6560

J. Phys. Chem. B 1997, 101, 6560-6565

Large Thermal Expansivity of Clathrate Hydrates
Hideki Tanaka* and Yoshinori Tamai
Department of Polymer Chemistry, Graduate School of Engineering, Kyoto UniVersity,
Sakyo, Kyoto 606-01, Japan

Kenichiro Koga
Department of Chemistry, 572 Hamilton Hall, UniVersity of NebraskasLincoln,
Lincoln, Nebraska 68588
ReceiVed: February 10, 1997; In Final Form: June 4, 1997X

The free energies of clathrates are calculated over a wide range of temperatures in order to explain a large
thermal expansivity of clathrate hydrates compared with that of ice. Several proton-disordered configurations
for hexagonal ice and type I chathrate hydrates are generated. The free energy is approximated to the sumof the minimum potential energy, the harmonic free energy, and the configurational entropy arising from thedisordered nature of protons. The free energy at a given temperature is minimized with respect to the volumeof the system. This enables us to evaluate the thermal expansivity from only intermolecular interaction
potentials. The larger thermal expansivity of clathrate hydrates than ice is successfully reproduced. It isfound that the large expansivity of clathrate hydrate structure I stems from the existence of guest molecules
and that a difference in oxygen atom arrangement between clathrate hydrates and ice plays a minor role.
The effective potential energy surface of a guest molecule becomes harmonic with an increase in temperature.
This seems to undermine the large difference in the thermal expansivity between clathrate hydrates and ice.

I. Introduction
Gas hydrates contain guest molecules in the cavities madefrom host water molecules which are firmly hydrogen-bondedwith each other.1,2 The clathrate hydrate is stable only when
the interaction between guest and water molecules dominates
over the sum of the two unfavorable terms: (1) entropy decreasearising from confinement of guest molecules in small void cagesand (2) free energy for formation of empty clathrate hydrate
structure from ice or liquid water. The thermodynamic stabilityof clathrate hydrates has long been accounted for by van der
Waals and Platteeuw (vdWP) theory.3 Recently, it has been
applied to various clathrate hydrates with a newly proposed
method of calculating the free energy of cage occupation fromintermolecular interactions. Thus, prediction of dissociationpressure was much improved without invoking empiricalparameters.4-6
It is not surprising that most thermodynamic and dynamic
properties of clathrate hydrates are similar to those of ice since
the interaction between guest and host water molecules is not
so strong as to alter those properties originating mostly fromthe hydrogen bonds. However, the thermal expansivity and thethermal conductivity of clathrate hydrates are exceptionallydifferent from those of ice.2 The thermal conductivity ofclathrate hydrates is roughly 20% of ice near the ice point andis 1 order of magnitude smaller at low temperatures below 150
K.7-10 The thermal conductivity of ice is inversely proportional
to temperature, as is the case of the normal crystal. On theother hand, that of hydrate is proportional to temperature,
although its temperature dependence is not so distinct. Molecular dynamics (MD) simulations have been performed toaccount for the anomalous conductivity and support the“resonance scattering model”.11-14* To whom all correspondence should be addressed. E-mail address:
tanaka@kuic.kyoto-u.ac.jp.
X Abstract published in AdVance ACS Abstracts, July 15, 1997.

S1089-5647(97)00511-7 CCC: $14.00

The thermal expansivity of ethylene oxide clathrate hydrate
is nearly twice as large as that of ice.15-17 This difference canbe accounted for by a difference in either host water or guest
molecules. The former is a difference in the arrangement ofhost water molecules; ice is made from hexagonal puckered
rings, while the clathrate hydrate is composed of planar
hexagonal and pentagonal rings. The latter is an effect of guest
molecules: the coupling of guest and host water or the guest
vibrations, or both. To clarify this point, a hypothetical clathrate
hydrate should be examined, which contains no guest moleculebut has the same molecular arrangement of water molecules as
the real hydrate. It is also important to examine whether the
potential energy or the vibrational free energy is the most crucial
factor in the large difference.
Tse et al. performed MD simulation in which the large
thermal expansivity was successfully reproduced.16 They
concluded that the structure difference contributes little to thedifference in the thermal expansivity between clathrate hydrates
and ice. In order to gain further insight into the origin of thelarge thermal expansivity, it is highly desirable to calculate thefree energy of clathrate hydrates as a function of both temperature and volume. This enables us to evaluate the various
contributions to the total free energy, separately. The equilibrium configuration at a given temperature can be obtained byminimizing the free energy of the system with respect to volume.
The free energy of ice and clathrate hydrate I is given by thesum of the minimum potential energy, the free energy ofintermolecular vibrations, and the configurational entropy, thelast of which arises from the proton-disordered nature of thosecrystallites. The configurational entropies per molecule for both
ice and clathrate hydrate forms are given by k ln(3/2) with great
accuracy, where k is the Boltzmann constant. The configurational entropies are independent of temperature and density andtherefore are ignored in the following calculations. Thus, thefree energy is approximated to the sum of the potential energy© 1997 American Chemical SocietyLarge Thermal Expansivity of Clathrate Hydrates

