Int J Thermophys (2012) 33:992–999DOI 10.1007/s10765-012-1238-5

Second Inflection Point of the Surface Tension of Water

Jana Kalova · Radim Mares

Received: 19 November 2011 / Accepted: 14 June 2012 / Published online: 24 June 2012© Springer Science+Business Media, LLC 2012

Abstract The theme of a second inflection point of the temperature dependenceof the surface tension of water remains a subject of controversy. Using data above273 K, it is difficult to get a proof of existence of the second inflection point, becauseof experimental uncertainties. Data for the surface tension of supercooled water andresults of a molecular dynamics study were included into the exploration of existenceof an inflection point. A new term was included into the IAPWS equation to describethe surface tension in the supercooled water region. The new equation describes thesurface tension values of ordinary water between 228 K and 647 K and leads to theinflection point value at a temperature of about 1.5 ◦C.

Keywords IAPWS · Inflection point · Molecular dynamics · Supercooled water ·Surface tension

1 Introduction

Much has been written about a second inflection point of the temperature dependenceof the surface tension of water, but this theme still remains a subject of controversy[1–3]. The same data can be used to support different hypotheses. When the surfacetension above 0 ◦C is explored, the inflection point cannot be reliably detected [1].

It is very difficult to get experimental data in the supercooled water region. Mea-surements of thermophysical properties of water are restricted to temperatures above

J. Kalova (B)Faculty of Science, University of South Bohemia, Ceske Budejovice, Czech Republice-mail: jkalova@prf.jcu.cz

R. MaresUniversity of West Bohemia, Pilsen, Czech Republic

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∼233 K, which is the limit of homogenous nucleation. Supercooled water exhibitsanomalous behavior, and a thermodynamically consistent view on its properties is stillsought [4–6]. Some of the theories are often used to predict thermophysical propertiesof supercooled water [7]. One of the effective methods to study properties of super-cooled water is the method of molecular dynamics (MD) [3]. For the surface tensionof the supercooled water region, there are three basic sets of experimental data [8–10].Also, the first published measurements below the freezing point are often cited [11].Because of the lower accuracy of the experimental data in this region, and because theexperimental data below 246 K are missing, it seems reasonable to include results ofa MD study [3,12] into our research.

The well-known Van der Waals equation [13] for the surface tension of liquids isas follows:

σ = Aτn,

in which the exponent n has the value of 1.25 for most liquids [14].The International Association for the Properties of Water and Steam (IAPWS)

issued the release on the surface tension of ordinary water substance [15,16]. Fortemperatures between 0.01 ◦C and 374 ◦C, this release contains critically evaluateddata for the surface tension σ , with estimated values of the uncertainty �σ .

Vargaftik et al. [15] first fitted data for the surface tension σ using the followingequation:

σ =9∑

i=1

ai(Tc − T )i

The equation has too many parameters and hence would not be well accepted.Vargaftik et al. have fitted the values approved by IAPWS subsequently to the

following well-known modification of the Van der Waals equation:

σ = Bτμ(1 + bτ), (1)

where B = 235.8 mN · m−1, τ = 1 − TTc

, μ = 1.256, b = −0.625, and Tc =647.096 K.

In the equation, T is the temperature in K, and the exponent μ is slightly differentfrom the universal exponent n used in the Van der Waals equation.

The range of validity of Eq. 1 is between the temperature of the triple point and thecritical temperature Tc = 647.096 K.

Resulting from Eq. 1, the inflection point is given by the condition,

d2σ

dτ 2 = μBτμ−2 ((μ − 1) + b (μ + 1) τ ) = 0.

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Such a formula yields only one value of the inflection point Ti for the equation,

Ti = Tc

(1 + μ − 1

b(μ + 1)

).

The calculated value of the first inflection point Ti = 530 K corresponds well with thevalue of 525 K presented in [3].

Many authors have searched for a weak inflection point [1,3,12] in the vicinity of thefreezing point. Using data above 273 K, it is difficult to get a proof of existence of thesecond inflection point, because of experimental uncertainties. Gittens [1] indicatedthat a new experimental method is needed to get an accuracy of measurements of atleast 0.001 mN ·m−1. Provided the IAPWS evaluated data are the best existing data forthe surface tension above the freezing point, the function of temperature dependenceis concave in this region.

That is why data below the freezing point of water are needed to test the secondinflection point.

