Thermodynamics of high-pressure ice polymorphs: ices III and V

V. Tchijova,*, R. Baltazar Ayalaa, G. Cruz Leona, O. Nagornovb

aFacultad de Estudios Superiores Cuautitlan, UNAM Centro de Investigaciones Teoricas Av. 1 de Mayo, s/n, Edif. A-1 Cuautitlan Izcalli, Edo. Mexico, C.P.

54700 MexicobMoscow Engineering Physics Institute (State University) Kashirskoe Shosse, 31, Moscow 115409, Russian Federation

Received 10 September 2003; revised 17 February 2004; accepted 17 February 2004

Available online 21 March 2004

Abstract

Thermodynamic properties of high-pressure ice polymorphs, ices III and V, are studied theoretically. The results of TIP4P molecular

dynamics simulations in the NPT ensemble are used to calculate the temperature dependence of the specific volume of ices III and V at

pressures 0.25 and 0.5 GPa, respectively. New P–V –T equations of state of ices III and V are derived using a method generalizing the one

proposed by Fei et al. [J. Chem. Phys. 99 (1993) 5369], and new results concerning the equilibrium phase transitions ice III-water and ice

V-water are presented.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: D. Thermodynamic properties; D. Phase transitions; C. High pressure

1. Introduction

Water substance is one of the most complex and

interesting compounds in the universe. It possesses a large

number of well-established solid-state polymorphs com-

monly known as ices, along with liquid, amorphous, and

recently discovered solid phases [1–3]. Thermodynamics of

water and ices, in particular their equations of state (EOS)

find applications in many areas, among them glaciology [4],

space and planetary science [5–8], and high-pressure food

processing [9,10]. Experimental studies of shock- wave

loading of ice [11–13] give evidence of multiple phase

changes behind the shock front. A kinetic model of multiple

phase transitions in ice [14–17] also requires P–V –T EOS

and thermodynamic quantities of liquid water and ice

polymorphs.

A variety of EOS of liquid water can be found in the

literature [18–24], some of them valid in very wide ranges

of pressure and temperature. The validity of the EOS of

water in the metastable region at high pressures was recently

investigated by Tchijov [25].

Thermodynamics of ice polymorphs has been the subject

of considerable experimental and theoretical research

[26–36]. In general, the P–V –T EOS of a substance in

the form V ¼ VðP;TÞ can be found by integration of the

total differential

dV

V¼ adT 2 bdP; ð1Þ

where V is the specific volume, T is temperature, P is

pressure, a and b are the coefficients of thermal expansion

and isothermal compressibility of a substance, respectively.

If the specific volume V is known as a function of T along a

line P ¼ P0ðTÞ on the P–T diagram, the P–V –T EOS can

be expressed as

V ¼ V0ðTÞexp 2ðP

P0ðTÞbðp;TÞdp

� �; ð2Þ

where V0ðTÞ ; VðP0ðTÞ; TÞ: Alternatively, if the specific

volume V is known as a function of P along a line T ¼ T0ðPÞ

(which, for example, can be an isotherm T ¼ T0), the P–V–

T EOS is given by the formula

V ¼ V0ðPÞexpðT

T0ðPÞaðP; tÞdt

� �; ð3Þ

where V0ðPÞ ; VðP;T0ðPÞÞ: In either case, b or a should be

known as functions of P and T at each point of the region of

thermodynamic stability of a substance.

Using Eq. (2) and additional assumptions on the behavior

of b as a function of T and on the jumps of the specific

volume of ices on the lines of phase transitions, Tchijov [32]

0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jpcs.2004.02.008

Journal of Physics and Chemistry of Solids 65 (2004) 1277–1283

www.elsevier.com/locate/jpcs

* Corresponding author. Fax: þ52-55-5873-2122.

E-mail address: tchijov@servidor.unam.mx (V. Tchijov).

derived a set of the P–V–T EOS of high-pressure ices for

pressures P # 1 GPa. In Ref. [15], the improved version of

the EOS of ice VI has been presented, valid for pressures up

to 2.1 GPa.

