Accurate description of phase diagram of clathrate hydrates at the molecular levelRodion V. Belosludov, Oleg S. Subbotin, Hiroshi Mizuseki, Yoshiyuki Kawazoe, and Vladimir R. Belosludov Citation: The Journal of Chemical Physics 131, 244510 (2009); doi: 10.1063/1.3276282 View online: http://dx.doi.org/10.1063/1.3276282 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/131/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Publisher’s Note: “Accurate description of phase diagram of clathrate hydrates at the molecular level” [J. Chem.Phys.131, 244510 (2009)] J. Chem. Phys. 132, 059901 (2010); 10.1063/1.3309961 Simulations of structure II H 2 and D 2 clathrates: Potentials incorporating quantum corrections J. Chem. Phys. 128, 064506 (2008); 10.1063/1.2825618 Fast synthesis method and phase diagram of hydrogen clathrate hydrate Appl. Phys. Lett. 88, 131909 (2006); 10.1063/1.2190273 Erratum: “Solid-fluid equilibrium in molecular models of n-alkanes” [J. Chem. Phys. 110, 664 (1999)] J. Chem. Phys. 118, 995 (2003); 10.1063/1.1528894 Phase Diagrams and Thermodynamic Properties of Binary Systems of Drugs J. Phys. Chem. Ref. Data 28, 889 (1999); 10.1063/1.556040

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Accurate description of phase diagram of clathrate hydratesat the molecular level

Rodion V. Belosludov,1,a� Oleg S. Subbotin,1,2 Hiroshi Mizuseki,1 Yoshiyuki Kawazoe,1

and Vladimir R. Belosludov1,2

1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan2Nikolaev Institute of Inorganic Chemistry, SB RAS, Novosibirsk 630090, Russia

�Received 19 October 2009; accepted 30 November 2009; published online 29 December 2009;corrected 19 January 2010�

In order to accurately estimate the thermodynamic properties of hydrogen clathrate hydrates, wedeveloped a method based on the solid solution theory of van der Waals and Platteeuw. This modelallows one to take into account the influence of guest molecules on the host lattice and guest-guestinteractions—especially when more than one guest molecule occupies a cage. The free energies,equations of state, and chemical potentials of hydrogen and mixed propane-hydrogen clathratehydrates of cubic structure II with different cage fillings have been estimated using this approach.Moreover, the proposed theory has been used for construction p−T phase diagrams of hydrogenhydrate and mixed hydrogen-propane hydrates in a wide range of pressures and temperatures. Forthe systems with well defined interactions the calculated curves of “guest gas-hydrate-ice Ih”equilibrium agree with the available experimental data. We also believe that the present modelallows one not only to calculate the hydrogen storage ability of known hydrogen hydrate but alsopredict this value for structures that have not yet been realized by experiment. © 2009 AmericanInstitute of Physics. �doi:10.1063/1.3276282�

I. INTRODUCTION

Dyadin et al.1,2 found the anomalous behavior of theH2O–H2 system at hydrogen pressures of 100–360 MPa andtemperatures of 263–283 K. It was hypothesized that a clath-rate phase of hydrogen hydrate was formed at these condi-tions. Interest in clathrate hydrates as potential hydrogenstorage materials has risen after the reports in which the au-thors not only confirmed the formation of the hydrogenclathrate hydrate with cubic structure II �CS-II� at 220 MPaand 234 K �Ref. 3� but also indicated the hydrogen contentwas 50 g/liter, which corresponds to 5.3 mass %.4 Moreover,the resulting hydrates remain stable at ambient pressures andliquid nitrogen temperatures T=77 K and decompose withhydrogen emission when heated to 140 K and hence may beused for hydrogen storage.

Following these works, many experimental and theoret-ical studies of hydrogen hydrates using different methodshave been conducted �see the recent reviews5,6�. Much of thework concentrated on accurate estimation of the hydrogencapacity of the CS-II clathrate hydrate and under what con-ditions the maximum capacity can be achieved. The cubicstructures CS-II have two types of cavities, the pentagonaldodecahedron �512� and hexakaidecahedron �51264�, com-monly classified as small and large cages, respectively. Dueto the small size of hydrogen molecules, single occupancy oflarge cavities was not expected. Thus, the possibility of four-fold filling of large cages and twofold filling of small cagesby hydrogen molecules proposed by Mao et al.3 have beenanalyzed in detail by several groups. The neutron diffraction

studies of the composition of CS-II hydrogen hydrate wereperformed and the single occupancy of small cavity wasfound,7 which was recently confirmed by other methods.8,9

Moreover, the hydrogen occupancy of large cavities can bereversibly varied and four molecules can be stored at highpressure or low temperature conditions.7 Other work focusedon the formation of hydrogen hydrates at lower pressuresince the extreme pressure required to stabilize pure hydro-gen clathrate hydrates makes it impractical for commercialuse. A significant reduction in the hydrate formation pres-sure, nearly two orders of magnitude, was found in the tet-rahydrofuran �THF�-hydrogen-water system.10,11 It wasshown that this binary CS-II THF+hydrogen hydrate is sta-bilized by the introduction of the THF molecules into thelarge cavities, reducing the hydrogen storage capacity of theCS-II hydrate. In light of these results, several studies havebeen performed regarding the hydrogen occupancy of thesmall cavity and the possibility of the large cage occupancyby hydrogen molecules at low concentrations of THF.12–15

Based on these results it was concluded that in the binaryTHF+H2 hydrate the hydrogen molecule exists only in smallcavities with single occupancy and the occupation of largecavities by hydrogen molecules at low THF concentrationsproposed in earlier study11 was not observed.

