Surface Tension of Supercooled Water: No Inflection Point down to

Letter pubs.acs.org/JPCL

Surface Tension of Supercooled Water: No Inflection Point down to −25 °C Jan Hrubý,*,† Václav Vinš,*,† Radim Mareš,*,‡ Jiří Hykl,† and Jana Kalov᧠†

Institute of Thermomechanics AS CR, v. v. i., Dolejškova 5, Prague 8, 182 00 Czech Republic Faculty of Mechanical Engineering, University of West Bohemia, Univerzitní 8, Pilsen, 306 14 Czech Republic § Faculty of Science, University of South Bohemia, Branišovská 31, 370 05 Č eské Budějovice, Czech Republic ‡

ABSTRACT: A dramatic increase in the surface tension of water with decreasing temperature in the supercooled liquid region has appeared as one of the many anomalies of water. This claimed anomaly characterized by the second inflection point at about +1.5 °C was observed in older surface tension data and was partially supported by some molecular simulations and theoretical considerations. In this study, two independent sets of experimental data for the surface tension of water in the temperature range between +33 and −25 °C are reported. The two data sets are mutually consistent, and they lie on a line smoothly extrapolating from the stable region. No second inflection point and no other anomalies in the course of the surface tension were observed. The new data lies very close to the extrapolated IAPWS correlation for the surface tension of ordinary water, which hence can be recommended for use, e.g., in atmospheric modeling. SECTION: Liquids; Chemical and Dynamical Processes in Solution

T

observed by Hacker. Computations by Feeney and Debenedetti14 based on the van der Waals theory of interfaces and a water model consistent with LLH indicate the possibility of SIP. However, due to crude approximations present in the mentioned theoretical approaches, the predicted anomalous behavior can either be an artifact or occur at lower temperatures. Experimental data for the surface tension of ordinary water between the triple point and the critical point was collected and correlated by Vargaftik et al.15 and later updated for the ITS-90 temperature scale and approved as a standard by the International Association for the Properties of Water and Steam (IAPWS).16 The IAPWS correlation is a modification of a universal scaling relation

he surface tension of supercooled water is a property that is of paramount importance for atmospheric studies. Cloud droplets may remain liquid down to −38 °C. Experiments indicate that liquid rather than crystalline nanodroplets are formed by homogeneous nucleation from supersaturated water vapor at least down to −73 °C.1 Water exhibits numerous anomalies at low temperatures. Historical (1895) data by Humphreys and Mohler2 for the surface tension of supercooled water down to −8 °C weakly indicated an anomaly of the interfacial properties of water: a second inflection point (SIP) in the dependence of the surface tension (σ) on temperature (T), corresponding to a progressive increase in surface tension with decreasing temperature below about 0 °C. The existence of this anomaly was later strongly supported by a highly internally consistent experimental data set by Hacker.3 Newer measurements by Floriano and Angell4 and Trinh and Ohsaka5 suffer from a scatter that precludes a definite judgment about the SIP. The existence of SIP has been supported by some molecular simulation results,6,7 while others8−10 have not shown this feature. Theoretical analysis and prediction of the σ(T) dependence rests in the understanding of water’s behavior in the “no-man’s land”, the region between observable supercooled water (down to about −38 °C at 0.1 MPa) and observable glassy forms of water (up to about −143 °C) in which the high ice nucleation rate prohibits experiments with bulk water. Various scenarios have been suggested by Mishima and Stanley,11 of which the liquid−liquid phase-transition hypothesis (LLH) involving a second critical point of water is particularly successful in modeling the strongly anomalous thermodynamic properties of supercooled water.12 Hrubý and Holten13 suggested a simple thermodynamic twostate model consistent with LLH, supporting the behavior © 2014 American Chemical Society

σcorr = Bτ μ(1 + bτ )

(1)

where σcorr is the correlated surface tension, τ = 1 − T/Tc is the dimensionless distance from the critical temperature Tc = 647.096 K, μ = 1.256 is a universal critical exponent, and coefficients B and b have values of 235.8 mN·m−1 and −0.625, respectively. Equation 1 shows just one inflection point at 256.46 °C. It extrapolates smoothly below the triple point temperature 0.01 °C. The goal of the present experimental study is to provide accurate data for the surface tension of supercooled water which would definitely decide the question of whether the surface tension shows an anomalous course, as seen in the data Received: November 26, 2013 Accepted: January 7, 2014 Published: January 7, 2014 425

