Surface Tension of Supercooled Water: Inflection Point-Free Course

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX-XXX

pubs.acs.org/jced

Surface Tension of Supercooled Water: Inflection Point-Free Course down to 250 K Confirmed Using a Horizontal Capillary Tube Václav Vinš,*,†,‡ Jan Hošek,† Jiří Hykl,† and Jan Hrubý† †

Institute of Thermomechanics of the CAS, Dolejškova 5, 182 00 Prague 8, Czech Republic Škoda Auto a.s., Tř. Václava Klementa 869, 293 01 Mladá Boleslav, Czech Republic

‡

ABSTRACT: The temperature course of the surface tension of supercooled water was suspected to exhibit an anomalous feature the so-called second inflection point (SIP). Besides some theoretical and molecular simulation studies, this hypothesis was primarily supported by experimental data by P. T. Hacker [NACA TN 2510, 1951]. Recently, the present group performed accurate surface tension measurements down to −26 °C using a modified capillary rise [Hrubý et al. J. Phys. Chem. Lett 2014, 5, 425 and Vinš et al. J. Phys. Chem. B 2015, 119, 5567] which, in contrast to Hacker’s data, showed no SIP anomaly. To confirm that the qualitatively different observations are not related to some fundamental phenomenon, we developed an experimental device employing basically the same method as Hacker with a horizontal capillary tube. New experimental data for the surface tension of supercooled water measured with the horizontal capillary setup down to −23 °C are presented in this study. The new data show a very good agreement with the previous capillary rise measurements. It was confirmed that the temperature dependence of the surface tension is free of SIP in a temperature range from −23 to 23 °C and can be well-represented by the IAPWS standard extrapolated below 0.01 °C. However, a small systematic deviation from the IAPWS correlation can be seen at temperatures below −15 °C.

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INTRODUCTION Liquid water at low temperatures exhibits numerous physical anomalies of which the density maximum at 4 °C is the most commonly known. During the last two decades, a large attention has been paid to investigating supercooled water, that is, liquid water under the thermodynamically metastable conditions at temperatures below the equilibrium freezing point (0 °C at atmospheric pressure). Many thermophysical properties of supercooled water, for example, isothermal compressibility and isobaric heat capacity, sharply increase at temperatures approaching −38 °C, which is the supercooling limit when macroscopic water samples freeze by homogeneous nucleation at atmospheric pressure.4 Many of these anomalies can be explained with the liquid−liquid phase transition hypothesis (LLPT) devised by Poole et al.5 on the basis of molecular simulations. The various scenarios of behavior of metastable water phases, their theoretical bases, and experimental evidence were presented by Mishima and Stanley.6 Since then, a broad debate has developed in the scientific community whether or not the strongly anomalous properties of supercooled water can be interpreted as a behavior of a mixture of two liquids with different densities. Many advanced experiments including measurements of speed of sound,7 viscosity,8 and other thermophysical properties aims to clarify this discussion. The LLPT hypothesis appears very successful in modeling the properties of liquid water at low temperatures and, in particular, the properties of supercooled water.9,10 © XXXX American Chemical Society

Similarly to other properties of the supercooled water, the surface tension is suspected to show an anomaly, which is usually referred to as the second inflection point (SIP), and which has the meaning of changing the curvature of the surface tension versus temperature dependency. We note that the first (regular) inflection point in the surface tension of water appears within the stable region at temperature of 256 °C. The surface tension of supercooled water is an important property especially in the atmospheric research where the formation and subsequent growth of water droplets and ice crystals are investigated.11−13 In some cases, water in clouds exists in the form of supercooled nanodroplets rather than ice crystals. Recent experiments by Manka et al.14 showed that liquid water droplets can form by homogeneous nucleation at temperatures even down to −73 °C. The SIP anomaly in the surface tension of water was primarily supported by the comprehensive experimental work conducted by Hacker1 in 1951 in the temperature range from −22.5 to 27.5 °C. Hacker’s data for the surface tension showed a noticeable change in the slope at temperatures around −8 °C. A similar trend could partly be seen already in the limited historical data by Humphreys and Mohler15 from 1895, who measured the surface tension down to −8 °C. More recent experimental data by Floriano and Angell16 and Trinh and Received: June 8, 2017 Accepted: September 22, 2017

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DOI: 10.1021/acs.jced.7b00519 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Ohsaka17 could neither verify nor refute the SIP anomaly as they were considerably scattered compared to the Hacker’s measurements. To clarify the SIP anomaly, our team developed a unique experimental apparatus based on the modified capillary rise method18 similar to that employed by Floriano and Angell.16 The new data sets2,3 showed a good internal consistency. The SIP anomaly could not be identified as the new data followed a quite different temperature trend than Hacker’s measurements. In this work, the surface tension of supercooled water has been further investigated using a new experimental apparatus. The measuring technique is rather different from the previous capillary rise experiments.2,3 The new apparatus allows for measurement of surface tension within horizontal capillary tubes, which is a similar method to that employed by Hacker.1 The main motivation for the current study was to verify our previous measurements2,3 using a significantly different experimental technique, which, in basic aspects, was similar to the Hacker’s experiment. In this way we wanted to make sure that the anomalous course of Hacker’s data is to be attributed to an unidentified experimental artifact rather than to the properties of supercooled water which, supposingly, might behave differently in the horizontal capillary compared to the standard vertical orientation.

