17960
J. Phys. Chem. C 2010, 114, 17960–17963
Generalized Cage Occupancy Behavior in the Binary Clathrate Hydrates Jiwoong Seol,† Jong-Won Lee,‡ Woong-Chul Shin,† Dong-Yeun Koh,† Jaehyoung Lee,† and Huen Lee*,† Department of Chemical and Biomolecular Engineering (BK21 program) and Graduate School of EEWS, KAIST, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea, Department of EnVironmental Engineering, Kongju National UniVersity, 275 Budae-dong, Cheonan, Chungnam 331-717, Republic of Korea, and Korea Institute of Geoscience and Mineral Resources, Gwahang-no 92, Yuseong-gu, Daejeon, 305-350, Republic of Korea ReceiVed: June 12, 2010; ReVised Manuscript ReceiVed: September 15, 2010
The competitive inclusion behavior of multiguests in the cages of clathrate hydrates is not yet well understood in spite of its significant importance in both scientific and technological fields. Here, we derive simple and generalized expressions related to cage occupancy ratios of binary clathrate hydrates, measure the cage occupancy ratios, and finally compare the proposed expressions with the experimental results. The present approaches cover three independent categories: the binary guests competitively occupy (1) both small and large cages, (2) only small cages, and (3) only large cages. In addition, we demonstrate that cage occupancy ratio is a simple but powerful variable that indicates the guest behaviors. More importantly, the present approaches only need to have the precise composition measurements to reveal the general nature of guest popularity in cages, while the original van der Waals-Platteeuw model requires several complex variables such as Langmuir constants and fugacities as indispensable prerequisites. The present outcomes might play a significant role in understanding guest occupancy details and, furthermore, provide clues for designing and synthesizing the most efficient hydrate structures to store gaseous molecules for a specific purpose. Introduction Versatile types of polyhedral water cages existing in the clathrate hydrates are determined by unique physicochemical characteristics of enclosed guest molecules. Particularly, we can see that the ionic clathrate hydrates comprising ionic guest species exhibit more complex guest-host interaction behavior than nonionic ones having common guests of hydrogen, methane, and carbon dioxide.1 The guest interaction with hydrogen-bonded cage frameworks strongly affects the P-T equilibrium, hydration number, chemical potential, absolute cage occupancy fraction, and relative ratio of small to large cages. We have also attempted to expand the cage dimensions via thermal stimulation of consecutive cooling/heating processes and have found that thermally expanded cages never shrink again in common plasticlike deformation.2 This structural change resulting from the sensitive tuning of highly flexible cage lattices might greatly enhance the storage capacity of gaseous guest molecules in clathrate hydrate materials. In the previous study we first attempted to thermodynamically derive the hydratecage occupancy relationship of binary guests at the infinitely diluted state.3 However, the inherent limitation of the present approach is that the binary guests compete with each other in order to occupy the empty cages and should be both truly and stably encaged with substantial fractions. Here, we propose a more generalized approach by greatly extending the narrowly limited infinite-dilution to the full concentration range irrespective of guest molecular details. In order to further identify gaseous guest population in structure I (sI) and structure II (sII) * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +82-42-350-3917. Fax: +82-42-350-3910. † KAIST. ‡ Kongju National University. † Korea Institute of Geoscience and Mineral Resources.