2 Supercooled Water

As listed in [7,17–19], the first published measurements of the vapor–liquid surfacetension of water for supercooled water were two data points at −5 ◦C and −8 ◦C [11].The declared precision of the measurement is higher than 0.4 %. Some doubts existedabout the validity of the two data points because the data showed a weak inflectionpoint in the vicinity of 273 K. Further measurements [9] confirmed the possibility ofthe weak inflection point.

Improvements of experimental techniques resulted in new measurements in 1951,when Hacker [9] reported experimental values for supercooled water in the temper-ature range from 251 K to 300 K. He measured 702 points, with 404 measurementsat temperatures below the freezing point. The obtained measurements were consis-tent with the existence of the inflection point in the vicinity of 0 ◦C, as was predictedin [11]. Hacker measured the surface tension of water in air. Usually, the differencebetween a measurement in air and a measurement of the liquid–vapor surface tensionis negligible, but for supercooled water the effect may be significant [7]. It may becaused by an anomalous gas solubility in the supercooled regime. The uncertainty ofHacker’s measurement is estimated to be less than ±0.17 % for the surface tension.

Floriano and Angell [8] measured the surface tension of water under its own vaporpressure, in the temperature range from 246 K to 333 K. The data seem to indicatea dependency of the surface tension on the capillary diameter [18]. Floriano andAngell data show higher dispersion mainly in the region of lower temperatures. Theuncertainty of their measurement was not estimated in the article. Floriano and Angellmentioned that the surface tension is not like other properties of water which displayanomalies when the temperature is lowered into the supercooled region.

Ohsaka and Trinh [10] published results of their measurements of the surface tensionof the supercooled water levitating in air [10]. The temperature range was from 252 Kto 292 K. The uncertainty is estimated to be less than ±1 % for the surface tension.

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Fig. 1 Experimental and simulation data for the surface tension of water: circles [8], x-marks [9], asterisks[10], squares [11], diamonds data from MD study [12], solid line Eq. 4

The measurements of Ohsaka and Trinh show systematic deviations from the otherdata.

We have used a MD study [12] on surface properties of supercooled water. In thearticle, the extended simple point charge (SPC/E) potential was used to calculate thesurface tension in the temperature range from 228 K to 293 K. Because the resultsof simulations in the supercooled region correspond well with experimental data, wehave assumed that we can use the data down to 228 K from this study. Experimentaland simulated data for supercooled water [8–12] are presented in Fig. 1.

In many applications, the thermodynamic properties of supercooled water areneeded [20], for example, in atmospheric boundary layer studies [21,22]. Data gainedby Hacker [9] are therefore used in the book by Pruppacher and Klett [21]. The authorshave also taken into account an assumption of the singular behavior of liquid waternear 228 K. An equation for the surface tension σ(mN · m−1) was presented in [21],the range of the equation validity is from 228 K to 273.15 K:

σ =6∑

n=0

antn, (2)

where t is the temperature in ◦C, a0 = 75.93, a1 = 0.115, a2 = 6.818×10−2, a3 =6.511 × 10−3, a4 = 2.933 × 10−4, a5 = 6.283 × 10−6, and a6 = 5.285 × 10−8.

Unfortunately, Eq. 2 does not meet the values of IAPWS at 273.15 K. The polyno-mial approximation was fitted only for temperatures below the freezing point, and itis not possible to use it for any temperatures above 0 ◦C. The singularity assumptionof the surface tension near 228 K as it was used in [21] is not valid [17].

The polynomial approximation is not suitable for searching for an inflection point;Eq. 2 gives two points—at 266 K and 257 K.

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Holten [19] published a nice equation for the temperature dependence of the surfacetension of supercooled water in his thesis:

σw(T ) = σI(T ) − σa tanh

(T − Ta

Tb

)+ σb, (3)

where σa = 2.854 mN · m−1, σb = 1.666 mN · m−1, Ta = 243.9 K, Tb = 35.35 K,and σI is an extrapolation of the IAPWS surface tension (Eq. 1) into the temperaturerange of validity of Eq. 3 from 100 K to 267.5 K. The equation is based on the modelused by Hruby and Holten [18] and is fitted to experimental data of Hacker [9]. It isnot possible to use it for temperatures above 267.5 K. Because the equation is convexat temperatures below 267.5 K and IAPWS is a concave function for temperaturesabove 273.16 K, Eq. 3 is not possible to be used for finding of the inflection point.The inflection point must lie between the two temperatures. Equation 3 shows otherinflection points in temperatures 246 K and 243 K.