supercooled water can still be reproduced by the extrapolated IAPWS correlation within the range of their uncertainty. The correlation by Pátek et al.21 is currently being revised by one of the IAPWS task groups22 as a potential candidate for a replacement of the current IAPWS standard. We note that the current IAPWS standard19 given by eq 1 is used as the only reference equation for the surface tension of water in this study to keep consistency with the previous experiments.2,3 In the nucleation studies related mostly to the formation of water droplets and ice crystals in atmosphere, a quadratic correlation developed by Viisanen et al.23 is commonly used. σ = 93.6635 + 9.133 × 10−3T − 2.750 × 10−4T 2

In eq 2, σ stands for the surface tension in mN·m−1, and T is the absolute temperature in Kelvin. The correlation by Viisanen et al. provides a reasonable prediction for the surface tension at low temperatures up to approximately 90 °C, where it starts to significantly deviate both from the IAPWS standard19 (eq 1) and the correlation by Pátek et al.21 Besides the experimental research, the modern tools of molecular simulations are being used for the prediction of the surface tension of supercooled water. In 2006, Lü and Wei24 used the SPC/E molecular model for water and modeled the surface tension at temperatures from 125 °C down to −80 °C. The authors claimed the SIP anomaly at the temperature of around +30 °C based on their simulated data. However, the reported results suffer of several discrepancies. At 0.01 °C, Lü and Wei predicted the surface tension of around 75.4 mN·m−1, which is remarkably close to the experimental surface tension19 of 75.65 mN·m−1, while the other molecular simulations25−27 with the SPC/E model provided mutually consistent values of around 65.6 ± 0.8 mN·m−1. Moreover, the overall temperature trend of the surface tension predicted by Lü and Wei differs strongly from the temperature trends of both the experimental surface tension19 and the results of other molecular simulations.25−27 In case of Lü and Wei’s simulations, the slope of the simulated data considerably decreases with increasing temperature at temperatures above 50 °C. Last but not least, the melting temperature for the SPC/E model is around −60 °C.26 Consequently, the application of the SPC/E molecular model on the prediction of supercooled water properties is rather questionable. Recently, Rogers et al.28 used the WAIL potential for water (Water potential from Adaptive force matching for Ice and Liquid)29 in the modeling of surface tension of water in the temperature range between −60 and 25 °C. The simulations showed very good agreement both with the experimental data within the stable region represented by the IAPWS standard19 and with the recent supercooled data measured down to −26 °C.2,3 On the other hand, the WAIL simulations deviate significantly from the IAPWS standard extrapolated below 0.01 °C as a continuous increase in the surface tension of supercooled water appeared at temperatures below −30 °C. The simulations by Rogers et al. point at a possible SIP-like anomaly in the surface tension of water, however at significantly lower temperatures than originally detected in the experiments by Hacker1 or in the molecular simulations by Lü and Wei,24 which provided values for SIP of around −8 °C and +30 °C, respectively. Because of noticeably good agreement between the WAIL simulations and the experimental data, Rogers et al. could directly compare their simulation results with the IAPWS standard based solely on the experimental data. Consequently, Rogers et al. proposed an additional exponential term in the

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CORRELATIONS FOR SURFACE TENSION OF SUPERCOOLED WATER The surface tension values measured in the previous work2,3 down to −26 °C could be well-represented by the international standard for the surface tension of ordinary water19,20 extrapolated below the triple point of water. The IAPWS (International Association for Properties of Water and Steam) standard is based on the correlation developed by Vargaftik et al.20 having the following form σIAPWS(T ) = Bτ μ(1 + bτ )

(2)

(1)

where τ = (Tc − T)/(Tc + 273.15) is the dimensionless distance from the critical point temperature Tc = 373.946 °C, B and b are coefficients having values of 235.8 mN·m−1 and −0.625, respectively, and μ = 1.256 is a universal critical exponent. The IAPWS standard was originally valid in the temperature range from the triple point of water 0.01 °C up to the critical point Tc; no data in the supercooled region were considered during its development. Pátek et al.21 have recently published new measurements for the surface tension of water in the range from −0.51 to 70.34 °C. The authors have revised a large set of experimental data for the surface tension of water including 1620 data points available in the temperature range covering the supercooled region and the stable liquid region up to the critical point temperature. A new correlation was developed on the basis of a nonparametric regression of 1055 selected high-quality data points. The correlation has the same form as the IAPWS standard19 given by eq 1. However, the critical exponent μ and coefficients B and b have slightly different values of 1.2527, 233.58 mN·m−1, and −0.61594, respectively. As Pátek et al. considered the supercooled data obtained with the capillary rise method,2,3 the proposed range of validity is from −25 °C to the critical point temperature Tc. However, as it was shown in the previous studies,2,3 the IAPWS correlation can also be successfully extrapolated to subzero temperatures down to −25 °C. Even though a small systematic offset was detected at temperatures below −17 °C,3 the experimental data for the surface tension of B



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amount of data with very good internal consistency in the temperature range from −22.2 to 27 °C. However, as already mentioned in the introduction, Hacker’s data show a systematic increase in the surface tension at temperatures below −8 °C, which became an important argument in the subsequent discussion about the existence of the SIP anomaly.33,34 Our team has therefore decided to use similar method with a horizontal capillary tube to verify previous capillary rise measurements,2,3 which are in disagreement with the Hacker’s data. Figure 1 shows a simplified scheme of the experimental apparatus with a horizontal capillary tube viewed from the top.

IAPWS standard (eq 1) correcting for the SIP anomaly in the supercooled region. The extended correlation is referred as IAPWS-E where E stands for Exponential. σIAPWS ‐ E(T ) = BE τ μ(1 + bEτ ) + σs exp[−c(T − Te)] (3)

In eq 3, τ is defined in the same way as in eq 1. The critical temperature Tc and exponent μ have the same values as in the original IAPWS correlation, while coefficients BE and bE are slightly different; BE = 235.697 mN·m−1 and bE = −0.624. The remaining parameters in eq 3 are σs = 1 mN·m−1, c = 0.116 K−1, and Te = 234.669 − 273.15 °C. Te is the so-called emergence temperature at which the surface tension of supercooled water predicted with the WAIL potential substantially deviates from the IAPWS correlation extrapolated below 0.01 °C.