Basing on Eq. (3), Fei et al. [33] proposed a general

scheme for the derivation of the P–V–T EOS of the solid.

According to this scheme, the coefficient of thermal

expansion of the solid can be recast as a function of P and

T as

aðP; TÞ ¼ a0ðTÞ 1 þK 0

T0

KT0

P

� �2h

; ð4Þ

where a0ðTÞ is the zero-pressure thermal expansion

coefficient commonly expressed as a linear function of

temperature, KT0 and K 0T0 are the isothermal bulk modulus

and its pressure derivative, respectively, and h is an

adjustable parameter related to the so-called Anderson-

Gruneisen parameter dT [33]. After substituting (4) into (3)

and calculating the integral term one gets the P–V–T EOS

of the solid. Fei et al. applied their scheme to derive the P–

V–T equation of state of ice VII.

In the present paper, we first generalize the method of Fei

et al. [33] and show that in order to reconstruct the P–V–T

EOS of the solid it suffices to know the values of the specific

volume of the solid along some isobar and along some

isotherm. Then we apply the generalized scheme to derive

the EOS of ices III and V. The density data along the

isotherms are available for many high-pressure ice poly-

morphs [28]. Since no measurements of the specific volume

of high-pressure ices have been performed along the isobars

(with the exception of D2O ice V [29]), we use our recent

results on molecular dynamics (MD) simulations of ices III

and V in the NPT ensemble [37] to find the specific volume

of these ices along the isobars P ¼ 0:25 GPa (ice III) and

P ¼ 0:5 GPa (ice V). (Note that MD simulations prove to be

useful in deriving the P–V–T EOS of a substance. Indeed,

density MD calculations of Baez and Clancy [38] were

recently used by Cruz Leon et al. [35] to derive the equation

of state of ice II). Finally, we apply new P–V–T EOS of ices

III and V to study the equilibrium ice III-water and ice V-

water phase transitions and compare the results with the

available experimental data.

1.1. P–V–T equation of state of the solid

Following [33] we suppose that (i) the specific volume

V0ðPÞ ; VðP; T0Þ of the solid is known along an isotherm

T ¼ T0 and (ii) the coefficient of thermal expansion can be

represented by the relationship (4). Introducing the function

FðPÞ ¼ 1 þK 0

T0

KT0

P

� �2h

;

we can rewrite Eq. (4) as

aðP; TÞ ¼ a0ðTÞFðPÞ: ð5Þ

Considering aðP; TÞ at pressure P ¼ P1 and dividing Eq. (5)

by aðP1;TÞ we get

aðP;TÞ ¼ aðP1;TÞFðPÞ

FðP1Þ: ð6Þ

Suppose that the specific volume V1ðTÞ ; VðP1;TÞ is

known along the isobar P ¼ P1; then aðP1; TÞ ¼1

V1ðTÞ�

dV1ðTÞ

dT: The coefficient of thermal expansion takes the

form

aðP;TÞ ¼1

V1ðTÞ

dV1ðTÞ

dT

FðPÞ

FðP1Þ: ð7Þ

The integral term in Eq. (3) readsðT

T0

aðP; TÞdT ¼ lnV1ðTÞ

V1ðT0Þ

FðPÞ

FðP1Þ: ð8Þ

Substituting Eqs. (8) into (3) we get the P–V–T EOS of the

solid:

V ¼ VðP;TÞ ¼ V0ðPÞexp lnV1ðTÞ

V1ðT0Þ

1 þK 0

T0

KT0

P

� �2h

1 þK 0

T0

KT0

P1

� �2h

26664

37775:ð9Þ

Eq. (9) shows that in order to find the P–V–T EOS of the

solid it suffices to know (i) its specific volume (or density)

along an isobar P ¼ P1 and an isotherm T ¼ T0; (ii) the

isothermal bulk modulus KT0 and its pressure derivative

K 0T0; and (iii) the value of the adjustable parameter h: Note

that the equation of state of the solid proposed in Ref. [33] is

a particular case of Eq. (9) corresponding to P1 ¼ 0: In the

next section, we apply the P–V–T EOS of the solid (9) to

ices III and V.