cages, we used solid-state nuclear magnetic resonance (NMR), gas chromatography, and direct gas measurements. We have realized through common experiences of spectroscopic analysis that the cage occupancy ratio is more readily and accurately measurable than the absolute cage occupancy and thus can assert that the relative guest population is a primary thermodynamic variable controlling the physical nature of clathrate hydrates at the molecular level. The present study covers the following: (1) the derivation of generalized expressions related to cage occupancy ratios of binary clathrate hydrates, (2) the spectroscopic measurements of cage occupancy ratio, and (3) the direct comparison between the proposed expression and experiment. Experimental Methods Pure ice was grated with a 100 µm sieve at 77 K and pressurized with a mixed gas of CO2 and CH4 at 4.0 MPa and 268 K. For the sII hydrate sample, a stoichiometric 5.56 mol % tetrahydrofuran (THF) solution was prepared, frozen, grated using a 100 µm sieve at 77 K, and then allowed to react with the CH4 and CO2 binary gas at 4.0 MPa. The binary gas hydrate with C2H6 and CH4 was also prepared from ground pure ice powder, which was pressurized at about 3.0 MPa and 268 K. All samples were matured for at least 7 days to attain the equilibrium state. After the reaction, powder X-ray diffraction analysis was done at 153 K for all samples to identify crystal structures and the conversion rate. We used the Rigaku D/maxRB with low-temperature equipment. The apparatus has a maximum capacity of 12 kW and can be operated at up to 83 K. We used ground ice powder to form hydrate samples, which is the same procedure as in our previous works.4 Lowtemperature powder X-ray diffraction method suggested that its conversion yield reaches more than 80% within 1 day. During the sample preparation, we kept pressurized ice powder for about
10.1021/jp105397k 2010 American Chemical Society Published on Web 09/29/2010
Cage Occupancy in Binary Clathrate Hydrates
J. Phys. Chem. C, Vol. 114, No. 41, 2010 17961
2 days until no pressure drops due to hydrate formation were observed. Then, thermal cycling across 273 K is used more than three times in order to convert unreacted water contents into gas hydrates. After 5 days, the samples reached a steady state without any further pressure drop. Powder X-ray diffraction patterns for those samples indicate that their conversion yields are more than 95%. The operating conditions of NMR and GC were the same as those used in our previous work.2 In addition, Raman spectra for binary hydrate of C2H6 and CH4 were recorded with a dispersive Raman spectrometer (LabRAM HR UV/vis/NIR, Horiba Jobin Yvon) equipped with a multichannel thermal electric cooled CCD detector in a laser diode backillumination system. The spectral resolution was 1 cm-1. The 514 nm line of a 40 mW Ar ion laser was used for spectral excitation. All the Raman spectra were measured at 123 K precooled using liquid N2 vapor in order to prevent sample dissociation. Obtained peak areas were used to calculate cage occupancies of each guest.5 Result and Discussion The present approaches cover three independent categories: the binary guests competitively occupy (1) both small and large cages, (2) only small cages, and (3) only large cages. First, we attempt to derive the generalized cage occupancy expression for category 1. Following the simple van der Waals-Platteeuw model,6 we attempted to derive the relationship between the cage occupancy ratios of enclosed guest components. The van der Waals-Platteeuw model is expressed as
θL2 θS2 CS1 - CS2 CL1 - CL2 ) dy1 (CS1 - CS2)y1 + CS2 (CL1 - CL2)y1 + CL2 (3)
dln
From eqs 2 and 3, we can conclude that
θL1 θL2 θL2 dln dln θS1 θS2 θS2 ) )or dy1 dy1 dy2 θL2 dln θS2 )0 dy2
dln
Cmlfl 1+
(1)
∑ Cmifi i
θL1 (CS1f1 + CS2f2)CL1f1 ) ln θS1 CS1f1(CL1f1 + CL2f2)
) ln[(CS1f1 + CS2f2)CL1] - ln[(CL1f1 + CL2f2)CS1] ) ln(CS1f1 + CS2f2) - ln(CL1f1 + CL2f2) + ln CL1 - ln CS1
where θml indicates the cage occupancy of component l in cage m, Cml the Langmuir constant of component l in cage m, fl the fugacity of component l, yl the mole fraction of component l in the vapor phase, and P the pressure. Assuming that the fugacities of each component are simply proportional to the mole fractions of the vapor phase, fl ) ylP, we can derive the following equations for binary guests at constant temperature and pressure.
ln
(const T,P) (4)
where dy1 ) -dy2 because y1 + y2 ) 1 for binary system. Equation 4 implies that the logarithmic curves of cage occupancy ratios between large and small cages have to exhibit identical differential changes at the same compositions of guest species 1 and 2, although both corresponding absolute cage occupancy ratios need not be the same. Of course, we might consider that more reliable and accurate results can be obtained by introducing the general fugacity relationship of fl ) φlylP, where φl indicates the fugacity coefficient of component l.