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EXPERIMENTAL METHODS The experimental technique employed in this study is based on the method developed originally by Ferguson and Kennedy30 for the measurement of liquids available in volumes of only several milli-/microliters and with an unknown density. In this method, a small liquid sample is sucked inside a capillary tube, which is, unlike in the capillary rise technique, placed in a horizontal position. One end of the capillary tube is connected to a pressure setup with gas, while the other end is left open to atmosphere. The small counter pressure of gas pushes the liquid thread inside the capillary tube toward its open end, where the shape of the liquid meniscus is observed with an optical setup consisting of a light source and a microscope or a camera. Increasing the gas pressure causes variation of the liquid meniscus shape, which with increasing pressure gradually changes from concave to planar and, subsequently, convex forms. In the case of exactly planar meniscus, the pressure in the liquid is equal to the ambient pressure. The gas counter pressure applied to the other end of the tube is compensated solely by the surface forces at the liquid meniscus inside the capillary tube. Knowing the gas counter pressure Δp corresponding to the planar meniscus at the open end of a capillary tube with inner diameter d, the surface tension σ can be determined as follows σ=

Δpd 4 cos θ

Figure 1. Scheme of the experimental apparatus (top view).

The capillary tube is partly placed inside a glass temperaturecontrolled chamber with flowing ethanol circulated from a bath. The water thread inside the capillary tube has a length such that the inner meniscus is located inside the temperature-controlled chamber. The open end of the capillary tube with the observed meniscus is kept at an ambient temperature. The second end of the capillary tube is connected to the pressure setup with inert gas operating in our case with helium with a purity of 99.996%. The horizontal experimental apparatus was developed from the capillary rise apparatus employed in the previous mesurements.18,2,3 Most of the components could remain common for both systems, for example, the temperature-control unit consisting of two thermostatic baths and special switch valves, or the helium distribution setup generating well-defined and stable counter pressure. The counter pressure was increased from approximately 100 Pa used for our modified capillary rise measurement3 to compensate for the surface tension enhancement due to temperature jump, to approximately 1000 Pa needed to cover the full Laplace pressure for the present horizontal technique. A temperature-controlled chamber with slightly different dimensions had to be used for the horizontal capillary tube with the maximum length of around 80 mm. A great attention was paid to the development of the optical setup.35 The liquid meniscus image was observed with a digital camera equipped with a 6.3× magnifying objective.

(4)

Equation 4 represents the Young−Laplace equation valid in this form for a small-diameter capillary tube where, as in the present experiments, the meniscus is very closely spherical. In eq 4, θ is the contact angle between water and the inner wall of the capillary tube. In our case, the fused silica capillary tubes with inner diameter of around 0.33 mm were used. For water and fused silica, a constant contact angle θ = 3 ± 1° can be considered in the low temperature region according to Stepanov et al.31,32 An important advantage of the introduced method with a horizontal capillary tube is that the density of the investigated liquid sample does not need to be known as the surface tension can be directly evaluated from the measured counter pressure Δp. We note that in the capillary rise technique, employed for instance in our previous measurements,18,2,3 Δp consisted to some extent of the hydrostatic pressure for which the temperature dependence of the liquid density had to be known. The experimental method by Ferguson and Kennedy30 was modified by Hacker1 for the measurement of the surface tension of supercooled water. Hacker collected an impressive C



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The illumination and imaging of the meniscus surface was performed with 4f optical configuration and with the use of a beam splitter. In the initial design, a red laser was used for the illumination of the liquid meniscus.36 However, unfavorable speckle patterns occurred in the digital camera image. The red laser was therefore replaced with a fiber optic light source,37 which resulted in a clear and reproducible image of the meniscus. More details about the experimental setup together with description of its development are provided in previous studies.35−37 Similarly as in the capillary rise measurements, the experiments with horizontal capillary tube were performed with ultrapure water samples prepared with a help of a reverse osmosis unit followed by an analytical purification system. The water samples had a constant resistivity of 18.2 MΩ, total organic carbon content below 5 × 10−9, and were free of particles larger than 0.2 μm. The fused silica capillary tubes required a special treatment before they could be used on the horizontal apparatus. The open end of the capillary tube had to be carefully grinded and polished in order that the outer surface was perfectly flat and perpendicular to the axis of the capillary tube.38 The capillary tube was repeatedly cleaned with chromosulfuric acid and flushed with ultrapure water before the measurement. The inner diameter of the capillary tube d was determined from the measurements at the reference temperature Tref = 15 °C and the surface tension σIAPWS calculated from the IAPWS standard19 (eq 1). d=

1 N

N

∑

Figure 2. Typical temperature variation during the experiment with a constant reference temperature of 15 °C; example for Data Set 5.

