Sensitivity of predicted gas hydrate occupancies on treatment of intermolecular interactions

transcript

Sensitivity of predicted gas hydrate occupancies on treatmentof intermolecular interactions

Caroline Thomas, Sylvain Picaud,a! Vincent Ballenegger, and Olivier MousisInstitut UTINAM, Université de Franche-Comté, CNRS/INSU, UMR 6213, Besançon Cedex 25030, France

!Received 21 December 2009; accepted 10 February 2010; published online 12 March 2010"

The sensitivity of gas hydrate occupancies predicted on the basis of van der Waals–Platteeuw theoryis investigated, as a function of the intermolecular guest-water interaction potential model, and ofthe number of water molecules taken into account. Simple analytical correction terms that accountfor the interactions with the water molecules beyond the cutoff distance are introduced, and shownto improve significantly the convergence rate, and hence the efficiency of the computation of theLangmuir constants. The predicted cage occupancies in pure methane and pure carbon dioxideclathrates, calculated using different recent guest-water pair potentials models derived from ab initiocalculations, can vary significantly depending on the model. That sensitivity becomes especiallystrong in the case of multiple guest clathrates. It is shown that the abundances of coenclathratedmolecules in multiple guest clathrate hydrates potentially formed on the surface of Mars can vary bymore than two orders of magnitude depending on the model. These results underline the strong needfor experimental data on pure and multiple guest clathrate hydrates, in particular in the temperatureand pressure range that are relevant in extreme environment conditions, to discriminate among thetheoretical models. © 2010 American Institute of Physics. #doi:10.1063/1.3352570$

I. INTRODUCTION

Clathrate hydrates !also known as gas hydrates or, sim-ply, hydrates" have attracted increasing scientific and engi-neering attention since their discovery by Sir Humphry Davyin 1810.1 Indeed, they have been found in oil and natural gaspipelines,2 in the sea floor,3 in the permafrost,4 and they arealso suspected to be present in comets and certain planetsand/or their satellites.5 These compounds are usually stablein a limited range of temperatures and pressures, and theirdissociation on Earth might lead, for instance, to the releaseof huge amounts of methane which is one of the most im-portant contributor to the greenhouse effect.6,7 In contrast,clathrate hydrates could be used to sequester carbon dioxidein the ocean to reduce its impact on global warming.8 Thepresence of clathrates on Titan and Mars has also been in-voked to interpret the atmospheric depletion of some mol-ecules with respect to solar abundances.5,9–11

Clathrate hydrates are nonstoichiometric crystallinestructures formed by inclusion of small molecules into a net-work of hydrogen bonded water molecules.12 Indeed, thesestructures are characterized by cavities of different sizes thatcan accommodate !usually" one guest molecule per cavity,the size of which determines the specific equilibrium clath-rate structure. The molecules encaged in the clathrate cavi-ties are bonded to the water molecules by van der Waals-typeinteractions and, as a consequence, they are free to rotateand vibrate into the cavity, whereas they have limitedtranslation.13

Two distinct structures of clathrates are mainly encoun-tered in nature. Structure I is a body-centered cubic structure

characterized by a cubic cell that contains 46 water mol-ecules, arranged in two small !512" and six large !51262"cages, where 512 indicates a polyhedron of water moleculesmade up with 12 pentagons, while 51262 indicates a polyhe-dron made up with 12 pentagons and two hexagons. Struc-ture II is a body-centered cubic structure in which each cubiccell contains 136 water molecules arranged in eight large!51264" and 16 small !512" cages.

Assessing the role that clathrate hydrates may play in theEarth’s environment and climate as well as in the atmo-spheric compositions in the solar system requires however asolid understanding of their structural and thermodynamicalproperties which is unfortunately not easily achieved by rou-tine experimental techniques. As a consequence, thermody-namic models have been developed for predicting the phasebehavior of hydrates,4 most of them being derived from thestatistical theoretical approach proposed by van der Waalsand Platteeuw,14 and generalized by Parrish and Prausnitz15

for the calculations of dissociation pressures of multipleguest clathrates hydrates. The major ingredient of these mod-els is the description of the guest-clathrate interaction whichis often represented by a Kihara potential with parametersfitted on experimental equilibrium data.4 However, the trans-ferability of these parameters to temperatures and pressuresbeyond the range for which they have been fitted isquestionable.16,17 Moreover, inconsistencies have been evi-denced between Kihara parameters derived from differentsets of experimental data,18 and it has been claimed thatKihara potential may not accurately describe the interactionbetween guest and water molecules.19 Recent works havethus focused on the determination of accurate intermolecularpotentials for guest-clathrate interactions, in which effectiveparameters are fitted from results of ab initio quantum me-

a"Author to whom correspondence should be addressed. Electronic mail:sylvain.picaud@univ-fcomte.fr.

THE JOURNAL OF CHEMICAL PHYSICS 132, 104510 !2010"

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chanical methods.19–21 Although it has been proved that us-ing such parameters may improve the prediction of pure hy-drate properties in some cases,19 the question of thesensitivity of the predictions to details in the calculation ofthe guest-clathrate interactions needs to be investigated.