ln
θml )
θL1 θS1 + dy1
dln
) ln[CS1φ1y1P + CS2φ2(1 - y1)P] - ln[CL1φ1y1P + CL2φ2(1 - y1)P] + ln CL1 - ln CS1 ) ln[(CS1φ1 - CS2φ2)y1 + CS2φ2] - ln[(CL1φ1 - CL2φ2)y1 + CL2φ2] + ln CL1 - ln CS1
For simplicity, we set the functions as
(CS1f1 + CS2f2)CL1f1 θL1 ) ln θS1 CS1f1(CL1f1 + CL2f2)
(CS1φ1 - CS2φ2)y1 + CS2φ2 ) F1(y1) (CL1φ1 - CL2φ2)y1 + CL2φ2 ) F2(y1)
) ln[(CS1f1 + CS2f2)CL1] - ln[(CL1f1 + CL2f2)CS1]
Then
) ln(CS1f1 + CS2f2) - ln(CL1f1 + CL2f2) + ln CL1 - ln CS1 ) ln[CS1y1P + CS2(1 - y1)P] - ln[CL1y1P +
θL1 θS1 F1′(y1) F2′(y1) ) dy1 F1(y1) F2(y1)
dln
CL2(1 - y1)P] + ln CL1 - ln CS1 ) ln[(CS1 - CS2)y1 + CS2] - ln[(CL1 - CL2)y1 + CL2] + ln CL1 - ln CS1
(5)
In the same manner, the cage occupancy ratio of component 2 is also induced as follows
θL1 dln CS1 - CS2 CL1 - CL2 θS1 ∴ ) dy1 (CS1 - CS2)y1 + CS2 (CL1 - CL2)y1 + CL2
(2) Here, the Langmuir constants depend only on temperature and thus can be treated as independent of vapor composition. For guest component 2 the exact same expression is derived
θL2 θS2 F1′(y1) F2′(y1) ) dy1 F1(y1) F2(y1)
dln
Finally, we can get the same result as that in eq 4.
(6)
17962
J. Phys. Chem. C, Vol. 114, No. 41, 2010
θL1 θL2 θL2 dln dln θS1 θS2 θS2 ) )or dy1 dy1 dy2 θL2 dln θS2 )0 dy2
dln
Seol et al.
θL1 θS1 + dy1
dln
(const T,P) (7)
We note that eq 7 was derived without making any assumptions. Accordingly, there exist no thermodynamic constraints coming from eq 7, and thus this equation can easily be extended to the high-pressure region irrespective of the complexity of the participating fugacity coefficients. Another remark is that eq 7 can be generally adopted for clathrate hydrate systems including two specific cages and binary gaseous guests over the entire guest composition range. Figure 1 shows the logarithm cage occupancy ratio of the binary sI clathrate hydrate of CO2 (1) and CH4 (2) from highly diluted to large excess CO2 guest compositions, which were measured from the previous experiment.2 At infinitely diluted condition, the previous result2 is well expressed in the other form
ln
θL1 θS1
|
∞ 1
+ ln
θL2 θS2
|
∞ 2
) 0.086 ≈ 0
or
θL1 θS1
|
∞ 1
×
|
θL2 ∞ ) θS2 2 1.09 ≈ 1
We note that the two curves have almost the same shape and curvature, although the intercepts are different. From this figure and the unique cage occupancy pattern, we can see two interesting features. One is that cage occupancy ratios have unique relationship regardless of the Langmuir constant originating from the properties of guest molecules. Another important thing is that if the cage occupancy information is given for one guest, then the other guest’s corresponding information is automatically determined because of the constraint of eq 7. Subsequently, using the hydrate stability condition of θL1 + θL2 ) 0.98∼0.99,7,8 we can readily obtain all other cage occupancy values for each specific cage and guest. Accordingly, eq 7 might possess a significant implication on cage dynamics and further act as a key variable for establishing guest inclusion phenomenon in clathrate hydrate matrix. So far, we have focused only on relatively simple clathrate hydrates having two different guests occupied in two different types of cages. At the present stage, it will be interesting to see what really occurs in the clathrate hydrates, in which the binary gaseous guests occupy only small cages and liquid guests occupy large ones with nearly full occupancy. We chose THF (5.56 mol %) + CO2 + CH4 (sII), which satisfies such an occupancy condition with full occupancy of THF in the large cages. Then, we focused only on the competitive small cage occupation between CO2 (1) and CH4 (2). At constant pressure and temperature the cage occupancy ratio can be expressed as
ln
CS1f1 θS1 ) ln θS2 CS2f2
≈ ln
Figure 1. Logarithmic occupancy ratio between large and small cages vs CO2 mole fraction (y1).