could be continuously verified. We note that for the measurements introduced in this study an average inner diameter calculated from eq 5 was considered in the surface tension analysis as no effects resulting in the inner diameter variation were observed. We note that the momentum of water molecules evaporated from the meniscus causes a small enhancement of the pressure in the liquid.39 However, this pressure enhancement was the same for the measurements at the temperature of interest and at the reference temperature, because the temperature of the open end and the humidity of the surrounding air had the same values corresponding to the ambient conditions for both the measurement at the temperature of interest and for the measurement at the reference temperature. Consequently, this effect cancels in the evaluated relative surface tension. The evaporation at the inner meniscus is insignificant because of the long diffusion length along the capillary. The digital camera observing the liquid meniscus had a constant frame rate of 8 frames per second during all measurements. The counter pressure of helium was recorded using an automated data acquisition (DAQ) unit with a constant time step of 125 ms, that is, with same rate of 8 records per second as the camera frame rate. Data logged by the DAQ unit were synchronized with the recorded video using a light-emitting diode (LED) whose power voltage was also logged by the DAQ unit. More details about the data synchronization and type of the employed equipment are provided in refs 36 and 37. Figure 3 shows several images of the liquid meniscus taken by the digital camera. With increasing helium counter pressure, the meniscus illuminated with the fiber optic light source gradually changed its shape from concave to planar and subsequently to convex. The size of the analyzed images was 485 × 490 pixels. The dark circular spots were caused by dust particles on the camera chip that could not be avoided; the spots did not change during the experiment, and their influence on the analysis of the recorded video could be neglected. In the video analysis, an 8-bit RGB color model, that is, a model consisting of three color channels with values from 0 to 255, was used. Figure 4 shows a contour plot of a weighted sum over the three RGB channels (i.e., RGB converted to a grayscale), which is referred to further as the RGB intensity. The blue pixels in the corner areas of the image had the RGB intensity below 100, while the yellow pixels corresponding to the illuminated liquid meniscus had the RGB intensity typically above 200. The example in Figure 4 corresponds to the planar meniscus shown in Figure 3d, where the image

Δp(Tref, i)σIAPWS(Tref, i)

i=1

4 cos θ

(5)

Consequently, the horizontal measurement is, similarly as the capillary rise method employed in previous studies,18,2,3 a relative measurement. The main output quantity of this work is the relative surface tension Y defined as follows Y (T ) =

σ (T ) σref

(6)

where σref = σIAPWS(Tref) calculated from eq 1 equals to 73.49 mN·m−1 at Tref = 15 °C. The inner diameter of the capillary tubes was inspected also optically by using a digital microscope. However, similarly as in case of the capillary rise method,2,3 the results obtained from the measurements of surface tension at the reference temperature were found more reliable. The capillary tubes had to be placed in a special holder during the grinding and polishing,38 which could cause some unexpected changes of the inner diameter. Moreover, an effect of a possible slight variation of the inner diameter along the capillary tube length is avoided to a great extent by the present procedure. The temperature inside the temperature-controlled chamber was repeatedly cycled between the measured set point and the reference temperature. Figure 2 shows an example of the typical temperature variation during one experiment corresponding to Data Set 5 discussed further. A series of repeated measurements at temperature Tref allowed for detecting possible harmful effects such as occurrence of impurities in the liquid sample or undesirable variation of the inner diameter of the capillary tube along its length. The liquid sample slowly evaporated into the ambient air and the inner meniscus gradually approached the open end of the capillary tube. Due to the repeated measurements at the reference temperature, the inner diameter D



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intensity variation corresponds to light inhomogeneity of the illumination fiber bundle. The counter pressure of helium, initially set to conditions with concave meniscus at the open end of the capillary tube, was gradually increased and subsequently decreased in two loops for each data point measured at the given temperature. This approach helped to detect possible hysteresis effect in the variation of the meniscus shape depending on the helium counter pressure. Each data point consisted of two values corresponding to the increasing counter pressure and two values at the decreasing counter pressure. An example of the counter pressure variation for the data point measured at −10 °C is shown in Figure 5. We note that no hysteresis effect was

Figure 5. Example of pressure variation for measurement at −10.33 °C from Data Set 6; red , recorded pressure; blue ■, pressure corresponding to the planar meniscus determined from the minima in the deviation from the average RGB intensity of the illuminated area; Δp = 911.2 Pa.

detected in any of the final measurements presented in this work. However, several preliminary measurements suffered a remarkable pressure hysteresis pointing on the presence of impurities on the capillary tube wall, which hindered fluent variation of the meniscus shape, or problems in the video transfer and subsequent video analysis addressed in the previous work.37 Four values of counter pressure shown in Figure 5 corresponding to the planar meniscus lie within the range Δp = 911.2 ± 0.6 Pa in this case. The planarity of the liquid meniscus was determined from the recorded video using two different approaches depending on the settings of the optical setup and the quality of recorded video. In the first method, the planar meniscus was considered at the largest illuminated area in the recorded image. In this case, pixels with the RGB intensity higher than the zero level defined at the nonilluminated image corners were counted. The planar meniscus was found at the maximum number of the illuminated pixels. Figure 6 shows a variation of the number of illuminated pixels during two pressure loops depending on time represented by the number of camera frames with a constant rate of 8 frames per second. The number of pixels is shown relative to the first frame. The method of finding the maximum illuminated area worked well for most of the measurements. Figure 7 shows an image of the planar meniscus, which could be correctly analyzed using the method of the maximum illuminated area. In some cases, however, the illuminated area showed a relatively flat maximum with several video frames providing similar results for different counter pressure values. Consequently, the uncertainty of the evaluated surface tension would be unsatisfactorily

Figure 3. Variation of the liquid meniscus at the open end of the capillary tube from concave (a to c) to planar (d) and subsequently convex shape (e and f). Example for measurement at −10.33 °C from Data Set 6.

Figure 4. Contour plot showing the weighted sum over three channels of the RGB color model (RGB intensity) of the planar meniscus. Example for the measurement at −10.33 °C from Data Set 6. E



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Figure 6. Illuminated area (number of illuminated pixels) relative to the first video frame; magenta , illuminated area; black ●, regions with the maximum illuminated area; red ●, maxima of the illuminated area. Example for the measurement at 3.48 °C from Data Set 4.

