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J. Phys. Chem. B 2007, 111, 8830-8835
Thermal Expansivity for sI and sII Clathrate Hydrates K. C. Hester,†,# Z. Huo,‡ A. L. Ballard,§ C. A. Koh,† K. T. Miller,† and E. D. Sloan*,† Center for Hydrate Research, Colorado School of Mines, Golden, Colorado 80401, Westhollow Technology Center, Shell Global Solutions (U.S.), Inc., 3333 Highway 6 South, Houston, Texas 77082, and BP America, Inc., 200 Westlake Park BouleVard, Houston, Texas 77079 ReceiVed: February 26, 2007; In Final Form: May 10, 2007
Knowledge of thermal expansivity can aid in the understanding of both microscopic and macroscopic behavior of clathrate hydrates. Diffraction studies have shown that hydrate volume changes significantly (as much as 1.5% over 50 K) as a function of temperature. It has been demonstrated previously via statistical mechanics that a minor change in hydrate volume (e.g., a 1.5% change in volume or 0.5% change in lattice parameter) can lead to a major change in the predicted hydrate formation pressure (e.g., >15% at >100 MPa for methane). Because of this sensitivity, hydrate thermal expansivity measurements, for both Structures I and II with various guests, are needed help quantify volume distortions in hydrate lattices to ensure accurate hydrate phase equilibria predictions. In addition to macroscopic phase equilibria, the thermal expansion of different hydrates can give information about the interactions between the guest molecules and the host lattice. In this work, the hydrate lattice parameters for four Structure I (C2H6, CO2, 47% C2H6 + 53% CO2, and 85% CH4 + 15% CO2) and seven Structure II (C3H8, 60% CH4 + 40% C3H8, 30% C2H6 + 70% C3H8, 18% CO2 + 82% C3H8, 87.6% CH4 + 12.4% i-C4H10, 95% CH4 + 5% C5H10O, and a natural gas mixture) systems were measured as a function of temperature. The lattice parameter measurements were combined with existing literature values. Both sI and sII hydrates, with a few exceptions, had a common thermal expansivity, independent of hydrate guest. Many guest-dependent correlations for linear thermal expansivity have been proposed. However, we present two guest-independent, structure-dependent correlations for sI and sII lattices, which have been developed to express the normalized hydrate lattice parameters (and therefore volume) as a function of temperature.
1. Introduction Clathrate hydrates are inclusion compounds consisting of water cages stabilized by small guest molecules such as CH4 and CO2.1 Structure I (sI), II (sII), and H (sH) are the most common of these water clathrate structures. While many properties of hydrates are similar to hexagonal water ice (ice Ih), the interactions of the guest molecules with the host water lattice are responsible for significant deviations. For example, the thermal conductivity of hydrates has been shown to be five times lower than that of ice Ih,2 attributed to scattered phonons caused by the guest molecules.3 Along with large differences in hydrate thermal conductivity, Tse et al.4 reported that hydrates have a thermal expansivity much greater than that of ice Ih,5 especially below 200 K, due to greater anharmonicity of the guest vibrations in the hydrate. The origin of this greater anharmonicity was attributed to interactions between the guest molecules and host water lattice and the vibrational motions of the guest molecules. Using constant pressure molecular dynamics simulations (NMD), the volume expansion of the lattice with temperature was found to be a minor function of the difference in structure between ice Ih and hydrate. Only after the inclusion of a hydrate guest (in * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: (303) 273 3723. Fax: (303) 273 3730. † Colorado School of Mines. ‡ Shell Global Solutions (U.S.), Inc. § BP America, Inc. # Current address: MBARI, 7700 Sandholdt Road, Moss Landing, CA 95039.
