thermal expansivities of cubic ice i and ice vii

Ž . Ž .Journal of Molecular Structure Theochem 461]462 1999 561]567

Thermal expansivities of cubic ice I and ice VII q

Hideki TanakaU

Department of Chemistry, Faculty of Science, Okayama Uni ersity, 3-1-1 Tsushima-naka, Okayama 700-8530, Japan

Received 25 June 1998; accepted 23 July 1998

Abstract

The free energies of low pressure cubic ice I and high pressure ice VII are calculated over a wide range oftemperature in order to examine whether the negative thermal expansivity observed in ice I is a common featureacross various ice morphologies. The Gibbs free energies for several proton disordered structures are minimized withrespect to volume variation. This enables us to evaluate the thermal expansivity at fixed temperature and pressurefrom only intermolecular interaction potential. While the negative thermal expansivity of the low pressure ice I inlow temperature regime is successfully reproduced, it is found that the expansivity is always positive for the highpressure ice VII. Q 1999 Elsevier Science B.V. All rights reserved.

Keywords: Thermal expansivity; Cubic ice; Ice VII

1. Introduction

Most of the crystalline ices under pressure suf-fer from strain and hydrogen bonds are more or

w xless distorted 1 . The counterpart of those highpressure ices is known as ice I. There exist twosorts of ice I forms, which are either stable andmetastable at atmospheric pressure and called

Ž . Ž .hexagonal ice ice Ih and cubic ice ice Ic . Mostof the quantities concerning the hydrogen bondsfor ice Ic are similar to those of ice Ih but aredifferent from those for other ice polymorphs

q Dedicated to Professor Keiji Morokuma in celebration ofhis 65th birthday.

U E-mail: [email protected]

w xstable under high pressure 1 ; the deviation fromthe linear hydrogen bond is the least in the lowpressure ices.

Each water molecule in ice Ic phase has fourneighbors firmly hydrogen bonded with the cen-tral one. It is composed of hexagonal rings andthe oxygen atoms in ice Ic form a diamond struc-ture. At pressure above 2 GPa, a high pressureice phase, called ice VII, emerges. The structureof ice VII is made from two interpenetrating iceIc lattices, each of which occupies vacant space ofthe other lattice. A central water molecule haseight neighbors but is connected to only half ofthem by hydrogen bonds. Both types of ice struc-tures are displayed in Fig. 1.

The thermal expansivity of ice Ih is negativew xbelow a temperature of 60 K 2 , which is also

0166-1280r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 1 6 6 - 1 2 8 0 9 8 0 0 4 4 4 - 8

( ) ( )H. Tanaka r Journal of Molecular Structure Theochem 461]462 1999 561]567562

Ž . Ž .Fig. 1. Structure of a ice Ic and b ice VII.

observed for tetrahedrally coordinated com-w xpounds such as silica 3 . Ice Ic is metastable and

observed in limited conditions but has similarproperties to ice Ih which is stable under lowpressure. Even though the thermal expansivity ofice Ic has not been measured, it must exhibit thesame behavior as ice Ih. In theoretical treatmentvia the Gruneisen relation, frequencies of some¨vibrational modes must modulate to the higherfrequency side when the crystalline solid is di-lated as explained below. Moreover, the heatcapacity of such a mode must vary significantlyagainst temperature change. Therefore, a stan-dard classical statistical mechanical treatmentseems to be inadequate to account for this un-usual thermal expansivity.

We have successfully reproduced the negativeŽ . w xexpansivity of ice Ic as well as Ih 4 whose

w xinteraction is described by TIP4P potential 5 .The origin of the negative thermal expansivity isfound to be the bending motions of the threehydrogen bonded molecules, which arises, in turn,from the tetrahedral coordination. To ourknowledge, no detailed measurement on the ther-

Ž .mal expansion of ice VII or VIII in low temper-

ature has been reported. While the coordinationnumber is 8 in ice VII, the number of hydrogen

w xbonds for a molecule is only four 1 . Hence, it isintriguing to examine whether ice VII shares thisunique property in the low temperature regimewith low pressure ice. In the present study, weexamine temperature dependence of the equilib-rium volume of ice VII at 2.5 GPa in the same

w xmanner as previously reported 4 . At low temper-ature, proton-ordered ice VIII is more stable

w xthan ice VII 6,7 . We adopt, however, ice VII inorder to compare thermal expansivity with ice Ic,which has a proton-disordered form.