On contrary to Eqs. 2 and 3 that are based on Hacker’s data, the extrapolation ofEq. 1 below the freezing point satisfactorily describes the Floriano and Angell data [8].Equation 1 does not yield any inflection point for temperatures below 529 K. Even anabsolute value of the second derivative of the temperature dependence of the surfacetension is growing, as the temperatures goes down. It means that the IAPWS functionEq. 1 is a concave one for supercooled water. The IAPWS equation of the temperaturedependence of the surface tension is very accurate for the description of values above0 ◦C. In the negative temperature region, Eq. 1 only fits experimental data [8]. If wewant to use the equation outside the range of validity, for temperatures below thefreezing point, a correction of this equation is necessary. We used the equation,

σ = Bτμ(1 + bτ + cτ n) (4)

Here, c is a new parameter (probably positive) and n is an unknown power. We wantedto keep the IAPWS Eq. 1 for temperatures above 0 ◦C, and fix the parameters B andb in Eq. 4. To find the correction, we used experimental and calculated data [3,8–11]from supercooled water and data of IAPWS [16] for temperatures between 273.16 Kand 373.15 K. The last term of Eq. 4 is negligible for higher temperatures. First, weestimated the optimal exponent n = 33 (only a natural exponent n was the target). Inthat case, we get for the coefficient c = (50.3 ± 6.5) × 103 (95 % confidence level).Equation 4 gives the second inflection point at the temperature of about (1.5±1.4) ◦C.

One can see that the Prupacher and Klett approximation (Eq. 2) and Holten approxi-mation (Eq. 3) give a good fit to the Hacker data [9]. The Pruppacher and Klett equationis not possible to be used for temperatures below 235 K and above 273 K, the Holtenequation (Eq. 3) is not possible to be used for temperatures above 267.5 K. One cansee that the extrapolation of the IAPWS equation (Eq. 1) satisfactorily describes theFloriano and Angell data [8].

Deviations of the surface-tension values calculated using Eq. 4 from IAPWS eval-uated data [16], for temperatures above 273 K, are plotted in Fig. 2. All deviationsare in the range of uncertainties tabulated in [16]. For example, for a temperature of273.16 K, the tabulated uncertainty is 0.38 mN · m−1.

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Fig. 2 Deviations of Eq. 1 (crosses) and of new Eq. 4 (open circles) from IAPWS evaluated data [16]

Fig. 3 Experimental and simulation data for the surface tension of water: circles [8], x-marks [9], squares[11], asterisks [10], diamonds data from MD study [12], solid line Eq. 4, dashed line Eq. 3, dotted line Eq. 1(IAPWS), dashed-dotted line Eq. 2

3 Conclusion

The possibility of the existence of a second inflection point in the temperature depen-dence of the surface tension is discussed in this article. The overview of experimentaldata for the surface tension of supercooled water is made. The experimental data werecompleted with values calculated using MD [12]. Another term has been added toEq. 1 to fit all data. Whence follows Eq. 4. As shown in the presented study, Eq. 4leads to the inflection point value at a temperature of about 1.5 ◦C.

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Based on Fig. 3, we can see that the problem of the second inflection point is stillopen. If only the data of Hacker are used (as Holten and Koop et al. did), the existenceof the second inflection point seems to be evident. If the data of Floriano and Angellor Ohsaka and Trinh are taken into account, the situation is not so clear. In that case,the data are possible to be described by extrapolation of the IAPWS equation belowthe freezing point, and the equation has no inflection point in the region. To make thedecision, new experimental data are needed.

The new equation (Eq. 4) describes all known experimental data for the surfacetension of water, for temperatures between 228 K and 647 K. We believe that theequation can be useful in many applications, for example, in atmospheric boundarylayer studies [21,22], because it describes data both for the supercooled water and forthe temperatures above the freezing point. The contribution of the added term into theIAPWS equation for temperatures above the freezing point is negligible, and resultsare comparable with the original IAPWS equation (within the tolerances of the tablevalues published in the IAPWS release).

Acknowledgments This study was supported by the Grant Agency of the Academy of Sciences of theCzech Republic under Grant IAA200760905 and by the Ministry of Education, Youth and Sports underGrant LA09011.

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