2. P–V–T equations of state of ices III and V

Ice III occupies the region on the P–T diagram of water

substance smallest of all known stable ice polymorphs. It is

thermodynamically stable between 240 and 260 K and 0.22

and 0.34 GPa. The structure of ice III is based on a

tetrahedral unit cell and comprises 12 water molecules. Ice

III is partially proton disordered [29]; the proton ordered

version of ice III is known as ice IX [1].

The structure of ice V is the most complicated of all ice

phases, with a monoclinic unit cell of 28 water molecules.

Ice V is partially proton disordered [29]; no ordered version

of this ice has been identified. The region of thermodynamic

stability of ice V on the P–T diagram extends roughly

between 210 and 270 K and 0.34 and 0.63 GPa.

In order to apply Eq. (9) to ices III and V we first need to

find the specific volume V1ðTÞ of these ices along the

corresponding isobars. To this end, we use the results of our

MD simulations [37]. The calculations were performed in

V. Tchijov et al. / Journal of Physics and Chemistry of Solids 65 (2004) 1277–12831278

the NPT ensemble, with the proton disordered simulation

cell consisting of 768 water molecules (ice III) and 756 water

molecules (ice V). For ice III, the simulations were carried out

along the isobar P1 ¼ 0:25 GPa, whereas for ice V the

simulations were performed along the isobar P1 ¼ 0:5 GPa.

In both cases, TIP4P and TIP5P water models were used. It

was found that given the identical initial configurations, at the

end of the simulation runs the TIP5P model produced

noticeably denser structures than those of TIP4P model. In

contrast, TIP4P model gave quite satisfactory volumetric

results along the isotherms as compared to experimental data

of Gagnon et al. [28]. Therefore, in the present paper we use

the following values of the specific volume of ices III and V

obtained in our TIP4P MD simulations [37]:

V1ðTÞ ¼ 0:4642 £ 1023 þ 2:885 £ 1026T 2 5:348

£ 1029T2ðice III at P1

¼ 0:25 GPaÞ ð10Þ

V1ðTÞ ¼ 0:9109 £ 1023 2 1:275 £ 1026T þ 3:061

£ 1029T2ðice V at P1

¼ 0:5 GPaÞ ð11Þ

In Eqs. (10) and (11), T is given in K and V1ðTÞ is given

in m3/kg.

The values of the isothermal bulk modulus KT0 and its

pressure derivative K 0T0 were found by Shaw [26]. For ice

III, KT0 ¼ 8:5 GPa y K 0T0 ¼ 5:7; whereas for ice V

KT0 ¼ 13:3 GPa and K 0T0 ¼ 5:2:

Gagnon et al. [28] measured the density r of ice V along

the isotherm T0 ¼ 237:65 K ( ¼ 235.5 8C) and fitted its

values to a quadratic equation in pressure. The equation

obtained is

rðP;T0Þ ¼ 1:1974 þ 0:01963P 2 0:001098P2: ð12Þ

In this equation, P is given in kbar and rðP; T0Þ is given in g/

cm3. Given Eq. (12), the specific volume of ice V

along the isotherm T0 ¼ 237:65 K can be calculated as

V0ðPÞ ¼ 1=rðP;T0Þ: The value of the adjustable parameter h

is found by fitting the specific volume VðP;TÞ of ice V to

experimental data of Bridgman [39] on the transition line ice

V-water. The least-square fitting gives h ¼ 7:86: The results

are presented in Fig. 1 in the area marked ‘Ice V’. Solid

boxes are the experimental data of Bridgman [39]. Solid line

represents the values of the specific volume of ice V

calculated along the transition line ice V-water. One can see

that the data of Bridgman are well reproduced by Eq. (9);

the relative error is less than 0.37%. It is also possible to

compare the density data of Lobban et al. [29] (recalculated

for H2O ice V) with the values of the density of ice V

computed using Eq. (9). The results are in good agreement;

the relative error does not exceed 0.9%.