In this work, we assess the sensitivity of predictions forcage occupancies in pure and multiple guest gas hydrates,using the statistical van der Waals–Platteeuw model togetherwith recent parameterization of interatomic pair potentialsderived from ab initio calculations for methane and carbondioxide clathrates.19–21 We consider first the sensitivity withrespect to the number of water molecules taken into account,as it has been shown that at least three layers of water mol-ecules around the guest molecules have to be included foraccurate calculations of hydrate properties.20 We then per-form a comparative study between the different potentialmodels available in the literature to describe the H2O–CH4and H2O–CO2 interactions. The predicted cage occupanciesin pure methane and pure carbon dioxide clathrates are seento vary quite significantly depending of the chosen intermo-lecular potential. In the case of multiple guest clathrates, thatsensitivity can become even dramatic: Abundances of coen-clathrated molecules can vary by more than two orders ofmagnitude depending on the model! This point is exempli-fied in the case of clathrates potentially formed on Mars, incontact with an atmosphere containing CO2, CH4, N2, andAr, a case for which trapping efficiencies of gaseous mol-ecules by clathrates have been calculated previously using asimpler description of the guest-hydrate interactions in termsof a spherically averaged Kihara potential.11

The paper is organized as follows. In Sec. II, we brieflyrecall the theoretical basis of the thermodynamical modelused for the calculations of the clathrates occupancies, with aspecial emphasis on the calculations of the guest-clathratelattice interactions. Then, in Sec. III, we calculate the theo-retical occupancies in pure methane and pure carbon dioxideclathrates, and present in Sec. IV the results obtained in mul-tiple guest clathrates potentially formed on Mars. Finally, thelast section is devoted to a summary and a short discussionof the results.

II. THEORETICAL BACKGROUND

A. Thermodynamic model for clathrate hydrates

The calculation of the relative abundance of guest spe-cies incorporated in a clathrate hydrate from a coexisting gasphase of specified composition at given temperature andpressure is made following the original formalism developedby Lunine and Stevenson5 which is based on the statisticalmechanics approach of van der Waals and Platteeuw.14 Thatapproach relies on four key assumptions: The guest speciesdo not distort the cages, the cages are singly occupied, thereare no interactions between guest species in neighboringcages, and classical statistics is valid, i.e., quantum effectsare negligible.4

In this formalism, the relative abundance fG of a guestspecies G in a clathrate !of structure I or II" is defined as the

ratio of the average number of guest molecules of species Gin the clathrate over the average total number of enclathratedmolecules, as

fG =b!yG,! + bsyG,s

b!%JyJ,! + bs%JyJ,s, !2.1"

where the sums in the denominator run over all speciespresent in the system, and bs and b! are the number of small!s" and large !!" cages per unit cell, respectively.

The fraction yG,x !x=! or s" of cages occupied by theguest species G !for a given type of cage and for a given typeof clathrate" are determined from the Langmuir constants as

yG,x =CG,xPG

1 + %JCJ,xPJ, !2.2"

where CG,x is the Langmuir constant of guest species G lo-cated in the cage of type x, and PG is the partial pressure ofguest species G assuming here that the sample behaves as anideal gas !otherwise the fugacity instead of the partial pres-sure of the guest species in the vapor phase should be used".The sum, %J, in the denominator runs over all species Jwhich are present in the initial gas phase. The Langmuirconstants CG,x are related to the strength of the interactionbetween the guest species and the surrounding lattice, as

CG,x =1

kBT& & exp'!

wG,x!r,!"kBT

(drd! , !2.3"

where wG,x is the interaction potential energy between theguest molecule located in the cage of type x and the wholeclathrate hydrate lattice. This interaction depends on the po-sition r and orientation ! vectors of the guest in the cage. Trepresents the temperature and kB the Boltzmann constant.Note that Eq. !2.3" relies on the assumption that the internalmodes of the guest !i.e., internal vibrations and electronicdegrees of freedom" are the same as in the ideal gas phase,that is they are not modified by the guest-lattice interaction.

Thus, this statistical approach relies on the accurate de-termination of the interactions between the guest species Gand the water molecules forming the clathrate hydrate lattice.

B. Guest-lattice interaction energy

The interaction energy between the guest molecules con-sidered here and the clathrate lattice can be represented by asum of pairwise site-site dispersion-repulsion wdisp-rep andelectrostatic welec contributions as

wG,x = %i=1

!wi,ljdisp-rep + wi,lj

elec" , !2.4"

where NG is the number of sites describing the guest mol-ecule and Nx

max represents the maximum number of watermolecules considered around the guest molecule located inthe cage of type x. These water molecules are described byNW sites of interaction.

Two different types of potential are usually considered torepresent the dispersion-repulsion contribution: the Lennard-Jones !LJ" 6–12 potential

104510-2 Thomas et al. J. Chem. Phys. 132, 104510 "2010!

wi,ljLJ = 4!ij)"ij

ri,lj12 !

ri,lj6 * !2.5"

and the Kihara potential !K"

wi,ljK = 4!ij) "ij

!ri,lj ! 2aij"12 !"ij

!ri,lj ! 2aij"6* when ri,lj # 2aij

!2.6"wi,lj

K = $ otherwise

in which ri,lj represents the distance between the ith site ofthe guest molecule and the jth site of the lth surroundingwater molecule and aij is a hard-core radius for the pairunder consideration.

The electrostatic contribution to the total interaction en-ergy between the guest molecule and the clathrate lattice hasto be taken into account when considering a polar moleculeinteracting with water. It can be described by the sum ofCoulombic interactions between partial charges located ondifferent sites of the interacting molecules as22

wi,ljelec =

ri,lj, !2.7"

where qi is the charge located on the ith site of the guestmolecule and qj is the charge located on the jth site of awater molecule.

