THE JOURNAL OF CHEMICAL PHYSICS 136, 094507 (2012)

Thermodynamics of supercooled waterV. Holten, C. E. Bertrand,a) M. A. Anisimov,b) and J. V. SengersInstitute for Physical Science and Technology and Department of Chemical and Biomolecular Engineering,University of Maryland, College Park, Maryland 20742, USA

(Received 7 December 2011; accepted 10 February 2012; published online 2 March 2012)

We review the available experimental information on the thermodynamic properties of supercooledwater and demonstrate the possibility of modeling these thermodynamic properties on a theoreticalbasis. We show that by assuming the existence of a liquid–liquid critical point in supercooled water,the theory of critical phenomena can give an accurate account of the experimental thermodynamic-property data up to a pressure of 150 MPa. In addition, we show that a phenomenological extensionof the theoretical model can account for all currently available experimental data in the supercooledregion, up to 400 MPa. The stability limit of the liquid state and possible coupling between crys-tallization and liquid–liquid separation are also discussed. It is concluded that critical-point ther-modynamics describes the available thermodynamic data for supercooled water within experimentalaccuracy, thus establishing a benchmark for further developments in this area. © 2012 AmericanInstitute of Physics. [http://dx.doi.org/10.1063/1.3690497]

I. INTRODUCTION

The peculiar thermodynamic behavior of supercooledwater continues to receive considerable attention, despiteseveral decades of work in this field. Upon supercooling,water exhibits an anomalous increase of its isobaric heatcapacity and its isothermal compressibility, and an anomalousdecrease of its expansivity coefficient.1 One explanation ofthese anomalies, originally proposed by Poole et al.,2 isbased on the presumed existence of a liquid–liquid criticalpoint (LLCP) in water in the deeply supercooled region. Thehypothesis of the existence of a critical point in metastablewater has been considered by many authors, as recentlyreviewed by Bertrand and Anisimov.3 In particular, severalauthors have made attempts to develop a thermodynamicmodel for the thermodynamic properties of supercooledwater based on the LLCP scenario.3–8 The existence of aliquid–liquid critical point in supercooled water is still beingdebated, especially in view of some recent simulations.9, 10

The purpose of this article is to demonstrate that a theoreticalmodel based on the presumed existence of a second criticalpoint in water is capable of representing, accurately and con-sistently, all available experimental thermodynamic-propertydata for supercooled ordinary and heavy water. While beingconceptually close to the previous works by Fuentevilla andAnisimov7 and Bertrand and Anisimov,3 this work is the firstcomprehensive correlation of thermodynamic properties ofsupercooled water, incorporating noncritical backgrounds ina thermodynamically consistent way.

This article is organized as follows. In Sec. II, we reviewthe currently available experimental information for the ther-modynamic properties of supercooled water. We also provide

a)Present address: Department of Nuclear Science and Engineering,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,USA.

b)Author to whom correspondence should be addressed. Electronic mail:anisimov@umd.edu.

an assessment of the IAPWS-95 formulation11, 12 for the ther-modynamic properties of H2O (designed for water at temper-atures above the melting temperature) when extrapolated intothe supercooled region, confirming the practical need for animproved equation of state for supercooled water. In Sec. III,we formulate a thermodynamic model for supercooled waterby adopting suitable physical scaling fields with respect to thelocation of a liquid–liquid critical point in supercooled water.This model only covers supercooled water above the homoge-neous ice nucleation temperature; that is, it does not describeglassy water, which exists below about 150 K.13 In Sec. IV,we show that this theoretical model yields an accurate repre-sentation of the thermodynamic property data of supercooledH2O up to pressures of 150 MPa. In Sec. V, we present aless-restricted phenomenological extension of the theoreticalmodel and show that this extension allows a representation ofall currently available experimental data for supercooled H2Oup to the pressure of 400 MPa. The article concludes with adiscussion of the results and of some unresolved theoreticalissues in Secs. VI and VII.

II. REVIEW OF EXPERIMENTAL DATA

Experimental data on the properties of supercooled wa-ter have been reviewed by Angell in 1982,14, 15 by Sato et al.in 1991,16 and by Debenedetti in 2003.1 Therefore, in thisreview, we focus on data published after 2003. For a com-plete overview of the older data the reader is referred to theearlier reviews. We restrict ourselves to bulk thermodynamicproperties; specifically, we do consider properties measuredin emulsions and capillary tubes, but exclude experiments innanopores. Besides reviewing experiments, we also assess theperformance of the current reference correlation for the prop-erties of water and steam, the “IAPWS Formulation 1995 forthe Thermodynamic Properties of Ordinary Water Substancefor General and Scientific Use,” or IAPWS-95 for short,11, 12

0021-9606/2012/136(9)/094507/18/$30.00 © 2012 American Institute of Physics136, 094507-1

094507-2 Holten et al. J. Chem. Phys. 136, 094507 (2012)

(a)0

100

200

300

400

Pre

ssur

e(M

Pa)

180 200 220 240 260 280

Temperature (K)

Hare and Sorensen (1987)Sotani et al. (2000)

Asada et al. (2002)Guignon et al. (2010)

(b)

180 200 220 240 260 280

Temperature (K)

Speedy and Angell (1976)Kanno and Angell (1979)

Ter Minassian et al. (1981)Mishima (2010)

FIG. 1. (a) Location of the experimental H2O density data.19–23 The thick solid curve is the solid–liquid phase boundary,24 the dashed curve is the homogeneousice nucleation limit,25, 26 and the thin solid curves are the solid–solid phase boundaries.27 The location of the dashed curve above 300 MPa is uncertain. At0.1 MPa and in the stable-liquid region, data from several older sources have been omitted for clarity. (b) Location of the experimental H2O density-derivativedata. Ter Minassian et al.28 have measured the expansivity coefficient, other authors22, 29, 30 have measured the isothermal compressibility.

when extrapolated into the supercooled region. Such an as-sessment has been carried out before,12, 17 but not with allproperty data that are now available. Most of the data dis-cussed here are provided in tabular form in the supplementarymaterial of this article.18 For supercooled heavy water, thereare fewer data than for ordinary water. A review of the datafor heavy water is in the supplementary material.18

A. Density

Since the review by Debenedetti,1 new data for thedensity of supercooled water have been reported.21–23 Mostnotable is the recent work of Mishima,22 who measuredthe density and compressibility down to 200 K and up to400 MPa; see Fig. 1(a). More accurate density measurementshave been published by Sotani et al.,20 Asada et al.,21 andGuignon et al.,23 but their lowest temperature is 253 K, so ina larger temperature range Mishima’s data are the only dataavailable. The data of Guignon et al. are in the same range asthe data of Sotani et al. and Asada et al.; the maximum den-sity difference between the data is 0.25%, which is within theexperimental uncertainty.

At atmospheric pressure, the density of supercooled wa-ter has been measured by several experimentalists.1 We con-sider the measurements of Hare and Sorensen19 of 1987 themost accurate. They showed that their measurements were notaffected by the “excess density” effect, which occurs in thincapillary tubes and caused too large densities in their 1986experiments31 and in experiments of others.

A comparison of the densities calculated from theIAPWS-95 formulation with the experimental density dataof Hare and Sorensen,19 of Sotani et al.,20 and of Mishima22

is shown in Fig. 2. While the IAPWS-95 formulation repro-duces the experimental density data at ambient pressure, thedeviations from the formulation become larger and larger

with increasing pressure. Especially at higher pressures, thereis a sizable discrepancy between the IAPWS-95 formulationand the experimental data; the slope (or the expansivity)even has a different sign. It must be stressed that significantdifferences between IAPWS-95 and the experimental data

0.1

40

80

120

160

200

241

281

320

359

399P/MPa

TM

950

1000

1050

1100

1150

1200

Den

sity

(kg/

m3)

200 220 240 260 280 300

Temperature (K)

Hare and Sorensen (1987)Sotani et al. (2000)

Mishima (2010)IAPWS-95

FIG. 2. Densities according to IAPWS-95 (curves). IAPWS-95 is valid tothe right of the melting curve24 TM; the IAPWS-95 values left of the melt-ing curve are extrapolations. The symbols represent experimental data ofMishima,22 Sotani et al.,20 and Hare and Sorensen.19 Mishima’s data havebeen adjusted as described in the Appendix of the supplementary material.18

094507-3 Thermodynamics of supercooled water J. Chem. Phys. 136, 094507 (2012)

only occur when IAPWS-95 is extrapolated outside its rangeof validity. Furthermore, there were no high-pressure densitydata of supercooled water available when the IAPWS-95formulation was developed.

B. Derivatives of the density

Both the isothermal compressibility and the expansiv-ity coefficient of supercooled water have been measured; seeFig. 1(b). The most accurate compressibility data are fromKanno and Angell,30 whereas Mishima’s22 data cover thelargest temperature range. The only expansivity measure-ments are from Ter Minassian et al.28 Hare and Sorensen19, 31

also have published expansivities (at 0.1 MPa), but these wereobtained from the derivative of a fit to their density data.