this case, ethylene oxide (EtO) in the sI lattice), a major difference was found in the thermal expansivity of the hydrate versus ice Ih. Along with the differences in thermal expansion between ice Ih and hydrate, clathrate hydrate thermal expansivity has been shown to be a function of a particular hydrate structure.4 The sI and sII lattice types were found to be comparable, but the sI lattice has a greater thermal expansivity compared to sII. The volume increase from 20 to 273.15 K for the EtO hydrate (sI) was 4.4%, while tetrahydrofuran (THF) hydrate (sI) expanded by 3.4%. Tanaka et al.6 used mode analysis to investigate the thermal expansion of sI Xe hydrate compared to that of ice Ih, investigating the various contributions to the free energy. The results agreed with the earlier work by Tse et al.7 in that the hydrate had a greater thermal expansion compared to that of ice Ih. This effect was again attributed to the guest molecules, not the difference in structure between hydrate and ice Ih. The guest free energy was divided into contributions from the guesthost interaction energy and the guest vibrational energy. Only the anharmonic term of the guest vibrational energy contributed to the increased thermal expansivity. In order to investigate the effect of the guest size, an artificial molecule was simulated with its Lennard-Jones size parameter 10% larger than that of Xe. The resulting thermal expansivity was found to be lower than that when Xe was the guest. The larger guest size decreases the vibrational anharmonicity, lowering the thermal expansivity. However, the authors6 suggest that the rotational motions in asymmetric guests will act in an opposite way, increasing the vibrational anharmonicity. Therefore, even with an increase in
10.1021/jp0715880 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/11/2007
Thermal Expansivity for sI and sII Clathrate Hydrates
J. Phys. Chem. B, Vol. 111, No. 30, 2007 8831
Figure 1. Lattice parameters versus temperature for various sI hydrates.
size, guests such as EtO should increase the hydrate thermal expansivity compared to Xe due to their rotational motions. Along with the volume changes due to temperature, other hydrate lattice distortions have been shown for systems under pressure and with different guest mixtures, within the same structure. However, in the statistical thermodynamics model developed by van der Waals and Platteeuw8 for clathrate hydrates, it was assumed that there was no hydrate lattice distortion caused by either guest type/composition, temperature, or pressure. Work has been done to account for these lattice distortions. Holder and co-workers9,10 have used molecular dynamics to gain an understanding of lattice distortions due to different guests. Even with the assumption of no lattice distortions, the van der Waals and Platteeuw8 (vdWP) model has been shown to work very well for phase equilibria predictions in hydrate systems at low pressures. However, Ballard11 demonstrated that small volume changes in a hydrate lattice (e.g., 1.5%) can lead to significant differences in the predicted hydrate formation conditions. For methane hydrate, this difference could be 15% at higher pressures (>100MPa). The limitations of the vdWP model appear to be, at least in part, due to the fact that it does not account for hydrate volume (lattice parameter) variations for different guests, pressures, or temperatures. With this in mind, Ballard11 modified the vdWP model to incorporate the variable hydrate volume by introducing a secondary model for hydrate volume, which accounts for hydrate guest(s), pressure, and temperature effects. With one of the ultimate goals being better thermodynamic predictions of hydrate phase equilibria over a wide temperature and pressure range, systematic measurements on gas hydrate lattice parameters are required. Since hydrates can be a mixture of many components (e.g., methane, ethane, propane) at any composition, it would be a staggering task to measure the lattice parameter for each hydrate system at every composition. Therefore, the objective of this work was to take measurements on select systems, combined with existing literature, and model gas hydrate lattice parameters as a function of temperature (i.e., thermal expansivity). In addition to the contribution of increased hydrate prediction accuracy, knowledge of hydrate thermal expansivity is of scientific interest because thermal expansivity provides information about the intermolecular interactions between the host waters and guest molecules and the vibrational motions of the guests. As hydrates become of increasing interest as a potential
