Ž .Fluid Phase Equilibria 144 1998 361–368

A novel approach to the stability of clathrate hydrates: grandcanonicalMC simulation

Hideki Tanaka )

Department of Polymer Chemistry, Graduate School of Engineering, Kyoto UniÕersity, Sakyo, Kyoto 606-01, Japan

Received 13 January 1997; accepted 1 June 1997

Abstract

The thermodynamic stability of clathrate hydrates encaging nonspherical ethane and ethylene molecules asŽ .well as spherical Xe atoms has been investigated by grandcanonical Monte Carlo GCMC simulation. In

addition to most of the static properties obtained by a standard Monte Carlo simulation, cage occupation ratiosat various pressures are calculated by GCMC simulation where guest molecules are allowed to be created ordeleted. Thus, we can evaluate the free energy of cage occupation at given guest pressure. This indirect methodprovides an alternative way to predict the stability of a variety of clathrate hydrates with simple GCMCsimulation runs without invoking normal-mode analysis and thermodynamic integration when the chemicalpotential difference of water between ice and empty hydrate is known. q 1998 Elsevier Science B.V.

Keywords: Clathrate hydrate; Grandcanonical Monte Carlo

1. Introduction

Gas hydrate consists of guest molecules and host water molecules which form a hydrogen-bondednetwork whose constituents are planar pentagonal and hexagonal rings. The clathrate hydrate is stableowing to interactions between guests and host water molecules. These interactions, though weak, playa crucial role for clathrate hydrates to be stabilized; it should be noted that empty hydrate ismetastable and has not been prepared but is useful as a reference system in discussing thethermodynamic stability. The thermodynamic stability of the clathrate hydrates has long been

Ž . w xexplained by the van der Waals and Platteeuw vdWP theory 1 . This is the only rigorous statisticalmechanical theory to treat clathrate hydrate stability and has been widely used to predict the phasebehavior of many hydrates. However, some empirical parameters are always required in practical

) Corresponding author.

0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0378-3812 97 00280-X

( )H. TanakarFluid Phase Equilibria 144 1998 361–368362

w xcalculation 2,3 . A direct application of this theory sometimes leads to an incorrect phase diagram, inparticular for large guest species.

Several computer simulation studies on clathrate hydrates have been reported in connection withw xthe stability of clathrate hydrates 4–9 . Some important quantities have been calculated other than

thermodynamic properties; especially, stress has been placed on the difference in the thermalw xconductivity between ice and clathrate hydrate and on its anomalous temperature dependence 10,11 .

In conjunction with dielectric relaxation, the hydrogen-bond rearrangement dynamics of clathratew xhydrates was investigated in detail 12 . It was found that there are two kinds of processes in the

connectivity changes; one is caused by thermal excitation in a perfect network region, which can beregarded as Poisson process and another is induced by defects in the environmental network, which isresponsible for power-law like behavior. The separation of two time scales is also found insupercooled liquids and glasses.

We extended the original vdWP theory to treat a hydrate encaging a comparable size of a moleculew xwith the vacant space of the larger cages 13 , or a hydrate in which both types of cages are occupied

w xby a single guest species 14 . The fixed lattice approximation, which is the most serious shortcomingin the original vdWP theory, was removed thereby introducing the coupling of the host with guest.Thus, a more accurate evaluation could be made in terms of only intermolecular interactions. It wasrevealed that the free energy term arising from the coupling of the host with the guest molecules is

w xsignificant for large guest molecules such as propane 13,14 .The thermodynamic stability of clathrate hydrates encaging nonspherical propane and ethane

molecules was also investigated by examining the free energy of cage occupation. The vdWP theorywas further extended in order to treat the accommodation of nonspherical guest molecules in the

w xclathrate hydrate cages 15 . The vibrational free energy of both guest and host molecules was dividedinto harmonic and anharmonic contributions. The free energy associated with the rotational degrees offreedom of the guest molecules was evaluated as a perturbation from the spherical guest. However,evaluation of the anharmonic free energy by the thermodynamic integration demands CPU timeheavily.

Ž .A grandcanonical Monte Carlo GCMC simulation is an alternative way to calculate a cage-oc-w xcupation ratio from which dissociation pressure can be estimated 15 . Once a cage-occupation ratio at

given temperature and guest gas pressure is known, the free energy of cage occupation is evaluatedimmediately. In addition to the cage-occupation ratio, GCMC simulation provides many propertiesthat are accessible by a standard MC simulation. In the present study, we will take advantage of thismerit of GCMC simulation and evaluate the thermodynamic stability of Xe, ethane and ethylenehydrates.