According to Wagner and Pruß,12 the behavior ofthe expansivity coefficient calculated from the IAPWS-95formulation should be reasonable in the liquid region atlow temperature. However, from Fig. 3 we see that theIAPWS-95 expansivity exhibits deviations up to 50% in thelow-temperature region, even at temperatures above the melt-ing temperature where the IAPWS-95 formulation shouldbe valid. The isothermal compressibility calculated from theIAPWS-95 formulation agrees with the experimental datadown to about 250 K and up to 400 MPa, as shown in Fig. 4.However, at lower temperatures, the extrapolated IAPWS-95compressibilities do not even agree qualitatively with the data.

Exp

ansi

vity

104α V

(K1)

0.150100200380

P/MPa380 MPa

200 MPa

2

0

2

4

6

240 260 280 300

Ter Minassian et al. (1981)Hare and Sorensen (1986)Hare and Sorensen (1987)

0.1MPa

20

10

0

240 260 280 300

Temperature (K)

FIG. 3. Expansivity coefficient according to IAPWS-95 (solid curves: withinregion of validity, dashed curves: extrapolation). Symbols represent experi-mental data of Ter Minassian et al.28 and Hare and Sorensen.19, 31

0.11050

100150190

300400

P/MPa

TM

2

3

4

5

6

7

Com

pres

sibi

lity

104

T(M

Pa1)

200 220 240 260 280 300

Temperature (K)

Speedy and Angell (1976)Kanno and Angell (1979)

Mishima (2010)

FIG. 4. Isothermal compressibility according to IAPWS-95 (curves).IAPWS-95 is valid to the right of the melting curve TM; the IAPWS-95 valuesleft of the melting curve are extrapolations. Symbols represent experimentaldata of Speedy and Angell,29 Kanno and Angell,30 and Mishima.22 Solid andopen symbols with the same shape correspond to the same pressure.

C. Heat capacity

The isobaric heat capacity CP of supercooled water hasbeen measured only at atmospheric pressure.1 Old measure-ments by Anisimov et al.32 down to 266 K already showedan anomalous increase of the heat capacity at moderate su-percooling. In breakthrough experiments, Angell et al.33–35

extended the range of measurements down to 236 K, anddemonstrated that CP keeps increasing with decreasing tem-perature (Fig. 5). More recent measurements by Archer andCarter,36 also down to 236 K, do not perfectly agree with thoseof Angell et al.35 Archer and Carter suggest that the tempera-ture calibration procedure of Angell et al. might cause a sys-tematic error in their measurements. Furthermore, Archer andCarter suspect that measurements of Tombari et al.37 (downto 245 K) could be affected by even more significant system-atic calibration errors. The measurements of Bertolini et al.38

down to 247 K agree with those of Angell et al. (after correc-tion, as described in the supplementary material18).

CP

CV

4.0

4.5

5.0

5.5

6.0

CP

,CV

(kJ

kg1

K1)

240 260 280 300

Temperature (K)

IAPWS-95 CP

Angell et al. (1982)

Bertolini et al. (1985)

Tombari et al. (1999)

Archer and Carter (2000)

IAPWS-95 CV

FIG. 5. Isobaric and isochoric heat capacity at 0.1 MPa according to IAPWS-95 (curves). Symbols represent experimental data.35–38

094507-4 Holten et al. J. Chem. Phys. 136, 094507 (2012)

Because there are no high-pressure measurements of CP

in the supercooled region, we mention here some experimentsat high pressure P in the stable region. Sirota et al.39 mea-sured CP up to 100 MPa and down to 272 K. Recently, Manyàet al.40 have measured CP at 4 MPa from 298 K to 465 K.It turns out that the results of Manyà et al. imply that thederivative (∂CP/∂P)T at constant temperature T is positive forpressures lower than 4 MPa, which contradicts the thermo-dynamic relation (∂CP/∂P)T = −T(∂2V/∂T2)P, with V beingthe molar volume. Hence, the data of Manyà et al. will not beconsidered in this article.

To our knowledge, the isochoric heat capacity CV ofsupercooled water has not been measured. The values pre-sented by several investigators33, 34, 41 were calculated fromother thermodynamic properties.

The isobaric and isochoric heat capacities predicted byIAPWS-95 at 0.1 MPa are shown in Fig. 5. The CP curve com-puted from IAPWS-95 follows the data of Angell et al.,35 towhich it was fitted. Consequently, IAPWS-95 systematicallydeviates from the data of Archer and Carter,36 as discussedby Wagner and Pruß.12 The isochoric heat capacity CV doesnot exhibit appreciable anomalous behavior according to theIAPWS-95 prediction.

D. Surface tension

As listed by Debenedetti,1 the surface tension of super-cooled water with vapor and with air has been measured byHacker,42 Floriano and Angell,43 and Trinh and Ohsaka,44 alldown to about 250 K. Hacker’s measurements show an inflec-tion point at about 268 K, below which the surface tension hasa stronger temperature dependence (Fig. 6). The data of Flo-riano and Angell are less accurate, but the authors also notedan inflection. The measurements of Trinh and Ohsaka show asystematic deviation from the other data, but the trend agreeswith Hacker’s data. Furthermore, a molecular dynamics sim-ulation by Lü and Wei45 shows an inflection point as well.

IAPWS has recommended an equation to represent thesurface tension of liquid water in equilibrium with watervapor.46 For a comparison with experimental data, it is nec-

72

74

76

78

80

Sur

face

tens

ion

(mN

/m)

250 260 270 280 290 300

Temperature (K)

Hacker (1951)

Floriano and Angell (1990)

Trinh and Ohsaka (1995)

FIG. 6. Surface tension according to the IAPWS equation46 (solid curve:within region of validity, dashed curve: extrapolation). Symbols representexperimental data from Hacker,42 Floriano and Angell,43 and Trinh andOhsaka.44

essary to consider the conditions under which the surfacetension was measured. Hacker,42 Floriano and Angell,43 andTrinh and Ohsaka44 measured the surface tension at atmo-spheric pressure in air. The influence of air at atmosphericpressure on the surface tension of water is usually neglected,47

but for supercooled water the effect may be significant sincethe solubility of nitrogen and oxygen in water increaseswith decreasing temperature. From high-pressure data,48 it isknown that the surface tension of water with nitrogen or oxy-gen is lower than the pure-water surface tension. In Fig. 6,we show a comparison of the experimental surface-tensiondata with the values calculated from the IAPWS equation. Asnoted above, the experimental values of the surface tensionsuggest an inflection point at about 268 K. The extrapolationof the IAPWS equation does not show an inflection, and thedifference with Hacker’s accurate data42 increases with de-creasing temperature. The data of Floriano and Angell43 showmore scatter, but below the freezing point most of their data lieabove the IAPWS extrapolation. The measurements of Trinhand Ohsaka44 lie below all other data; the deviation is aboutthe same as their experimental uncertainty, which is 1%.

E. Speed of sound

The speed of sound in supercooled water has been mea-sured in a broad frequency range, from 54 kHz to about20 GHz. In stable water at atmospheric pressure, no signifi-cant dependence on the frequency is found in this range, butthe speed of sound in supercooled water shows a dispersionthat increases with cooling. In this article, we consider thethermodynamic (zero-frequency) limit of the speed of sound,which is associated with the adiabatic compressibility.14

Taschin et al.49 estimate that dispersion effects become no-ticeable at frequencies of 1 GHz and higher. We will thereforenot consider the measurements in the 1–10 GHz range ob-tained with Brillouin light scattering, which show a speed-of-sound minimum between 250 K and 280 K; the temperatureof the minimum increases with increasing frequency.50, 51 Thelower frequency (ultrasonic) measurements are shown in Fig.7. Below 255 K, there appears to be dispersion; however, thespeed of sound decreases with increasing frequency, whichsuggests an anomalous (negative) dispersion. While a nega-tive dispersion of the speed of sound very close to the vapor–liquid critical point is, in principle, possible due to the diver-gence of the thermal conductivity,52 there is not yet any exper-imental indication for an anomaly of the thermal conductivityof supercooled water.53, 54 Recent simulations of Kumar andStanley55 show a minimum for this property. Debenedetti1

argues that negative dispersion can be ruled out because theBrillouin experiments all show positive dispersion. Accordingto Taschin et al.,49 the apparent negative dispersion is a resultof systematic errors in the data of Trinh and Apfel56 and Bacriand Rajaonarison.57 The IAPWS-95 formulation agrees withthe recent data of Taschin et al.49 to within their uncertainty.