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−9 °C. As can be seen, the two independent data sets obtained in this study are highly consistent with each other. The data deviates only slightly from the extrapolated IAPWS correlation over the entire temperature range down to −25 °C. The absolute maximum difference between the experimental data and the IAPWS correlation does not exceed 0.16 mN·m−1, which corresponds to a relative deviation below 0.2%. This is in contrast with the data by Hacker, which shows a different slope at temperatures below −10 °C. The directly measured result in these experiments is the ratio Y of the surface tension at a given temperature to the surface tension at a reference temperature Tref = +20 °C,

by Hacker and in some simulation and theoretical results, or whether it rather follows a smooth extrapolation such as that provided by the IAPWS formulation. To enhance the reliability of the data, the measurements were performed independently by two groupsthe Prague group and the Pilsen groupusing different experimental setups. Both setups were based on the capillary elevation method modified for measurements under the metastable supercooled liquid condition. Because the probability of the appearance of an ice nucleus is proportional to the volume of the liquid, only a short section (a few centimeters) of the capillary in the vicinity of the meniscus was cooled to the desired temperature, while the rest of it remained at ambient temperature. The surface tension was determined based on measurements of the height of the liquid column as σ(Tin) =

gd [ρ(Tin)(h − hamb) + ρ(Tamb)hamb] 4 cos θ

Y (T ) =

σ (T ) σref

(3)

This fact follows from the way of determining the capillary diameters as discussed in the Experimental Methods. The reference surface tension obtained from the IAPWS correlation (1) as σref = 72.74 mN·m−1 was used to evaluate the capillary diameters. However, the experimental relative surface tension Y is independent of this choice. At ambient temperatures, standard precise methods such as the Wilhelmy plate19 can be used. If in future a more accurate value is available for the surface tension at the reference temperature, the new value of σref can be used for converting the present relative data into absolute surface tension values. Table 1 shows representative data for the surface tension of supercooled water measured with both setups. The full data set

(2)

Equation 2 follows from the balance of capillary and gravity forces assuming a stepwise temperature profile as discussed in Experimental Methods; Tin and Tamb stand, respectively, for temperature inside the temperature-controlled chamber and the ambient temperature, h denotes the total height of the water column, hamb is its part at ambient condition, g denotes the local gravity, d stands for the inner capillary diameter, and θ is the contact angle between the liquid meniscus and the capillary wall. Water density ρ at a given temperature and atmospheric pressure was evaluated from the IAPWS-95 equation of state17 with uncertainty of 0.03% and 0.0001% at temperatures below and above 0.01 °C, respectively. The Pilsen apparatus used gaseous nitrogen for maintaining the temperature of the water meniscus inside the capillary tube. The Prague apparatus employed ethanol as a heat transfer medium. Detailed description of the two experimental setups and the measuring technique is given in section Experimental Methods. Figure 1 shows the new experimental data for the surface tension of water compared to the extrapolated IAPWS correlation (eq 1). Also shown is the data by Hacker3 indicating a second inflection point or rather a kink around

Table 1. Relative Surface Tension of Supercooled Water Measured on Two Setupsa Prague setup

a

Pilsen setup

T (°C)

σ/σref

T (°C)

σ/σref

0.05 −1.95 −3.95 −5.95 −7.96 −9.95 −11.97 −13.97 −15.97 −17.93 −20.01 −22.00 −23.91

1.03970 1.04396 1.04822 1.05195 1.05595 1.05921 1.06264 1.06671 1.07036 1.07428 1.07774 1.08182 1.08514

−0.30 −2.18 −4.22 −6.07 −7.93 −10.00 −12.11 −14.34 −16.25 −17.97 −20.64 −23.57 −24.88

1.04112 1.04531 1.04778 1.05234 1.05543 1.05925 1.06308 1.06826 1.07122 1.07467 1.07887 1.08356 1.08504

To convert to absolute values, use σref = 72.74 mN·m−1.

including results obtained with alternative experimental approaches will be published separately. The relative combined standard uncertainty of the relative surface tension Y is below ur(Y) = 0.2% for both data sets. This estimate includes uncertainties of height measurements, uncertainties of density due to the equation of state and due to the uncertainty of temperature measurements, uncertainty of the shape of the temperature profile along the capillary, uncertainty of the surface tension due to the uncertainty of measured temperature, and statistically evaluated scatter of the experimental data. The relative uncertainty of the absolute surface tension σ is given as