TABLE 1: sI Hydrate Thermal Expansion Measurements CH4 + CO2
C2H6
CH4 + C2H6
CO2
CH4 ) 85% CO2 ) 15%
CH4 ) 47% C2H6 ) 53%
T/K
a/Å
T/K
a/Å
T/K
a/Å
T/K
a/Å
77 98 111 123 133 145 157 173 188 203 217
11.896 11.910 11.919 11.925 11.934 11.943 11.954 11.963 11.972 11.981 11.995
83 123 143 153
11.838 11.853 11.869 11.875
77 102 123 141 153 167 183 195 207
11.834 11.843 11.855 11.867 11.876 11.886 11.897 11.908 11.921
133 148
11.923 11.936
energy resource, hydrate thermophysical properties, including thermal expansivity, over a wide range of compositions, temperatures, and pressures will be needed. The results of this work have been incorporated into the hydrate prediction program, CSMGem, developed by Ballard.11 In this work, lattice parameter measurements of both sI and sII hydrate structures with different guest molecules will be presented. We also present hydrate thermal expansivity correlations, where structure-dependent volume changes due to guest and temperature can be incorporated into hydrate prediction programs. 2. Apparatus and Experimental Procedures A Siemens D-500 X-ray diffractometer with a cobalt X-ray tube (λ ) 1.78897 Å) was used. MDI DataScan was used for data collection along with analysis by Jade 5 software. The diffractometer alignment was checked weekly with a standard sample (SRM 1976, alumina plate) from the National Institute of Standards and Technology. Hydrates were formed in a 20 cm3 pressure cell from 0.15 Å, or 2.7% volume change). Additionally, a trend in the slopes of the lattice parameters as a function of temperature appears also to exist in the sII hydrate thermal expansivities. 4. Discussion 4.1. Expressions for sI and sII Thermal Expansivity. The linear thermal expansion coefficient is defined as26
R)
1 ∂a a ∂T P,n
( )
(1)
Thermal Expansivity for sI and sII Clathrate Hydrates
J. Phys. Chem. B, Vol. 111, No. 30, 2007 8833
Figure 3. Normalized lattice parameters versus temperature for various sI hydrates based on eq 4.
where a represents the lattice parameter. If R as a function of T is assumed to be
R ) a1 + a2(T - T0) + a3(T - T0)2
(2)
by separation of variables and integration, we get
(
)
a2 a3 a ) exp a1[T - T0] + [T - T0]2 + [T - T0]3 a0 2 3 or
a - a0 ) a0
(
exp a1[T - T0] +
)
a2 a3 [T - T0]2 + [T - T0]3 - 1 (3) 2 3
where a0 is the lattice parameter at a reference temperature T0. Note that if all hydrates of a given structure can be fit using eq 3, then a0 is the only guest-dependent parameter in the expression for hydrate lattice parameter/volume. Since a general trend was observed for both sI and sII hydrate lattice parameters as a function of temperature, it appeared that one equation for each structure could sufficiently describe the hydrate lattice parameter change with temperature. A leastsquare regression of all data from each structure was used to determine the a1, a2, and a3 coefficients from eq 3. Two equations were developed to describe structure-dependent hydrate thermal expansivity. With reference temperature T0 at 298.15 K, the relative sI lattice parameters can be expressed as
asI - asI,0 ) exp(1.128 × 10-4[T - T0] + 1.8003 × asI,0 10-7[T - T0]2 - 1.5898 × 10-11[T - T0]3) - 1 (4) As shown in Figure 3, eq 4 fits well for most of the hydrate systems both in this work and from the literature, with the exception of the Xe16 and C3H6O17 (TMO) systems. The coefficients in eq 4 are summarized in Table 3. The average difference between the experimental and calculated lattice parameters for all data sets, using eq 4, is only 0.004 Å (0.04%). This error is comparable to the experimental error for these measurements. The systems that showed the greatest deviation, Xe16 and TMO,17 have an average difference between experi-
TABLE 3: General Relative Lattice Parameter Correlation for sI and sII Hydrates coefficient
sI (eq 4)
sII (eq 5)
a1 a2/2 a3/3
1.1280E-04 1.8003E-07 -1.5898E-11
6.7659E-05 6.1706E-08 -6.2649E-11