J. Phys. Chem. B, Vol. 101, No. 33, 1997 6561

at zero temperature and the harmonic free energy, thereby
introducing explicit temperature and density dependence.
In the present study, the thermal expansivity is calculated,
and the most likely mechanism of large thermal expansivity isdelineated. The present paper is organized as follows. In
section II, methods to calculate the thermal expansivity aredescribed. In section III, results on the thermal expansivity arepresented and a difference in the free energy between ice andclathrate hydrates is examined. Conclusions are given in sectionIV.
II. Theory and Method
a. Intermolecular Interaction and Structure of the Unit
Cell. In the present study, all interactions are assumed to be
pairwise additive. The water-water intermolecular interaction
is described by the TIP4P potential.18 The Xe-Xe interaction
is given by a spherical Lennard-Jones (LJ) potential with thesize and energy parameters of 4.047 Å and 1.9205 kJ mol-1.19
For the water-guest interaction, we assume the LorentzBerthelot rule with the LJ parameters for the oxygen atom on
the water molecule set equal to those for TIP4P water: σOO )3.154 Å and OO ) 0.6487 kJ mol-1. The interaction potentials
for all pairs of molecules are truncated smoothly at 8.655 Å bymultiplying a switching function.4
The unit cell of the hydrate I is cubic and the experimental
lattice parameter is used in the following calculation; the sizeof the unit cell is 12.03 Å.1,2 There exist six large and two
small cages in the unit cell of clathrate hydrate I. The numberof water molecules is 368, which corresponds to eight unit cellsof type I clathrate hydrate, and the edge size of the cubic basic
cell is 24.06 Å. The guest Xe atoms occupy all 64 available
cages. The basic cell of low-pressure hexagonal ice (ice Ih) is
orthorhombic, containing 288 water molecules. The size of thecell is (a × b × c )) 23.67 × 18.23 × 22.33 Å.20 Those cellparameters are not fixed but are adjusted so that the free energyof the system has a minimum value at a given temperature as
described below.
Six configurations are generated for both clathrate hydrate I
and hexagonal ice; all the structures are of proton-disordered
form. The orientations of individual water molecules are so
assigned as to satisfy the Bernal-Fowler rule with a vanishingvalue of the dipole moment but otherwise random. The potential
energy of each system is minimized by a standard quenching
method prior to the normal mode analysis.
b. Free Energy Calculation. In order to obtain the densityof state, g(ω), for intermolecular vibrational motions, normal
mode analysis is performed by diagonalizing the mass-weighted
force constant matrix m-1/2 V m-1/2, where m is the appropriately defined mass tensor and V is calculated by direct
differentiation of the intermolecular interaction. The densityof state for intermolecular vibration is obtained by a simple
average of six configurations generated. Once the density ofstate is obtained, the calculation of the harmonic free energy fv
is straightforward. The harmonic free energy per molecule attemperature T is given via a quantum mechanical partition
function for a harmonic oscillator as∫fv ) kT ln[2 sinh(pω/2kT)]g(ω) dω

(1)where p is the Planck constant divided by 2π. The total freeenergy is approximated to the sum of the interaction energy
and the (intermolecular) vibrational free energy fv, neglecting
both the configurational entropy and the pV (pressure × volume)
terms, the latter of which is negligibly small at atmosphericpressure. That is, the free energy of the system, F, is given by