In parallel with the most studied THF+H2 binary hy-drate, other experimental studies on the various structures ofmixed hydrates of hydrogen with cyclohexanone,16 carbondioxide SO2,17,18 methane CH4,5,19 propane,20,21 tetra-n-butylammonium bromide and tetra-n-butylammoniumfluoride,22 methylcyclohexane, and methyl tert-butyl ether23

have been carried out and their stability ranges as well asa�Electronic mail: rodion@imr.edu.

THE JOURNAL OF CHEMICAL PHYSICS 131, 244510 �2009�

0021-9606/2009/131�24�/244510/12/$25.00 © 2009 American Institute of Physics131, 244510-1

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hydrogen compositions have been also determined. Many in-vestigated systems revealed similar thermodynamic behaviorwith that of the THF+H2 binary hydrate, some of whichshowed slightly higher stability but without improved hydro-gen storage ability. However, the possible formation of hy-drogen clathrate with different hydrate structures by addingvarious guest molecules suggests the practical feasibility ofthe binary hydrate as a hydrogen storage candidate,5 espe-cially, if the “composition tuning mechanism” can be real-ized in multicomponent systems with the guest moleculesother than THF. In this case, it is important to study thephase diagram of multicomponent clathrate hydrates in de-tail, which is still a challenging task due to their complexity.From this point of view, theoretical approaches can be usefulfor understanding the physical and chemical properties of thebinary hydrate and they can support the experimental explo-ration of novel hydrogen storage materials based on clathratehydrates.

To date the theoretical investigations of pure and mixedhydrogen hydrates with CS-II structure using different meth-ods have been performed. Density functional theory,24,25

quantum chemical,26–31 diffusion quantum Monte Carlo,32

statistical thermodynamics,33 grand canonical Monte Carlo,34

and classical molecular dynamics35–37 modeling have beenused. These works have mainly been related to the propertiesof the hydrogen molecules and additional guests inside watercavities and their effects on the energetic and dynamic sta-bilities of the hydrates. Thus, the possibility of filling of largecages by clusters of hydrogen molecules has been shownusing these models. However, various conclusions have beendrawn from these studies regarding the H2 occupancy ofsmall cages of the host lattice.

Currently, theoretical models used for description ofphase diagrams are based on the statistical model of idealsolid clathrate solutions of van der Waals–Platteeuw�vdWP�.38 In this approach, the agreement with experimentaldata for concrete systems is achieved using the phenomeno-logical parameters specified for each system. This theory andall of its subsequent derivatives embody four main assump-tions: �a� the cages contain at most one guest; �b� the guestmolecules do not interact with each other; �c� the host latticeis unaffected by the nature as well as by the number of en-caged guest molecules; and �d� classical statistics is valid.The first three assumptions are clearly violated in the case ofhydrogen clathrates with multiple occupancy by guest mol-ecules. Several studies were performed to generalize theoriginal vdWP in order to overcome these limitations. Theinfluence of the guest molecules on the host lattice in de-scription of thermodynamics properties of gas hydrates wastaken into account by Zele et al.39 Within this model, theequilibrium conditions of the CS-II hydrogen hydrate withmultiple occupancy were determined using the Langmuirconstants calculated for single guest molecules representedas rigid clusters of 4 and 2 hydrogen molecules inside thelarge and small cavities, respectively.40 Later, this approachwas extended to several multicomponent gas hydrates usingthe same approximation for calculation of Langmuirconstants.41 Recently, the phase behavior of various hydrates

containing THF and H2 was reproduced using the thermody-namic model based on the nonideal hydrate solid solutionmodel for multiphase equilibrium.42 In that work, instead ofthe variation of parameters used in different models, a singleset of consistent parameters within of framework of a singlemodel has been proposed which allow one to reproduce hy-drate formation conditions a number of hydrates containingTHF and/or H2.

The extension of the vdWP statistical thermodynamicalmodel without specific phenomenological parameters wasformulated by Tanaka and Kiyohara.43,44 Within the har-monic approximation, the vibrations of the guest moleculesin cavities and the effect of guest-host interactions onchanges in the host framework vibrations were taken intoaccount. However, since it allows no dependence of the hostlattice parameters on temperature and pressure and hencedoes not give a complete account of the influence of guestmolecules on the host lattice, this approximation can be con-sidered as only a first step. Moreover, the calculation ofLangmuir constants is relatively easy only in the case ofsingle occupancy. Further extension to multiple guest occu-pancy of a single cavity makes the integration difficult toperform.34,45

Another approach that allows the accurate estimation ofthe thermodynamic properties of clathrate hydrates with mul-tiple filling within the cages has been proposed recently byour group.46,47 This method is also based on the vdWP theorywith some modifications that account for multiple cage oc-cupancy, host lattice relaxation, and the description of thequantum nature of hydrogen behavior in the cavities. Thevalidity of the proposed approach was confirmed for meth-ane and xenon hydrates with single occupancy,46 as well asfor argon and krypton with both single and doubleoccupancy.47 The obtained results were in agreement withknown experimental data.