There are no measurements of the speed of sound ofsupercooled water at high pressure. In the stable region,Petitet et al.58 performed measurements at 10 MHz up to460 MPa, including the region close to the melting curve,

094507-5 Thermodynamics of supercooled water J. Chem. Phys. 136, 094507 (2012)

1.1

1.2

1.3

1.4

1.5

Spe

edof

soun

d(k

m/s

)

240 260 280

Temperature (K)

Trinh and Apfel (1978), 3 MHzBacri and Rajaonarison (1979), 925 MHzTrinh and Apfel (1980), 54 kHzTaschin et al. (2011), 140 MHz

FIG. 7. Speed of sound at 0.1 MPa. Solid curve: prediction of the IAPWS-95formulation. Dashed curve: model of Sec. IV; dotted curve: model of Sec. V.Symbols represent experimental data.49, 56, 57, 60, 61

down to 253 K. Petitet et al. have also published data down to253 K at atmospheric pressure, but these data were taken fromConde et al.59 Since Conde et al. measured at 5 GHz, thesedata do not represent the zero-frequency limit of the speed ofsound.

F. Spinodal

In the supercooled region, IAPWS-95 predicts a re-entrant liquid spinodal, as shown in Fig. 8. The spinodal pres-sure becomes positive at 233.6 K, which is a few degreesbelow the homogeneous nucleation limit. At about 195 Kand 175 MPa, the spinodal curve crosses the homogeneousnucleation limit and enters the region where supercooled wa-ter can be experimentally observed. Up to about 290 MPa,the spinodal curve stays in the experimentally accessiblerange. Since a spinodal has not been observed there, the

TH TM

IAPWS-95 spinodal

binodal

IAPWS-95 spinodal

C

200

100

0

100

200

300

Pre

ssur

e(M

Pa)

200 300 400 500 600

Temperature (K)

FIG. 8. Location of the liquid spinodal according to the IAPWS-95 for-mulation. The curved marked with TH is the homogeneous ice nucleationlimit,25, 26 and TM denotes the melting curve.24 Thin solid lines are the solid–solid phase boundaries.27 Also shown is the location of the binodal, the vaporpressure curve, and its extension into the metastable region. The vapor–liquidcritical point is denoted by C.

spinodal of IAPWS-95 contradicts experimental evidence.While a re-entrant spinodal is thermodynamically possible,Debenedetti has argued that the re-entrant spinodal sce-nario suggested by Speedy62 is implausible for supercooledwater.1, 63 Debenedetti’s argument may be summarized as fol-lows. In the pressure–temperature plane (Fig. 8), a re-entrantspinodal must intersect the metastable continuation of the bin-odal, the vapor-pressure curve. At the intersection, liquid wa-ter is simultaneously in equilibrium with water vapor and un-stable with respect to infinitesimal density fluctuations. Sincethese two conditions are mutually exclusive, such an intersec-tion cannot exist. On the other hand, at the vapor–liquid criti-cal point the binodal and spinodal do coincide, which is possi-ble because the vapor and liquid phases are indistinguishablethere.

G. Liquid–liquid coexistence curve

Both the existence of a second critical point and its lo-cation are still being debated in the literature. If the sec-ond critical point exists, there should be a liquid–liquid tran-sition (LLT) curve – separating a hypothetical high-densityliquid and low-density liquid – which ends at the criticalpoint. The assumed LLT curve extends upwards in pressurefrom the second critical point, not downwards as is the casefor the vapor–liquid critical point. At pressures below the crit-ical pressure, water’s response functions exhibit an extremumnear the “Widom line,” which is the extension of the LLTcurve into the one-phase region and the locus of maximumfluctuations of the order parameter.

While the location of the critical point obtained by dif-ferent simulations varies greatly, different attempts to locatethe LLT and the Widom line from experimental data haveyielded approximately the same result. Kanno and Angell30

fitted empirical power laws to their compressibility measure-ments and obtained singular temperatures located from 5 Kto 12 K below the homogeneous nucleation temperature TH

TH

0

50

100

150

200

Pre

ssur

e(M

Pa)

180 200 220 240

Temperature (K)

Kanno et al. (1975)Kanno and Angell (1979)Mishima and Stanley (1998)Kanno and Miyata (2006)Mishima (2010)Bertrand and Anisimov (2011)

FIG. 9. Homogeneous ice nucleation temperatures (open circles25 andsquares26 and fitted solid curve), Mishima’s22 conjectured liquid–liquid co-existence curve (dotted) and liquid–liquid critical point (large solid diamond),and Kanno and Angell’s curve (dashed) connecting the fitted singular temper-atures (solid circles).30 Open diamond: bend in the melting curve of ice IV;64

large solid square: the liquid–liquid critical point suggested by Bertrand andAnisimov.3

094507-6 Holten et al. J. Chem. Phys. 136, 094507 (2012)

(Fig. 9), suggesting a LLT that mimics the TH curve butshifted to lower temperature. Mishima measured metastablemelting curves of H2O ice IV (Ref. 64) and D2O ices IV andV,65 and found that they suddenly bent at temperatures of 4 Kto 7 K below TH. According to Mishima, this is indirect evi-dence for the location of the LLT, but a one-to-one correspon-dence between a break in the melting curve and the LLT hasbeen questioned by Imre and Rzoska.66

Mishima22 approximated the LLT by a quadratic functionof T with approximately the same shape as the TH curve, butallowing a shift to lower temperature. His final result, shownin Fig. 9, is close to Kanno and Angell’s curve for pressuresup to 100 MPa, albeit with a different curvature.

III. THERMODYNAMIC MODEL FORSUPERCOOLED WATER

In this section, we further develop a scaling model forbulk thermodynamic properties supercooled water, which wasearlier suggested by Fuentevilla and Anisimov7 and, more re-cently, modified and elaborated by Bertrand and Anisimov.3

A. Scaling fields and thermodynamic properties

Fluids belong to the universality class of Ising-like sys-tems whose critical behavior is characterized by two indepen-dent scaling fields, a “strong” scaling field h1 (ordering field)and a “weak” scaling field h2, and by a dependent scaling fieldh3 which asymptotically close to the critical point becomes ageneralized homogeneous function of h1 and h2,67–69

h3(h1, h2) ≈ |h2|2−α f ±(

h1

|h2|2−α−β

). (1)

In this expression α � 0.110 and β � 0.326 are universal crit-ical exponents70, 71 and f±, with the superscripts ± referringto h2 > 0 and h2 < 0, is a universal scaling function exceptfor two system-dependent amplitudes. Associated with thesescaling fields are two conjugate scaling densities, a stronglyfluctuating scaling density φ1 (order parameter) and a weaklyfluctuating scaling density φ2, such that

dh3 = φ1 dh1 + φ2 dh2 (2)

with

φ1 =(

∂h3

∂h1

)h2

, φ2 =(

∂h3

∂h2

)h1

. (3)

In addition, one can define three susceptibilities, a “strong”susceptibility χ1, a “weak” susceptibility χ2, and a “cross”susceptibility χ12,

χ1 =(

∂φ1

∂h1

)h2

, χ2 =(

∂φ2

∂h2

)h1

, (4)

χ12 =(

∂φ1

∂h2

)h1

=(

∂φ2

∂h1

)h2

. (5)

In fluids and fluid mixtures one encounters a large variety ofdifferent types of critical phenomena.72 The asymptotic ther-modynamic behavior near all kinds of critical points can bedescribed in terms of Eq. (1). The differences arise from the

actual relationships between the scaling fields and the phys-ical fields,73 subject to the condition that at the critical point

h1 = h2 = h3 = 0. (6)

In one-component fluids the relevant physical fields are thechemical potential μ (Gibbs energy per mole), the tempera-ture T, and the pressure P. To satisfy condition (6), one defines�μ = μ − μc, �T = T − Tc, and �P = P − Pc. In this ar-ticle, we adopt the usual convention that a subscript c refersto the value of the property at the critical point. There are twospecial models for critical behavior that deserve some atten-tion. The first is the lattice gas in which the ordering field h1 isasymptotically proportional to �μ and the weak scaling fieldproportional to �T.74–76 Hence, in the lattice gas φ1 is propor-tional to �ρ = ρ − ρc and φ2 is proportional to �s = s − sc,where ρ is the molar density and s is the entropy density. Thelattice gas provides a model for the vapor–liquid critical pointwhere the molar density yields the major contribution to theorder parameter. In practice, the asymptotic critical behaviorof a fluid near the vapor–liquid critical point, including that ofH2O,77 can be described by a slight modification of the lattice-gas model to account for some lack of vapor–liquid symmetryin real fluids. Another special model, introduced by Bertrandand Anisimov,3 is a “lattice liquid” in which the ordering fieldis asymptotically proportional to �T and in which the weakscaling field is proportional to �μ.3 Near the liquid–liquidcritical point in weakly compressible supercooled water theentropy yields the major contribution to the order parameterand not the mass density, as first pointed out by Fuentevillaand Anisimov.7 Thus the thermodynamic properties near thisliquid–liquid critical point can be described by a slight mod-ification of the lattice-liquid model to account for some lackof symmetry in the order parameter.3

To implement a scaled thermodynamic representation itis convenient to make all thermodynamic properties dimen-sionless in terms of the critical parameters Tc and ρc orVc = ρ−1

c ,

T = T

Tc, μ = μ

RTc, P = PVc

RTc, (7)

where R is the molar gas constant. For the dimensionlessphysical densities we define

V = V

Vc, S = S

R, CP = CP

R, (8)

where V is the molar volume, S is the molar entropy, andCP is the isobaric molar heat capacity. The thermodynamicmodel of Bertrand and Anisimov was formulated in terms ofP (μ, T ) for which

dP = V −1dμ + V −1S dT . (9)

We have found it more convenient, and practically equivalent,to formulate this thermodynamic model in terms of μ(P , T )for which

dμ = V dP − S dT . (10)

Thus in this formulation, similar to that suggested earlier byFuentevilla and Anisimov,7 we identify the order parameter

094507-7 Thermodynamics of supercooled water J. Chem. Phys. 136, 094507 (2012)

with the entropy itself instead of the entropy density. In ourmodel the scaling fields are related to the physical fields as

h1 = �T + a′�P , (11)

h2 = −�P + b′�T , (12)

h3 = �P − �μ + �μr (13)

with

�T = T − Tc

Tc, �P = (P − Pc)Vc

RTc, �μ = μ − μc

RTc.