Figure 8. Deviation from the average RGB intensity of the illuminated area relative to the first video frame; red , deviation from the average RGB intensity; black ●, regions with the maximum illuminated area; blue ■, minima in the deviation from the average RGB intensity. Example for measurement at −10.33 °C from Data Set 6.

capillary tubes used for the final measurements of the surface tension of supercooled water are provided in Table 1 together Table 1. Inner Diameter d of the Capillary Tubes with their Standard Uncertainty u(d) and Contributions to the Relative Standard Uncertainty ur(d) Corresponding to the Reference Surface Tension ur,ref and the Relative Surface Tension ur,Y

Figure 7. Planar meniscus determined from the maximum of illuminated area. Example for measurement at 3.48 °C from Data Set 4 depicted in Figure 6.

high. Therefore, another method was used for measurements with the flat maxima of the illuminated area. The planarity of the liquid meniscus was determined based on the fact that the flat meniscus provides a more uniform reflection compared to a curved (either concave or convex) meniscus. In the planar case, the pixels of the illuminated area had a similar RGB intensity, as illustrated in Figure 4. The uniform reflection of the planar meniscus could be expressed in terms of a deviation from the average RGB intensity of the illuminated area. The planar meniscus was identified as the minimum of this deviation. Figure 8 shows the deviation of the RGB intensity of the illuminated area from its average value. As can be seen, sharp minima in the deviation of RGB intensity were reached, which allowed for a more accurate detection of the planarity of the meniscus compared to the method of maximum illuminated area. The example provided in Figure 8 corresponds to the pressure loop shown in Figure 5. We note that the settings of the optical setup did not change during the experimental run performed with one liquid sample inside the capillary tube. Consequently, the data sets presented further in this work were analyzed in a consistent way using one of the two methods.

data set

d (mm)

u(d) (mm)

ur,ref(d)

ur,Y(d)

1 2 3 4 5 6

0.3325 0.3321 0.3303 0.3312 0.3313 0.3364

0.0021 0.0021 0.0020 0.0020 0.0021 0.0022

0.0050 0.0050 0.0050 0.0050 0.0050 0.0050

0.0037 0.0037 0.0036 0.0036 0.0039 0.0040

with the contributions to the relative standard uncertainty ur(d) = u(d)/d. The relative standard uncertainty of the inner diameter determined in terms of eq 5 consists of two contributions. 2 ur2(d) = ur,ref (d) + ur,2Y (d)

(7)

ur,ref(d) denotes the uncertainty coming from the reference surface tension σref = σIAPWS(Tref) calculated from the IAPWS standard,19 and ur,Y(d) is the contribution from the relative surface tension, which will be discussed further in more detail. We note that, for the horizontal technique, the contribution from the relative surface tension ur,Y(d) ≤ 0.004 is approximately four times higher than in the previous measurements with the capillary rise method3 with ur,Y(d) ≤ 0.001. The uncertainty coming from the reference surface tension ur,ref(d) is the same, as the IAPWS standard was considered as the reference correlation in both experiments. The only difference compared to the capillary rise measurements is the different value for the reference temperature Tref = 15.00 °C. In the previous work,3 the reference temperature of 20.00 and 30.00 °C was considered for the height measurement and the counterpressure measurement, respectively. Table 2 summarizes the final experimental data for the surface tension of ordinary water obtained with the horizontal method. Six data sets measured with horizontal capillary tubes summarized in Table 1 are provided. Measurements at the

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RESULTS AND DISCUSSION As it was discussed in the previous section, the experimental technique employed in this work represents a relative measurement since the capillary tube inner diameter was determined in terms of eq 5 from the comparative measurements at the reference temperature. Values of the inner diameter for six F



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Table 2. continued

Table 2. Relative Surface Tension Y of the Supercooled Water Measured with the Horizontal Capillary Tubes at Temperature T together with its Standard Uncertainty u(Y) and the Absolute Surface Tension σ Related to the Reference Surface Tension σrefa Tb (°C)

Y

−22.33 −22.33 −19.36 −19.34 −16.82 −16.81 −14.35 −14.34 −14.34 −9.89 −9.88 −4.94 −4.94 6.41 6.43 18.85 18.85 20.48 20.48

1.0711 1.0719 1.0660 1.0650 1.0602 1.0610 1.0564 1.0560 1.0560 1.0477 1.0479 1.0388 1.0383 1.0162 1.0170 0.9921 0.9917 0.9893 0.9888

−21.78 −21.78 −21.76 −19.79 −19.79 −17.86 −17.83 −17.82 −17.81 −13.85 −13.85 −10.88 −10.87 −6.92 −6.92 −3.97 −3.96 −3.79 −0.47 −0.46 4.94 4.96 10.35 10.44 10.45 10.52

1.0707 1.0704 1.0706 1.0664 1.0668 1.0618 1.0632 1.0634 1.0627 1.0558 1.0559 1.0509 1.0506 1.0423 1.0421 1.0373 1.0374 1.0368 1.0303 1.0308 1.0200 1.0203 1.0092 1.0092 1.0090 1.0090

−22.35 −22.34 −22.33 −20.27 −20.27 −20.27 −17.81 −17.81 −17.81 −15.32

1.0712 1.0711 1.0708 1.0665 1.0668 1.0670 1.0623 1.0621 1.0622 1.0575

u(Y) Data Set 1 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0052 0.0052 0.0052 0.0052 0.0051 0.0051 0.0051 0.0051 0.0052 0.0052 Data Set 2 0.0054 0.0054 0.0054 0.0053 0.0053 0.0054 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 Data Set 3 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053