2. Theory and method

The thermodynamic stability of clathrate hydrates is evaluated only with intermolecular potentials.In the present study, all interactions are assumed to be pairwise-additive. The water–water inter-

w xmolecular interaction is described by the TIP4P potential 16 . This potential has often been used inthe study of water. It is believed to be the most reliable within the framework of pair potential, at least

Ž .in reproducing various properties of pure water, in particular pressure the equation of state . TheTIP4P model consists of four interaction sites; a positive charge q on the hydrogen atom, a negativeH

( )H. TanakarFluid Phase Equilibria 144 1998 361–368 363

Table 1Ž .Intermolecular interaction parameters for spherical approximate and nonspherical ethane and ethylene molecules together

with those for xenon

Xenon Ethane Ethylene

Atom s e Group s e Group s e

Xenon 4.047 1.921 Ethane 4.418 1.704 Ethylene 4.232 1.571Methyl 3.775 0.866 Methylene 3.85 0.586Bond length 1.530 Bond length 1.340

˚ y1Size parameter s and bond length are in A and energy parameter e is in kJ mol .The nonspherical molecules consist of either ‘methyl’ and ‘methylene’ groups, each of which is treated as a single site.

Ž . Ž .charge y2 q on the bisector of two OH bonds, and a Lennard–Jones LJ interaction betweenH

oxygen atoms. The ethane and ethylene molecules consist of two interaction sites, which interact with˚each other via LJ potential. The bond lengths for ethane and ethylene are fixed to 1.53 and 1.324 A,

w xrespectively 17 . The references of both ethane and ethylene molecules are spherical and are of LJw xtype interaction 18 . The LJ parameters for both the real and reference molecules are given in Table

1. For the water–guest interaction, the Lorentz–Berthelot rule is assumed. The interaction potentials˚ w xfor all pairs of molecules are truncated smoothly at 8.655 A 13 . The interaction beyond the cut-off is

taken into account with fixed configuration at temperature 0 K.Six clathrate hydrate I structures generated previously are used. The unit cells of the hydrate I are

˚cubic and experimental lattice parameters are used in the following calculation: asbscs12.03 Aw xfor hydrate I with asbsgspr2 2,3 . The structures are of proton-disordered form and have zero

net dipole moment. These structures are generated in an ad hoc manner. There exist six larger and twosmaller cages in the unit cell of clathrate hydrate I. The potential energy of the system is minimized toobtain the exact local minimum structure by the steepest descent method in a similar way as used in

w xthe analysis of water 19 .The accommodation of the guest molecules can be regarded as an adsorption of the guest in the

Žcavity. The number of guest molecules at a given pressure at a given chemical potential of the gas.phase of guest species can be evaluated in the same fashion as the usual adsorption process by

GCMC simulation. This simulation is carried out with the fixed parameters of temperature, volume ofthe hydrate, and the chemical potential of the guest species, m. The chemical potential of the guestmolecule is calculated from the pressure of the gas phase taking into account the second virialcoefficient.

Since guest molecules are adsorbed in the distinct cavity, a method in the present study is not thatw xused in fluid phase simulation where an arbitrary position is available to insert a particle 20 . Instead,

we apply GCMC simulation for fixed adsorption points, which is similar to a method treating Isingspin model. The adsorption points are, however, not restricted to the centers of the cavities but a guestmolecule is allowed to be inserted in some space inside the cavities. The practical method of GCMCsimulation with fixed adsorption cavities is as follows. One of the cavities in the system is chosenrandomly. If it is empty, the position and orientation are assigned with a probability distributionŽ .f D r, V , where D r is the displacement from the center of the cage and V denotes the Euler angle.

The trial creation is accepted with a probability,

min 1, zÕ exp yb w D r ,V rf D r ,V 1Ž . Ž . Ž .Ž .