F ) nwuw + nsus + fv(nw + ns)(2)

where nw and ns are the numbers of water and guest molecules,
and uw and us are the potential energies of water-waterinteractions and the sum of water-guest and guest-guest
interactions, respectively. The interaction energy is dependent
only on density, but the vibrational free energy is a function ofboth density and temperature. In order to eliminate the effectof the fixed volume on thermodynamic properties and to takethe thermal expansion into account, the normal mode analyses
are also performed by elongating and shrinking the edges ofthe basic cell by 1.5%; the cell size parameters, ξ, are 1 and 1( 0.015. The free energy F in eq 2 is minimized with respect
to volume at each temperature. In ice Ih, the ratios a/c and b/c
are fixed to the constant values and the basic cell is uniformly
expanded or shrunk.
The force constant matrix V is decomposed into submatrices,
Vww for water-water, Vws, Vsw for water-guest, and Vss forguest-guest contributions in the case of the occupied hydrates.
As reported previously,4 Vsw and Vws play a rather minor rolein the free energy calculation. The submatrix Vww for occupied
hydrate is different from that for empty hydrate since elementsof the former matrix include the guest-water interaction
differentiated twice with respect to the coordinate of a watermolecule. The difference in Vww between empty and occupied
hydrates accounts mostly for the discrepancy between themeasured dissociation pressure and the predicted one based on
the original vdWP theory.4
III. Results and Discussion
a. Difference in Structural and Energetic Properties.
There are several distinct hydrogen bond patterns in the
crystalline structures arising from proton-disordered forms ofice and clathrate hydrate I. The pattern is represented by thedihedral angles for a hydrogen-bonded pair, an angle for a bondsequence of HO(H)OH neglecting the central hydrogen, (H),between two oxygen atoms since O(H)O is nearly linear for astrongly hydrogen-bonded pair. Clathrate hydrate I has only
an eclipsed form. In contrast, only a staggered form is allowedin cubic ice. Both staggered and eclipsed forms are allowed in
ice Ih; the population ratio of the staggered form relative to theeclipsed is exactly 3. We examine distributions of the dihedral
angle for a hydrogen-bonded pair in ice Ih and clathrate hydrate
structures generated. In clathrate hydrate I, there are indeed
only two distinct dihedral angles; the dihedral angle is either 0or 2π/3. The dihedral angles in ice Ih range from 0 to π, withan interval of π/3; there are four distinct angles. The populations
in dihedral angles are given in Figure 1 and Table 1. For acomplete set to classify the conformation for a pair of hydrogenbonded molecules, we must specify a pair of dihedral angles.
In the case of clathrate hydrate I, there are only two possiblecombinations of two dihedral angles, which are (0, 2π/3) and(2π/3, 2π/3), while there are four patterns in ice Ih; they are(0,2π/3), (π/3, π/3), (2π/3, 2π/3) and (π/3, π). The number ofoccurrences for each pattern is also given in Table 1. The
relative occurrence in clathrate hydrate I is (2:0:1:0) and thatin ice Ih is (2:3:1:6).
Each hydrogen bond pattern expressed by a combination ofdihedral angles has a different interaction energy. For fourcombinations of dihedral angles given above, the interactionenergies in ice Ih are -21, -19, -26, and -24 kJ mol-1,
respectively. In clathrate hydrate I, the interaction energy ofthe hydrogen-bonded pair is either -21 or -26 kJ mol-1. These6562 J. Phys. Chem. B, Vol. 101, No. 33, 1997

Tanaka et al.

Figure 1. Dihedral angle distribution for individual water moleculesin hexagonal ice (solid line) and empty hydrate I (dotted line).

Figure 2. Pair interaction energy distribution for individual watermolecules in hexagonal ice (solid line) and empty hydrate I (dottedline).

TABLE 1: Percentages of a Single Dihedral Angle (Above)
and Combination of Two Dihedral Angles (Below) for a
Hydrogen-Bonded Pair for Ice Ih and Clathrate Hydrate I
angle
Ihhydrate I

0

π/3

2π/3

π

8
33

50
0

17
67

25
0

combination of angles

Ihhydrate I(0, 2π/3)
eclipsed(π/3, π/3)
staggered(2π/3, 2π/3)
eclipsed(π/3,π)staggered

17
67

25
0

8
33

50
0

energy differences must be reflected in a distribution ofhydrogen bond energy. The pair interaction energy distributions
at 0 K for ice Ih and clathrate hydrate I are shown in Figure 2,
which are defined as
N