C. Interaction potential models for CO2 and CH4clathrates

Various parameterizations for the CO2 and CH4-waterinteractions have been derived recently for gas hydrates onthe basis of ab initio calculations.19,20 The effective LJ orKihara parameters proposed by these authors are summa-rized in Table I, where models taken from Klauda and

Sandler20,21 are labeled by KS or models from Sun andDuan19 are labeled by SD.

The dispersion-repulsion interactions can be calculatedwith different levels of atomistic detail, i.e., different num-bers of interacting sites on the CO2, CH4, and H2O mol-ecules. In the SD models, all sites on all molecules are con-sidered: three sites for H2O !labeled Ow and Hw", three sitesfor CO2 !labeled C and O", and four sites for CH4 !labeled Cand H". In the KS-CO2 model, dispersion-repulsion interac-tions for atomic pairs C–Hw and O–Hw are neglected, i.e.,dispersion-repulsion interactions are calculated only with siteOw. The parameters for the various interactions dependmoreover in this model on the type of cage occupied by theCO2 molecule.

In the simplest KS models for the CH4–H2O interac-tions, the methane molecule is represented by a single sitelocated on the carbon atom, while the water molecule is rep-resented similarly with the single site Ow. The interaction iscalculated then either by a LJ potential !model KS-CH4

!1""which depends on the type of cage that encloses the methanemolecule, or by a Kihara potential !model KS-CH4

!4"". In themodels KS-CH4

!2" and KS-CH4!3", only one of the two mol-

ecules !methane or water" is represented by a single site.Note that some of these models also consider electro-

static interactions between the guest and the surrounding wa-ter molecules. Because these electrostatic interactions arehowever supposed to be weak in the case of methane hy-drate, only model SD-CH4 takes them into account, by dis-tributing atomic charges on C and H atoms !see Table II",and by considering the charge distribution of the TIP4Pmodel for water,23 in which a negative charge is carried by asite Mw slightly displaced with respect to the position of theOw atom. These electrostatic interactions can no longer beneglected in the case of CO2 hydrate, and in both KS and SDmodels for CO2, the electrostatic interactions between a CO2

TABLE I. Parameters used to calculate the different guest-water dispersion-repulsion interactions !see text forthe definition of each type of models". " is the LJ diameter, ! is the depth of the potential well at equilibriumdistance, and a is the radius of the impenetrable core of the Kihara-type potentials. Note that some parametersalso depend on the type of cage occupied by the guest molecule.

Model Site of guest Site of water

! /kB !K" " !Å"

a !Å"Small cage Large cage Small cage Large cage

KS-CH4!1" C Ow 123.062 123.747 3.501 3.512 ¯

KS-CH4!2" C Ow 123.976 3.451 ¯

H Ow 1.494 3.207 ¯KS-CH4

!3" C Ow 113.574 3.525 ¯C Hw 33.014 2.236 ¯

KS-CH4!4" C Ow 123.900 3.119 0.383

SD-CH4 C Ow 61.400 3.627 ¯C Hw 22.400 2.200 ¯H Ow 40.790 2.777 ¯H Hw 15.100 1.500 ¯

KS-CO2 C Ow 103.976 120.664 3.480 2.716 ¯O Ow 17.211 78.482 3.129 3.071 ¯

SD-CO2 C Ow 53.400 2.955 ¯C Hw 25.160 2.400 ¯O Ow 90.230 3.034 ¯O Hw 42.560 2.480 ¯

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and a H2O molecule are calculated on the basis of partialcharges distributed on the interacting sites, with the valueslisted in Table II. Again, the TIP4P model is used to repre-sented water molecules in the SD-CO2 model, whereas par-tial charges are distributed on Ow and Hw atoms in theKS-CO2 model.

D. Convergence of calculations of guest-latticeinteractions

Most of the previous clathrates studies are based on thespherically averaged Kihara potential for the calculation ofthe guest-lattice interactions, in which the clathrate lattice isrepresented by one layer of water molecules, only. However,it has been shown that the contribution to the guest-latticeinteraction energy from water molecules beyond the firstshell can be significant.20,24,25 We determine in this sectionthe number of water molecules that has to be taken intoaccount to reach converged results, at minimal computationalcost, for the methane and carbon dioxide clathrates understudy. In the following calculations, the whole clathrate isconsidered as a rigid fixed body. The locations of the oxygenatoms in the water lattice were determined from diffractionexperiments, whereas the positions of the hydrogen atoms ofthe water molecules are randomly chosen to minimize thedipole moment of the clathrate structure26,27 in a manner con-sistent with the Bernal–Fowler ice rules.28

Successive water shells around a guest molecule in aclathrate can be defined based on the distribution functionsof the water molecules around the center of the occupiedcage for structures I and II. These distribution functions, forsmall and large cages, are characterized by quite well-defined peaks allowing the definition of up to four waterlayers located, for a small cage, at around 4 Å from thecenter !first shell", between 6.5 and 7 Å !second shell", be-tween 8 and 9 Å !third shell" and between 9.5 and 12 Å!fourth shell". Around a large cage, these shells are locatedbelow 4.7 Å from the center of the cage !first shell", between7 and 8 Å !second shell", between 8 and 10 Å !third shell",and between 10 and 13 Å !fourth shell". Beyond these dis-tances, a nearly homogeneous distribution of water mol-

ecules is seen from the center of the cage. As a consequence,the water molecules beyond the fourth shell can be well rep-resented by means of a continuous approach which leads to acorrection to the potential energy of a guest molecule com-puted using a spherical cutoff of radius rc. For thedispersion-repulsion contribution, and for condensed phasessuch as clathrates considered here, this correction reads29

