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ELSEVIER JcNan%id-LipoiQ*65166(1995)285-288 0

The Stability and Dynamics of Clathrate Hydrates

Eldeki Tanaka

Wvlsloa of I’olyaer Chemistry, Cradaate School of Engineering Kyoto Uniiersity, Sakyo, Kyoto 6oQ-Ql Japan

Abstract The thermodynamic stability of a clatbrate hydrate encaging nonspherical

ethane molecule has been investigated by examining the free energy of cage occupancy. A generalized van der Wasls aud Platteeuw theory is extended in order to treat rotational motions of guest molecules in the clathrate hydrate cages.

1. Introduction Gas hydrate consists of guest and the host water molecules which form a

hydrogen bonded network. The clathrate hydrate is stable only when guest molecules exist in the cages of the hydrate. The tberntodyaamic stability of the chubrate hydratea has been explained by van der Waals and Platteeuw (vdWP).I wbbaomeempirieal parameteta, thistheory isapplieabkovelawide~ range.%3 Howewr, ita applicatiad sometimes predicts an irteomct phase hebavior for large pest species. We generalixed the original vdWP theory for t&e hydrate encagit~g a comparable size of a molecule with the larger cage.*s It was revcded that this gotwaliion is very important to aecoamt for the di$cmpascy between the predicth from the original vdWP theory aud the experiuteutal ma&t. In the present study, we examine the magnitude of the anharmoaic contrib&oa from the host water molecules and also the anh srmotlic free energy arisilkg from the rotational degrees of freedom of guest molecule by Monte Carlo (MC!) simulation with the Gaussian statistics.

II. Theory and method The water-water intermolecular interaction is described by the TIP4P potential.*

The ethaae molecule consists of two interactioo sites. each of which interacts with each other via Lentmrd-Jones (Ll) potential.’ Tbe nfexence of e&me molecule is spherical and is of W type interaction with size aad energy parameters of 4.18 A and 1.72 wrooi.* m w paramete n for methyl group of ethaue are 3.78 Aaod 0.866 kJ/mol. For the water-guest intemctiw, the Loretm-Benbelua mle is assmed. The interaction potentials for all pairs of moleculea are truftcated smoothly at IMSS A?

A ;,eerrertdizstio~ of the vdWP theory was made to deal with the couplii of tbe host molecular motions witb those of guestx. The origid theory attd ita geaeralixution are detailed elsewhere.” WOObt%iRthC&ltlW&d~~R2iRiQlUQ3 structure of clathrate hydrates encagiag spherical guest molecules. The potential ertergy is expanded to a polywosial of the displacement from the eqailihrium position. The expansion is truncated at the ,quadratic order for the small diap~mtutt. This tnmeatiatt is ttot appmptiato for the hydrate of a Q al guoatm&culubeca~itrotateawithaIowenergybarrierud&e SC ~oftbepoceatidisno#negIiib&.

286

The ataharmonic contribution to the free euergy is evaluated bya thermodynamic integration method with a reference system of hartnouic oseiilatore. This free energy difference between the real and the reference system A - A,, is giveu by

A-&,=- RT In < exp[-B(S - @@)I b ,

where 9 and 6 are the real and reference system potential. respectiveiy and the average c >8is taken over the reference harmonic oscillators. Since the potential of the harmonic oscillator system having the potential minimum value iIo is written by

@0=4?+&~Q?n. 0

the probability for the system (ice or empty hydrate) to have a set of normal mode coordinates q 3: (4,. qt. . . . qwmr) is given by

This method provides a much more efficient sampling way for a harmonic system than the usual Metropolis schemeto This is because the distribution of the amplitudes is the Gaussian and each mode is independent of other modes; the generated coufigurations have no correlations. In the case of occupation of nonspherical ethane molecules, the reference system is chosen to be the hydrate of spherical guest molecules. The orientations of the guest ethane motecules in the resl system are assigned randomiy.

Ilf. Results and disenssion The guest moiecnlar ‘motions are examined by performing a molecular dynamics

(MD) simulation. T&linear and angular velocity correlation functions of guest etbne are depicted in Figure 1 together with their power spoctra. The spectra of the translational motions have distinct peaks which shift to lower frequency region with decreasing the temperature. The rotational spectra have peaks at about 0 cm-t. There are small peakm in non-zero frequency region indicating that the rotation of ethane is somewhat bidered. The rotational motions have been studied for mostly polar aad some of apolar molecules.J Those suggest that guests m&c&a rotate rather freely, which agrees with our observation.

The anharmonic free energy is evaluated for empty hydrate and cubic ice (ice

f&n in Table 1 The calculated free energy due to the anharmonic potential energy surface is

The anharmonic contribotion to the free energy of empty hydrate is the same’ as ice. which is as large as -0.61 k3 per mole of water.