CS1y1P CS1y1 ) ln CS1 - ln CS2 + ln y1 - ln y2 ) ln CS2y2P CS2y2
Here, the assumption of fl ) ylP can be quite reasonable because of the favorable enclathration of gaseous guests in small
Figure 2. First derivative of logarithmic cage occupancy ratio vs CO2 mole fraction (y1). The red curve is fitted using eq 8.
cages of pure THF hydrate even at relatively low pressure. Thus, we have the following equation
θS1 θS2 d(ln y1 - ln y2) 1 1 1 ) ) + ) dy1 dy1 y1 y2 y1y2 (const T,P) (8)
dln
Equation 8 indicates that the derivative of logarithmic cage occupancy ratio is simply expressed as a function of mole fraction. Figure 2 shows the deviations between eq 8 and the experimental data based on the previous work.3 Both agree well for the overall composition range. Finally, we attempted to analyze the C2H6 + CH4 binary hydrate, in which the CH4 molecules occupy both small and large cages, while C2H6 molecules occupy only the large cages. C2H6 molecule is assumed to only occupy the large cage, which is true on the basis of the spectroscopic results and molecular size. Again, we note that competitive large cage occupancy occurs between C2H6 and CH4. Experimentally, the C2H6 + CH4 binary hydrate can be also easily formed at relatively low pressure and 268 K. By using the previous assumption of fl ) ylP, we can simply derive the following expressions for the system having two guests of C2H6 (1) and CH4 (2) at constant pressure and temperature
Cage Occupancy in Binary Clathrate Hydrates
ln
J. Phys. Chem. C, Vol. 114, No. 41, 2010 17963
θL1 CL1f1 ) ln θL2 CL2f2
≈ ln
CL1y1P CL1y1 ) ln CL1 - ln CL2 + ln y1 - ln y2 ) ln CL2y2P CL2y2
θL1 θL2 d(ln y1 - ln y2) 1 1 1 ) ) + ) dy1 dy1 y1 y2 y1y2 (const T,P) (9)
dln
The identical results of eqs 8 and 9 imply that the basic guest inclusion mechanism appears to be exactly the same for the competitive occupation in either small or large cages. As long as both guests have a preference for one specific cage, which can be, of course, either the small or the large one, the overall macroscopic guest popularity can be described as the inverse of the mole fraction product. As shown in Figure 3, both experimental (measured with Raman spectroscopy) and calculated results match well. It is well-known that the C2H6 + CH4 clathrate hydrate exhibits structural transition at a certain gas phase composition.5 It is quite surprising that the differential coefficient is fixed at a constant value, even though a complete structure change of sI to sII occurs at a specific mole fraction (y1 ≈ 0.27) without discontinuity due to cage restructuring. The appearance of structure transition does not affect the cage occupancy pattern as also confirmed by the same functional dependency of eqs 8 and 9. In order to evaluate nonideality of the guests, we checked the compressibility factor of these guests at their experimental conditions. The compressibility factors for all the guests were larger than 0.91 (especially about 0.97 for CH4) at their experimental conditions (up to about 40.0 bar for pure CO2 and up to about 20.0 bar for C2H6). As such, even though guest species can be treated as ideal gas at the experimental conditions, some deviations between the prediction model and the experimental values could be explained with the nonideality. Because pure CH4 hydrate requires higher formation pressure than pure CO2 hydrate, CH4 + CO2 hydrate at lower yCO2 would generally have higher formation pressure, which leads to increased nonideality due to increased pressure. In the same manner, we can explain the nonideality of C2H6 + CH4 hydrate at lower yC2H6 conditions. Conclusions This study focused on the cage occupancy ratio of binary gas hydrate at various gas compositions. For three representative hydrate systems of CO2 + CH4 (sI), THF + CO2 + CH4 (sII), and C2H6 + CH4 (sI and sII), the cage occupancy ratios were measured and compared with the predicted values. The simple and generalized relationships can be applied over the whole composition range regardless of structural transition. The original van der Waals-Platteeuw model requires several complex variables such as Langmuir constants and fugacities as indispensable prerequisites in order to estimate molecular properties. However, the prediction approaches suggested in this study only need to have the precise composition measurements irrespective of molecular details in order to reveal the general nature of guest population in cages. Moreover, the cage occupancy of each guest naturally varies according to formation temperature and pressure conditions; however, the derived
Figure 3. First derivative of logarithmic cage occupancy ratio vs C2H6 mole fraction (y1). The structure transition from sI to sII appears below the composition of y1 ≈ 0.278.