2. Theory and method

2.1. Intermolecular interaction and structure of unitcell

Kataoka has shown in terms of interaction en-ergy at temperature 0 K that Carravetta]Cle-

Ž . w xmenti CC potential 8 is the best to describephase behaviors of ice Ic and VIII among simplepair potentials derived from ab initio MO calcula-

w xtion 9 . We extend this kind of calculation to

( ) ( )H. Tanaka r Journal of Molecular Structure Theochem 461]462 1999 561]567 563

finite temperature range for CC water in thepresent study. The interaction potentials for all

w xpairs of molecules are truncated smoothly 10,11 .The basic cell of ice VII is cubic containing 432

Žwater molecules. The sizes of the cell are a sb˚.sc s20.271, 20.494 and 20.727 A, which corre-

spond to densities of 1.55, 1.50 and 1.45 g cmy3,respectively. Five proton-disordered configura-tions are generated for ice VII structure. Tenproton-disordered configurations for ice Ic aregenerated, each containing 216 water molecules.The density ranges from 0.775 to 0.950 g cmy3

with an interval of 0.025 g cmy3. All ice struc-tures generated are proton-disordered and havezero net dipole moment since the periodicboundary condition imposed leads to an infinitedipole moment of the whole system. Here, only asmall number of configurations are examinedcompared to the total number of allowed config-urations. The method adopted here does not,however, cause any serious problems since dif-ferent configurations have similar free energy val-ues. In practical calculation, a minimizing processof the potential energy is required for the specificintermolecular interaction since each generatedconfiguration corresponds only roughly to theminimum energy structure. The crystalline con-figurations of the minimum potential energy areobtained by applying the steepest descent method.

2.2. Gruneisen relation¨

The thermal expansivity of crystalline solid isevaluated via calculation of its Helmholtz free

Ž .energy, A T ,V , which is a function of the tem-perature T and the volume V. The free energy isexpressed by

Ž . Ž . Ž . Ž .A T ,V sU V qF T ,V yTS , 1c

Ž . Ž .where U V and F T ,V are the potential energyof the solid at equilibrium position at tempera-ture 0 K and the vibrational free energy. Theconfigurational entropy S for an N moleculec

Ž .system for both ice forms is given by kN ln 3r2with great accuracy, where k is the Boltzmann

w xconstant 1 . In order for the free energy in Eq.

Ž .1 to be a more tractable form, the anharmonicŽ .vibration free energy is removed and F T ,V is

replaced by the harmonic vibrational free energyŽ .F T ,V . This does not mean the crystalline solid0

is treated as a harmonic system but the anhar-monicity is partially incorporated in the free en-

Ž .ergy expression since A T ,V is dependent on theŽ .volume through Eq. 1 and the anharmonic na-

Ž .ture of U V is reflected to the frequencies ofvibrational modes.

Thermodynamics relates the linear thermal ex-pansivity a with the free energy

Ž . 2 Ž .3as lnVr T syk ArT V , 2p T

where k is the isothermal compressibility, whichTis always positive for a stable system. The har-monic vibrational free energy is given by

Ž . w Ž .x Ž .F T ,V skT ln 2sin h b "v r2 , 3Ý0 ii

where " is the Planck constant divided by 2p andb stands for 1rkT. The linear thermal expansivityis given by

Ž .asg C k r3V , 4¨ T

where C and g are the heat capacity and the¨Gruneisen parameter, respectively. The heat ca-¨pacity is given by the sum of the heat capacity ofthe individual mode as

Ž .C s C . 5Ý¨ ii

Here, C is the heat capacity of the ith mode,iwhich is

y22Ž . Ž . w Ž . xC sk b "v exp b "v exp b "v y1 .i i i i

Ž .6

The Gruneisen parameter is given using C by¨ i

Ž .gs g C r C , 7Ý Ýi i ii i


where g is a mode Gruneisen parameter defined¨ias

Ž . Ž .g sy ln v r lnV . 8i i

Instead of the quantum mechanical partitionŽ .function for a harmonic oscillator, Eq. 3 may be

replaced by the classical partition as

c Ž . Ž . Ž .F T ,V skT ln b "v , 9Ý0 ii

However, this gives rise to a constant heatcapacity, k, for all vibrational motions. In view of

Ž .Eq. 4 , the thermal expansivity cannot change itssign in the classical system since C , k and V¨ Tare all positive and g is no longer dependent on

Ž .temperature. Therefore, Eq. 3 is essential toaccount for the change in the sign of a in the lowtemperature regime.