(14)In Eq. (11) a′ represents the slope −dT /dP of the phase-coexistence or Widom line at the critical point. In Eq. (12)b′ is a so-called mixing coefficient which accounts for thefact that the critical phase transition in supercooled water isnot completely symmetric in terms of the entropy order pa-rameter. Introduction of mixing of this type is also knownin the literature as the revised-scaling approximation.78

Equation (1) only represents the asymptotic behavior of theso-called singular critical contributions to the thermodynamicproperties. To obtain a complete representation of the thermo-dynamic properties we need to add a regular (i.e., analytic)background contribution. As has been common practice indeveloping scaled equations of state in fluids near the vapor–liquid critical point,77, 78 the regular background contributionis represented by a truncated Taylor-series expansion aroundthe critical point

�μr =∑m, n

cmn(�T )m(�P )n with c00 = c10 = c01 = 0.

(15)The first two terms in the temperature expansion of �μr de-pend on the choice of zero entropy and energy and do notappear in the expressions of any of the physically observ-able thermodynamic properties. Hence, these coefficients maybe set to zero. Furthermore, the coefficient c01 = Vc − 1 = 0.Strictly speaking, critical fluctuations also yield an analyticcontribution to h3.79, 80 In this article, we incorporate this con-tribution into the linear background contribution as has alsobeen done often in the past.

The current treatment of the background contributiondiffers from that in earlier publications.3, 7 Previously, atemperature-dependent background was added to the criticalpart of each property. Because the background of each prop-erty was treated separately, the resulting property values werenot mutually consistent. In this work, the background is addedto the chemical potential, and the backgrounds in the derivedproperties follow, ensuring thermodynamic consistency.

From the fundamental thermodynamic differentialrelation (10) it follows that

V =(

∂μ

∂P

)T

= 1 − a′φ1 + φ2 + �μrP, (16)

S = −(

∂μ

∂T

)P

= φ1 + b′φ2 − �μrT. (17)

In this article, we adopt the convention that a subscript P in-dicates a derivative with respect to P at constant T and a sub-script T a derivative with respect to T at constant P . Finally,the dimensionless isothermal compressibility κT , expansivitycoefficient αV , and isobaric heat capacity CP can be expressedin terms of the scaling susceptibilities χ1, χ2, and χ12,

κT = − 1

V

(∂V

∂P

)T

= 1

V

[(a′)2χ1 + χ2 − 2a′χ12 − �μr

P P

],

(18)

αV = 1

V

(∂V

∂T

)P

= 1

V

[ − a′χ1 + b′χ2 + (1 − a′b′)χ12 + �μrT P

], (19)

CP = T

(∂S

∂T

)P

= T[χ1 + (b′)2χ2 + 2b′χ12 − �μr

T T

].

(20)

B. Parametric equation of state

It is not possible to write the scaled expression (1) forh3 as an explicit function of h1 and h2. Such attempts alwayscause singular behavior of the thermodynamic potential in theone-phase region either at h1 = 0 or at h2 = 0. This problem issolved by replacing the two independent scaling fields, h1 andh2, with two parametric variables: a variable r which measuresa “distance” from the critical point and an angular variable θ

which measures the location on a contour of constant r. Atransformation most frequently adopted has the form

h1 = ar2−α−βθ (1 − θ2), h2 = r(1 − b2θ2). (21)

From Eqs. (1) and (3) it then follows that the order parameterφ1 must have the form81

φ1 = krβM(θ ), (22)

where M(θ ) is a universal analytic function of θ . In princi-ple, this function can be calculated from the renormalization-group theory of critical phenomena.82 In practice, one adoptsan analytic approximant for M(θ ), the simplest one beingM(θ ) = θ ,83

φ1 = krβθ. (23)

Equations (21) and (23) define what is known as the “linearmodel” parametric equation of state. In these equations a andk are two system-dependent amplitudes related to the twosystem-dependent amplitudes in Eq. (1), while b2 is a uni-versal constant which is often approximated by84

b2 = 2 − α − 4β

(2 − α − 2β)(1 − 2β)� 1.361. (24)

Equations (21) and (23) with the specific choice (Eq. (24)) forb2 is known as the “restricted” linear model.76 The resultingparametric equations for the various thermodynamic proper-ties can be found in many publications.76–78, 80, 85, 86 In this ar-ticle, we are using the “restricted” linear model to describe

094507-8 Holten et al. J. Chem. Phys. 136, 094507 (2012)

the supercooled-water anomalies. The parametric equationsneeded for the analysis in this article are listed in the Ap-pendix of the supplementary material.18

IV. COMPARISON WITH EXPERIMENTAL DATA

The scaling theory, formulated in Sec. III, represents thethermodynamic behavior asymptotically close to the criticalpoint. Specifically, the liquid–liquid (LLT) curve that followsfrom Eq. (11) is a straight line, while the actual LLT curveshould exhibit curvature (as shown in Fig. 9). In this section,we investigate the thermodynamic properties of supercooledliquid water in a range of pressures and temperatures, wherethe asymptotic theory appears to be adequate. Issues relatedto nonasymptotic features of the scaling theory will be ad-dressed in Sec. V. Hence, we restrict the asymptotic theoreti-cal model to pressures not exceeding 150 MPa, where the LLTcan be reasonably approximated by a single straight line. Theslope of the LLT was constrained to values that are close to theslopes of the curves of Kanno and Angell30 and Mishima22 inthe range of 0 MPa to 150 MPa. Specifically, the value of a′ inEq. (11) was restricted to the range of 0.065 to 0.090. Becausethe position of the LLT is not precisely known, the criticalpoint was allowed to deviate up to 3 K from Mishima’s curve.It was found that the results of the model were rather insen-sitive to the critical pressure Pc, so Pc was constrained to thevalue of 27.5 MPa obtained by Bertrand and Anisimov.3 Weshall further discuss this issue in Sec. VI B. It was also foundthat a nonzero mixing coefficient b′ did not significantly im-prove the fit, so b′ was set to zero. This means physically thatthe liquid-critical behavior in supercooled water exhibits littleasymmetry in the order parameter and is indeed very close tolattice-liquid behavior.

Changes in the third decimal place of the values of thecritical exponents α and β result in small density changesthat are of the order of 0.1%. However, some of the densitymeasurements for water are more accurate than 0.1%; for ex-ample, the accuracy of the data of Hare and Sorensen19 andSotani et al.20 is 0.01%. Therefore, the values of the criticalexponents must be given with at least four decimal places. Wehave adopted the values of Pelissetto and Vicari70 and have setα = 0.1100 and β = 0.3265. The value for the molar mass ofH2O (18.015 268 g/mol) was taken from Wagner and Pruß12

and the molar gas constant R (8.314 4621 J mol−1 K−1) wastaken from Mohr et al.87

The number of terms in the background �μr [Eq. (15)]was increased step by step until the experimental data couldbe accurately represented. The final background contains 14free parameters. The reason for the many background terms ofhigher order in temperature and pressure is that the responsefunctions are second derivatives of the thermodynamic poten-tial. To obtain, for example, a background term in the com-pressibility of second order in pressure, it is necessary to havea fourth-order pressure-dependent term in the potential. Theterms in the backgrounds for each property are at most thirdorder in temperature or in pressure. We want to emphasizethat the observed anomalies are indeed due to the critical partof the equation of state since the nonlinear contributions tothe regular part are needed only when the maximum pressure

considered is higher than about 100 MPa. Besides the back-ground parameters, there are five additional parameters to bedetermined: the critical temperature Tc and volume Vc, thelinear-model amplitudes a and k, and the slope of the LLTline a′. As noted, the values of Tc and a′ were constrained toa limited range.