The empirical equations for the density of ice III along

the isotherm T0 ¼ 237:65 K ( ¼ 235.5 8C) reported by

the Canadian group [28,40,41] are somewhat confusing.

Indeed, the pressure dependence of the density of ice III at

T0 ¼ 237:65 K is first presented as

rðP; T0Þ ¼ 1:1321 þ 0:01206P2 ð13Þ

(Ref. [28]), then as

rðP; T0Þ ¼ 1:1321 þ 1:2057 £ 1023P2 ð14Þ

(Ref. [40]), and finally as

rðP; T0Þ ¼ 1:1321 þ 1:2057 £ 1022P ð15Þ

(Ref. [41]). In Eqs. (13)–(15) P is given in kbar and rðP; T0Þ

in g/cm3. From these equations, the specific volume of ice

III on the isotherm T0 ¼ 237:65 can be found as V0ðPÞ ¼

1=rðP;T0Þ: In order to find the equation that best represents

the function V0ðPÞ for ice III, we calculate the specific

volume of this ice on the transition line ice III-water. In each

of the three cases above we use the value h ¼ 1 since our

computational experiments show little influence of h on the

values of the specific volume of ice III on the line ice III-

water. The results are presented in Fig. 1 in the area marked

‘Ice III’. Solid boxes are the experimental data of Bridgman

[39]. Dashed lines 1, 2, and 3 correspond to Eqs. (13), (15),

and (14), respectively, substituted into Eq. (9). One can see

that the usage of Eq. (13) in the P–V–T EOS (9) leads to

significant errors in the values of the specific volume

(dashed line 1). We choose Eq. (15) as the most recent of the

three and use the following formula for V0ðPÞ of ice III:

V0ðPÞ ¼1

rðP;T0Þþ DV ; ð16Þ

where DV ¼ 0:0058 £ 1023 m3/kg. The value of DV is

added in order to adjust the values of the specific volume

Fig. 1. The specific volume of ices III and V along the transition lines ice

III-water and ice V-water. Solid boxes are the experimental data of

Bridgman [39]. Solid lines represent the values of the specific volume

calculated according to the derived P-V-T EOS (9). Dashed lines 1, 2, and 3

correspond to the usage, in Eq. (9), of the density data taken from Refs. 28,

41, and 40, respectively (see text).

V. Tchijov et al. / Journal of Physics and Chemistry of Solids 65 (2004) 1277–1283 1279

of ice III on transition line ice III-water to the data of

Bridgman [39]. In Fig. 1, solid line in the area marked

‘Ice III’ represents the specific volume of ice III

calculated using Eq. (9) with the selected function V0ðPÞ

given by Eq. (16). One can see that the experimental and

the calculated values of the specific volume are in good

agreement; the relative error is less than 0.13%. The

density data of Lobban et al. [29] (recalculated for H2O

ice III) are also in good agreement with the values of the

density of ice III computed using Eq. (9); the relative

error does not exceed 0.8%.

We also compare the jumps DV of the specific volume

across the transition line ice III-ice V with experimental data

on DV reported by Bridgman [39]. The calculated values are

5.7–8.6% larger than those of Bridgman.

A complete thermodynamic knowledge of any substance

is afforded by a knowledge of its P–V–T EOS and a

knowledge of the isobaric heat capacity CP along some

curve of the P–T diagram not an isothermal [39]. Recently,

Tchijov [42] proposed a scheme for the calculation of the

isobaric heat capacity of high-pressure ice polymorphs (ices

VII, VI, V, and III). Given the P–V–T EOS of ices III and V

and the isobaric heat capacity CP in the regions of their

thermodynamic stability, other thermodynamic quantities of

these ices can be calculated using standard thermodynamic

relationships [43].

3. Equilibrium ice III-water and ice V-water phasetransitions

In this section, we use the following notation. Let C be a

set of indices 1, 3, 5, 6, and w which refer to ices Ih, III, V,

VI, and liquid water, respectively. The thermodynamic

quantities of a phase i [ C are labeled by the corresponding

subscript, e.g., Vi: The transition line between the coexisting

phases i and jði; j [ CÞ is denoted by Lij: The triple points

are denoted by {ijk}. Pijk and Tijk denote the values of

pressure and temperature at the triple point {ijk}: The

coordinates of the triple points and the discussion of the

phase diagram of stable phases of ice-water system can be

found elsewhere [1].