Figure 1. Surface tension of supercooled water. Experimental data: ○, Pilsen setup (this work); +, Prague setup (this work); ★, Humphreys and Mohler;2 ◊, Hacker;3 Δ, Floriano and Angell;4 □, Trinh and Ohsaka.5 × , Lü and Wei molecular simulation;7 −·, Dutcher et al. correlation;18 −, eq 1 426

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The Journal of Physical Chemistry Letters ur(σ ) = [ur2(Y ) + ur2(σref )]1/2

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The two reported data sets are in mutual agreement, and they exclude the existence of the second inflection point or other anomalous behavior of the surface tension of water down to −25 °C. The data sets were obtained using different setups, independent calibrations and different experimenters to enhance the reliability of the data. The existence of the second inflection point discussed, e.g., by Lü and Wei,7 Holten et al.,12 and by Kalová and Mareš,23 was mostly supported by Hacker’s data.3 The anomalous course of Hacker’s data is most likely an experimental artifact.

(4)

The uncertainty of the reference surface tension as given by IAPWS16 is comparatively high, u(σref) = 0.36 mN·m−1, corresponding to ur(σref) = 0.5%. Much smaller uncertainties were estimated by Pallas and Harrison.20 For surface tension at 20 °C, they estimate 72.86 ± 0.05 mN·m−1 based on literature data. Their own measurements resulted in 72.869 ± 0.035 mN· m−1. Here we stick to the IAPWS-recommended reference value and uncertainty yielding combined standard relative uncertainty ur(σ) = 0.54% according to eq 4. Figure 2 shows the deviations of the experimental surface tensions from the IAPWS correlation (1) extrapolated to

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EXPERIMENTAL METHODS The measurements were performed with fresh ultrapure water with resistivity 18.2 MΩ·cm, total organic carbon 1 to 5 ppb, and free of particles larger than 0.2 μm. The uncertainty of the measured temperatures was below 0.006 °C for both setups. The transition of the capillary temperature from Tamb to Tin occurred at length of at most 4 mm. This deviation from the ideal stepwise temperature profile assumed in eq 2 resulted in an error of the surface tension smaller than 0.0170%. The cooling path was isobaric (at atmospheric pressure). Similarly as in the study by Floriano and Angell,4 the surface tension was measured with a descending meniscus. For the ascending direction, the motion of the meniscus was irregular probably due to the microscopic roughness of the inner capillary surface. Since water almost perfectly wets a clean fused silica surface, contact angle θ is close to zero. A temperature independent value of θ = 3° determined by Stepanov et al.24 for the receding contact angle, i.e., the case relevant for the descending meniscus, was considered in eq 2. The capillary inner diameters were first measured optically using fractured capillaries. However, more reliable values reported below were obtained when the diameter was determined from a set of capillary rise measurements close to the reference temperature 20 °C by inverting eq 2. Consequently, the present data are relative measurements. In the case of the Pilsen Setup, the temperature of the upper part of the capillary tube placed inside a small rectangular glass chamber equipped with two Pt100 temperature sensors was maintained by gaseous nitrogen. The gas flowed through a heat exchanger, where it was cooled a few degrees below the required temperature, and fine temperature control was achieved using a subsequent resistive heater. The upper end of the capillary tube was freely open to the atmosphere. Consequently, the water meniscus inside the capillary tube was in contact with the ambient air. The height of the elevated water column was detected by a high precision cathetometer with uncertainty of uh = 0.01 mm. The inner diameter of the capillary tube was evaluated as 0.364 ± 0.002 mm. Forty-five data points at temperatures below the triple point were collected. In the case of the Prague Setup, the capillary tube was placed inside a cylindrical glass chamber equipped with two optical glasses placed on its bases. The glass chamber was connected to two thermostatic baths with circulating ethanol. A set of specially designed switch valves together with the good heat transfer characteristics of the liquid allowed rapid temperature changes and short equilibration times. Two high-precision Pt100 temperature sensors measured the ethanol temperature close to the capillary tube. The height of the water column was measured using a cathetometer with an uncertainty of uh = 0.04 mm. Unlike the Pilsen setup, the open end of the capillary tube was connected to a pure helium distribution setup, which