The model was fitted to heat-capacity data of Archerand Carter36 and IAPWS-95, expansivity data of Hare andSorensen,19 IAPWS-95 and Ter Minassian,28 compressibil-ity data of Speedy and Angell,29 Kanno and Angell30 andMishima,22 density data of Hare and Sorensen,19 Sotaniet al.,20 IAPWS-95 and Mishima,22 and speed-of-sound dataof Taschin et al.49 We have made small adjustments to the dataof Mishima as described in the supplementary material.18 Forall quantities except the heat capacity, values from IAPWS-95 were only used at 0.1 MPa and above 273 K. For the heatcapacity, IAPWS-95 values in the range of 273 K–305 K upto 100 MPa were used. (We considered heat-capacity valuescalculated from IAPWS-95 more reliable than high-pressureheat-capacity data of Sirota et al.39) To reduce the time neededfor optimization, not all data points were used in the fittingprocess; about 60 points were selected for each of the quan-tities heat capacity, expansivity, compressibility, and density.The selected data are given in the supplementary material.18

The model was optimized by minimizing the sum ofsquared residuals, where the residual is the difference be-tween model and experiment, divided by the experimentaluncertainty.88 For some data the uncertainty was not givenand had to be estimated. The resulting optimized parametersare listed in Table I. The value of a′ for H2O is exactly 0.09because a′ was restricted to the range of 0.065–0.090, and theoptimum is located at the edge of this range. The fitted modelsare valid up to 300 K and from 0 to 150 MPa.

As can be seen in Fig. 10, the model represents the ex-perimental density data well. The density jumps at low tem-perature because the isobars cross the LLT curve there. InFig. 11, the temperature of maximum density is plotted as afunction of pressure, both for the model and for an extrapola-tion of the IAPWS-95 formulation. At pressures higher thanabout 60 MPa, the extrapolation of IAPWS-95 deviates fromthe experimental data, while the current model agrees with thedata (except at negative pressures, where the model was not

TABLE I. Parameter values for H2O in the model of Sec. IV.

Parameter Valuea Parameter Value

Tc/K 224.23 c11 1.536 3 × 10−1

Pc/MPa 27.5 c12 − 6.487 9 × 10−3

ρc/(kg m−3) 948.77 c13 7.709 0 × 10−3

a 0.229 24 c20 − 3.888 8 × 100

k 0.377 04 c21 1.734 7 × 10−1

a′ 0.090 c22 − 6.415 7 × 10−2

c02 7.177 9 × 10−2 c23 − 6.985 0 × 10−3

c03 − 4.093 6 × 10−4 c30 6.981 3 × 10−1

c04 − 1.099 6 × 10−3 c31 − 1.145 9 × 10−1

c05 2.949 7 × 10−4 c32 7.500 6 × 10−2

aFinal digits of parameter values are given to allow reproducing the values of propertieswith the model but do not have physical significance.

094507-9 Thermodynamics of supercooled water J. Chem. Phys. 136, 094507 (2012)

0.120406080100120140

P/MPa

950

1000

1050D

ensi

ty(k

g/m

3)

200 220 240 260 280 300

Temperature (K)

Hare and Sorensen (1987)

Sotani et al. (2000)

Mishima (2010)

FIG. 10. Densities of H2O according to the model (curves). The symbolsrepresent experimental data of Mishima,22 Sotani et al.,20 and Hare andSorensen.19 The symbols for Mishima’s densities on different isobars are al-ternatingly open and filled to guide the eye.

fitted to any experimental data). The curve of the temperatureof maximum density does not intersect the homogeneous nu-cleation curve at its break point at about 200 MPa and 180 K.Such an intersection was expected by Dougherty.92

The agreement of the compressibility data with valuesof the model is shown Fig. 12. An interesting feature is theintersection of the isobars of 0.1 MPa and 10 MPa at about250 K. The experimental data do not confirm or rule out suchan intersection because of the scatter and the lack of data be-low 245 K. However, the intersection implies that the pressurederivative of the compressibility, (∂κT/∂P)T, is positive at lowtemperature and ordinary pressures.

TM

TH

0

50

100

150

Pre

ssur

e(M

Pa)

200 220 240 260 280

Temperature (K)

This workThis workIAPWS-95Caldwell (1978)

Ter Minassian et al. (1981)Henderson and Speedy (1987)Sotani et al. (2000)Mishima (2010)

FIG. 11. Temperature of maximum density of H2O as a function of pressureaccording to the model of Sec. IV (thick solid curve), the model of Sec. V(dashed), and IAPWS-95 (dotted). To the left of the melting curve TM,24, 89

the IAPWS-95 formulation is extrapolated outside its range of validity. THdenotes the homogeneous nucleation limit.25 Symbols represent experimen-tal data.20, 22, 28, 90, 91 The temperatures of maximum density for Mishima’sdata22 were determined by locating the maxima of fits to his density data.

0.11050100

150

P/MPa

3

4

5

6

7

8

200 220 240 260 280 300

Temperature (K)

Speedy and Angell (1976)Kanno and Angell (1979)

Mishima (2010)C

ompr

essi

bili

ty10

4T

(MPa

1)

0.11050100150

P/MPa

0

5

10

15

200 220 240 260 280 300

FIG. 12. Isothermal compressibility of H2O according to the model (curves).For clarity, the curves are not shown for temperatures below the LLT linein the bottom graph. Symbols represent experimental data of Speedy andAngell,29 Kanno and Angell,30 and Mishima.22 Solid and open symbols withthe same shape correspond to the same pressure.

Figure 13 shows experimental data for the expansivitycoefficient and the values predicted by the model for five pres-sures. The model follows the experimental data, contrary tothe extrapolation of the IAPWS-95 formulation. At 240 K,where the difference between Hare and Sorensen’s data of1986 and 1987 is largest, the expansivity predicted by themodel lies between them.

Heat-capacity data are compared with the model’s pre-dictions in Figs. 14 and 15. In Fig. 14, it is seen that the modelfollows the data of Archer and Carter,36 whereas IAPWS-95 follows the data of Angell et al.,35 to which it was fit-ted. However, the curvature of the 0.1 MPa isobar of themodel is slightly higher than that suggested by the data ofArcher and Carter. Murphy and Koop93 proposed a heat-capacity curve with a broader peak than that of our model, butwith about the same maximum value. The predicted isochoricheat capacity CV diverges, as will be discussed in Sec. VI C.Figure 15 shows the heat capacity as a function of pressure.There is a systematic difference between the data of Sirotaet al.39 in the stable region and the values of IAPWS-95, andthe data of Sirota et al. were not selected for the fit of the cur-rent model. At 250 K and pressures above about 50 MPa, themodel predicts a smaller pressure dependence of the heat ca-pacity than IAPWS-95. The pressure dependence of the heatcapacity is thermodynamically related to the expansivity co-efficient, and we have seen that the extrapolated expansivity

094507-10 Holten et al. J. Chem. Phys. 136, 094507 (2012)

0.13060100150

P/MPaE

xpan

sivi

ty10

4α V

(K1)

2

0

2

4

220 240 260 280 300

0.1 MPa

30

60100150

30

20

10

0

220 240 260 280 300

Temperature (K)

Ter Minassian et al. (1981)

Hare and Sorensen (1986)

Hare and Sorensen (1987)

FIG. 13. Expansivity coefficient of H2O according to the model (solidcurves) and IAPWS-95 (dashed: within region of validity, dotted: extrapo-lations). Symbols represent experimental data of Ter Minassian et al.28 andHare and Sorensen.19, 31

coefficient of IAPWS-95 does not agree with experimen-tal data at low temperature and high pressure (see Fig. 13).Therefore, differences between the heat-capacity values of thecurrent model and IAPWS-95 are to be expected.

The speed of sound predicted by the model is shown inFig. 7. The prediction agrees fairly well with the experimentaldata. At 240 K, the model shows a minimum in the speed ofsound, whereas IAPWS-95 predicts a monotonically decreas-ing speed of sound with decreasing temperature. The diver-

CP

CV

CV

3

4

5

6

7

8

CP

,CV

(kJ

kg1

K1)

200 220 240 260 280 300

Temperature (K)Angell et al. (1982) Archer and Carter (2000)

FIG. 14. Isobaric and isochoric heat capacity of H2O versus temperature at0.1 MPa according to the model (solid curves), IAPWS-95 (dashed), and theprediction of Murphy and Koop (dotted). Symbols represent experimentaldata of Angell et al.35 and Archer and Carter.36 The dashed-dotted line is theestimated regular part of CP. The predicted thermodynamic behavior of CV

is discussed in Sec. VI C.

273 K288 K306 K

250 K250 K3.8

4.0

4.2

4.4

CP

(kJ

kg1

K1)

0 50 100 150

Pressure (MPa)

This work

IAPWS-95

Sirota et al. (1970):

306 K

288 K

273 K

FIG. 15. Isobaric heat capacity of H2O versus pressure according to themodel (solid curves) and IAPWS-95 (dashed). Symbols represent experimen-tal data of Sirota et al.39

gence of the speed of sound of the model at low temperatureis related to a stability limit as will be discussed in Sec. VI C.