In this work, we use this model to calculate the curves of“guest gas-hydrate-ice Ih” phase equilibrium and the degreeof filling of the large and small cavities for pure hydrogenand mixed propane+hydrogen hydrates of CS-II in variousrange of pressures and temperatures. For the binary C3H8

+H2 hydrate, we also analyze the possibility to realize the“composition tuning mechanism” proposed earlier forTHF+H2 hydrate but which has not yet been confirmed.6

II. THEORY AND COMPUTATIONAL DETAILS

The mathematical formalism of the present model for thegeneral case and in the case of clathrate hydrates with twotypes of cavities and one type of guest was described in ourprevious studies46 and Ref. 47. Here, we have formulated ourapproach for the hydrate having two types of cavities, large�L� and small �S�, and the possibility of single occupancy oflarge cavities by a type guests and multiple occupancy ofsmall and large cavities by b type guests. In this case, theexpression for free energy F can be written in the followingform:

244510-2 Belosludov et al. J. Chem. Phys. 131, 244510 �2009�

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F = F1�V,T,�y�� + kTNS

���1 − i=1

kS

ySibln�1 −

i=1

kS

ySib +

i=1

kS

ySib ln

ySib

i! �+ kTNL��1 − yL

a − i�=1

kL

yLi�bln�1 − yL

a − i�=1

kL

yLi�b

+ yLa ln yL

a + i�=1

kL

yLi�b ln

yLi�b

i�! � , �1�

where F1 is the part of free energy at a given degree of fillingfor the S and L cavities, �y�= �yL

a ,ysb ,ys

kSb ,yLb , . . . ,yL

kLb� andthe second and third terms are the entropic parts of the freeenergy arising from the guest subsystems. The small cavitiescan hold a number, i �i=1, . . . ,kS�, of guest molecules oftype b and the large ones can be occupied by one guestmolecule of type a or a number, i� �i�=1, . . . ,kL�, of guestmolecules of type b. The degree of occupation of cavities oftype L by guest molecules of type a, yL

a =NLa /NL, is the frac-

tion of filled, NLa, to the total number, NL, of L cavities. In the

case of guest molecules of type b, the degree of occupation

of the S �L� cavities, ySib=NS

ib /Ns �yLi�b=NL

i�b /NL�, is the frac-

tions of filled, NSib �NL

i�b�, to the total number, NS �NL� of S�L� cavities, respectively.

The free energy of a crystal can be calculated within theframework of the quasiharmonic lattice dynamicsmethod48,49 and gives

F1 = U��y�� + Fvib, �2�

where U is the potential energy of clathrate hydrate, Fvib isthe vibrational contribution,

Fvib =1

2jq�

�� j�q�� + k�Tjq�

ln�1 − e−��j�q��k�T� , �3�

where � j�q�� is the jth frequency of crystal vibration and q� isthe wave vector. The free energy calculated within this ap-proach takes into account the contributions of both guest-host and guest-guest interactions.

In the quasiharmonic approximation, the free energy ofcrystal has the same form as in the harmonic approximationbut the structural parameters at fixed volume depend on thetemperature. This dependence is determined self-consistentlyduring calculation of the free energy of the system underinvestigation. The equation of state can be found by numeri-cal differentiation of the free energy,

P�V,T� = − � �F

�V

0. �4�

The subscript “0” implies constancy of all thermodynamicparameters except those relative to which the differentiationis performed.

After obtaining the free energy values, the chemical po-tentials, �L

a, of the guest molecules of type a, located in the

large cavities, and the chemical potentials, �Sib and �L

i�b, ofguest molecules of type b, located as a clusters i and i� in thesmall and large cavities, can be expressed, respectively, as

�La�P,T,�y�� = � �F�V,T,�y��

�NLa

0

= �̃La + kT ln

yLa

1 − yLa −

i�=1

kL

yLi�b

, �5�

�Sib�P,T,�y�� = � �F�V,T,�y��

�NSib

0

= �̃Sib + kT ln

ySib

i ! �1 − i=1

KS

ySib , �6�

�Li�b�P,T,�y�� = � �F�V,T,�y��

�NLi�b

0

= �̃Li�b + kT ln

yLi�b

i� ! �1 − yLa −

i�=1

kL

yLi�b , �7�

�̃La �

�F1�V�P�,T,�y���NL

a , �̃Sib �

�F1�V�P�,T,�y���NS

ib ,

�8�

�̃Li�b �

�F1�V�P�,T,�y��

�NLi�b

.

The last derivatives can be found by numerical calculationusing the following approximation:

� �F1�V�P�,T,�y���NL

a 0

F1�NL

a� − F1�NLa − NL

anLa�

NLanL

a , �9�

� �F1�V�P�,T,�y���NS

ib 0

F1�NS

ib� − F1�NSib − NS

ibnSib�

NSibnS

ib , �10�

� �F1�V�P�,T,�y��

�NLi�b

0

F1�NL

i�b� − F1�NLi�b − NL

i�bnLi�b�

NLi�bnL

i�b,

�11�

where NLanL

a, NLi�bnL

i�b, and NSibnS

ib are numbers of guest mol-ecules of type a, i� clusters of b-type guest molecules, and iclusters of b-type guest molecules that have been removedfrom the L and S cavities in the clathrate hydrate, respec-tively.

Knowing the Helmholtz free energy F and equation ofstate of the system, the relation between the Gibbs free en-ergy ��P ,T , �y�� and the chemical potentials of the host andguest molecules is expressed by the following:

��P,T,�y�� = NQ�Q + NLa�L

a + i=1

2

NSib�S

ib + i=1

4

NLi�b�L

i�b

= F�V�P�,T,�y�� + PV�P� . �12�

Substituting the expression �1� for F into Eq. �12� allows oneto obtain the chemical potential of the host molecules �Q,

244510-3 Phase diagram of clathrate hydrates J. Chem. Phys. 131, 244510 �2009�

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�Q�P,T,�y�� =1

NQ��F�V�P�,T�y�� + PV�P�� − NL

a�La −

i=1

kS

NSib�S

ib − i�=1

kL

NLi�b�L

i�b�= �̃Q + kT�NL��1 − yL

a − i�=1

kL

yLi�bln�1 − yL

a − i�=1

kL

yLi�b + yL

a ln yLa +

i�=1

kL

yLi�b ln

yLi�b

i�! �+ NS��1 −

i=1

kS

ySibln�1 −

i=1

kS

ySib +

i=1

kS

ySib ln

ySib

i! �� , �13�

�̃Q �PV�P�

NQ+

1

NQF1�V�P��,�T,�y��

− �LyLa�̃L

a − �Si=1

kS

ySib�̃S

ib − �L i�=1

kL

yLi�b�̃L

i�b, �14�

where �t=Nt /NQ, �t=S ,L� is the ratio between number ofcavities of type t, Nt and the number of water molecules inthe original hydrate, NQ.