&wdisp-rep = !163

rc3 , !2.8"

where ' is the number of water molecules per unit volume.The corresponding correction for the electrostatic contri-

bution to the guest-lattice interaction is based on the reactionfield method30 which assumes a constant dielectric environ-ment beyond the cutoff rc characterized by the dielectric con-stant !. This correction for the condensed phases consideredhere is thus written as30

&welec =1

4%!0rc3

!! ! 1"2! + 1 %

i,jqiqjri,lj

2 . !2.9"

For the clathrates considered in the present study, !=58.12

As an illustration of the importance of taking into ac-count the interaction energies with distant water molecules#Eqs. !2.8" and !2.9"$, we consider here the case of a methaneclathrate of structure I and calculate the corresponding valuesof the Langmuir constant CCH4,x as a function of rc #see Eq.!2.3"$. Figure 1 shows the results obtained when the guest-lattice interactions are represented by the KS-CH4

!1" model,for a methane molecule located either in a small !x=s" or ina large !x=!" cage. The corresponding results indicate thatlarge values of rc !i.e., a large number of water molecules"are required to obtain the convergence in the calculated val-ues of CCH4,x. Moreover, they show that considering only thefirst water shell around the guest molecule is a very badapproximation leading to a value of the Langmuir constantwhich can be typically one order of magnitude smaller thanthe converged value. However, including the correctionterms #Eqs. !2.8" and !2.9"$ in the calculations of the guest-lattice interaction energy greatly improves the convergenceof the CCH4,x values, which is obtained for rc values of about

TABLE II. Partial charges used to calculate the different guest-water elec-trostatic interactions !see text for the definition of each type of models". KSand SD indicate whether these parameters are taken from Klauda and San-dler !2003" or from Sun and Duan !2005".

Model Molecule Site

Charges

Small cage Large cage

KS H2O Hw 0.458 0.458Ow (0.916 (0.916

CO2 C 0.649 0.652O (0.3245 (0.326

SD H2O Hw 0.52Mw (1.04

CH4 C (0.48H 0.12

CO2 C 0.652O (0.326

FIG. 1. Values of the Langmuir constant obtained for a methane clathrate ofstructure I when using the KS-CH4

!1" model to represent the CH4–H2O in-teractions. The results are given for different values of rc and for a methanemolecule located either in !a" a small cage or !b" in a large cage. Trianglescorrespond to values obtained with an atom-atom approach including onlythe water molecules located within a sphere of radius rc with respect to thecenter of the cage, whereas squares correspond to values calculated by tak-ing into account the additional corrections due to the interaction with watermolecules located beyond rc. Note that dashed and dotted lines are guide foreyes, only.

typically 12–13 Å. In other words accurate calculations ofthe Langmuir constants require to take explicitly into ac-count at least four water shells around the guest molecule,with additional corrections to the potential interaction energyrepresenting the water layers beyond the fourth shell. Verysimilar results are also obtained when considering the otherpotential models considered here, and the present conclu-sions are also valid for carbon dioxide clathrates.

III. CAGE OCCUPANCIES FOR PURE METHANE ANDPURE CARBON DIOXIDE CLATHRATES

To assess the reliability of calculations made with themethods presented above, we calculate, for the various inter-action potential models introduced in Sec. II C, a physicalobservable of interest, namely the cage occupancy yG,x in thecase of a pure methane !G=CH4" or pure carbon dioxide

!G=CO2" clathrate of structure I for large !x=!" or small!x=s" cages. All the calculations are performed at 273 K andat the corresponding dissociation pressure. The results for thedifferent interaction models are given in Tables III and IV forthe methane and carbon dioxide clathrates, respectively, to-gether with the corresponding values of the Langmuir con-stants. We also report in these tables the results obtainedwhen increasing the number of water layers taken into ac-count in the calculations.

There are two ways for reading these results. First, if weconsider the occupation of small cages calculation with agiven model, for instance the KS-CH4

!3" model !third line ofTable III", the different columns of Table III show that thevalues of the Langmuir constant vary by more than a factorof 5 when taking into account in the interaction potential aninfinite number of water molecules #i.e., four layers of water

TABLE III. Values of the occupancies yK,x and of the Langmuir constants CK,x !Pa!1" calculated for the different models considered here, for a methanemolecule located in a clathrate of structure I !x=s ,! for small and large cages, respectively". Different numbers of water shells around the methane moleculehave been taken into account in the calculations. The last column corresponds to the most accurate calculations, taking explicitly into account four water shellsand adding corrections to the interaction energy due to the water molecules beyond the fourth shell. All the calculations have been performed at 273 K andat the corresponding dissociation pressure.