Using P–V–T EOS and thermodynamic quantities of

ices III, V, and liquid water, it is possible to study the

equilibrium ice III-water and ice V-water phase transitions.

For liquid water, we use the IAPWS Formulation 1995 for

the thermodynamic properties of ordinary water substance

[24] (see Appendix A).

The method we apply has been previously used to study

equilibrium solid-solid phase transitions [35]. Consider an

equilibrium mixture of two coexisting phases i and

jði; j [ CÞ: Let xð0 # x # 1Þ be a fraction of the phase i

in a unit of mass of a two-phase mixture. The specific

volume V and the entropy S of a mixture are given by

V ¼ xVi þ ð1 2 xÞVj; S ¼ xSi þ ð1 2 xÞSj: ð17Þ

The following thermodynamic equations describe equili-

brium phase transitions and phase composition of a mixture

of two phases:

(i) The Clapeyron–Clausuis equation

dT

dP¼

TðVi 2 VjÞ

Qij

; ð18Þ

(ii) The equation for the latent heat of phase transition Qij

[43]

dQij

dP¼ CPi 2 CPj þ

Qij

T2

QijðViaTi 2 VjajÞ

Vi 2 Vj

" #

�TðVi 2 VjÞ

Qij

; ð19Þ

(iii) The equation for adiabatic compressibility of a two-

phase mixture [44]

›V

›P

� �S¼x

›Vi

›P

� �Tþ

2T

Qij

›Vi

›T

� �PðVi 2VjÞ

"

2CPiTðVi 2VjÞ

2

Q2ij

#þð12 xÞ

›Vj

›P

� �T

�

þ2T

Qij

›Vj

›T

� �PðVi 2VjÞ2CPjT

ðVi 2VjÞ2

Q2ij

#: ð20Þ

For isentropic processes (S ¼ const), Eqs. (18)–(20) form a

system of ordinary differential equations with pressure P as

independent variable and T ; Qij; and V as the functions of P:

Solving the initial-value problem for the system Eqs. (18)–

(20), one can find the isentropes V ¼VðPÞ of the equilibrium

two-phase mixture, the dependence of the latent heat Qij on

pressure, and the line of phase transition Lij : T ¼ TðPÞ on

the P–T diagram. Since experimental data on the lines Lw3

and Lw5 are available [39], this method provides an indirect

check of the consistency of P–V–T EOS and thermodyn-

amic properties of ices III and V by comparing the measured

and the calculated transition temperatures. The system

(18)–(20) is integrated numerically by the 4-th order

Runge-Kutta method. At each pressure step, after the new

values of P;T ;Qij; and V have been calculated the algebraic

Eq. (17) is used to find a new phase composition of a

mixture.

For ice III-water phase transition (i ¼ w; j ¼ 3), the

initial values of P, T, and Qw3 are chosen at the triple point

{13w}: P ¼ P13w ¼ 0:2099 GPa, T ¼ T13w ¼ 251:165 K

[45], Qw3 ¼ 212:8 £ 103 J/kg [46]. Let x be a fraction of

water in a two-phase mixture. Five calculations are

performed with initial values of x from 0.2 to 1.0 with a

step of 0.2. The calculations are held until the pressure

P35w ¼ 0:3501 GPa at the triple point {35w} is reached. The

results are presented in Figs. 2 and 3.