σc (mN/m) 78.71 78.77 78.34 78.26 77.91 77.97 77.63 77.60 77.60 76.99 77.00 76.34 76.30 74.68 74.74 72.91 72.87 72.70 72.66 78.68 78.66 78.68 78.37 78.39 78.02 78.13 78.14 78.09 77.58 77.59 77.23 77.20 76.60 76.58 76.23 76.24 76.19 75.72 75.75 74.96 74.98 74.16 74.16 74.15 74.15 78.71 78.71 78.69 78.37 78.39 78.41 78.06 78.05 78.06 77.71 G

Tb (°C)

Y

−15.32 −15.31 −11.35 −11.34 −7.39 −7.39 −7.39 −3.46 0.41 0.49 8.40 20.51 20.51

1.0576 1.0579 1.0510 1.0505 1.0425 1.0434 1.0428 1.0355 1.0290 1.0284 1.0125 0.9888 0.9892

−20.34 −20.33 −16.28 −16.28 −12.32 −12.31 −7.41 −7.40 −2.50 −2.48 3.46 3.48 3.48 7.43 7.44 7.44 22.48 22.50

1.0667 1.0678 1.0594 1.0596 1.0525 1.0526 1.0438 1.0444 1.0344 1.0344 1.0232 1.0231 1.0227 1.0150 1.0146 1.0148 0.9846 0.9842

−22.36 −20.29 −20.29 −20.29 −17.31 −17.31 −17.30 −13.35 −13.35 −9.39 −9.38 −9.38 −6.41 −6.41 −2.98 2.47 2.48 2.48 8.91 8.91 11.46 11.47

1.0729 1.0683 1.0679 1.0678 1.0620 1.0622 1.0623 1.0551 1.0545 1.0468 1.0466 1.0460 1.0416 1.0415 1.0349 1.0238 1.0243 1.0238 1.0123 1.0120 1.0088 1.0085

−23.25 −23.25 −23.24 −21.24 −21.23

1.0734 1.0732 1.0738 1.0702 1.0702

u(Y) Data Set 3 0.0053 0.0053 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0051 0.0051 0.0051 0.0051 0.0051 Data Set 4 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0052 0.0052 0.0052 0.0052 0.0052 0.0051 0.0051 0.0051 0.0050 0.0050 Data Set 5 0.0054 0.0054 0.0054 0.0054 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0051 0.0051 0.0051 0.0051 Data Set 6 0.0054 0.0054 0.0054 0.0054 0.0054

σc (mN/m) 77.72 77.74 77.24 77.20 76.61 76.68 76.63 76.09 75.62 75.57 74.40 72.66 72.69 78.39 78.47 77.85 77.86 77.35 77.35 76.71 76.75 76.02 76.01 75.19 75.18 75.15 74.59 74.56 74.58 72.35 72.33 78.84 78.51 78.47 78.47 78.04 78.05 78.07 77.54 77.49 76.93 76.91 76.86 76.55 76.53 76.05 75.23 75.27 75.23 74.39 74.37 74.13 74.11 78.88 78.86 78.91 78.65 78.64



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absolute surface tension σ(T) can be easily recalculated from the values for the relative surface tension Y(T) given in Table 2 by using eq 6. Figure 9 shows the new data for the surface tension of supercooled water measured with the horizontal capillary tubes

Table 2. continued b

T (°C) −18.25 −18.24 −15.28 −15.27 −10.33 −10.33 −10.32 −7.38 −7.37 −7.37 −4.45 0.46 0.54 0.54

Y 1.0654 1.0652 1.0606 1.0603 1.0483 1.0486 1.0487 1.0423 1.0425 1.0424 1.0372 1.0282 1.0288 1.0293

u(Y) Data Set 6 0.0054 0.0054 0.0054 0.0054 0.0053 0.0053 0.0053 0.0053 0.0053 0.0053 0.0054 0.0053 0.0053 0.0053

σ (mN/m) c

78.29 78.28 77.94 77.92 77.04 77.05 77.06 76.60 76.61 76.60 76.22 75.56 75.60 75.64

Reference temperature Tref = 15.00 °C is the same for all data sets. Standard uncertainty of temperature is better than u(T) = 0.040 K. c σ = Yσref, σref = σIAPWS(15.00 °C) = 73.49 mN/m. a b

reference temperature of 15 °C are not given as they represent more than 120 data points, which were used for the calibration of the inner diameter of the capillary tube. As already mentioned in Experimental Methods, the main result of the present experiment is the relative surface tension Y. Hence, only the standard uncertainty for the relative surface tension is given in Table 2. Main contributions to the standard uncertainty of the relative surface tension u(Y) were the standard uncertainty of temperature including also its stability u(T) ≤ 0.040 °C and the standard uncertainty of counter pressure of helium u(Δp) ≤ 3.5 Pa consisting of two contributions coming from the direct measurement using two independent pressure transducers37 and from the subsequent synchronization of the recorded video with the measured counter pressure. The standard uncertainty of temperature was slightly better compared to the previous capillary rise measurements3 with u(T) ≤ 0.060 °C as more accurate calibrations of the temperature probes were employed in the new experiment. On the other hand, the uncertainty of counter pressure corresponding to the planar liquid meniscus was significantly higher compared to the uncertainties related to the height measurement in the capillary rise technique. As a result, the standard uncertainty of the relative surface tension achieved with the horizontal method is up to 3.5 times higher than in the capillary rise measurements.3 The standard uncertainty of the absolute surface tension σ(T) in mN·m−1 can be determined from the following equation for the relative standard uncertainty ur(σ) = u(σ)/σ ur2(σ ) = ur2(Y ) + ur2(σref )