Figure 2. Deviation of the new and literature data from the IAPWS correlation (1); ○, Pilsen setup (this work); +, Prague setup (this work); ★, Humphreys and Mohler;2 ◊, Hacker;3 Δ, Floriano and Angell.4

subzero temperatures. The Pilsen data and the Prague data are very consistent internally and mutually. Both new data sets are very close to the smoothly extrapolating IAPWS correlation. Hence, no anomaly is observed. Also shown is the questionable data by Hacker,3 which has been considered the most accurate until now due to its high internal consistency and good agreement with published data above the triple point. The data by Floriano and Angell4 shows a large scatter. Nevertheless, it seems to support the smooth extrapolation as provided by the IAPWS correlation rather than the anomalous course of Hacker’s data. The data by Humphreys and Mohler2 seems to support Hacker, but the “trend” of this historical data can actually reflect the experimental scatter. The data by Trinh and Ohsaka5 is not shown, as it suffers from large systematic deviations exceeding the scale of Figure 2. One important difference between the two setups was that for the Prague setup the liquid meniscus was in contact with helium, while for the Pilsen setup it was in contact with air. At ambient temperature and atmospheric pressure, helium enhances the surface tension by 0.002%21 and air decreases it by about 0.1%.22 The effect of helium is completely negligible and this was the reason for its choice. The influence of gas on the relative surface tension Y reported here cancels out except for possible temperature dependence in case of air. The good agreement of the Prague and Pilsen data sets indicates that the effect of air at atmospheric pressure on the surface tension of water remains small down to −25 °C. 427

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(14) Feeney, M. R.; Debenedetti, P. G. A Theoretical Study of the Interfacial Properties of Supercooled Water. Ind. Eng. Chem. Res. 2003, 42, 6396−6405. (15) Vargaftik, N. B.; Volkov, B. N.; Voljak, L. D. International Tables of the Surface Tension of Water. J. Phys. Chem. Ref. Data 1983, 12, 817−820. (16) IAPWS Release on Surface Tension of Ordinary Water Substance. 1994; http://www.iapws.org/. (17) Wagner, W.; Pruss, A. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. J. Phys. Chem. Ref. Data 2002, 31, 387−535. (18) Dutcher, C. S.; Wexler, A. S.; Clegg, S. L. Surface Tension of Inorganic Multicomponent Aqueous Electrolyte Solutions and Melts. J. Phys. Chem. A 2010, 114, 12216−12230. (19) Wilhelmy, L. Ü ber die Abhängigkeit der CapillaritätsConstanten des Alkohols von Substanz und Gestalt des benetzten festen Körpers. Ann. Phys. 1863, 195, 177−217. (20) Pallas, N. R.; Harrison, Y. An Automated Drop Shape Apparatus and the Surface Tension of Pure Water. Colloids Surf. 1990, 43, 169− 194. (21) Wiegand, G.; Franck, E. U. Interfacial Tension between Water and Non-polar Fluids up to 473 K and 2800 bar. Ber. Bunsen-Ges. Phys. Chem. 1994, 98, 809−817. (22) Massoudi, R.; A. D. King, J. Effect of Pressure on the Surface Tension of Water. Adsorption of Low Molecular Weight Gases on Water at 25°. J. Phys. Chem. 1974, 78, 2262−2266. (23) Kalová, J.; Mareš, R. Second Inflection Point of the Surface Tension of Water. Int. J. Thermophys. 2012, 33, 992−999. (24) Stepanov, V. G.; Volyak, L. D.; Tarlakov, Y. V. Wetting Contact Angles of Certain Systems. J. Eng. Phys. Thermophys. 1977, 32, 646− 648; Translated from Inzhenerno-Fizicheskii Zhurnal 1977, 32, 1000− 1003.

facilitated elevation or depression of the water column inside the capillary tube and flushing of the tube with pure helium before the measurement. Each measurement was performed with a fresh water column to avoid possible artifacts due to dissolution of the capillary wall constituents and diffusion of the surface active traces toward the meniscus. The inner diameter of the capillary tube was evaluated as 0.3229 ± 0.0016 mm. In total, 25 data points were measured at temperatures below the triple point.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]; Phone: +420 266 053 762; Fax: +420 286 584 695. *E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was supported by Grants IAA200760905 and M100761201 from the Academy of Sciences CR, Czech Science Foundation Grant GPP101/11/P046, Ministry of Education CR grant LG13056, and institutional support RVO:61388998. Many thanks go to I. Kašı ́k of the IPE AS CR for his help with capillaries, V. Holten from the University of Maryland for hints on the literature, and to our technicians, J. Blaha and B. Šmı ́d.

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REFERENCES

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