relationships of their occupancy ratio eliminate T and P variables, only including vapor composition. Further researches should be needed on how to treat (1) hydrates of other structures, (2) liquid guest hydrates, and (3) other organic host materials besides water using the unique cage occupancy ratio. However, the most significant feature of this study is that cage occupancy ratio strongly provides the characteristic behavior of guest molecules enclathrated in host molecules. Thus, we suggest that the cage occupancy ratio could be a simple and representative thermodynamic variable to explain the thermodynamics for many other inclusion compounds as well as clathrate hydrate. Acknowledgment. This research was funded from the Ministry of Knowledge Economy through “Recovery/Production of Natural Gas Hydrate using Swapping Technique” [KIGAM - Gas Hydrate R&D Organization]. This research also supported by the National Research Foundation of Korea grant [WCU program: R31-2008-000-10055-0, and NRL program: R0A2005-000-10074-0(2009)] funded by the Ministry of Education, Science and Technology (MEST). The authors would like to acknowledge funding from the Ministry of Knowledge Economy through “Energy Technology Innovation Program”. The authors would also like to thank the Korea Basic Science Institute (Daegu) for assistance with 600 MHz solid-state NMR. References and Notes (1) Shin, K.; Cha, J.-H.; Seo, Y.; Lee, H. Physicochemical Properties of Ionic Clathrate Hydrates. Chem. Asian J. 2010, 5, 22–34. (2) Park, Y.; Choi, Y. N.; Yeon, S.-H.; Lee, H. Thermal Expansivity of Tetrahydrofuran Clathrate Hydrate with Diatomic Guest Molecules. J. Phys. Chem. B 2008, 112, 6897–6899. (3) Seol, J.; Lee, J.-H.; Kim, D.-Y.; Takeya, S.; Ripmeester, J. A.; Lee, H. Molecular Cage Occupancy of Clathrate Hydrates at Infinite Dilution: Experimental Determination and Thermodynamic Significance. J. Phys. Chem. B 2010, 114, 804–808. (4) Lee, H.; Lee, J.-W.; Kim, D.-Y.; Park, J.; Seo, Y.-T.; Zeng, H.; Moudrakovski, I. L.; Ratcliffe, C. I.; Ripmeester, J. A. Tuning Clathrate Hydrates for Hydrogen Storage. Nature 2005, 434, 743–746. (5) Subramanian, S.; Kini, R. A.; Dec, S. F.; Sloan, E. D., Jr. Evidence of Structure II Hydrate Formation from Methane + Ethane Mixtures. Chem. Eng. Sci. 2000, 55, 1981–1999. (6) van der Waals, J. H.; Platteeuw, J. C. Clathrate Solutions. AdV. Chem. Phys. 1959, 2, 1–57. (7) Uchida, T.; Hirano, T.; Ebinuma, T.; Narita, H.; Gohara, K.; Mae, S.; Matsmoto, R. Raman Spectroscopic Determination of Hydration Number of Methane Hydrates. AIChE J. 1999, 45, 2641. (8) Zhang, C.; Duan, Z.; Zhang, Z. Molecular Dynamics Simulation of the CH4 and CH4-H2O Systems up to 10 GPa and 2573 K. Geochim. Cosmochim. Acta 2005, 69, 4411.
JP105397K