∑i ∑
j*i

xp(ν) ) 〈1/Nδ(ν - φ)〉(3)

where 〈 〉 indicates the ensemble average and φ is the TIP4P
pair potential function. There are two peaks corresponding todistinct hydrogen bond patterns in clathrate hydrate I. On theother hand, there are three distinct hydrogen bonds in ice Ih; apeak around -20 kJ mol-1 may be further decomposed into
two peaks. The peak height at the lowest interaction energy ishigher in clathrate hydrate I than in ice Ih. However, the total
potential energy for ice Ih is lower than that for clathrate hydrate
I because there are a large number of hydrogen bonds ofintermediate strength at around -24 kJ mol-1 in ice Ih.
The arrangement of oxygens in clathrate hydrate I is different
from that of ice Ih beyond the second nearest neighbors. Thisis viewed from the radial distribution functions (RDFs). The
RDFs at 0 K are shown in Figure 3. The first peaks
corresponding to the hydrogen-bonded pair are omitted since
their heights and locations are different from each other only
slightly. All the peaks are well separated from each other unlike
liquid water. A difference emerges in the location of the thirdpeak, as expected from the difference in the dihedral angle
distribution. Although those structures from which RDFs arecalculated have no thermal energy, the peaks are rather wide
due to the random distribution of protons.
The densities of state of intermolecular vibration for ice Ihand empty clathrate hydrates I averaged over six configurations
are shown in Figure 4a. The intermolecular vibrational motions
are split completely into translational and rotational motions of

Figure 3. Radial distribution functions for the oxygen pair in hexagonalice (solid line) and empty hydrate I (dotted line).

individual molecules. The density of state for empty clathrate
hydrate I has several peaks at different positions from ice Ih in
the lower frequency regions corresponding to the translationalmotions of the individual molecules. However, the overall
patterns are similar to each other. A main difference appears
in the higher frequency region composed of rotational motionsof individual water molecules: The single peak in the higher
frequency region for ice Ih is split into two parts in clathrate
hydrate I (both for empty and occupied hydrates).
The density of state calculated from only Vww for the occupied
hydrate is different from that for the empty hydrate. Since guest
molecules have only translational degrees of freedom, thecoupling between guest and water is limited to translationalmotions. The densities of state for empty and occupied hydrate
I calculated from only Vww are shown in Figure 4b. Agreementwith experiment is semiqualitative.21 As expected, occupationof cages gives rise to the blue shift of the peaks in the lower
frequency region (0-300 cm-1), which in turn gives higher freeenergy of cage occupation than that evaluated according to theoriginal vdWP theory.
b. Free Energy and Thermal Expansivity. We calculate
the free energy values with three different densities at a given
temperature. The intermediate (ξ ) 1) cell size has, in most of
the cases, the lowest free energy among the three in thetemperature range we are interested in (153-273 K). Thus,
the minimum free energy is accurately calculated by fitting those
three values to a quadratic function against cell volume. The
equilibrium volume at a given temperature is calculated together
with the free energy. It should be noted that even empty hydrate
has a minimum free energy against volume change, which

Large Thermal Expansivity of Clathrate Hydrates

J. Phys. Chem. B, Vol. 101, No. 33, 1997 6563

Figure 6. Temperature dependence of thermal expansivity for hexagonal ice (solid line), empty hydrate I (dotted line), and occupied
hydrate I by xenon (dash-dot line).

Figure 4. Densities of state for intermolecular vibrational motions (a)for hexagonal ice (solid line) and empty hydrate I (dotted line) and (b)for occupied hydrate I by xenon (solid line) and empty hydrate I (dottedline) as a function of wave number.

0.72 × 10-4 K-1, at 200 K (average over a, b, and c axes) is
fairly larger than the experimental value, 0.56 × 10-4 K-1. It
is interesting that the calculated thermal expansivity for empty
hydrate, 0.73 × 10-4 K-1, is almost the same as that for ice Ih.
Although a difference in thermal expansivity between experimental and theoretical values is not so small, the calculated value
for occupied hydrate is larger than that for ice Ih and empty
hydrate. This suggests that our calculation is reliable inexamining the origin of the large discrepancy of the thermal
expansivity (a part of anharmonicity is taken into account in
the potential energy at its minimum structure when the volume
is changed to find the free energy minimum).
The large thermal expansivity is originated from the existenceof guest molecules since the thermal expansivity of empty
hydrate is much smaller than that of occupied hydrate and isalmost the same as that of ice. It is important to examine whichpart of guest contributions plays a major role in increasing thethermal expansivity: (1) vibrational free energy change of thehost lattice, (2) potential term of the guest-water interaction,(3) vibrational free energy of guest molecules in the cages. The
appropriate, though approximate, free energy of the system, F1,
is written as