Type of cage Model 1 shell 2 shells 3 shells 4 shells 4 shells with correction

Small KS-CH4!1" CK,s 3.1817)10!7 6.8275)10!7 1.0623)10!6 1.4231)10!6 1.7539)10!6

yK,s 44.5% 63.3% 72.8% 78.2% 81.6%KS-CH4

!2" CK,s 3.6251)10!7 7.5371)10!7 1.1502)10!6 1.5205)10!6 1.8555)10!6

yK,s 47.8% 65.5% 74.4% 79.3% 82.4%KS-CH4

!3" CK,s 3.1831)10!7 6.8540)10!7 1.0666)10!6 1.4286)10!6 1.7600)10!6

yK,s 44.5% 63.3% 72.9% 78.3% 81.6%KS-CH4

!4" CK,s 3.7758)10!7 6.6506)10!7 9.0359)10!7 1.0930)10!6 1.2173)10!6

yK,s 48.8% 62.6% 69.5% 73.4% 75.4%SD-CH4 CK,s 6.0507)10!7 1.3248)10!6 2.0689)10!6 2.7675)10!6 3.3932)10!6

yK,s 60.4% 77.0% 83.9% 87.5% 89.5%Large KS-CH4

!1" CK,! 1.7425)10!6 3.4036)10!6 5.2912)10!6 6.8891)10!6 8.1245)10!6

yK,! 81.5% 89.6% 93.0% 94.6% 95.3%KS-CH4

!2" CK,! 1.6031)10!6 3.0001)10!6 4.5290)10!6 5.7917)10!6 6.7520)10!6

yK,! 80.2% 88.3% 91.9% 93.6% 94.4%KS-CH4

!3" CK,! 1.6293)10!6 3.1455)10!6 4.8404)10!6 6.2618)10!6 7.3530)10!6

yK,! 80.4% 88.8% 92.4% 94.0% 94.9%KS-CH4

!4" CK,! 1.1671)10!6 1.8788)10!6 2.5209)10!6 2.9731)10!6 3.2304)10!6

yK,! 74.6% 82.6% 86.4% 88.2% 89.1%SD-CH4 CK,! 3.5570)10!6 6.9349)10!6 1.0682)10!5 1.3791)10!5 1.6135)10!5

yK,! 90.0% 94.6% 96.4% 97.2% 97.6%

TABLE IV. Values of the occupancies yK,x and of the Langmuir constants CK,x !Pa!1" calculated for the different model considered here, for a CO2 clathrateof structure I !x=s ,! for small and large cages, respectively". Different numbers of water shells around the methane molecule have been taken into accountin the calculations. The last column corresponds to the most accurate calculations, taking explicitly into account four water shells and adding corrections tothe interaction energy due to the water molecules beyond the fourth shell. All the calculations have been performed at 273 K and at the correspondingdissociation pressure.

Type of cage Model 1 shell 2 shells 3 shells 4 shells 4 shells with correction

Small KS-CO2 CK,s 3.1635)10!7 8.3285)10!7 1.4553)10!6 2.1021)10!6 2.5677)10!6

yK,s 25.2% 47.0% 60.7% 69.1% 73.2%SD-CO2 CK,s 3.1236)10!6 9.2936)10!6 1.6798)10!5 2.4652)10!5 3.0352)10!5

yK,s 76.9% 90.8% 94.7% 96.3% 97.0%Large KS-CO2 CK,! 6.2343)10!7 1.2316)10!6 1.8188)10!6 2.3874)10!6 2.7851)10!6

yK,! 39.9% 56.7% 65.9% 71.7% 74.8%SD-CO2 CK,! 1.0330)10!5 2.2742)10!5 3.5415)10!5 4.7286)10!5 5.5504)10!5

yK,! 91.7% 96.1% 97.4% 98.1% 98.3%

molecules+correcting terms of Eqs. !2.8" and !2.9"$ insteadof only one water shell around the methane molecule. As aconsequence, the value of the cage occupancy yCH4,s in-creases from 44.5% to 81.6% when the number of watermolecules considered in the interaction energy calculationsincreases. Even larger variations are obtained for the CO2clathrate !Table IV" with the KS-CO2 model !i.e., a modelsimilar to the KS-CH4

!3" model" because yCO2,s increases from25.2% !one layer of water molecules in the calculations" to73.2% !whole clathrate considered". Similar results are alsoobtained for the other models considered in the present study,with cage occupancy values which depend on the number ofwater molecules taken into account. Reliable predictions forphysical observables can therefore only be made with accu-rate values of the Langmuir constant, which requires, asshown in Sec. II D, to consider at least four water shells andcorrection terms for the cutoff errors.

If we consider now the last columns of Tables III and IV,i.e., the most accurate calculations of the cage occupancyvalues, the present results also show large variations in theLangmuir constant and of the cage occupancy values de-pending on the interaction model used. Indeed, for example,yCH4,! increases from 89.1% to 97.6% when considering theSD-CH4 model instead of the KS-CH4

!3" model, for a meth-ane clathrate with CH4 molecules located in large cages. In asimilar way, for a carbon dioxide clathrate !with large cageoccupancy, last column of Table IV" yCO2,! increases from74.8% with the KS-CO2 model to 98.3% with the SD-CO2model.

It is however interesting to note that the variations in thevalues of yG,x are less important when changing the guest-clathrate interaction model !i.e., last columns of Tables IIIand IV" than when varying the number of water moleculestaken into account in the calculation of this interaction. Inother words, calculations of the clathrate cage occupanciesbased on a rather simple model allowing to account for avery large number of water molecules are certainly as accu-rate as calculations based on a very sophisticated model !ob-tained for instance from quantum calculations" but includinga limited number of water molecules only.