We have also fitted the model to experimental data onheavy water. The quality of the description of heavy water issimilar to that of ordinary water. Details of the fit and graphsare presented in the supplementary material.18

We have also compared the predictions of the model withexperimental data obtained for water confined in nanopores.Ice formation is suppressed in nanoscale confinement, thusenabling one to study liquid water below the temperatureof bulk spontaneous nucleation TH. However, the propertiesof water in such confinement are very different from those ofbulk water, being strongly affected by the geometry of con-finement and the specifics of liquid–surface interaction. Nev-ertheless, a comparison with the properties of confined wa-ter may provide a new insight into the thermodynamics ofbulk supercooled water. As we discuss in the supplementarymaterial,18 many of the results of experiments on confinedwater can still be interpreted in the framework of the secondcritical point hypothesis.

V. SEMI-EMPIRICAL EXTENSIONOF SCALING MODEL

As mentioned in Sec. IV, the asymptotic theoreticalmodel implies a linear LLT line. An attempt to include anadditional term accounting for curvature of the LLT curve hasbeen made by Fuentevilla and Anisimov7 by introducing apressure-dependent coefficient a′ in Eq. (11). However, sucha procedure yields terms proportional to φ1 in some responsefunctions which do not vanish far away from the critical point,where the critical part should not play a role anymore. Inprinciple, this problem can be solved by including crossoverfrom singular critical behavior asymptotically close to thecritical point to analytic behavior far away from the criticalpoint, as has been done in the equation of state for H2O nearthe vapor–liquid critical point,94 and has also been suggestedby Kiselev.5, 6 However, it turns out that the theoretical modelcan represent all experimental data for H2O up to the max-imum available pressure of 400 MPa, if we simply remove

094507-11 Thermodynamics of supercooled water J. Chem. Phys. 136, 094507 (2012)

TABLE II. Parameter values for the extended model of H2O.

Parameter Value Parameter Value

Tc/K 213.89 c12 − 4.656 9 × 10−3

Pc/MPa 56.989 c13 2.362 7 × 10−3

ρc/(kg m−3) 949.87 c14 − 2.869 7 × 10−4

a 0.116 24 c20 − 3.614 4 × 100

k 0.432 80 c21 − 1.500 9 × 10−2

a′ 0.108 98 c22 − 2.460 9 × 10−2

c02 4.079 3 × 10−2 c23 9.867 9 × 10−4

c03 − 6.791 2 × 10−4 c30 5.426 7 × 10−1

c04 − 7.566 9 × 10−6 c31 1.062 0 × 10−1

c05 1.092 2 × 10−5 c32 1.275 9 × 10−2

c11 1.954 7 × 10−1 c41 − 7.997 0 × 10−2

any constraints on the slope of the LLT and on the criticalparameters in the unstable region which cannot be measuredexperimentally. With the addition of only two backgroundterms, we were then able to fit almost all experimental datafor supercooled water with this semi-empirical extension ofthe theoretical model. This extension does not supersede thedescription limited to pressures up to 150 MPa, as discussedabove, because the data above 150 MPa are so far away fromthe asymptotic critical region that the thermodynamic prop-erties are dominated by the regular noncritical backgrounds.Moreover, the liquid–liquid transition line (which is locatedin the experimentally inaccessible, unstable region) that isimposed by this extension seems to be located away fromplausible predictions.

After fitting the model to a selection of experimentaldata18 at pressures up to 400 MPa, the parameters listed inTable II were obtained. Some parameters are significantlydifferent from the earlier values. In particular, the criticalpressure is about a factor of two higher than in Sec. IV, andthe critical temperature is 10 K lower. Nevertheless, the ex-tended model does represent almost all available experimentalthermodynamic data for supercooled water. A comparisonbetween the density values predicted by the model is pre-sented in Fig. 16. The model reproduces the data, exceptfor Mishima’s points between 160 MPa and 300 MPa below230 K. The temperature of maximum density is plotted inFig. 11. The calculated behavior of the extended model issimilar to that of the asymptotic model in Sec. IV in the rangefrom 0 MPa to 120 MPa. At higher pressures, the extendedmodel predicts higher temperatures of maximum densitythan the previous model, but the results are still within theexperimental uncertainty.

Compressibility data are compared with values of the ex-tended model in Fig. 17. As in the previous model, the isobarsof 0.1 MPa and 10 MPa intersect, but the intersection is lo-cated at a lower temperature. The data of Speedy and Angell29

and Kanno and Angell30 are well represented. The extendedmodel also reproduces most of Mishima’s data, which havea lower accuracy than the data of Angell and co-workers.Figure 18 shows the expansivity coefficient predicted by theextended model, which agrees with the data of Ter Minassianet al.28 Below 250 K, the model agrees better with the data ofHare and Sorensen of 1986 than with their data of 1987.

0.1

20

40

60

80100

120

140160180200219241260281300320340359380399

P/MPa

950

1000

1050

1100

1150

1200

Den

sity

(kg/

m3)

200 220 240 260 280 300

Temperature (K)

Hare and Sorensen (1987)Sotani et al. (2000)Mishima (2010)

FIG. 16. Densities of H2O according to the extended model (curves). Thesymbols represent experimental data of Mishima,22 Sotani et al.,20 and Hareand Sorensen.19 The symbols for Mishima’s densities on different isobarsare alternatingly open and filled to guide the eye. The vertical lines throughMishima’s points are uncertainties given by Mishima.

0.11050100150190300400

P/MPa

0

2

4

6

8

10

12

Com

pres

sibi

lity

104

T(M

Pa1)

Speedy and Angell (1976)Kanno and Angell (1979)

Mishima (2010)

10

8

6

200 220 240 260 280 300

Temperature (K)

FIG. 17. Isothermal compressibility of H2O according to the extended model(curves). Symbols represent experimental data of Speedy and Angell,29

Kanno and Angell,30 and Mishima.22 Solid and open symbols with the sameshape correspond to the same pressure. The regular backgrounds for the com-pressibility are shown by dashed-dotted lines. The negative values of the reg-ular parts are due to incorporating the fluctuation-induced critical backgroundin χ2 as noted in the supplementary material.18

094507-12 Holten et al. J. Chem. Phys. 136, 094507 (2012)

0.1 MPa

50

100

160240380

30

20

10

0

10

Exp

ansi

vity

104α V

(K1)

220 240 260 280 300

Temperature (K)

Ter Minassian et al. (1981)Hare and Sorensen (1986)Hare and Sorensen (1987)

FIG. 18. Expansivity coefficient of H2O to the extended model (curves).Symbols represent experimental data of Ter Minassian et al.28 and Hareand Sorensen.19, 31 The dashed-dotted line is the estimated regular part at0.1 MPa.

The calculated heat capacity at 0.1 MPa is compared withexperimental data in Fig. 19. For the extended model, themaximum of the heat capacity is lower than for the asymptoticmodel, and the curvature of the data of Archer and Carter36

is better represented. The predicted heat capacity is shownas a function of pressure in Fig. 20. There are large differ-ences between the results of the model and those of IAPWS-95; the model predicts a minimum in the 250 K heat-capacityisotherm at about 240 MPa. A minimum in the heat capacityat this location was also predicted by Ter Minassian et al.28

based on their measurements of the expansivity coefficient.The speed of sound predicted by the model, shown in

Fig. 7, agrees fairly well with the experimental data ofTaschin et al.49

VI. PHYSICAL INTERPRETATION OF THE MODEL

A. Two states in supercooled water

Two features make the second critical point in water phe-nomenologically different from the well-known vapor–liquid

CP

CV

CV

3

4

5

6

7

8

CP

,CV

(kJ

kg1

K1)

200 220 240 260 280 300

Temperature (K)Angell et al. (1982) Archer and Carter (2000)

FIG. 19. Isobaric and isochoric heat capacity of H2O versus temperatureat 0.1 MPa according to the extended model (solid curves), IAPWS-95(dashed), and the prediction of Murphy and Koop (dotted). Symbols representexperimental data of Angell et al.35 and Archer and Carter.36 The dashed-dotted line is the estimated regular part of CP. The predicted thermodynamicbehavior of CV is discussed in Sec. VI C.

273 K288 K

306 K

250 K3.8

4.0

4.2

4.4

CP

(kJ

kg1

K1)

0 100 200 300 400

Pressure (MPa)

This workIAPWS-95

Sirota et al. (1970):306 K288 K273 K

FIG. 20. Isobaric heat capacity of H2O versus pressure according to the ex-tended model (solid curves) and IAPWS-95 (dashed). Symbols represent ex-perimental data of Sirota et al.39

critical point. The negative slope of the liquid–liquid phasetransition line in the P–T plane means that high-density liq-uid water is the phase with larger entropy. The relatively largevalue of this slope at the liquid–liquid critical point (about 25times greater than for the vapor–liquid transition at the crit-ical point) indicates the significance of the entropy changerelative to the density change and, correspondingly, the im-portance of the entropy fluctuations. These features suggestthat liquid–liquid phase separation in water is mostly drivenby entropy rather than by energy, thus supporting the “lattice-liquid” choice for the scaling fields given by Eqs. (11)–(13)with b′ = 0.