mental and calculated relative lattice parameters of 0.014 (0.11%) and -0.017 Å (-0.14%), respectively. The deviations for these two systems will be discussed in section 4.2. Similarly, the relative sII lattice parameters can be expressed as (T0 ) 298.15 K)
asII - asII,0 ) exp(6.7659 × 10-5[T - T0] + 6.1706 × asII,0 10-8[T - T0]2 - 6.2649 × 10-11[T - T0]3) - 1 (5) Again, as shown in Figure 4, this general correlation fits the data very well for most of the systems. The coefficients in eq 5 are summarized in Table 3. The average difference between experimental and calculated lattice parameter is 0.004 Å (0.02%) using all of the data sets. The main deviation comes at temperatures below 100 K for some of the systems. In the systems that deviated below 100 K, the CH4 + C2H624 and the THF25 systems agree with the general correlation given by eq 5, except at 7 K. The maximum deviation between the experimental and calculated lattice parameter at 7 K is 0.02 Å (0.1%). The systems that showed the greatest deviation, TMO17 and C5H10O (THP) + CH4, have an average difference between experimental and calculated lattice parameters of 0.012 (0.07%) and -0.012 Å (-0.07%), respectively. The deviations for these two systems will be discussed in section 4.2. 4.2. Hydrate Thermal Expansion Related to Hydrate Structure and Guest-Host Interactions. On the basis of eq 1, the linear thermal expansion coefficient R can be calculated using the general relative lattice parameter expressions given in eqs 4 and 5 for sI and sII, respectively. As shown in Figure 5, the thermal expansion coefficient of sI and sII hydrates is comparable over the entire temperature range (20-273.15 K). However, the difference between the structures is more pronounced at the higher temperatures. The thermal expansivity for sI is almost twice that of sII at 273.15 K. The reason behind the difference in thermal expansivity between sI and sII is still unclear and an outstanding question. Both hydrate structures
8834 J. Phys. Chem. B, Vol. 111, No. 30, 2007
Hester et al.
Figure 4. Normalized lattice parameters versus temperature for various sII hydrates based on eq 5.
Figure 5. Linear thermal expansion coefficient versus temperature for sI and sII hydrates and ice Ih.5
have thermal expansion coefficients much larger than that of ice Ih5 below 200 K. This result is in agreement with previous findings.7 The free energy of the guest molecules has been attributed to be the major reason for large thermal expansion of hydrates compared to ice Ih. Interestingly though, as shown in Figures 3 and 4, many different hydrate guest types for a given structure have a very similar thermal expansivity. For sI, the Xe hydrate deviates from the general trend with a smaller thermal expansivity. NMD simulations7 showed that an empty sI lattice had a smaller thermal expansivity versus an EtO hydrate. The lower Xe hydrate thermal expansivity16 has been attributed to it being a closer representation of a hypothetical empty lattice compared to other guest molecules. From simulations, the empty lattice thermal expansivity is very similar to ice Ih. TMO(sI)17 deviates from the general trend with a larger thermal expansivity. TMO(sI) is the largest sI former examined in this study (Figure 1). This could imply that the guest-host interaction energy becomes more dominant as the larger guests stretch the hydrate lattice. As with the above reasoning for sI TMO as the guest molecule, the system with the largest measured sII guest, THP + CH4, deviates from the general sII trend with a higher thermal expansivity. It should be noted that while TMO had a higher thermal expansivity in sI, it has a slightly lower thermal expansivity than the general sII thermal expansivity. This could be due to its small size in the large cage of the sII lattice, similar to Xe in sI. TMO can form both sI and sII based on the