( )H. TanakarFluid Phase Equilibria 144 1998 361–368364

Ž .where z is the fugacity of the guest and w D r, V is the potential of the guest with the surroundingwater molecules and also with guests. The probability distribution is normalized for linear guest as

sy1HHf D r ,V d rdVs2p Õ 2Ž . Ž .where s is the symmetry number of the guest molecule and Õ is the volume of the cavity. If the

w xchosen cage is occupied, the guest is deleted with a probability 21 ,

min 1,f D r ,V exp b w D r ,V rzÕ 3Ž . Ž . Ž .Ž .The difference from the GCMC simulation for fluid systems consists in the available insertion

volume of the guest and the existence of distinguishable cages. Instead of the unbiased distribution onŽ .the position, which is uniform in space fs1 inside a cage , an introduction of a more effective

probability distribution is preferable, which is

3r2 2f D r ,V sÕ kbr2p exp ybkD r r2 4Ž . Ž . Ž .Ž .where k is a force constant, which is determined based on the frequency of the spherical guest in the

Ž w x. y1clathrate hydrate see Refs. 14,15 . In the present simulation, we adopt a k value of 5 kJ mol˚ y2A . The barrier height of rotation is low as shown in Fig. 1, and so the distribution on the orientationof the inserting guest is not biased. The long range correction is important in each creation or deletiontrial. The correction is made using the LJ parameters of the spherical guest.

The GCMC simulations are performed when the systems are in equilibrium with the gas phaseguest at several pressure values; 0.01, 0.04, 0.1, 0.2 and 0.4 MPa. The chemical potential is calculatedtaking into account the second virial coefficient of the spherical guests. We obtain the mean

² :occupation ratio js n rN at each gas pressure, where N is either the number of the larger cage Nc l

Ž . Ž .Fig. 1. The potential energy curves of ethane left and ethylene right molecules in a large cavity of the clathrate hydrate IŽ .around two orthogonal axes perpendicular to the molecular symmetry axis, solid and dashed lines , where the molecular

Ž .axis is chosen to coincide with the minor axis of the ellipsoidal cage tetrakaidecahedron . The dash–dot line indicates theŽ .potential energy curve obtained from rotation around the minor z axis after rotation of ethane around the x axis by p r2.

The unit of energy is kJ moly1. No long-range correction term is added.

( )H. TanakarFluid Phase Equilibria 144 1998 361–368 365

² :or the smaller cage N and the ensemble average of occupied cages is denoted by n for each types c

of cage. The cage-occupation ratios for small and large cages are calculated separately for xenonhydrate. The free energy of cage occupation is written in terms of the mean occupation ratio and thechemical potential as

fsmqkT ln 1rjy1 5Ž . Ž .

This provides the free energy of cage occupation. Thus, we can predict dissociation pressure if theŽ .chemical potential difference between water ice and empty hydrate is known.

3. Results and discussion

Guest ethane and ethylene are placed at the center of the cage in order to examine a potentialenergy barrier of rotation. The potential curves for the nonspherical ethane and ethylene againstangles of rotation are given in Fig. 1. The C symmetry axis of ethane or ethylene molecule is so`

Ž .chosen to coincide with the minor z axis of the oblately shaped larger cage in the clathrate I. TheŽ .two curves show the potential surfaces by rotating along the two orthogonal vectors x and y axes

perpendicular to the molecular C axis. The other energy curve is also plotted, where molecule is`

rotated around the x axis by 908 and then the potential energy is calculated by rotating along theŽ . y1minor z axis. The potential barrier in the last plot is very low, 1 kJ mol . However, the potential

energy barrier is as high as 6 kJ moly1 around the x or y axis due to the oblate shape of the cage.The GCMC simulations for six ethane and six ethylene hydrate structures at four pressure values

are performed. The pressure values of the gas phase guest molecules are converted to the chemicalpotential values. In usual GCMC simulations of fluid systems, the creation and deletion trials areattempted with the same probability as the trial move; each sort of trials is usually attempted with theprobability of 1r3. The creation step seems to be accepted with a much higher probability in thepresent GCMC method since a location of a guest molecule to be inserted is biased to increase theprobability of acceptance. Therefore, a cycle of our GCMC simulation is composed of one creation ordeletion trial of an arbitrarily chosen cage and the subsequent ten trial moves of either host or guestmolecule. Equilibrium states of six lattice structures are achieved by the initial 50,000 cycles of usualMC simulations with no creation or deletion trial. Then, GCMC simulations are carried out for3,000,000 cycles, in which the first 1,000,000 cycles are used to equilibrate the systems with respectto the occupation of guest molecules.