V. Tchijov et al. / Journal of Physics and Chemistry of Solids 65 (2004) 1277–12831280

Fig. 2 shows a mixed-phase region of ice III and water on

the P–V diagram. Solid lines are the calculated boundaries

of the two-phase region, solid boxes are the experimental

data of Bridgman [39], and dashed lines are the calculated

isentropes of the equilibrium ice III-water mixture. For the

initial values of the fraction of water in the mixture x ¼ 0:2;

0.4, 0.6, 0.8, and 1.0 at P ¼ P13w; the corresponding final

values of x at P ¼ P35w are 0.161, 0.361, 0.561, 0.760, and

0.961, respectively. Fig. 3 shows the calculated ice III-water

melting curve Lw3 (solid line) and the experimental

data of Bridgman [39] (solid boxes). The maximum

deviation between the measured and the calculated values

of transition temperature along Lw3 is less than 0.33 K.

For ice V-water transition (i ¼ w; j ¼ 5), the initial

values of P, T, and Qw5 are chosen at the triple point

{35w}: P ¼ P35w ¼ 0:3501 GPa, T ¼ T35w ¼ 256:164 K

[45], Qw5 ¼ 260:3 £ 103 J/kg [46]. Five calculations are

performed with initial values of the fraction of water x

varying from 0.2 to 1.0 with a step of 0.2. The calculations

are held until the pressure P56w ¼ 0:6324 GPa at the

triple point {56w} is reached. The results are presented

in Figs. 4 and 5.

Fig. 4 depicts a mixed-phase region of ice V and water on

the P–V diagram. Solid lines are the calculated boundaries

of a two-phase region. Solid boxes are the experimental data

Fig. 3. Calculated and experimental values of ice III-water transition

pressure as functions of temperature. Solid boxes represent experimental

data of Bridgman [39]. Solid line is the calculated line of phase transition in

the interval of pressure P13w # P # P35w; according to Eqs. (18)–(20).

Fig. 4. Mixed-phase region of ice V and liquid water on the P–V diagram.

Solid lines are the calculated boundaries of the mixed-phase region. Solid

boxes are the experimental data of Bridgman [39]. Dashed lines are the

calculated isentropes. The numbers 0.2, 0.4, 0.6, 0.8, and 1.0 indicate the

initial values of the fraction of liquid water in an equilibrium mixture.

Fig. 5. Calculated and experimental values of ice V-water transition

pressure as functions of temperature. Solid boxes represent experimental

data of Bridgman [39]. Solid line is the calculated line of phase transition in

the interval of pressure P35w # P # P56w; according to Eqs. (18)–(20).

Fig. 2. Mixed-phase region of ice III and liquid water on the P-V diagram.

Solid boxes are the experimental data of Bridgman [39]. Solid lines are the

calculated boundaries of mixed region. Dashed lines are the calculated

isentropes. The numbers 0.2, 0.4, 0.6, 0.8 and 1.0 indicate the initial values

of the fraction of liquid water in an equilibrium mixture.

V. Tchijov et al. / Journal of Physics and Chemistry of Solids 65 (2004) 1277–1283 1281

of Bridgman [39]. Dashed lines are the calculated isentropes

of ice V-water mixture. For initial values of the fraction of

water in the mixture x ¼ 0:4; 0.6, 0.8, and 1.0 at P ¼ P35w;

the final values of x at P ¼ P56w are 0.181, 0.407, 0.633, and

0.859, respectively. For initial value of x ¼ 0:2; liquid water

completely transforms into ice V along the corresponding

isentrope. Fig. 5 shows the calculated ice V-water melting

curve Lw5 (solid line) and the experimental data of

Bridgman [39] (solid boxes). The measured and the

calculated values of ice V-water transition temperatures

are in good agreement; the maximum deviation does not

exceed 0.4 K.

As it was mentioned in the Introduction, the P–V–T

EOS and thermodynamic properties of high-pressure ices

are used in the kinetic model of multiple phase transitions in

ice compressed by shock waves [14–17]. The forms of the

isentropes of two-phase mixtures permit to predict

the properties of shock waves [47]. Figs. 2 and 4 show

that the computed isentropes of ice III-water and ice V-

water mixtures are convex downward, as are the isentropes

of solid-solid phase transitions ice II-ice III, ice Ih-ice II,

and ice II-ice V [35]. This indicates ‘normal’ compressi-

bility of a two-phase mixture and the stability of shock

wave. On the contrast, the isentropes of ice Ih-water mixture

are convex upward [31] and thus exhibit anomalous

compressibility. In this case, shock wave is unstable and

transforms into a centered compression fan [47].