We also compare in Table V the most accurate values ofyG,x obtained with the different models considered here withthe corresponding values obtained using the standard ap-proach based on a spherically averaged Kihara potential us-ing the Parrish and Prausnitz parameters11,15 !denoted as thePP-CH4 and the PP-CO2 models". For both methane and car-bon dioxide clathrates, the cage occupancies calculated withthe spherically averaged Kihara potentials are within therange of values calculated with the various more sophisti-cated atom-atom models considered here. This can be under-stood by the fact that the Kihara potential parameters pro-posed by Parrish and Prausnitz are based on experimentalresults and thus they take into account the interaction withthe whole clathrate through effective parameters.15

Let us finally mention that the comparison of the calcu-lated yG,x values with experimental data should in principlediscriminate between the different potential models usedhere. Unfortunately, there are only a few experimental resultsavailable in the literature and, in addition, they are not fully

consistent with each other.31–33 Nevertheless, we report onFig. 2 the cage occupancies for methane clathrate measuredfrom Raman spectroscopy near32 or above33 the equilibriumpressure around 273 K together with the calculated valuesobtained using the different models considered here and tak-ing into account the whole clathrate in the calculations of theguest-clathrate interactions. These values have been calcu-lated at different pressures for the comparison with the ex-perimental data. Figure 2 shows that all the calculated valuesfall within the experimental error bars. However, if theKS-CH4

!4" model appears quite good at low pressures and forthe small cages, it is rather bad at higher pressures and itseems even worse for large cages. In contrast, the other mod-els !KS-CH4

!1", KS-CH4!2", KS-CH4

!3", and SD-CH4" appearrather well suited to describe methane in large cages, but lessaccurate for occupancy of small cages. It is interesting tonote again that the results obtained with the spherically av-eraged Kihara potential PP-CH4 proposed by Parrish andPrausnitz15 are also compatible with the reported experimen-tal data.

As far as we know, the experimental data available in theliterature are unfortunately limited to conditions that are rep-resentative of the clathrate formation on Earth only. How-ever, clathrates may also be stable at lower temperatures andpressures on other bodies of the solar system, and it is offundamental interest to also assess the accuracy of the inter-action potential calculations in these conditions. We thus cal-culate the cage occupancies for pure methane and pure car-bon dioxide clathrates as a function of the temperature, andfor pressures along the corresponding dissociation curves.The results are given in Fig. 3 for two different models of

TABLE V. Comparison between the most accurate values of the Langmuirconstants and of the occupancies yK,x !x=s ,! for small and large cages,respectively" obtained for pure CH4 and CO2 clathrates of structure I !seelast column of Tables III and IV" and the corresponding values obtainedwhen using a spherically averaged Kihara potential based on the Parrish andPrausnitz !1972" parameters. All the calculations have been performed at273 K and at the corresponding dissociation pressure.

Molecule Model Small cages !x=s" Large cages !x=!"

CH4 KS-CH4!1" CK,x 1.7539)10!6 8.1245)10!6

yK,x 81.6% 95.3%KS-CH4

!2" CK,x 1.7600)10!6 7.3530)10!6

yK,x 81.6% 94.9%KS-CH4

!3" CK,x 1.8555)10!6 6.7520)10!6

yK,x 82.4% 94.4%KS-CH4

!4" CK,x 1.2173)10!6 3.2304)10!6

yK,x 75.4% 89.1%SD-CH4 CK,x 3.3932)10!6 1.6135)10!5

yK,x 89.5% 97.6%PP-CH4 CK,x 3.1142)10!6 1.5054)10!5

yK,x 88.7% 97.4%CO2 KS-CO2 CK,x 2.5677)10!6 2.7851)10!6

yK,x 73.2% 74.8%SD-CO2 CK,x 3.0352)10!5 5.5504)10!5

yK,x 97.0% 98.3%PP-CO2 CK,x 3.9194)10!6 4.1994)10!5

yK,x 80.6% 97.8%

interactions, namely KS-CH4!2" and SD-CH4 for methane, and

KS-CO2 and SD-CO2 for carbon dioxide clathrates, respec-tively. This figure shows that the clathrate occupancy in-creases when decreasing the temperature, irrespective of themodel and of the clathrate considered in the calculations.However, the most interesting feature is that the two modelsfor methane lead to very similar clathrate occupancy!#0.9" below 200 K, indicating that the modeling of meth-ane clathrates at low temperature would not strongly dependon the interaction potential considered in the calculations.Such a conclusion is however not valid for carbon dioxideclathrates because the calculated occupancies strongly de-pend on the interaction model, even at low temperature ofastrophysical interest.

To conclude this section, it is worth noting that becausedifferent models lead to different values of the cage occu-pancy, careful experimental measurements in the temperaturerange of interest are actually necessary to assess the accuracyof the theoretical approaches.

IV. CAGE OCCUPANCIES IN A MULTIPLE GUESTCLATHRATE: APPLICATION TO MARS

When a multiple guest clathrate forms from a coexistinggas phase of specified composition, the calculated cage oc-cupancies can be even more sensitive to small differences inthe calculations of the intermolecular potentials than in pureclathrates. As an illustration, we characterize the compositionof clathrate hydrates potentially formed in the near subsur-face of Mars as a function of the gas phase composition,reinvestigating published results11 obtained using Langmuirconstants calculated from simple averages of Kihara poten-tials on spherical cavities.

