supersaturated solution can be studied far from equilibrium with
0022-3654/87/2091-4193$OlSO/O 0 1987 American Chemical Society