As a first step to understand a relation between wa-ter’s polyamorphism and the behavior of cold aqueous so-lutions, Bertrand and Anisimov3 have introduced a mean-field “two-state” model that clarifies the nature of the phaseseparation in a polyamorphic single-component liquid. Two-fluid-states models trace their lineage back to a 19th cen-tury paper by Röntgen.95 Relatively recently, Ponyatovskiıet al.96 and, more quantitatively, Moynihan97 have describedthe emergence of a LLCP in supercooled water as resultingfrom the effects of nonideality in a mixture of two “compo-nents,” with their fraction being controlled by thermodynamicequilibrium. However, while Moynihan assumed a “regular-solution” type of nonideality, which implies an energy-drivenphase separation, such as the vapor–liquid transition or theconventional liquid–liquid transition in binary solutions, webelieve that a near “athermal-solution” type of nonideality ismainly responsible for the entropy-driven phase separation inmetastable water near the LLCP.

It is assumed that liquid water is a “mixture” of twostates, A and B, of the same molecular species. For instance,these two states could represent two different arrangementsof the hydrogen-bond network in water and correspond tothe low-density and high-density states of water. The frac-tion of water molecules, involved in either structure, denotedφ for state A and 1 − φ for state B, is controlled by ther-modynamic equilibrium between these two structures. Unlikea binary fluid, the fraction φ is not an independent variable,but is determined as a function of pressure P and tempera-ture T from the condition of thermodynamic equilibrium. The

094507-13 Thermodynamics of supercooled water J. Chem. Phys. 136, 094507 (2012)

simplest “athermal solution model”98 predicts a symmetricliquid–liquid phase separation for any temperature, with thecritical fraction φc = 1/2, if the interaction parameter, whichcontrols the excess (nonideal) entropy of mixing is higherthan its critical value. However, unlike a purely athermal non-ideal binary fluid, the entropy-driven phase separation in apolyamorphic single-component liquid will not be present atany temperature. On the contrary, the critical temperature Tc

is to be specified through the temperature dependence of theequilibrium constant K by

ln K = λ

(1

T− 1

Tc

), (25)

where λ is the heat (enthalpy change) of “reaction” betweenA and B. A finite slope of liquid–liquid coexistence in theP–T plane can be incorporated into the two-state lattice liquidmodel if one assumes that the Gibbs energy change of the“reaction” also depends on pressure.

A conceptually similar two-state model suggested byHrubý and Holten99 has explained the inflection point in thesurface tension (Fig. 6) by a rapid increase of the fraction ofwater molecules in the low-density state as water is cooleddown.

B. Liquid–liquid phase separation

A consequence of the second critical point is a phase sep-aration into a high-density and a low-density liquid at tem-peratures below the critical temperature. From the “lattice-liquid” two-state model with a steep slope of the LLT line inthe P–T phase diagram it follows that the phase separation ismostly driven by entropy rather than density. The LLT line isnot exactly vertical in the phase diagram due to a small den-sity difference between the two liquid phases, as pointed outby Bertrand and Anisimov.3 In Fig. 21, a temperature–densityphase diagram is shown including both the vapor–liquid andliquid–liquid critical points. The region below the second crit-ical point is shown in more detail in Fig. 22, where predictions

C1

Liquid

200

300

400

500

600

Tem

pera

ture

(K)

0 500 1000

Density (kg /m3)

C2

200

220

240

900 1000 1100

FIG. 21. Temperature–density diagram. C1 and C2 indicate the first and sec-ond critical point. The line left of C1 is the saturated vapor density accordingto IAPWS-95. Down to 250 K, the line marked by “Liquid” is the saturatedliquid density computed with IAPWS-95; below 250 K, the line representsthe liquid density at 0.1 MPa predicted by our model. The dashed lines showthe densities of the two liquid phases in equilibrium on the LLT line.

ADHADL140

160

180

200

220

240

Tem

pera

ture

(K)

900 1000 1100 1200

Density (kg /m3)

FIG. 22. Temperature–density diagram of the two liquid phases in equilib-rium on the LLT line. Solid line: asymptotic model, dashed line: extendedmodel, dotted line: prediction of the two-state regular-solution model ofMoynihan.97 Solid dots mark the second critical points. Open circles arethe densities of the relaxed low-density amorphous (LDA) and high-densityamorphous (HDA) phases of water at 200 MPa, taken from Fig. 5 of Loertinget al.100

by several models of the densities of the two liquid phases arecompared.

In the second-critical-point scenario it is assumed thatthe LLT line is connected to the phase-transition line betweentwo amorphous phases found below about 130 K.102 Sincethe amorphous phase-transition line is nearly horizontal in theP–T diagram, as shown in Fig. 23, it follows that the LLTmust be strongly curved to account for such a connection.In our model, we have linearized the LLT, so the predictedequilibrium densities at 135 K differ from the experimentaldensities of the amorphous phases, as seen in Fig. 22.The two-state model of Moynihan97 was fitted to the exper-imental amorphous-phase densities, as shown in Fig. 22. InMoynihan’s model, the LLT is also a straight line, but itsslope, dP/dT, is less than half of the slope of our LLT lines(Fig. 23). Comparing our models for the 0–150 MPa and

0

100

200

300

Pre

ssur

e(M

Pa)

120 160 200 240

Temperature (K)

a

TH

b

cHDALDA

FIG. 23. Different linearizations of the LLT curve. The thick solid curve isthe LLT curve suggested by Mishima.22 The lines marked a and b are thelinearized LLT lines of our models restricted to 150 MPa and 400 MPa, re-spectively; the line marked c is the LLT line of Moynihan.97 Symbols markthe liquid–liquid critical points; the solid diamond is the critical point ofMishima,22 the solid square is that of Bertrand and Anisimov,3 the opencircle is that of our asymptotic model, the crossed circle is that of our ex-tended model, and the cross is that of Moynihan.97 The dotted line is thephase-transition line between the two amorphous phases LDA and HDA asestimated by Whalley et al.101

094507-14 Holten et al. J. Chem. Phys. 136, 094507 (2012)

FIG. 24. Reduced sum of squared residuals as a function of the location of the liquid–liquid critical point for H2O. (a) Asymptotic model, fitted to experimentaldata up to 150 MPa. (b) Extended model, fitted to experimental data up to 400 MPa.

0–400 MPa ranges, we see that a lower LLT slope is neededfor the larger pressure range. In conclusion, with a linearizedLLT, the larger the desired temperature and pressure range ofthe model, the smaller dP/dT slope of the linearized LLT lineis required.

Based on the fit of the model to the experimental data, theliquid–liquid critical point can be located in a certain range oftemperatures and pressures, depending on the pressure limitfor experimental data involved in the fit. Without any prelim-inary idea on the location of the LLT curve, the position ofthe liquid–liquid critical point is uncertain. In principle, wecannot exclude even a negative value for the critical pressure.The reason for the uncertainty is that the experimental dataare located in a nonasymptotic temperature range and pres-sure range, not closer than 5% (in relative temperature andpressure) from the assumed location of the critical point, andonly on the high-temperature side of the liquid–liquid transi-tion. Our assumption on the location of the LLT curve is basedon the predictions of different authors22, 30, 65, 103 which giveapproximately the same result. We have linearized this curvefor a certain range of pressures. If the data are restricted up to100 MPa, the critical pressure is optimized to 28 MPa and thecritical temperature is found to be 227 K.3 If we include morehigh-pressure data and linearize the LLT curve in the pressurerange up to 150 MPa, the optimal value for the critical pres-sure becomes less certain and moves up. This tendency con-tinues when we include high-pressure data and linearize theLLT curve up to 400 MPa. As illustrated in Fig. 24(a), bestfits for the 150 MPa model are obtained with critical-pressurevalues between about 25 MPa and 70 MPa. For the extended,400 MPa, model, the optimum critical-pressure values moveto a higher range from 40 MPa to 85 MPa [Fig. 24(b)].

C. Absolute stability limit of the liquid state

For the supercooled liquid state to be thermally and me-chanically stable, both the isochoric heat capacity CV and theisothermal compressibility κT must be positive. The scalingmodel predicts regions in the phase diagram where CV or κT

is negative, as shown in Fig. 25. As noted by Bertrand andAnisimov,3 the stability locus is located close to the Widomline, but its exact location depends on the parameters of themodel. Our model predicts that the thermal stability condi-tion is violated before the mechanical stability condition aswater is cooled down, contrary to the violation order shownby a cubic equation of state that describes the vapor–liquidtransition.104, 105 At pressures slightly above the critical pres-sure, the mechanical stability limit coincides with the LLTline; that is, one of the liquid phases (the low-density liquid)is unstable. We note that a spinodal for the LLT is not definedin the scaled equation of state.106

The isochoric heat capacity CV is related to the isobaricheat capacity by1

CV = CP − T α2V

ρκT

. (26)

TH

C2

T = 0 Widom line

CV = 0T < 0

0

10

20

30

40

Pre

ssur

e(M

Pa)

220 225 230 235

Temperature (K)

C2

26

28

30

224.0 224.5

FIG. 25. Absolute stability limit of the liquid state, predicted by the model.Solid line: limit of mechanical stability, where the isothermal compressibilityκT is zero. Long dashed line: limit of thermal stability, where the isochoricheat capacity CV is zero. TH marks the experimental homogeneous nucleationlimit, and C2 indicates the second critical point. The dotted line, at which h1= 0 [Eq. (11)], indicates the liquid–liquid transition line and its extensionbelow the critical pressure, the Widom line. The inset shows the stabilitylimits in the vicinity of the critical point.