All the calculations are performed at the particular pointon the dissociation curve that corresponds to the present av-erage atmospheric pressure on Mars, i.e., P=7 mbar and at atemperature that is thus equal to 152.9 K. The gas phasecontains carbon dioxide, nitrogen, and argon,34 together withmethane because it has also been recently detected in theMartian atmosphere.35

Following Ref. 11, four different initial abundances ofmethane in the gas phase are considered, namely 0.01%,

FIG. 2. Comparison between the occupancies yCH4,x !x=s ,! for small andlarge cages, respectively" calculated with different models for describing themethane/clathrate interaction potential and the corresponding experimentaldata for a pure methane clathrate. !a" Occupancy of small cages and !b"occupancy of large cages. The experimental data are taken from Uchida etal. !Ref. 33" !blue circles and pink diamonds correspond to measurements at273.6 and 273.8 K, respectively" and from Sum et al. !Ref. 32" !measure-ments at 273.6 K, green squares". The different models used for the calcu-lations of the methane/clathrate interaction are !see text": KS-CH4

!1" !blackcircles", KS-CH4

!2" !white empty circles", KS-CH4!3" !black triangles",

KS-CH4!4" !white empty triangles", SD-CH4 !black squares", and PP-CH4

!black diamonds" models.

FIG. 3. !a" Variation with temperature of the occupancies yCH4,x !x=s ,! forsmall and large cages, respectively" calculated with the KS-CH4

!2" !blackcurves" and the SD-CH4 !blue curves" models for a pure methane clathrateof structure I. !b" Variation with temperature of the occupancies yCO2,x cal-culated with the KS-CO2 !black curves" and the SD-CO2 !blue curves" mod-els for a pure methane clathrate of structure I. Results for small !x=s" andlarge !x=!" cages are represented by solid and dashed lines, respectively.

10%, 50%, and 90%. The largest values are typical ofmethane-rich conditions in which CH4 might be suppliedfrom the subsurface of Mars by microbial or geological pro-cesses or typical of ancient atmospheres. In contrast, the low-est values are more typical of recent atmospheric composi-tions. In each case, we assume that the ratios CO2 /N2,CO2 /Ar, and N2 /Ar are equal to those measured in thepresent Martian atmosphere, and xCH4

+xAr+xCO+xN2=1.

For the particular point considered on the dissociationcurve, we thus calculate the relative abundance fG #Eq. !2.1"$for the different gases trapped in the multiple guest clathrate,for each initial methane abundance in the gas phase, as wellas the corresponding abundance ratio which is defined as theratio between the relative abundance fG of a given gas in themultiple guest clathrate hydrate and its initial gas phaseabundance xG.9,10 As an illustration, three different modelshave been considered for CH4 and CO2 in the calculations ofthese relative abundances and abundance ratios !KS-CH4

!2",SD-CH4, KS-CO2, and SD-CO2 for atom-atom potentialmodel, as well as spherically averaged Kihara potentials usedin our previous study of clathrate on Mars11", whereas theguest-clathrate interactions for N2 and Ar have been repre-sented by spherically averaged Kihara potentials, only, to beconsistent with our previous study on Mars11 and because, asfar as we know, other potentials are unfortunately missing inthe literature.

Let us first recall the main conclusions obtained previ-ously when using spherically averaged Kihara potentials:11

!i" the relative abundance of Ar and N2 is almost negligiblein multiple guest clathrates on Mars; !ii" although the abun-dance ratio !fCH4

/xCH4" of CH4 increases with xCH4

, as a con-

sequence of the larger trapping of methane in the corre-sponding multiple guest clathrate hydrates, CH4 is poorlytrapped in clathrate hydrates on Mars; !iii" CO2 is efficientlytrapped in clathrate hydrates, with an abundance ratio alwayslarger than 1; !iv" finally, although the trapping of CH4 ismore and more efficient when the initial gas phase is en-riched in CH4, it remains much less efficient than the trap-ping of CO2.

As shown by the results given in Table VI, these conclu-sions may change when using more refined potential modelsto calculate the guest-clathrate interactions. Indeed, the quan-titative values of the relative abundances, and hence of theabundance ratios, exhibit significant differences when usingdifferent potential models, especially when the gas phasecontains a large proportion of methane. For instance, thepresent results show that for a gas phase containing 90% ofmethane, the calculated abundance of methane in the corre-sponding multiple-guest clathrate would decrease from 91%to 26% when using the SD model instead of the KS one, withan intermediate value of 50% of methane when using the PP!i.e., spherically averaged Kihara" potential.11 As a conse-quence, the relative abundance of carbon dioxide would in-crease from about 8% to more than 73% when changing theinteraction potentials.

Moreover, different interaction models lead to signifi-cantly different values of the calculated abundance ratios.Indeed, this ratio for methane remains lower than 1 in allsituations when using SD and PP potentials, whereas it isalways greater than 1 when using the KS potentials, meaningthat the efficiency of the methane trapping is strongly depen-dent on the interaction model used in the calculations. A

TABLE VI. Relative abundances fG of CH4, CO2, N2, and Ar calculated in multiple guest clathrate as a functionof the initial gas phase abundances !xK". These values together with the abundance ratios !fK /xK" are calculatedat 7 mbar and 152.9 K !i.e., on the dissociation curve" for different atom-atom models describing the CH4 andCO2-clathrate interactions. The KS column gives the results obtained when using the KS-CH4

!2" and the KS-CO2

models together, whereas the SD column corresponds to calculations performed when using together theSD-CH4 and the SD-CO2 models. In all cases the calculations have considered spherically averaged Kiharapotentials based on the Parrish and Prausnitz parameters !Ref. 15" for N2-clathrate and Ar-clathrate interactions.All these results are also compared with those previously obtained when using spherically averaged Kiharapotentials for all species !PP column" !Ref. 11".