Based on this formalism, the different phase equilibria ofhydrates can be obtained. The p�T� curve for “gas phase-hydrate phase” equilibrium can be found from the equalitycondition of the chemical potentials of both types of guestmolecules in the original hydrate and gas phases,

�La�P,T,�y�� = �a

gas�P,T� , �15�

�tb�P,T,�y�� = �b

gas�P,T� . �16�

For the case of multiple filling of the cages, it is important toaccount for the interaction between the i� number of guest

molecules of type b in the cavity of type t, Uti�b �i� equal to i

or i� for the S or L cavities, respectively�. Thus, the chemicalpotential of a b-type guest can be approximately written as

�tb

1

i��t

i�b + Uti�b� . �17�

The gas phase of the guests has been considered as a non-ideal gas for calculation of the chemical potential,50

�igas�P,T� = kT ln� xiP

kT�2��2

mikT3/2� +

2kT

Vj=1

2

Bijxj , �18�

where xi-concentration of i-type guest in gas phase and Bij

are the second virtual coefficients,

Bij = 2��

0 �1 − exp�− Wij�r�kT

r2dr , �19�

and Wij�r� is the atom-atom interaction. Substituting Eqs. �5�and �18� into Eq. �15� and Eqs. �6�, �7�, �17�, and �18� intoEq. �16� the following expressions for the degree of cavityfilling by guests of types a and b can be obtained, respec-tively,

yLa =

La

1 + La +

i�=1

kL

i� ! Li�b

, La = exp� 1

kT���a

gas − �̃La���

�20�

and

ySib =

i ! Sib

1 + i=1

kS

i ! Sib

, yLi�b =

i� ! Lib

1 + La +

i�=1

kL

i� ! Li�b

,

�21�

ti�b = exp� 1

kT��i��gas − Ut

i�b� − �̃ti�b��� .

The p�T� curve for “gas phase-ice Ih-hydrate phase” equilib-rium can be found separately for the guest and host mol-ecules in different phases from the equality condition of thechemical potentials,

�Q�P,T� = �̃Q + kT��S ln�1 − i=1

kS

ySib

+ �L ln�1 − yLa −

i�=1

kL

yLi�b�

= �Qice�P,T� , �22�

where yLa and yS

ib or yLi�b are found from Eqs. �20� and �21�,

respectively.In the present approach we do not take into account the

free motion of guest molecules inside cavities, assuming thatthe guest molecules form a sublattice inside the host frame-work. We suggest that this motion will be insignificant in thecase of multiple occupancy of the water cavity by smallguest molecules due to the increasing role of guest-guestinteractions inside the cavity, limiting their free motion. Thisallows us to consider that the location of guest moleculeswill be determined as potential minima of both guest-guestand host-guest interactions. The temperature dependence ofthe free energy of the clathrate phase is determined self-consistently during calculations using Eqs. �2� and �3� asexplained above. Thus the effect of temperature on thechemical potentials both the guest and host molecules hasbeen considered. Obviously, the role of free motion on thepotential energy of small molecules will be increased in the

244510-4 Belosludov et al. J. Chem. Phys. 131, 244510 �2009�

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case of single occupancy of the large cavity. Within the pro-posed approach we could not exactly estimate this changeand we assume that at single occupancy and temperatureslower than 273 K the hydrogen molecule will not move farfrom its local minimum determined by selected host-guestinteractions and hence the effect of free motion on thechemical potential can be neglected.

The present formalism was used to calculate the p�T�equilibrium curves of the “gas-hydrate-ice Ih” phases and thedegree of filling of the large and small cavities for pure andmixed propane-hydrogen hydrates with the CS-II structure.A single unit cell of CS-II hydrate with 136 water moleculeswas selected in the present calculations. Multiple filling ofboth the large �L� and small �S� cavities by hydrogen andsingle filling of large cavities by propane molecules wereconsidered. In the case of ice Ih, the calculations were per-formed using a supercell of the 128 water molecules. Theinitial positions of the water oxygen atoms of the CS-II hy-drate lattice were taken from the x-ray analysis of the doublehydrate of THF and hydrogen sulfide performed by Mak andMcMullan.51 Considering both the ice Ih and clathrate hy-drate phases, the interaction between water molecules wasdescribed by the modified simple point charge extended�SPCE� E potential and its parameters were presented in ourprevious study.47 It was shown that the values of the unit-cellvolume calculated with the modified SPCE water-water in-teraction potential agree quantitatively with experimentaldata52 at low temperatures, while the results of calculationsusing the standard single point charge-extended �SPCE�potential53 exceed the experimental values.47 The protonswere placed according to the Bernal–Fowler rules and thewater molecules were oriented such that the total dipole mo-ments of the simulation cells of ice and hydrate were zero54

with a precision of better than 0.1% of the magnitude of thedipole moment of a single water molecule. The long-rangeelectrostatic interactions were computed by the Ewaldmethod.55

The hydrogen molecules were considered as sphericallysymmetric particles and their interaction potential was for-mulated as

UH2–H2=

D0

� − 6�6 exp�1 − r

�� − ���

r6� , �23�

where r is intermolecular separation, D0 is the potential welldepth, � is the intermolecular distance at the minimum of thepotential �i.e., where UH2–H2

=−D0�, and � is a dimensionless“steepness factor.” For the H2–H2 interaction, the long-distance dispersion interaction was taken from Ref. 56 andthe short-range repulsion was estimated within density func-tional theory using the all-electron mixed-basis method.57