F1 ) nw(uw + f ′w) + ns(us + f ′s)

Figure 5. Temperature dependence of volume for hexagonal ice (solidline), empty hydrate I (dotted line), and occupied hydrate I by xenon(dash-dot line). The volumes are scaled so that they are unity at zerotemperature.

indicates that empty hydrate could be metastable. The relative
volume with reference to the volume at 0 K is plotted in Figure
5. The volumes for ice Ih, empty hydrate I, and occupied
hydrate increase with temperature. The volume for occupied
hydrate I increases more abruptly than that for either empty
hydrate or ice Ih.
The linear thermal expansivities for ice Ih, empty hydrate I,
and occupied hydrate I are shown as a function of temperaturein Figure 6. The thermal expansivity for occupied hydrate at
200 K is 1.04 × 10-4 K-1, which is to be compared with theexperimental value, 0.77 × 10-4 K-1 (for ethylene oxide hydrate
structure I).16 The calculated thermal expansivity for ice Ih,(4)where f w′ and f s′ are the harmonic vibrational free energies forwater and guest, which are calculated from force constantsubmatrices Vww and Vss separately, neglecting Vws and Vsw. It
is known that the vibrational free energy is well separated into
water and guest terms since Vws and Vsw contribute little to thetotal free energy.4 The vibrational free energy difference ofwater (f w′ - f 0w) is missing in the original vdWP treatment
where f 0w is the vibrational free energy of empty hydrate I.
This amounts to 0.2 kJ mol-1 at 200 K and may be significant
in larger thermal expansivity of occupied hydrate. Therefore,
the free energy of only water,

F2 ) nw(uw + f ′w)(5)

is minimized to calculate the thermal expansivity, which isdifferent from the free energy of empty hydrate,

F0 ) nw(u0w + f 0w)(6)

The thermal expansivity by minimizing F2 in eq 5 is plotted in
Figure 7 together with those for empty hydrate and occupied
hydrate by minimizing F0 and F1. Since the approximate freeenergy (eq 4) is actually the same as that according to eq 2, the6564 J. Phys. Chem. B, Vol. 101, No. 33, 1997

Tanaka et al.
TABLE 2: Einstein Frequencies of Empty and Occupied
Hydrates (in cm-1) and Approximate Gru1 neisen Constant at
200 K γi, Where i Specifies Mode Character: The Modes
Are Classified into Translational (trans), Rotational (rot)Motions of Water and Guest Translations (guest), Which
Are Averages of the Diagonal Terms of the Force ConstantMatrix; ζ Is the Scale Factor of the Guest Size with
Reference to Xenon; In Parentheses, the Heat Capacity
Weighted Gru1 neisen Constant Is Given
type
empty

Figure 7. Temperature dependence of thermal expansivities foroccupied hydrate by minimizing the free energy in eq 2 (solid line), in
eq 4 (dotted line), and in eq 5 (dash-dot line). The free energy in eq 2
takes account of all the matrix elements in diagonalizing the force
constant matrix. In eq 4, the submatrices are diagonalized separately.
In the free energy according to eq 5, only the submatrix Vww is included.
See also text.