2.3. Free energy calculation

Ž .We obtain the density of state g v for inter-molecular vibrational motions from normal modeanalysis by diagonalizing the mass weighted forceconstant matrix. The density of state forintermolecular vibration is obtained by simple

Ž . Ž .average of 5 ice VII and 10 ice Ic configura-tions generated. Once the density of state isobtained, evaluation of the harmonic free energy

Ž .F is straightforward. The potential energy U V0Ž .is dependent only on density but the F T ,V is a0

function of both density and temperature.If we can calculate the isothermal compressibil-

ity, it is a simple task to evaluate the thermalŽ .expansivity via Eq. 2 . Actually, the isothermal

compressibility at given temperature is hard tocalculate since the pressure is one of the leastreliable thermodynamic properties derived fromsimulations. Instead, we obtain the equilibriumvolume by minimizing the free energy at a giventemperature. Here, the free energy to be

Ž .minimized is the Gibbs free energy, G T , p sŽ .A T ,V qpV: the Gibbs free energy takes a mini-

mum value against volume variation at a givetemperature when pressure, p, is fixed to a con-

stant value, 2.5 GPa for ice VII and 0.1 MPa forice Ic. The Gibbs free energies at three cell sizesare calculated, which are fitted to a quadraticfunction against volume. The equilibrium molarvolume at a given temperature is calculatedtogether with the Gibbs free energy. Once thevolume is obtained as a function of temperature,it is straightforward to calculate the thermal ex-pansivity.

3. Results and analysis

It is found that even incorporating the vibratio-nal free energy, CC potential has local free en-ergy minima corresponding to ice Ic and VII andboth ice structures do not collapse spontaneouslyat the densities in the local free energy minima.

Each hydrogen bond pattern expressed by acombination of dihedral angles has a differentinteraction energy. For combinations of dihedral

Ž . Ž .angles of pr3, pr3 and pr3, p , the interac-tion energies are y20 and y24 kJ moly1, respec-tively. The pair interaction energy distributions attemperature 0 K for ice Ic and ice VII are shownin Fig. 2, which is defined as

¦ ;Ž . Ž . Ž .x ¨ s 1rN d ¨ y f , 10Ý Ýpi j/i

²:where indicates the ensemble average and f isthe pair potential function. There are two distinctpeaks in both ice Ic and VII. Furthermore, thereare many pairs whose interactions are weak orrepulsive in ice VII. Individual peaks lower thany15 kJ moly1 in Fig. 2 are in good correspon-dence to two conformations represented by thecombination of two dihedral angles.

The densities of state for ice Ic and ice VII areshown in Fig. 3. The intermolecular vibrationalmotions are split completely into translation-dominant and rotation-dominant motions in bothice forms. In ice Ic, there are two main peaks at70 cmy1 and 200 cmy1. The former correspondsto a bending motion of three hydrogen bondedmolecules and the latter is associated with astretching motion of a hydrogen bonded pair. The


Fig. 2. Pair interaction energy distribution for individual wa-Ž . Ž .ter molecules in ice VII solid line and ice Ic dotted line .

peak at 70 cmy1 observed in ice Ic disappears inice VII. The higher frequency region in ice VIIexhibits a broader distribution than that of ice Ic.

We calculate the free energy values with dif-ferent densities at a given temperature. The mini-mum Gibbs free energy is accurately calculatedby fitting the three lowest free values to aquadratic function of the cell volume. The equi-librium volume at a given temperature is obtainedtogether with the minimum free energy. Finally,we calculate the thermal expansivities for ice Icand VII, which are plotted against temperature inFig. 4. The calculated thermal expansivity for iceIc, 1.2=10y4 Ky1, at 200 K is fairly larger thanthe experimental value of ice Ih, either 0.36=

y4 y1 w x y4 y1 w x10 K 2 or 0.56=10 K 12 . It is nega-

Fig. 3. Densities of state for intermolecular vibrational mo-Ž . Ž .tions for ice VII solid line and ice Ic dotted line as a

function of wave number.