The exponential-6 potential used here gives the correctfunctional form at small intermolecular separations.58 Theintermolecular distance at the minimum of the potential�=0.2967 nm is shorter than the minimum of H2–H2 poten-tial surface �0.345 nm� obtained by ab initio calculations forthe dimer.59 The present empirical potential better describesthe interaction between hydrogen molecules in water cavitiesbecause the experimentally determined distance between thefour tetrahedrally arranged D2 molecules in the large cage of

the CS-II clathrate hydrate is found to be �=0.293 nm.7 Thedetails of this potential as well as the modeling the H2O–H2

interaction can be found elsewhere.33

The Lennard-Jones potential was used for propane

UC3H8–C3H8= 4 ���

r12

− ��

r6� , �24�

with parameters �=5.637 Å and =2.0129 kJ /mol forspherically symmetric particles.60 From the hydrogen storageapplication point of view, the storage media should be stableat relatively high temperatures. At temperatures higher than160 K, the effect of the nonspherical nature of guest mol-ecules is not significant for description of the thermodynamicphase diagram and hence can be neglected.

III. RESULTS AND DISCUSSION

As a first step, we have examined the CS-II hydrogenhydrate using our approach. In this case, the chemical poten-tials of the hydrogen and water molecules and the degree ofcavity filling have been calculated using Eqs. �6�, �7�, �13�,�14�, and �21�. Since only one type of guest has been con-sidered, the terms related to the a-type guest have been omit-ted in these equations. The calculations of the free energiesof the CS-II hydrates with different hydrogen occupationhave been performed after full structure optimization of boththe host framework and the position of the hydrogen mol-ecules inside the cavities. It has been found that the presentempirical potential well describes the interaction between thehydrogen molecules in the water cavities. Thus, in the largecavities, four H2 molecules have a tetrahedral arrangementwith an average distance of 2.96 Å from each other and adistance of 1.81 Å from the center of the cage. This agreeswell with the value of experimentally measured the samedistances for a cluster of four deuterium molecules �2.93�1�and 1.80�1� Å, respectively�.7 The tetrahedral configurationof the hydrogen molecules in the large cavity was also ob-served in earlier first-principles calculations24–26 as well as inrecent calculations using diffusion Monte Carlo methods.32

The hydrogen-hydrogen distance realized in the CS-II hy-drate is shorter than the equilibrium distance �3.78 � be-tween molecules in solid hydrogen61 and can be attributed tothe unique feature of hydrogen hydrate related to guest-guestand guest-host interactions that allowed the high local den-sity of hydrogen inside hydrates. The results of optimizationhave also shown that the inclusion of the two hydrogen mol-ecules in a small cavity is unfavorable energetically due tovery short intermolecular distance �2.62 �.

The calculated pressure dependence of the chemicalpotentials �̃Q of the water molecules of hydrogen hydratesof CS-II with multiple occupations of S cavities ��̃Q

iH2S,

i=1, .. ,kS� and L cavities ��̃Qi�H2L, i�=1, .. ,kL� by H2 at

T=200 K excluding the entropy term and �̃Q0 of the water

molecules of the empty host lattice of C-II hydrate is dis-played in Fig. 1. The effect of a guest on the water hostframework has been observed even in the case of the accom-

244510-5 Phase diagram of clathrate hydrates J. Chem. Phys. 131, 244510 �2009�

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modation of a single hydrogen molecule within the hydratecavities and becomes more significant for multiple degree offilling.

Based on these results, the following linear approxima-tion for the change of the water molecule chemical potentialsin the empty hydrate lattice under the influence of hydrogenmolecules can be determined in order to estimate the watermolecule chemical potential in hydrate in the case when theoccupation of large cage is nonintegral,

�̃Q��yS��,��yL�� = �Q0 +

i=1

kS

aiySiH2 +

i�=1

kL

bi�yLi�H2, �25�

where the coefficients ai and bi� are the difference ai

= �̃QiH2S−�Q

0 ; bi= �̃QiH2L− �̃Q

H2S, between chemical potential atvarious integer occupancies.

It has been found that this difference increases with in-creasing occupancy and at a high degree of filling; it can becompared to the difference ��Q= �̃Q

0 −�Qice between chemical

potentials of the host lattice and ice that is commonly used inthe construction of phase diagrams within framework of thevdWP theory.38 This indicates again the importance of bothguest-host and guest-guest interactions for the description ofthe thermodynamic properties of clathrates with multiple fill-ing of their cavities by even small guest molecules.

The inclusion of only the guest-host interactions is notadequate to reproduce the formation pressure and the regionof stable phase of hydrogen hydrate observed experimentallyand hence the entropic terms of the chemical potentials ofhost molecules in Eqs. �13� and �14� related to the guestsubsystem need to be considered. To determine these termswe need to know the degree of filling of the large and smallhydrate cavities by hydrogen molecules. The degree of fillingby both the large and small cavities can be estimated at vari-ous pressures and temperatures using the chemical potentialsof the hydrogen molecules �Eqs. �6�, �7�, and �18��. In allcases, it has been shown that with increasing pressure thefilling of the large cages increases consecutively from one to

FIG. 1. Pressure dependence of the chemical potentials of water moleculesat T=200 K for empty and hydrogen CS-II hydrates excluding the entropyterms at various degrees of cavity filling by hydrogen molecules.

FIG. 2. Degree of filling of the small �1H2 dashed, 2H2 solid lines� and the large �1H2 dash-dotted, 2H2 short dashed, 3H2 dash-dot-dotted, and 4H2 dottedlines� cavities of the hydrogen CS-II hydrate at �a� 160 K, �b� 200 K, �c� 230 K, and �d� 260 K.