thermal expansivity is also the same, the former, 1.04 × 10-4
K-1, the latter, 1.04 × 10-4 K-1 at 200 K. However, the thermal
expansivity from eq 5, 0.81 × 10-4 K-1, is only slightly larger
than that from eq 6, 0.73 × 10-4 K-1, suggesting the guest
free energy term us + f s′ is important to obtain the large thermal
expansivity. A further calculation as to which component, us
or f ′s, contributes more significantly is also made. The thermal
expansivity from eq 4 but dropping the guest interaction energy
us is 1.22 × 10-4 K-1, which is fairly large compared with that
including the guest interaction. Therefore, a main source ofthe larger thermal expansivity is the vibrational motions of guest
molecules inside the cages which dominates over the guest
interaction energy us. The guest interaction term diminishes
the large thermal expansion.
The large thermal expansivity is reproduced via the freeenergy-volume relation. Then, a question is raised as to theguest size dependence of the thermal expansivity: which guest
alters the thermal expansivity more seriously, larger or smaller?
As has been discussed by Tse et al., a larger guest reduces thefree volume of the void space in clathrate hydrates and increases
the collision effect with cages. This consequently gives rise toa larger cell size at a given temperature. The thermal expansivity is a further thermal effect on the volume, which is alreadyexpanded even at very low temperatures by accommodating alarge guest. To examine the size effect, the same procedure isapplied to a clathrate hydrate encaging an artificial guest, whose
LJ size parameter is 10% larger (ζ ) 1.1) than the original LJ
size parameter for xenon (ζ ) 1). The equilibrium cell volume
at 200 K is larger by 2% than that encaging normal xenon (ζ )1). However, the thermal expansivity, 0.82 × 10-4 K-1, isnearly the same as that of the empty hydrate value.
This implies that the larger guest molecule reduces theanharmonicity which is responsible for the large thermal
expansivity. The potential surface of a guest with surrounding
water molecules becomes harmonic with an increase in guest
size, but the harmonic potential does not contribute to thethermal expansion. The difference in thermal expansivity
between two kinds of guests (i.e., ζ ) 1 and 1.1) becomes
smaller with decreasing temperature. An effective potential
surface of a guest molecule in a cage becomes quadratic at hightemperatures, unlike normal solids. This is examined in detailin conjunction with the anomalous thermal conductivity ofclathrate hydrates.14 The anharmonicity of guest vibrations,which is the main source of the large thermal expansivity,

occupied (ζ ) 1)

occupied (ζ ) 1.1)

trans
rot
guest

187
682

Einstein Frequency
189
682
24

193
683
37

trans
rot
guest

2.27 (1.87)
0.71 (0.13)

Grüneisen Constant
2.32 (1.64)
0.72 (0.11)
3.34 (0.47)

2.33 (1.65)
0.72 (0.11)
2.94 (0.42)

diminishes at high temperatures. It has been shown experimentally that there is no discernible difference in lattice constantfor a large range of hydrates with guest molecules of different
sizes.22 Anharmonic vibrational motions arising from guest
rotational motions will make a large contribution to the thermal
expansivity, which are not dealt with in the present study.
We examine which kind of vibrational motion is the key tothe large thermal expansivity, by separating vibrations into threeterms depending on mode character, host translations, hostrotations, and guest translations. To do this, approximateGrüneisen constants averaged over modes belonging to the samemode character are evaluated separately. Instead of normal
modes in which three kinds of motions are mixed, it is preferable
to classify vibrational modes into three categories by its
character as explained above. A trace of the matrix, tr(m-1/2
V m-1/2), is invariant under unitary transformation, and thesquare root of an average of diagonal elements in each molecule
gives its Einstein frequency. In the same sense, the square rootof an average in each mode character gives an approximate
Einstein frequency specific to each character of mode. Those
Einstein modes are calculated by diagonal elements of the massweighted force constant matrices: The diagonal terms areaveraged separately, depending on mode character; translationaland rotational degrees of freedom for water and translational
degree of freedom for guest. Each frequency is listed in Table
2. Although the translational frequency is affected little in Xe
with ζ ) 1, both frequencies of water and guest translationalmotions are shifted to the higher side in the case of ζ ) 1.1.
The shifts are remarkably larger. In contrast, rotational motionsof water are not affected. The Grüneisen constants calculated
at 200 K are also given in Table 2, which agrees reasonably
with that by experimental work23 (it is approximately unity overa wide range of temperatures). The Grüneisen constant fromguest vibrations becomes smaller for larger guests. This is again
consistent with the results obtained in the last paragraph.
IV. Concluding Remarks
The thermal expansivity of xenon clathrate hydrate has been
calculated by evaluating the free energies at various temperaturesin order to explain the origin of the large thermal expansivityof clathrate hydrates compared with that of ice. Several protondisordered configurations for hexagonal ice and type I clathrate
hydrates are generated. The free energy of the system isapproximated to the minimum potential energy, the harmonic
free energy and the configurational entropy arising from thedisordered nature of protons. The free energy at a given
temperature is minimized with respect to the volume of theLarge Thermal Expansivity of Clathrate Hydrates
system. Thus, the equilibrium volume is obtained. This
provides an alternative way to evaluate the thermal expansivity
from only intermolecular interaction potentials.
The experimental observation that the larger thermal expansivity of clathrate hydrates than ice is successfully reproduced.
It is found that the large expansivity of clathrate (structure I)
arises from the existence of guest molecules and that a differencein oxygen arrangement between clathrate hydrates and ice plays
a minor role. An effective potential surface of a smaller guest
is more anharmonic and therefore causes larger thermal expansivity. Raising temperature has the same effect as increasing
the guest size and diminishes the expansivity relative to ice.
It should be noted that rotational motion of an aspherical guest
molecule, which is not treated in the present study, may
contribute to the anharmonic free energy, and therefore a further
difference from ice is expected in clathrate hydrates such as
ethylene oxide hydrate. A further study is required to examine
the effect of rotational motions of guests on thermal expansivity
and also conductivity.
Acknowledgment. The authors thank K. Kiyohara and I.
Okabe for providing proton-disordered ice Ih and clathrate
hydrate I structures. This work is supported partially by a Grantin-Aid from the Ministry of Education, Science and Culture.
Most of the calculations were carried out with the use of
supercomputers at the Institute for Molecular Science.
References and Notes(1) Davidson, D. W. WatersA ComprehensiVe Treatise Franks, F., Ed.;
Plenum: New York, 1973; Vol. 2.
(2) Sloan, E. D. Clathrate Hydrate of Natural Gases; Marcel-Dekker:
New York, 1990.