concentration used to form the hydrate. This indicates that it can fit in both the sI and sII large cage. From Figure 2, at 220 K, the sII TMO hydrate has a smaller volume than that of THF hydrate by 1.9%. Note that air, containing very small sII formers, appears to follow the general sII thermal expansivity, but the data set is limited between 126 and 216 K. One may venture to say that the dominant factor in the hydrate thermal expansivity is the existence of a guest in the hydrate structure, with only a weak dependence on the guest type. Similar trends are observed for the sII propane and sII methane + propane data, in which methane occupies the small cages in the latter case. Likewise, little difference in the thermal expansivity of sI methane and sI ethane hydrates was seen, suggesting that occupation of small hydrate cages or the guest type present was not a dominant factor. While the thermal expansion of the hydrate appears to be a very minor function of the guest type present, the absolute volume expansion (implied from Figures 1 and 2), at a given temperature, is primarily a function of the large guest in the hydrate. A supporting and interesting feature of both the sI and sII data shows that occupation of the small cages of the structures does not greatly affect the absolute lattice parameter. A comparison of sII hydrates of pure C3H8 with C2H6 + C3H8 and CH4 + C3H8 showed similar lattice parameters (Figure 2). This can also be seen more clearly when comparing pure CH4 and C2H6 hydrate and the binary CH4 + C2H6 hydrate (Figure 1). While pure CH4 hydrate has a 0.8% difference in lattice parameter from pure C2H6 hydrate at 130 K, the lattice parameters of a binary CH4 + C2H6 hydrate is only different by 0.1% from pure C2H6 hydrate. Hydrate isothermal compressibility shows different behavior than thermal expansivity. As indicated by Raman measurements27 of hydrate lattice modes, a large hydrate guest, for example, C2H6, causes the isothermal compressibility of the hydrate to decrease compared to that with a smaller guest, for example, CH4. This guest-dependent isothermal compressibility has also been shown for many hydrate guests with neutron diffraction studies.28 Unlike hydrate isothermal compressibility, at least under ambient pressure conditions, hydrate thermal expansion across a large temperature range (20-273.15 K) largely appears to be a much stronger function of structure compared to guest type. However, more thermal expansivity measurements, especially for systems at the upper/lower guest size limits for a particular structure and under pressure, are needed to complete this picture.
Thermal Expansivity for sI and sII Clathrate Hydrates 5. Conclusions In this work, hydrate thermal expansivities were measured for both sI and sII hydrates, including thermal expansivity measurements for four sI- and seven sII-forming systems. It has been shown that both sI and sII hydrate thermal expansivities can be expressed by general correlations for most of the systems measured. These general correlations in thermal expansivities for a given hydrate structure have been incorporated into a modified vdWP hydrate model and should allow for enhanced predictions when volume data for a given system are unavailable. The thermal expansion of sI and sII hydrates is different from ice Ih below 200 K. It was shown that the absolute volume of a hydrate at a given temperature was highly dependent on the hydrate guest, with the hydrate volume dominated by the large guest in a mixture. However, more importantly, this work showed that the hydrate thermal expansivity was not a strong function of the particular guest type in the structure. With a few exceptions, a wide range of different hydrate guest molecules have very similar hydrate thermal expansivity behavior. Acknowledgment. K. Hester was supported through National Undersea Research Program Grant UAF03-0098. The author thanks Chevron for their donation of the Siemens D-500 X-ray diffractometer. The authors also thank our industrial consortium members involved in funding this work: BP, Chevron, U.S. Department of Energy, Halliburton, Petrobras, ConocoPhillips, ExxonMobil, Petrobras, Schlumberger, and Unocal. References and Notes (1) Sloan, E. D. Clathrate Hydrates of Natural Gases, 2nd ed.; Marcel Dekker: New York, 1998. (2) Stoll, R. D.; Bryan, G. M. J. Geophys. Res. 1979, 84, 1629. (3) Tse, J. S.; Shpakov, V. P.; Belosludov, V. R.; Trouw, F.; Handa, Y. P.; Press, W. Europhys. Lett. 2001, 54, 354. (4) Tse, J. S. J. Phys. 1987, 48, 543. (5) Rottger, K.; Endriss, A.; Ihringer, J.; Doyle, S.; Kuhs, W. F. Acta Crystallogr., Sect. B 1994, 50, 644. (6) Tanaka, H.; Tamai, Y.; Koga, K. J. Phys. Chem. B 1997, 101, 6560.
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