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four hydrogen molecules �Fig. 2�. The degree of cage occu-pancy is also dependent on the operating temperature. Attemperatures lower than 200 K, the degree of filling of thelarge cavities differs notably from the maximum numbers atlow pressure and approaches four with increasing pressure asshown in Figs. 2�a� and 2�b�. At low temperature, almost100% of the large cavities can be filled by four H2 moleculesat high pressure. In the case of the small cavities, the sametemperature and pressure behaviors have been observed. Asmall number of small cavities with double H2 occupancyare found in the CS-II hydrogen hydrate at high pressureswithin our approach. However, the maximum value ofdouble occupancy of the small cages does not exceed 5% at160 K and lies in the experimentally defined range of 0.9�1�–1.1�1� for the small cavities.7 It can be expected that athigher pressures and low temperatures the number of smallcages with two hydrogen molecules will be increased and inprinciple the twofold filling originally proposed by Maoet al.3 could be realized for a limited number of small cavi-ties. At high temperature �T=260 K�, the degree of smallcavity filling has achieved value of 0.6 �see Fig. 2�d�� indi-cating that hydrogen molecules occupied only 60% of thesmall cavities in single form. This effect can be explained bythe high hydrogen diffusion and migration at high tempera-tures that was observed experimentally7 and also confirmedrecently by first-principles calculations.62 The ratio of thelow occupied/unoccupied small cavities can also be attrib-uted to the effect of guest-guest interaction on the dynamicsof the hydrogen molecules inside the cavities. In the case ofmultiple occupancy the dynamics of the guest molecules isgoverned by the highly anisotropic intermolecular potentialresulting from both the guest-host and guest-guest interac-tions. The absence of additional guests inside the same cavitydecreases the anisotropy of the intermolecular potential andhence decreases the energy barrier of free hydrogen motion.

Taking into account the entropic terms of the chemicalpotential of the host molecules related to the guest sub-system, and the p−T curve for “gas phase-ice Ih-hydratephase” equilibrium of hydrogen hydrate can be constructedusing Eq. �22�. It is well-known that the empty hydrate struc-tures are metastable as compared to hexagonal ice and ourapproach reproduced this behavior as shown in Fig. 3. Theintersection of the chemical potential curves of ice Ih and thehydrate host lattice has been found �see Fig. 3� that definesthe formation pressure of the hydrogen hydrate at a giventemperature in agreement with the experimental phase dia-gram of hydrogen-water system related to hydrogenclathrate.63 At this temperature, the hydrogen hydrate existsin a metastable phase at low pressure and stabilizes withpressures greater than p=330 bar. The calculated p−T curvefor “gas phase-ice Ih-hydrate phase” equilibrium is displayedin Fig. 4. The cage occupancy at which the CS-II hydrogenhydrate becomes stable can be obtained from data analysisillustrated in Figs. 2 and 4. For example, at T=200 K, thestable phase of hydrogen hydrate is realized when hydrogenoccupies 100% of the large and 80% of the small cages. Inthe case of the large cages 90% are filled by four hydrogenmolecules and others are occupied equally by 2 and 3 hydro-gen molecules. 80% of the small cages are filled by one

hydrogen molecule and double occupancy is not observed. Itis important to note that the stabilization of the CS-II hydro-gen clathrate occurs after the large cavities show a high frac-tion �almost 90%� of quadruple occupancy.

The calculated curve of the phase transformation be-tween hexagonal ice and hydrogen hydrate agrees well withthe available experiments.63,64 Reasonable correlation hasbeen achieved with results obtained for the D2–D2O systemby high pressure neutron diffraction.63 Moreover, in the pres-sure region between 0.6 and 1.1 kbar our results lie betweenthe points of decomposition �white circles� and formation�filled circles� for hydrogen hydrates obtained recently forthe H2–H2O system by another experimental group.64 Thediversity of the experimental64 and calculated results can beconnected to the low concentration of hydrogen �1.2 wt %�in the CS-II phase achieved by Barkalov et al.64 which ismore than three times lower than the values estimated in thepresent study as well as that proposed by Lokshin et al.7 Theexperimentally observed pressure hysteresis in ice-hydratetransformation may be related to the kinetic difference be-tween the formation and decomposition of hydrate. The

FIG. 3. Pressure dependence of the chemical potentials of water moleculesat T=200 K for Ih �solid line�, empty CS-II hydrate �dotted line� and hy-drogen CS-II hydrates including �dashed line� and excluding �dash-dottedline� the entropy terms.

FIG. 4. The p−T phase diagram of H2O–H2 system related to Ih-CS-IItransition. Experimental data were taken from Ref. 63 �crosses� and Ref. 64�open and filled circles�. The dotted line presents ice Ih-liquid water equi-librium phase transition.

244510-7 Phase diagram of clathrate hydrates J. Chem. Phys. 131, 244510 �2009�

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slower route for decomposition of the hydrogen hydrate canbe the subject of additional studies from the viewpoint ofhydrogen storage applications. Thus, the exact determinationof the phase diagram of this clathrate system requires furtherexperimental studies.5

Figure 5 shows the hydrogen storage capacity of theCS-II hydrogen hydrate at various temperatures. The hydro-gen content continues to increase due to multiple filling ofthe large cages. At low temperature the mass percentage ofhydrogen in the CS-II hydrogen hydrate can reach amount upto 3.8 wt % for pressures of 160–180 bar. This value corre-sponds to configuration of four hydrogen molecules in thelarge cavities and one molecule in the small cages of theCS-II hydrate structure. At higher pressure the hydrogenstorage capacity can increase up to 4 wt % indicating thechanges from one to twofold filling for a limited number ofsmall cavities. Increasing the temperature up to 260 K sig-nificantly reduces the amount of stored hydrogen. The maxi-mum amount of hydrogen stored at this temperature and highpressures is about 3.5 wt %.