The free energy differences between the real and the reference clathrate hydrates are nlso given in Tablu 1. The mharmnnic free energy change from the spherical with harmonic approximation to the nonspherical guests is -0.25 kJ /mol. Thu8, we can calculate the total free energy change upon accommodation of no2ispherical guest molecules. The di~iatii pressurep, at 273.15 K is obmined from the intersection between tbe chemical potential curve and the horizontal line corresponding to the difference in chemical potential between ice and empty hydrate t& - flov. which is calculated to be - 0.72 kJ/mol, using the previous calculation and the anharmonic free energy obtained in the present study. The chemical potetttial differeDcea between occupied and empty hy&ates are plotted in Figure 2 for the nonspherical (harmonic + anharmonic terms) and the spherical (harmonic temt) guest molecules together with thut caladated from the original vdWP theory.

267 267

1 1 1 1

0.5 0.5

0 0 .5 .5

-0.5 -0.5 0 0 250 500 250 500 0 Q 50 50 100 150 100 150

time/2.5fs time/2.5fs frequency/cm-‘~ frequency/cm-‘~ Figure 1, The translationai (thin) and rotational (heavy) velocity autocorrelation functions md power spectra at 273.15 K(mlid line), 223.15 K(d&ed line) and 173.15 K(dash-dot line). Left (time correlations). right (power .spectra).

-0.2

p/O.1 MPa Figure 2. Diiiatiia pressme of, ethme hydrate at 273.15 K. Soiid. d&e& sn8 dash-dot iines show the dissociation presmres for the aoasp4erical gaest evabated hy a&unxmic aud banmonic free energy. for spherical gaea m&mle evaluated by only hammnie, free energy, aad for sqhericnt guest according to t&e original vdWP theory. respectively. The horizontal Iiaes low the chanicA potential diffemwe hetweea ice and empty hydrate, sdoo]rad; hantmnk+uWrmmic. dotted; harmonic.

Table 1. Free energy due to the anharmonic contributions. Free energy is in kJ/mol. The anharmonic free energy of the guest molecule is denoted by A (kJ per mole of guest). The reference systems for ice, empty hydrate and the hydrate encaging spherical guest molecules are corresponding harmonic oscillators. The reference system for the hydrate encaging nonspherical guest molecules is the hydrate occupied by spherical guest molecules.

ftee energy mode energy

ice -0.50 6.80 -0.61 6.80

0-w -0.25 6.35 A +2.53

Tbe experimental dissociation pressure pd is 0.53 MPas which is very close to our present result, p&!iO MPa, but is different, from the previous harmonic oscillator approximation, p#=O.24 MPa, and the original vdWP theory p,d.t6 IbfPa The occupation number of the cage per unit lattice is ranging fromS.5 to 5.6 in those methods. There is some variation in experimental results, 5.6 lo 6.0.” Although the present method is approximate one and the chemical potential difference between ice and empty hydrate does not agree perfectly with the experimental observation, this provides a way to evaluate the free energy of the cage occupation taking account of aouspherical nature of guest molecules.

IV. Concluding Remarks In the preseut study, a generalized vdWP theory is further extended in order to

treat rotational motions in the clathrate hydrate cages. The vibrational free energy of both guest and host molecules is divided into harmonic and anharmonic contributions. The anharmonic free energy associated with the rotational degrees of freedom of tbe guest molecules is evaluated as a perturbation from the spherical guest. ‘fly aabarutonic term is found to be essential in determining the free energy of the hindered rotation ibr the guests. It is revealed that thermodynamic properties according to the present method enable us to evaluate the phase behavior of the clathrate hydrate more accurately.

References 1 J. H. van der Waals and J. C. Platteeuw. Adv. Chem. Phys.. 2, 1 (1959). 2 E. D. Sloan, Clathrate Hydrates of Natural Gases, (Marcel Dekker. New York,

1990). 3 D. W. Davidson, Water - A Comprehensive Treatise, Vol.2, edited by F. Franks,

(Plenum; New York, 1973). 4 H. Tanaka and K. Kiyohara. J. Chem. Phys. 98, 4086 (1993). 5 H. Tanaka and K. Kiyohara, J. Cktm. Phys. 98, 8110 (1993). 6 W. L. Jorgensen, j. Chattdrasekhar. 3. D. Madura, R. W. lmpey. and M. L. Klein. J.

, c. s. Swenson, J. Am. C&cm. Sot. 106.6638 (1984). 8 J. 0. Hirshfelder, C. F, Curtiss, and R, B. Bird. Motecular Theory of Gases and

Liiuids, (Wiley, New York, 1954). 9 1. ohmine, H. Tanaka, and P. G. Wolynes. I. Chem. Phys. 89. 5852 (1988) 10 A. Pohorille. L. R. Pratt, R. A. LaViolette, WI. A. Wilson, and R. D. MacElroy, J.

Chem. Pbys. 87, 6070 (1987). II Y. P. Ha&, J. Chem. Thenaodya. 18, 915 (1986).