Ž .Fig. 4. Thermal expansivities of ice VII at 2.5 GPa solid line ,Ž .ice Ic at atmospheric pressure dotted line and experimental

w x Ž .measurement 2 filled circle as a function of temperature.

tive below a temperature of 50 K, which is ingood agreement with the experimental observa-tion. Ice VII has positive thermal expansivity inthe whole temperature range and its magnitude isfairly larger than that of ice Ic at high tempera-ture.

Since the sign of the thermal expansivitychanges at around 50 K in ice Ic, there are twonecessary conditions to be satisfied. Firstly, theheat capacity of individual vibrational modes issubject to significant variation against tempera-

Ž . Ž .ture in view of Eqs. 4 and 7 . This is notsatisfied when the partition function according tothe classical mechanics is used because it gives a

Žconstant temperature and frequency indepen-.dent heat capacity. Instead, use of a quantum

mechanical partition function is essential. Sec-ondly, there must be unusual modes which havenegative g . Those two conditions stem from theichange in the sign of a under positive C and k¨ Tfor any stable system.

The dilation of the volume from the equilib-rium position at 0 K induces normally the shift ofmode frequency to lower side so that the vibratio-nal free energy becomes lower while the interac-tion energy becomes higher. The volume is de-termined by those two different contributions.However, frequencies of some modes must havedifferent volume dependences in order for a to


change its sign in ice Ic. We calculate modeGruneisen parameters to examine their positivity.¨Let K be the mass weighted force constant matrix

Ž y1r2 y1r2 .to be diagonalized Ksm Vm where Vand m are the second derivative matrix of theintermolecular interaction and the mass matrix.The diagonalized force constant matrix K isdobtained by

† Ž .K sUKU , 11d

where U is the unitary matrix for diagonalization.First, we obtain U at density 0.80 g cmy3 for ice

y3 Ž .Ic and 1.5 g cm for ice VII. Next, Eq. 11 isapplied to two sets of ice structures which are

Ž .elongated or shrunk by linear factors of 1"0.005 using the matrix, U. The small change ofthe volume does not alter the characters of indi-vidual modes. Hence, diagonalization is expectedto be performed by the common U for threedifferent volumes. Indeed, this procedure is suc-cessfully applied and all the modes are real.

The frequency dependence of g is defined asi

¦ ;Ž . Ž . Ž .r v Dvs g rg v 12Ý i vi

for modes of frequency range between v andvqDv. Here the average is taken over all con-

Fig. 5. Frequency dependence of the Gruneisen parameter¨Ž . Ž .for ice VII solid line and ice Ic dotted line .

Ž .figurations generated. That is, r v is proportio-nal to the Gruneisen parameter whose frequency¨lies between v and vqDv. As plotted in Fig. 5,the contribution from the low frequency region isnegative in ice Ic. The negative g is not observediin ice VII and the thermal expansivity is alwayspositive.

The negative thermal expansivity of ice Ic atlow temperature arises from the unusual negativeg and from the larger heat capacity of theseimodes relative to those of other modes havingpositive g . We have shown the modes havinginegative g correspond to the bending motions ofi

w xthree water molecules hydrogen bonded 4 . Thatthe frequencies of these modes are low is a key tothe negative thermal expansivity. The negative g istems from the tetrahedral coordination. In thecase of higher coordination in ice VII, thoseunusual modes having negative g disappear.i

4. Conclusion

The free energies of low pressure cubic ice Iand high pressure ice VII are calculated over awide range of temperature. The Gibbs free ener-gies for several proton-disordered structures gen-erated are minimized with respect to volume vari-ation. The thermal expansivity is calculated atfixed temperature and pressure for two ice formsinteracting via CC potential. While the negativethermal expansivity of ice in the low temperatureregime is successfully reproduced in the case ofthe low pressure ice, it is found that the expansiv-ity is always positive for the high pressure ice VII.

Acknowledgements

The author is grateful to I. Okabe for providingproton-disordered ice Ic structures. Most of thecalculation was carried out with the use of super-computers in the Institute for Molecular Science.

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thermal expansivities of cubic ice i and ice vii

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