Figure 9. Surface tension of the supercooled water; red ○, new experimental data; blue , IAPWS correlation19 (eq 1) extrapolated below 0.01 °C; green ---, correlation by Pátek et al.;21 black − · −, IAPWS-E correlation by Rogers et al.28 (eq 3).

together with the IAPWS correlation extrapolated below 0.01 °C (eq 1) and recent correlations by Pátek et al.21 and by Rogers et al.28 (eq 3). All three correlations provide comparable predictions in the temperature range of the new experimental data. The correlation by Pátek et al. provides slightly higher surface tension at low temperatures below −15 °C compared to the IAPWS correlation. In case of the correlation by Rogers et al., the difference is more pronounced as it considers significant increase in the surface tension at very low temperatures around −30 °C. Figure 10 compares new experimental data with the literature data including historical data by Humphreys and

(8)

where σref = σIAPWS (Tref) is the reference surface tension, that is, in our case the surface tension obtained from the IAPWS standard (eq 1) at the reference temperature 15 °C. A weak point of the currently valid IAPWS standard for the surface tension of water is that it provides relatively large uncertainties at temperatures close to the room temperature.3,19 For instance, the uncertainty of the reference surface tension at 15 °C has the value u(σref) = 0.37 mN·m−1. The IAPWS standard is therefore currently being revised22 to improve its uncertainty estimates and to extend its range of validity in the supercooled region. In case a more accurate value for the reference surface tension σcorr(15 °C) becomes available, the

Figure 10. Deviation of the relative surface tension Y from the IAPWS correlation19 (eq 1) σIAPWS divided by the reference surface tension σref = σIAPWS(Tref), Tref = 15 °C; black ▶, Humphreys and Mohler;15 gray ◆, Hacker;1 black *, Floriano and Angell;16 gray ■, Trinh and Ohsaka;17 blue ●, Hrubý et al.;2 blue ▲, Vinš et al.3 counterpressure data; blue ▼, Vinš et al.3 height data; red ○, this study.

Mohler15 from 1895, Hacker’s data1 supporting the SIP anomaly, other relatively scattered data by Floriano and H



Article

region at temperatures above 90 °C. The correlation by Rogers et al28 (eq 3) slightly overpredicts the new data at temperatures below −15 °C as it considers an exponential increase in the surface tension of supercooled water at low temperatures.

Angell16 and Trinh and Ohsaka,17 and our previous measurements with the modified capillary rise technique.2,3 The experimental data are shown as the deviation of the relative surface tension Y from the IAPWS correlation19 extrapolated below the triple point of water σIAPWS/σref. The reference surface tension is taken at 15 °C and equals to 73.49 mN·m−1. As can be seen, the new measurements show a considerably better internal consistency compared to the older data taken from the literature. Moreover, the new data are in good agreement with the previous capillary rise measurements.2,3 We note that red circles at zero line at 15 °C represent more than 120 data points measured at temperatures between 15.34 and 15.55 °C, which were used in the evaluation of the capillary tube inner diameter. Figure 11 shows the uncertainty ranges of the new data obtained with the horizontal capillary tubes and of the previous

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CONCLUSIONS A new measuring technique for investigation of the surface tension of supercooled water was developed in this study. The new method similar to that employed by Hacker1 uses a horizontal capillary tube together with a precise optics for observing the liquid meniscus whose shape is gradually changed through well-defined counter pressure of inert gas. The experimental apparatus with horizontal capillary tube was successfully used for the measurement of surface tension of pure water in the temperature range from −23 to 23 °C. Although the standard uncertainty of new data, given as the relative surface tension Y related to the reference surface tension at 15 °C, is almost 3.5 times higher than in the previous capillary rise measurements,3 the data are internally consistent and are in a very good agreement with the capillary rise data. This fact reassures the correctness of the capillary rise measurements under the metastable conditions. This represents an important conclusion especially with respect to the planned revision and extension of the range of validity of the currently valid international standard for the surface tension of ordinary water.22 The new data do not show the SIP anomaly in the surface tension of water in the considered experimental range from −23 to 23 °C. Consequently, the SIP anomaly observed in the data by Hacker1 already at the temperature around −8 °C is considered as an experimental artifact. However, similarly as the capillary rise measurements, the horizontal capillary data show a slight systematic offset from the extrapolated IAPWS standard at temperatures below approximately −15 °C. It is obvious that measurements at even lower temperatures would be welcome to clarify the SIP anomaly. However, according to our experience, the lowest temperature limit for the horizontal technique is around −24 °C. The capillary tubes employed for the horizontal capillary apparatus require a rather time demanding treatment, and the risk of their damage due to the ice formation represents a serious obstacle. Lower temperatures could possibly be reached with the capillary rise method, for which less sensitive capillary tubes can be used.

Figure 11. Deviation of the relative surface tension Y from the IAPWS correlation19 (eq 1) σIAPWS divided by the reference surface tension σref = σIAPWS(Tref), Tref = 15 °C. Comparison of the new data with the previous capillary rise measurements including the standard uncertainty ranges and with the surface tension correlations; blue ●, Hrubý et al.;2 blue ▲, Vinš et al.3 counterpressure data; blue ▼, Vinš et al.3 height data; red ▶, Data Set 1; red ■, Data Set 2; red ▼, Data Set 3; red ▲, Data Set 4; red *, Data Set 5; red ●, Data Set 6; blue ---, standard uncertainty range of the data by Vinš et al.;3 red − · −, standard uncertainty range of the new data; cyan , correlation by Viisanen et al.;23 green ---, correlation by Pátek et al.;21 black − · −, IAPWS-E correlation by Rogers et al.28