094507-15 Thermodynamics of supercooled water J. Chem. Phys. 136, 094507 (2012)

The critical part of CV expressed through the scaling sus-ceptibilities is derived in the Appendix of the supplementarymaterial.18 As the compressibility κT becomes smaller withdecreasing temperature, CV first becomes negative and thendiverges to negative infinity as κT approaches zero. At tem-peratures below the mechanical stability limit, where κT isnegative, CV is larger than CP (Fig. 14). Another property thatdiverges is the speed of sound c, which is related to CP, CV,and κT according to1

c = (ρκT CV /CP )−1/2. (27)

The speed of sound diverges to positive infinity (Fig. 7)when CV approaches zero (at the thermal stability limit),which occurs at a slightly higher temperature than the zero-compressibility limit. In the unstable region, the speed ofsound is imaginary.

Kiselev5 has computed a kinetic spinodal of supercooledwater and found that this stability limit was located a fewkelvin below the homogeneous nucleation limit. Kiselev callsthe area in the phase diagram below the stability limit a “non-thermodynamic habitat,” not because liquid water is neces-sarily mechanically unstable there, but because the lifetimeof liquid water is smaller than the time needed for equili-bration. Using molecular simulations, Moore and Molinero107

have also found a stability limit just below the homogeneousnucleation temperature. They show that below this limit, thetime for crystallization is shorter than the relaxation time ofthe liquid, so liquid water cannot be equilibrated.

An important question arises: What terms in our modelare responsible for the instability of the liquid state?Figure 26 shows the contributions of the individual suscepti-bilities to the compressibility, as given by Eq. (18). The strongand weak susceptibilities χ1 and χ2 are positive in the en-tire temperature range, whereas the cross susceptibility χ12

changes sign on the Widom line. The sum of the contributionsof the three susceptibilities is positive in the entire range; it isthe contribution of the regular part, which is negative, thatresults in a negative compressibility below a certain tempera-ture.

ˆ TV (a )′ 2

1

2

total

2a′ 12

ˆ rP P

0.2

0.1

0.0

0.1

0.2

200 220 240 260

Temperature (K)

FIG. 26. Contribution of the susceptibilities, χ1, χ2, and χ12, and the regularpart to the compressibility, as given by Eq. (18), at 10 MPa. The thick linerepresents the total compressibility multiplied by the volume, κT V .

VII. DISCUSSION

We have demonstrated that a theoretical model based onthe assumption of a liquid–liquid critical point in supercooledwater can represent the thermodynamic properties of both su-percooled H2O and D2O to pressures of 150 MPa. Moreover,by allowing the slope of the LLT line and the critical pressureto be freely adjustable parameters, the model can representalmost all available thermodynamic property data for super-cooled water. Nevertheless, there are still a number of issuesthat need to be considered.

A principal issue is that the existence of a liquid–liquidcritical point is not the only possible explanation for theanomalous behavior of the thermodynamic properties of su-percooled water. Scenarios for a singularity-free or critical-point-free interpretation have also been proposed.108–110 Re-cently, Stokely et al.111 have shown, for a water-like latticemodel, how all such scenarios can be described by varyingtwo quantities, the strength of the hydrogen bonds and the co-operativity of the hydrogen bonds. An intriguing possibilityof the existence of multiple critical points in supercooled wa-ter, as predicted by some simulations,112, 113 is discussed byBrovchenko and Oleinikova.114 Another possibility is that re-sponse functions, such as the compressibility, do not divergeat a single temperature corresponding to a critical tempera-ture but at a range of pressure-dependent temperatures Ts(P )corresponding to a crystallization spinodal, the absolute limitof stability of the liquid phase. Most recently, the discussionon the nature of the anomalies observed in supercooled wa-ter received an additional impetus after Limmer and Chandlerreported new simulation results9 for two atomistic models ofwater, mW (Ref. 115) and mST2.116 They did not find twoliquid states in the supercooled region and excluded the pos-sibility of a critical point for the models studied. However, themost recent simulations by Sciortino et al.117 of the originalST2 model confirm the existence of a liquid–liquid criticalpoint for that particular model, with two distinct liquid states.They found no evidence of crystallization during their simula-tion time, while there was enough time for the liquid to equi-librate. It would be important to compare the anomalies ob-tained for various models with those exhibited by real water.

Recent simulations of Moore and Molinero107 indicatethat for simulation cells larger than the critical ice nucleus,spontaneous crystallization occurs before liquid–liquid sepa-ration can equilibrate. An approach to consider a coupling be-tween spontaneous crystallization near the absolute stabilitylimit of the liquid state and liquid–liquid separation is the the-ory of weak crystallization pioneered by Brazovskiı,118 fur-ther developed by Brazovskiı et al.,119 and reviewed by Katset al.120 According to this theory, the fluctuations of the trans-lational (short-wavelength) order parameter ψ renormalizethe mean-field distance �0 = (T − T0)/T between the tem-perature T and the mean-field absolute stability limit T0 of theliquid phase

� = �0 + κ�−1/2, (28)

where � = (T − Ts)/T , where Ts is the fluctuation-affectedstability-limit temperature, and κ is a molecular param-eter, similar to the Ginzburg number that defines the

094507-16 Holten et al. J. Chem. Phys. 136, 094507 (2012)

validity of the mean-field approximation in the theory of crit-ical phenomena.79, 121 The theory of weak crystallization re-quires κ�−1/2 � �0 and �0 � 1. There are two nontrivialconsequences of the effects of translational-order fluctuations:(1) The heat capacity would contain a fluctuation correctionδCP ∼ �−3/2; (2) the renormalized gap � may be positiveeven at negative �0, meaning that translational-order fluctua-tions stabilize the liquid phase.

The theory of weak crystallization was used to de-scribe a coupling between one-dimensional density mod-ulation and orientational fluctuations in liquid crystals; itwas supported by accurate light-scattering and heat-capacitymeasurements.121–123

Similarly, we could consider a coupling between thetranslational-order fluctuations and the critical fluctuations ofthe liquid–liquid order parameter φ1. A relationship betweenthe structural transformation in liquid water and its crystal-lization rate is suggested by recent simulations by Mooreand Molinero.107 The lowest order term associated with sucha coupling in the free-energy Landau expansion would be∼ψ2φ1. Such a coupling would be possible even if the virtualcritical point is below the stability limit of the liquid phase.

Formation of ice is not “weak crystallization.” This iswhy application of weak crystallization theory to supercooledwater is questionable. However, it may not be hopeless. Thegap � between the melting temperature Tm and the tempera-ture of spontaneous crystallization TH is of the order of 0.1.Hence, a further investigation of the possibility of such cou-pling would be worthwhile. The actual question is how theexperimental observations can be explained by a theory thataccounts for both crystallization and liquid–liquid separation.The final conclusion concerning the existence of the liquid–liquid critical point in water should be based on the ability toquantitatively describe the experimental data.

Because the liquid–liquid critical point in supercooledwater is not experimentally accessible, the experimental dataare not in the asymptotic critical region. Hence, a mean-fieldcritical-point equation may also be sufficient to describethe experimental data in practice.124 However, we haveverified that it is impossible to represent the experimentaldata in terms of a polynomial representation that does notinclude any divergent critical behavior of the thermodynamicresponse functions.

Assuming the existence of the LLCP in supercooled wa-ter, we confirm the finding of Fuentevilla and Anisimov7 andBertrand and Anisimov3 that the critical point is located atmuch lower pressure than predicted by most simulations, def-initely below 100 MPa, while the precise value of the criticalpressure is uncertain. However, regardless of all open ques-tions, we have shown that a critical-point parametric equationof state describes the available thermodynamic data for super-cooled water within experimental accuracy, thus establishinga benchmark for any further developments in this area.

ACKNOWLEDGMENTS

We acknowledge collaboration with D. Fuentevilla and J.Kalová at an early stage of this research. We thank R. Torrefor sending us the experimental data on the sound velocity,

C. A. Angell, A. H. Harvey, S. Sastry, H. E. Stanley, and W.Wagner for useful comments and suggestions, and S.-H. Chenand Y. Zhang for discussing their results on confined super-cooled water. We also thank J. P. O’Connell for clarifying theindependence of the mechanical and thermal instability crite-ria. The research has been supported by the Division of Chem-istry of the US National Science Foundation under Grant No.CHE-1012052. Additional support for V. Holten has been pro-vided by the Burgers Program of the University of Marylandand the J. M. Burgerscentrum in the Netherlands.

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