Ž 0The free energy of cage occupation the free energy contributed from the free rotor, f , isrot.subtracted is calculated and listed in Table 2. The cage occupation ratio from the GCMC simulation

is shown in Figs. 2 and 3 together with that obtained from the direct calculation for xenon previouslyw xreported 14 . The free energy of cage occupation decreases with increasing pressure. The magnitude

of error decreases as the occupation ratio approaches the limiting value, unity, as is evident from Eq.Ž .5 . The agreement of the GCMC result with the direct evaluation of the free energy of cageoccupation is not so good at low pressure. This indirect method results in an error of a few kJ moly1

at lower pressure of the guest species. However, this GCMC simulation provides a way to make arough estimation of the stability of clathrate hydrates. In the case of ethane and ethylene hydrates,agreement with the experiment values is reasonable. The error in dissociation pressure is small,

( )H. TanakarFluid Phase Equilibria 144 1998 361–368366

Table 2Ž . Ž .Free energy of cage occupation by xenon, spherical s and nonspherical l ethane and ethylene evaluated by the direct

Ž .calculation including inharmonic contributions and by the mean occupation according to Eq. 5

Pressure p Free energy

Ž . Ž . Ž . Ž . Ž . Ž .Xenon large Xenon small Ethane s Ethane l Ethylene s Ethylene l

From direct calculationy44.74 y36.52 y39.44 y38.02

From GCMC simulation0.01 y40.83 y32.76 y35.12 y34.71 y33.92 y31.100.04 y42.14 y34.98 y37.16 y35.91 y34.85 y31.040.1 y43.38 y36.39 y38.75 y37.43 y36.16 y31.360.2 y44.01 y36.91 y39.20 y38.07 y36.88 y31.960.4 y44.51 y37.04 y39.41 y38.46 y37.29 y32.81

The free energy of the corresponding free rotor is omitted.‘Large’ and ‘small’ denote type of cages.Free energy is in kJ moly1 and pressure is in MPa.

approximately 0.01 MPa, except for nonspherical ethylene, as expected from the free energycomparison between the present results and the direct evaluation. The energy parameter of ethylenemight be too small for the linear model.

There are several different size parameters of the approximated spherical LJ interaction. Theseparameters are determined from the second virial coefficients or the shear viscosities. Therefore,long-range attractive interaction may contribute more significantly. In some cases, the size parameter

Ž . Ž .Fig. 2. Mean occupation ratio of the larger open circle and solid line cage and smaller cage filled circle and dashed lineŽ . Ž .calculated from GCMC simulation circles for xenon hydrate together with those from direct calculation lines .

( )H. TanakarFluid Phase Equilibria 144 1998 361–368 367

Ž .Fig. 3. Mean occupation ratio of nonspherical ethane and ethylene filled circle and filled square and spherical ethane andŽ .ethylene open circle and open square calculated from GCMC simulations.

for ethylene is larger than that for ethane. However, both bond length and the LJ size parameter forŽ .the methyl or methylene group are crucial in a confined state such as in a larger cage of clathrate

hydrates. The longer bond length for ethane justifies the larger LJ size parameter describing anŽ .approximate spherical interaction for clathrate hydrates. With this larger size parameter for ethane

clathrate hydrate, the LJ interaction parameter must be larger than that for ethylene since theexperimental dissociation pressure for ethylene is higher than that for ethane. Previously, we haveused appropriately those combinations of parameters when the free energies of cage occupation werecalculated with the spherical approximation of guests.

4. Conclusion

The thermodynamic stability of clathrate hydrates encaging nonspherical ethane and ethylenemolecules as well as spherical Xe atoms has been investigated by GCMC simulation. A difficulty inevaluating the anharmonic free energy of cage occupation by a thermodynamic integration is avoided,which arises mainly from hindered guest rotations. Cage occupation ratios at various pressures arecalculated by GCMC simulation where guest molecules are allowed to be created or deleted. Thus, weobtain the free energy of cage occupation, which gives the dissociation pressure. It is shown thisindirect method provides a useful way to predict the stability of a variety of clathrate hydrates withsimple GCMC simulation runs without invoking normal mode analysis and thermodynamic integra-tion when the chemical potential difference of water between ice and empty hydrate is known. Thecalculated free energy in the present study is in fair agreement with the directly evaluated value. Theerror becomes smaller as the pressure of the guest species increases.

( )H. TanakarFluid Phase Equilibria 144 1998 361–368368

Acknowledgements

The author thanks K. Kiyohara for providing proton-disordered clathrate hydrate I structures.

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