As mentioned before, the formation of hydrogen hydratecan be achieved in binary hydrates by stabilizing the largecages of the water host framework with a second guest com-ponent, such as THF. However, despite the numerous studiesrelated to the binary CS-II THF+hydrogen hydrate there isno common conclusion concerning compositional tuning ofthe guest content in this hydrate. At equilibrium for a mixtureof gas phase guest molecules with water �ice Ih� at definiteconditions one could expect that a mixed hydrate could form.In this case increasing the hydrogen content in the hydratemay become possible by means of component ratio variationin the gas phase. Therefore, the thermodynamic properties ofthe binary hydrate systems can be studied using our model. Itis already established that THF can be classified as an “irri-tant” compound with a low toxicity category and thus can beused only in modest amounts. Here we have selected pro-pane as a second guest component because of the binaryCS-II propane+hydrogen hydrate has already beensynthesized20,21 propane as THF occupies only the largecavities of the CS-II hydrate and the formation of this hy-drate is very similar to the situation observed in the THF-

hydrogen-water system.21 Moreover, along with hydrogen,propane can be considered as an additional energy resource.

Here we have two types of guest molecules and hence allterms related to both types of guest, a and b, should be takeninto account. The calculations of the thermodynamic proper-ties of the CS-II propane+hydrogen hydrates with a singlepropane and different hydrogen occupations have been per-formed after full structure optimization of both the hostframework and the positions of the guest molecules insidethe cavities. The degree of filling of the large cavities by boththe propane and hydrogen molecules as a function of pres-sure for selected temperatures is shown in Figs. 6 and 7.Here, we have considered only a very small concentration�up to 2%� of propane in the gas phase and the high tempera-ture region, which seems to be most attractive from the view-point of hydrogen storage. At a low propane concentration inthe gas phase, it has been found that in the low pressureregion the large cavities are mainly filled by propane mol-ecules and hydrogen occupies only the small ones. The hy-drogen molecules singly occupied about 90% and 70% of thesmall cages in the CS-II hydrate at high pressure and 200 and230 K, respectively, which is same than the small cavityoccupation for the pure CS-II hydrogen clathrate �see Figs.2�b� and 2�c�� in the same temperature region. The filling ofthe large cages by hydrogen molecules increases consecu-tively from one to four hydrogen molecules with increasingpressure. In contrast to hydrogen, the propane filling de-creases with increasing pressure and in the high pressureregion the propane molecules are gradually expelled by hy-drogen molecules. The tuning of large cage occupancy canbe attributed to the “composition tuning mechanism” pro-posed by Lee et al.11 As seen in Figs. 6 and 7, the pressurevalue, at which the tuning can be observed, depends on thepropane concentration in the gas phase. Decreasing propaneconcentration from 2% to 0.01% allows one to reduce thecomposition turning pressure almost in half. At higher tem-perature, the tuning occurred at higher pressure as shown inFig. 7. Based on these results one can expect that the tuningpressure will rapidly increase with increasing of propaneconcentration and hence the tuning effect will occur at higherpressure. Since the thermodynamic behavior of propane-hydrogen-water system is similar to that of the THF-hydrogen-water system, the present results can explain thediscrepancy in experimental data related to the hydrogencomposition in binary THF+hydrogen hydrate.

The chemical potential curves of the water molecules inice Ih and in mixed propane+hydrogen CS-II hydrate as afunction of pressure at T=260 K are shown in Fig. 8. It hasbeen found that the occupation of large cavities by propanedrastically affect the stabilization of the hydrate structure,resulting in a substantially decreased formation pressure. Ata low propane concentration of 0.1% in the gas phase, thebinary hydrate becomes stable at a pressure around 380 bar.This pressure is about four times lower than that neededto form the pure hydrogen clathrate with CS-II structure�p=1.1 kbar� in the same temperature region. By analogywith THF, propane brings the hydrate formation conditionscloser to ambient conditions and it is undoubtedly favorablefor a hydrogen storage material. At propane concentration of

FIG. 5. Hydrogen storage capacity of the CS-II hydrogen hydrate as afunction of pressure for various temperatures.

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5% in the gas phase, the formation pressure of 50 bar forbinary propane+hydrogen hydrate can be achieved. Thisvalue is in a good agreement to experimentally reported for-mation conditions ��50 bar at 280 K� for binary THF+H2

hydrate.10

The calculated p−T curve for “gas phase-ice Ih-mixedpropane+hydrogen hydrate phase” equilibrium in tempera-ture region of 160–280 K is displayed in Fig. 9. Reasonablecorrelation has been achieved between the calculated andexperimental65 curves of the hexagonal ice-CS-II propanehydrate phase transformation. This confirmed the validity ofselected potential used for propane molecules. At high pro-pane concentrations the behavior of the phase transition lineis similar to the experimental one for CS-II propane hydrateespecially in the low temperature region. At lower concen-tration the line of the phase transition is shifted to higherpressure. However, in this case the formation pressure of thebinary CS-II propane+hydrogen hydrate is still significantlysmaller than that of the pure hydrogen clathrate hydrate. Thisindicates the dominant role of propane molecules in the for-mation of the binary propane+hydrogen hydrate even at lowpropane concentration. The changing of the slop of the curvecan be characterized by increasing hydrogen content in thegas as well as weaker hydrogen-water interactions in com-parison to the propane-water interactions.