4. Conclusions

We have generalized the equation of state of the solid

proposed by Fei et al. [33]. We have shown that in order to

find the P–V–T EOS of the solid it suffices to know its

specific volume along the intersecting isobar and isotherm

inside the region of its thermodynamic stability, as well as the

isothermal bulk modulus and its derivative with respect to

pressure. The P–V–T EOS contains one adjustable par-

ameter h: We have applied the generalized P–V–T EOS to

ices III and V. For high-pressure ice phases, the specific

volume along the isotherms is known from the experiments

[28]. The values of the specific volume of ices III and V along

the isobars were obtained from our TIP4P MD simulations in

the NPT ensemble [37]. The isothermal bulk modulus and its

pressure derivative were taken from Ref. [26]. For ice V, the

adjustable parameter h ¼ 7:86 minimizes the mean square

deviation between the experimental [39] and the calculated

specific volume on the ice V-water transition line. In contrast,

little influence of h (around its value h ¼ 1) on the mean

square deviation have been found for ice III. The calculated

values of the specific volume of ices III and V are in good

agreement both with experimental data of Bridgman on the

lines of phase transition ice III-water and ice V-water and

with the recent density data of Lobban at al. [29]. Basing

on these results, we conclude that in the absence

of experimental measurements the molecular dynamics

simulations of high-pressure ice polymorphs provide a useful

tool in obtaining the density data.

We have also studied the equilibrium isentropic ice III-

water and ice V-water phase transitions and found that the

results of our computer simulations are in good agreement

with the existing experimental data. In particular, the lines

of phase transitions on the P–T diagram of water substance

are well reproduced using new P–V–T EOS and thermo-

dynamic quantities of ices III and V. The calculated

isentropes of equilibrium two-phase mixtures studied here

exhibit normal (not anomalous) behavior, in contrast with

those of ice Ih-water mixture. This provides an insight into a

respond of ice to shock compression.

The authors are grateful to National University of

Mexico for financial support of this work by the PAPIIT

grant No. IN105401. One of the authors (VT) is indebted to

Prof. Dr-Ing. Wolfgang Wagner for the FORTRAN code of

IAPWS Formulation 1995.

Appendix A

In the present paper, we use the IAPWS Formulation

1995 for the thermodynamic properties of ordinary water

substance (IAPWS-95) [24] recommended for general and

scientific use by the International Association for the

Properties of Water and Steam. This formulation is the

fundamental equation for the specific Helmholtz free energy

f expressed as

f ðr; TÞ

RT¼ Fðd; tÞ ¼ F0ðd; tÞ þFrðd; tÞ ðA1Þ

where d ¼ r=rc; t ¼ Tc=T ;F0yFr are the ideal-gas part and

the residual part of f, respectively, Tc and rc are the

temperature and the density at the critical point, and R is the

specific gas constant. The formulation is valid in the water

region of the P–T diagram from the melting curve to

1273 K at pressures up to 1 GPa, though at least for basic

properties such as pressure and enthalpy, IAPWS-95 can be

extrapolated to extremely high pressures and temperatures

[24]. All thermodynamic properties of liquid water can be

derived from (A1); some of most commonly used quantities

are presented in Ref. [24]. In particular, the P–V–T EOS of

liquid water is given by the equation

P ¼ rRT 1 þ›Fr

›d

� �: ðA2Þ

The specific volume of liquid water V ¼ VðP;TÞ is found

from (A2) numerically; to this end, the Newton–Raphson

root-finding method is used. All necessary thermodynamic

quantities can then be expressed as functions of P and T. It is

worthwhile mentioning that numerical solution of non-

linear algebraic Eq. (A2) with respect to r ¼ 1=V is rather

sensitive to initial approximation, the situation similar to

that with the other P–V–T EOS of liquid water [25].

V. Tchijov et al. / Journal of Physics and Chemistry of Solids 65 (2004) 1277–12831282

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