J. Phys. Chem. B, Vol. 101, No. 33, 1997 6565
(3) van der Waals, J. H.; Platteeuw, J. C. AdV. Chem. Phys. 1959, 2,
1.
(4) Tanaka, H.; Kiyohara, K. J. Chem. Phys. 1993, 98, 4098.
(5) Tanaka, H.; Kiyohara; K. J. Chem. Phys. 1993, 98, 8110.
(6) Tanaka, H. J. Chem. Phys. 1994, 101, 10833.
(7) Andersson, P.; Ross, R. G. J. Phys. C: Solid State Phys. 1983, 16,1423.
(8) Ashworth, T.; Johnson, L. R.; Lai, L. High Temp.-High Press.
1985, 17, 413.
(9) Handa, Y. P.; Cook, J. G. J. Chem. Phys. 1987, 91, 6327.
(10) Tse, J. S.; White, M. A. J. Phys. Chem. 1988, 92, 5006.
(11) Tse, J. S.; Klein, M. L.; McDonald, I. R. J. Chem. Phys. 1983, 78,
2096.
(12) Tse, J. S.; Klein, M. L.; McDonald, I. R. J. Chem. Phys. 1984, 81,6146.
(13) Tse, J. S. J. Incl. Phenomena 1994, 17, 259.
(14) Inoue, R.; Tanaka, H.; Nakanishi, K. J. Chem. Phys. 1996, 104,9568.
(15) Roberts, R. S.; Andrikidis, C.; Tanish, R. J.; White, G. K.
Proceedings of the 10th International Cryogenics Engineering Conference;
Callan, H., Bergland, P., Krusius, M., Eds; Butterworths: Helsinki, 1984.
(16) Tse, J. S.; McKinnon, W. R.; Marchi, M. J. Phys. Chem. 1987,
91, 4188.
(17) Touloukian, Y. S.; Kirby, R. K.; Taylor, R. E. Lee, J. Y. R.
Thermophysical Properties of Matter; Plenum: New York, 1977; Vol. 13,
p 263.
(18) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.;
Klein, M. L. J. Chem. Phys. 1983, 79, 926.
(19) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory ofGases and Liquids; Wiley: New York, 1954.
(20) Eisenberg, D.; Kauzmann, W. The Structure and Properties ofWater; Oxford University Press: London, 1969.
(21) Tse, J. S. Chem. Phys. Lett. 1993, 215, 383.
(22) Davidson, D. W.; Handa, Y. P.; Ratcliffe, C. I.; Ripmeester, J. A.;
Tse, J. S.; Dahn, J. R.; Lee, F.; Calvert, L. D. Mol. Cryst. Liq. Cryst. 1986,
141, 141.
(23) Tse, J. S. J. Phys. C 1987, 1, 543.