Figure 10 shows the hydrogen storage capacity of themixed propane+hydrogen CS-II hydrate at various propaneconcentrations and temperatures at 230 and 260 K. The hy-drogen content continues to increase due to hydrogen fillingof the large cages. The mass percentage of hydrogen in themixed CS-II propane+hydrogen hydrate can amount up to3.5 wt % at T=260 K and lower concentrations of propanein the gas phase and a pressure of 1000 bar. This value iscomparable with the hydrogen content �3.5 wt %� estimatedfor pure hydrogen hydrate at the same temperature.

The introduction of propane as a second guest compo-nent in the binary hydrogen hydrate does not only reduce theformation pressure but also reproduces the hydrogen storageability of the pure clathrate hydrate. Therefore, it is impor-tant to determine the optimal balance between storage capac-ity and formation conditions for the practical feasibility ofthe binary hydrate as a hydrogen storage candidate.5 Thereare significant difficulties in experimental investigation andtreatment of phase diagrams of multicomponent clathrate hy-drates that could result in a loss of important informationrelated to the guest content in storage media. The proposedapproach could support experimentalists in the practicalrealization of hydrogen storage material based on clathratehydrate.

FIG. 6. Degree of filling of the small and large cavities of the propane+hydrogen CS-II hydrate at a propane concentrations of 0.01% �a�, 0.1% �b�, 1% �c�,and 2% �d� in the gas phase at T=200 K.

244510-9 Phase diagram of clathrate hydrates J. Chem. Phys. 131, 244510 �2009�

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IV. CONCLUSION

We have presented a formalism for calculating the ther-modynamic properties of clathrate hydrate with weak guest-host interactions. In distinction from the original vdWPtheory the suggested model accounted for the influence ofguest molecules on the host lattice as well as guest-guest

interaction. The mathematical formulas have been presentedin the case of clathrate hydrate having two types of cavitiesand two different types of guest molecules. Using this ap-proach, the phase diagrams of the hydrogen and propane+hydrogen hydrate with structure II have been constructedand they are in agreement with available experimental data.

FIG. 7. Degree of filling of the small and large cavities of the propane+hydrogen CS-II hydrate at a propane concentrations of 0.01% �a�, 0.1% �b�, 1% �c�,and 2% �d� in the gas phase at T=230 K.

FIG. 8. Pressure dependence of the chemical potentials of water moleculesat T=260 K for Ih, CS-II propane+H2 hydrates including the entropy termsat various propane concentration in the gas phase.

FIG. 9. The p−T phase diagram of the H2O–H2 system related to Ih-CS-IItransition at various propane concentrations in the gas phase. The experi-mental results are represented by circles �Ref. 65�.

244510-10 Belosludov et al. J. Chem. Phys. 131, 244510 �2009�

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High pressure conditions are necessary for the CS-II hydro-gen hydrate formation and the formation pressure can besignificantly reduced in the presence of propane as a secondguest in the binary hydrogen hydrate. The mass percentageof hydrogen in the CS-II hydrogen hydrate can reach amountup to 3.8 wt % for pressures of 160–180 bar and at 160 K.This value corresponds to a configuration of four hydrogenmolecules in the large cavities and one molecule in the smallcages of the CS-II hydrate structure proposed by Lokshin etal.7 At higher pressure the hydrogen storage capacity canincrease up to 4 wt % indicating the twofold filling for alimited number of small cavities. Increasing temperature re-duces the amount of stored hydrogen. However, in the caseof propane there is a possibility to archive the amount ofstored hydrogen �around 3.5 wt % of hydrogen at 260 K and1000 bar� at small concentrations of propane in the gasphase. These results indicate that the guest-host interaction isessential and should be accurately estimated in the calcula-tion of the phase diagram of hydrogen hydrate.

The proposed method is quite general and can be appliedto the various nonstoichiometric inclusion compounds withweak guest-host interactions. However, it is significant thatthe present model of inclusion compounds allows the calcu-lation of thermodynamic functions starting from the well-defined potentials of the intermolecular guest-host and guest-guest interactions. In principle the present formalism couldbe used for the study of hydrate formation from liquid water.In this case, instead of the chemical potential of hexagonal

ice used in Eq. �22� the chemical potential of the water mol-ecule in the liquid phase should be determined in a widerange of pressures and temperatures and the effect of guestmolecules soluble in water should be taken into account.However, the modeling of the water phase is a separate taskand is not possible to perform only within the lattice dynam-ics approach. The supercell of water representing the liquidphase needs to be constructed using Monte Carlo and mo-lecular dynamics methods. Therefore, in our study we se-lected the temperature region in which hexagonal ice exists.Moreover, it was experimentally shown that the formation ofhydrogen clathrate from H2 gas and Ih ice occurs at least 100times faster compared to the reaction with water.63 Fromviewpoint of storage applications, this aspect is very impor-tant and hence the results presented will be useful for experi-mentalists.

ACKNOWLEDGMENTS

This work has been supported by New Energy and In-dustrial Technology Development Organization �NEDO� un-der “Advanced Fundamental Research Project on HydrogenStorage Materials.” We are also thankful to Dr. T. M. Brierefor carefully reading this manuscript. The authors also aregrateful for the continuous support of the HITACHISR11000-K2/51 supercomputing facility by the ComputerScience Group at the Institute for Materials Research, To-

FIG. 10. Hydrogen storage capacity of the CS-II propane+hydrogen hydrate as a function of pressure for various propane concentrations at �a� T=160 K,�b� T=200 K, �c� T=230 K, and �d� T=260 K.

244510-11 Phase diagram of clathrate hydrates J. Chem. Phys. 131, 244510 �2009�

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hoku University. R.V.B. also thanks the Ministry of Educa-tion, Culture, Sports, Science, and Technology of Japan�Grant No. 19651039� for financial support. Moreover,V.R.B. and O.S.S. are greatly appreciative of support fromthe Russian Fund for Basic Research through Grant No.08-03-00191.

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