Molecule xK

fK fK /xK

KS SD PP KS SD PP

CH4 1)10!4 2.27)10!4 4.29)10!6 1.66)10!5 2.2702 4.2883)10!2 0.1660.1 0.1937 4.74)10!3 1.77)10!2 1.9370 4.7394)10!2 0.1770.5 0.6298 4.08)10!2 0.127 1.2596 8.1660)10!2 0.2540.9 0.9155 0.2646 0.496 1.0172 0.2940 0.551

CO2 0.957 0.9783 0.9999 0.999 1.0223 1.0449 1.0440.861 0.7900 0.9952 0.982 0.9175 1.1559 1.1060.478 0.3641 0.9591 0.873 0.7617 2.0065 1.8260.096 8.35)10!2 0.7353 0.504 0.8694 7.6596 5.250

N2 2.72)10!2 1.74)10!2 5.16)10!5 2.20)10!4 0.6412 1.90)10!3 0.81)10!2

2.40)10!2 1.31)10!2 5.03)10!5 2.05)10!4 0.5441 2.10)10!3 0.85)10!2

1.36)10!2 4.67)10!2 4.91)10!5 1.54)10!4 0.3433 3.61)10!3 1.13)10!2

0.27)10!2 7.04)10!2 3.49)10!5 5.86)10!5 0.2609 1.29)10!3 2.15)10!2

Ar 1.61)10!2 3.98)10!3 1.39)10!5 2.71)10!4 0.2473 8.62)10!4 1.68)10!2

1.50)10!2 3.29)10!3 1.43)10!5 2.62)10!4 0.2196 9.54)10!4 1.75)10!2

0.81)10!2 1.41)10!3 1.35)10!5 1.69)10!4 0.1739 1.66)10!3 2.10)10!2

0.16)10!2 3.01)10!4 1.05)10!5 4.56)10!5 0.1883 6.54)10!3 2.83)10!2

reverse situation is obtained when considering CO2, theabundance ratios of which being always larger than 1 whenusing PP and SD models and between 1 and 0.87 when usingKS potentials.

These differences can be related to the results presentedin Fig. 3 where it is shown that KS and SD models do notexhibit the same behavior with respect to carbon dioxidetrapping, whereas they give similar results for CH4. Indeed,around 150 K, only 80% of the large cages and 90% of thesmall cages of a pure CO2 clathrate are filled when using theKS potential, whereas the occupancy of both large and smallcages calculated with the SD potential is very close to one.As a consequence, when considering a multiple guest clath-rate dominated by CO2, there is certainly much more emptyspace to allow the trapping of large amount of methane whenusing the KS instead of the SD potential model.

V. CONCLUSIONS

We have presented a detailed investigation of the sensi-tivity of theoretically predicted gas clathrate occupancies onthe way of calculating guest-water interactions. The calcula-tions are based on the van der Waals and Platteeuw theory inwhich the clathrate occupancies can be directly computedthrough the knowledge of the intermolecular interaction po-tential between the guest and the water molecules of theclathrate. This interaction is often described by very simplemodels such as the spherically averaged Kihara potential, butmore sophisticated empirical forms based on atom-atom ap-proaches with interaction parameters fitted on ab initio cal-culations have recently appeared in the literature.19–21 How-ever, when using such atom-atom potential models, theaccuracy of the predicted values is very sensitive to the num-ber of water molecules taken into account in the calculationsof the Langmuir constants. We have thus determined theminimum number of water molecules that need to be takeninto account explicitly to get converged results for the Lang-muir constants of a pure methane and a pure carbon dioxideclathrate. Furthermore, we used in our approach simple ana-lytical correction terms that account for the interactions withthe water molecules beyond the cutoff distance. These termsimprove significantly the convergence rate, and hence theefficiency of the computation of the Langmuir constants.

We have also performed a comparative study betweenthe different potential models available in the literature todescribe the H2O–CH4 and H2O–CO2 interactions, and wehave shown that these different models can lead to signifi-cant variations of the calculated pure clathrate occupancies.Moreover, we have shown that these variations can be a keypoint when considering multiple guest clathrates because ofthe competition that exists between the difference species tofill the cages. Indeed, considering the specific case of mul-tiple guest clathrates that may be formed on Mars, weshowed that different interaction models lead to differentconclusions concerning the calculated abundance ratios andthus the trapping efficiency of clathrates at the surface ofMars.

The sensitivity of the results to the treatment of the in-termolecular interaction should thus be emphasized in all thestudies devoted to the prediction of multiple guest clathrateoccupancies. Moreover, the present conclusions support astrong need for experimental data on pure and multiple guestclathrate hydrates, hopefully in the temperature and pressurerange that are typical of extreme environment conditions.

ACKNOWLEDGMENTS

We thank Dr. S. Alavi for providing us the coordinates ofthe water molecules in the cubic cell of clathrates of struc-tures I and II.

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Sensitivity of predicted gas hydrate occupancies on treatment of intermolecular interactions

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