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data measured with the capillary rise technique.2,3 The data are depicted as the deviation of the relative surface tension Y from the IAPWS standard extrapolated below 0.01 °C. As can be seen, most of the new data lie in the uncertainty range of the capillary rise measurements with u(Y) ≤ 0.0020. However, in the case of future analysis of the new data, we strongly recommend to consider the uncertainties provided in Table 2. Differences of above-discussed correlations applicable in the supercooled region of water are also provided in Figure 11. All three correlations by Viisanen et al.,23 Pátek et al.,21 and Rogers et al.28 provide comparable estimates and their results lie within the uncertainty range of the new measurements. The best agreement with the new experimental data is achieved in case of correlation by Pátek et al.21 Quadratic correlation by Viisanen et al.23 (eq 2) provides comparable predictions with the extrapolated IAPWS correlation in the supercooled region. However, as discussed in Correlations for Surface Tension of Supercooled Water, the correlation by Viisanen et al. strongly deviates from the surface tension of water within the stable

AUTHOR INFORMATION

Corresponding Author

*Phone: +420 608 514 106. Fax: +420 286 584 695. E-mail: [email protected]. ORCID

Václav Vinš: 0000-0002-6250-1420 Funding

The study was supported from the Czech Science Foundation grant no. GJ15-07129Y and the institutional support RVO: 61388998. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Karel Studenovský from the Czech Technical University in Prague for his help with the polishing and grinding of the fused silica capillary tubes and to Wolfgang Skopalik from Thorlabs GmbH for his help with the detection of errors in the video transfer from the digital camera. I



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(22) Minutes from 2016 IAPWS Meeting, TPWS/SCSW Working Group, 2016, Dresden, Germany, September 11−16, 2016; http:// www.iapws.org. (23) Viisanen, Y.; Strey, R.; Reiss, H. Homogeneous Nucleation Rates for Water. J. Chem. Phys. 1993, 99, 4680−4692. (24) Lü, Y. J.; Wei, B. Second Inflection Point of Water Surface Tension. Appl. Phys. Lett. 2006, 89, 164106−1−164106−3. (25) Chen, F.; Smith, P. E. Simulated Surface Tensions of Common Water Models. J. Chem. Phys. 2007, 126, 221101. (26) Sakamaki, R.; Sum, A. K.; Narumi, T.; Yasuoka, K. Molecular Dynamics Simulations of Vapor/Liquid Coexistence Using the Nonpolarizable Water Models. J. Chem. Phys. 2011, 134, 124708. (27) Vinš, V.; Celný, D.; Planková, B.; Němec, T.; Duška, M.; Hrubý, J. Molecular Simulations of the Vapor−Liquid Phase Interfaces of Pure Water Modeled with the SPC/E and the TIP4P/2005 Molecular Models. EPJ Web Conf. 2016, 114, 02136. (28) Rogers, T. R.; Leong, K.-Y.; Wang, F. Possible Evidence for a New Form of Liquid Buried in the Surface Tension of Supercooled Water. Sci. Rep. 2016, 6, 33284. (29) Pinnick, E. R.; Erramilli, S.; Wang, F. Predicting the Melting Temperature of Ice-Ih with Only Electronic Structure Information as Input. J. Chem. Phys. 2012, 137, 014510. (30) Ferguson, A.; Kennedy, S. J. Notes on Surface Tension Measurement. Proc. Phys. Soc. 1932, 44, 511−520. (31) Stepanov, V. G.; Volyak, L. D.; Tarlakov, Y. V. Wetting Contact Angles of Certain Systems. J. Eng. Phys. 1977, 32, 646−648; Translated from Inzhenerno-Fizicheskii Zhurnal 1977, 32, 1000−1003. (32) Volyak, L. D.; Stepanov, V. G.; Tarlakov, Y. V. Experimental Investigation of the Temperature Dependence of Wetting Angle of Water (H2O and D2O) on Quartz and Sapphire. Zh. Fiz. Khim. 1975, 49, 2931−2933. (33) Hrubý, J.; Holten, V. A Two-Structure Model of Thermodynamic Properties and Surface Tension of Supercooled Water. Proceedings of the 14th International Conference on the Properties of Water and Steam,2005; http://www.iapws.jp/proceedings.html; pp 241−246. (34) Kalová, J.; Mareš, R. Second Inflection Point of the Surface Tension of Water. Int. J. Thermophys. 2012, 33, 992−999. (35) Hošek, J.; Vinš, V.; Hykl, J. Influence of the Light Source on the Liquid Optical Element Planarity Measurement. Proc. SPIE 2014, 9442, 94420E. (36) Vinš, V.; Hošek, J.; Hykl, J.; Hrubý, J. An Apparatus with a Horizontal Capillary Tube Intended for Measurement of the Surface Tension of Supercooled Liquids. EPJ Web Conf. 2015, 92, 02108. (37) Vinš, V.; Hošek, J.; Hykl, J.; Hrubý, J. Improvements of the Experimental Apparatus for Measurement of the Surface Tension of Supercooled Liquids Using Horizontal Capillary Tube. EPJ Web Conf. 2016, 114, 02135. (38) Hošek, J.; Studenovský, K. Reducing the Edge Chipping for Capillary End Face Grinding and Polishing. EPJ Web Conf. 2013, 48, 00005. (39) Pérez-Díaz, J. L.; Á lvarez-Valenzuela, M. A.; García-Prada, J. C. The effect of the partial pressure of water vapor on the surface tension of the liquid water−air interface. J. Colloid Interface Sci. 2012, 381, 180−182.

ABBREVIATIONS IAPWS, International Association for Properties of Water and Steam; LLPT, liquid−liquid phase transition hypothesis; RGB, 8-bits color model red−green−blue; SIP, second inflection point

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REFERENCES

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