C.
Qiu
a,
Y.
Krüger‡
a,
M.
Wilke§
b,
D.
Marti
ac,
J.
Rička
a and
M.
Frenz
*a
aInstitute of Applied Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland. E-mail: martin.frenz@iap.unibe.ch
bGeoForschungsZentrum Potsdam, Telegrafenberg, 14473 Potsdam, Germany
cDepartment of Photonics Engineering, Technical University of Denmark, Frederiksborgvej 399, Himmelev, 4000, Roskilde, Denmark
First published on 1st September 2016
We present new experimental data of the low-temperature metastable region of liquid water derived from high-density synthetic fluid inclusions (996–916 kg m−3) in quartz. Microthermometric measurements include: (i) prograde (upon heating) and retrograde (upon cooling) liquid–vapour homogenisation. We used single ultrashort laser pulses to stimulate vapour bubble nucleation in initially monophase liquid inclusions. Water densities were calculated based on prograde homogenisation temperatures using the IAPWS-95 formulation. We found retrograde liquid–vapour homogenisation temperatures in excellent agreement with IAPWS-95. (ii) Retrograde ice nucleation. Raman spectroscopy was used to determine the nucleation of ice in the absence of the vapour bubble. Our ice nucleation data in the doubly metastable region are inconsistent with the low-temperature trend of the spinodal predicted by IAPWS-95, as liquid water with a density of 921 kg m−3 remains in a homogeneous state during cooling down to a temperature of −30.5 °C, where it is transformed into ice whose density corresponds to zero pressure. (iii) Ice melting. Ice melting temperatures of up to 6.8 °C were measured in the absence of the vapour bubble, i.e. in the negative pressure region. (iv) Spontaneous retrograde and, for the first time, prograde vapour bubble nucleation. Prograde bubble nucleation occurred upon heating at temperatures above ice melting. The occurrence of prograde and retrograde vapour bubble nucleation in the same inclusions indicates a maximum of the bubble nucleation curve in the ϱ–T plane at around 40 °C. The new experimental data represent valuable benchmarks to evaluate and further improve theoretical models describing the p–V–T properties of metastable water in the low-temperature region.
Speedy and Angell3 observed that the behaviour of the response functions in the supercooled region resembles the behaviour of κT when approaching the spinodal line, where κT ∝ ∂ϱ/∂p|T → ∞. The liquid–vapour spinodal delimits the portions of the thermodynamic p–ϱ–T surface where fluid water can exist in a stable or metastable state. The liquid branch of the spinodal starts at the critical point and runs to negative pressures (tensile stress) with decreasing temperature. However, to explain the divergences in the supercooled water at atmospheric pressure, it would have to run through a pressure minimum and then increase again, as shown in Fig. 1a, to reach zero pressure at around 228 K. Speedy suggested that such re-entrant behaviour is indeed possible, because of the well-known density anomaly of water, the density maximum found at 4.0 °C when moving along an isobar at atmospheric pressure. When moving along isochores p(T|ϱ) with different densities ϱ, one finds the corresponding pressure minima, whose loci form the so-called temperature of maxima density (TMD) line, the dotted line in Fig. 1. The TMD line extends to the negative pressure region where liquid water is in a stretched metastable state. If the TMD line hits the spinodal, then it must be in a spinodal pressure minimum, as can be shown by a simple thermodynamic argument. Thus, the spinodal is re-entrant, possibly reaching zero pressure at the desired temperature of −45 °C.
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Fig. 1 Two models proposed for the stability limit of metastable liquid water. (a) The “stability limit conjecture”5 postulates a single continuous spinodal curve that exhibits a pressure minimum and returns to positive pressures at low temperatures. The TMD line remains negatively sloped in the metastable region. The p–T diagram additionally displays the liquid–solid (L + S) and liquid–vapour (L + V) equilibrium curves. At low temperatures, the spinodal is hidden behind the Ticenr curve indicating homogeneous ice nucleation. (b) The model proposed by Poole et al.6 displays a TMD line that bends over to a positive slope in the metastable region and does not intersect the spinodal. As a consequence of this, the spinodal remains positively sloped and extends to higher negative pressures with decreasing temperature. The model postulates a second critical point (cp2) terminating the HDA–LDA coexistence line (HDA = high-density amorphous ice, LDA = low-density amorphous ice). Note, the liquid–solid coexistence curve and the ice nucleation curve are not shown. |
Speedy's stability limit conjecture was soon challenged, but direct experimental verification remains impossible because in stretched water spontaneous bubble nucleation occurs way before reaching the spinodal, and in the supercooled region the accessible range is limited by ice nucleation. However, a strong indication against Speedy's conjecture came from molecular dynamic simulations using realistic water models. Poole and coworkers6 found evidence that with decreasing density the TMD line in the p–T representation bends to a positive slope and never meets the spinodal, as indicated in Fig. 1b. Thus, the simulated liquid spinodal keeps monotonically decreasing with decreasing temperature. In this scenario, another reason for the anomalies of supercooled water must be sought, for example a Widom line7 emanating from a novel critical point associated with two states of amorphous ice, or a novel state of low-density liquid.8 Moreover, the number of simulations and possible scenarios has increased,4 but the debate still suffers from the lack of experimental data.
The major problem that frustrates experimental studies in the metastable region of liquid water is heterogeneous nucleation of the vapour bubble. Different experimental techniques, such as Berthelot tubes, shock waves, or acoustic cavitation have been used to approach the liquid-spinodal (see the report by Caupin and Herbert9 and references therein), but by far the highest negative pressures of up to −140 MPa have been achieved by (pseudo-)isochoric cooling of synthetic fluid inclusions in quartz crystals (e.g. Zheng et al.10). Due to their microscopic size and strong hydrogen bonds between water and quartz, spontaneous heterogeneous nucleation of both the vapour and solid phases is strongly hampered and the water remains in a stretched metastable state down to large negative pressures. A disadvantage of the fluid inclusion approach, however, is that the fluid pressure cannot be directly measured, but has to be calculated from an adequate equation of state that provides a reasonable extrapolation into the metastable region (e.g. IAPWS-9511).
Previous applications of fluid inclusions in this field have focused on the measurements of spontaneous retrograde bubble nucleation temperatures Tvapnr to determine the maximum negative pressures along different fluid isochores.10,12,13 The density of the water in the inclusions as well as the corresponding liquid-isochores has been calculated from an equation of state, based on the measurements of the liquid–vapour homogenisation temperature Th. Because spontaneous nucleation of the vapour bubble was a prerequisite for the measurements of both Th and Tvapnr, these studies were restricted to inclusion densities lower than 943 kg m−3, corresponding to homogenisation temperatures greater than 120 °C. At higher densities, spontaneous nucleation usually fails to occur, particularly in small inclusions.
A different approach to determine fluid pressures inside the inclusions has been used by Alvarenga et al.14 and recently by Pallares et al.4,15 They measured Brillouin scattering of metastable liquid fluid inclusions at different temperatures and internal pressures and compared the data with reference spectra measured along the liquid–vapour equilibrium curve. From the frequency shift with respect to the reference spectra, they calculated the sound velocity in the stretched water, and finally the pressure inside the inclusions.
In the present study, we report new microthermometric data from synthetic high-density water inclusions that, in most cases, do not show spontaneous vapour bubble nucleation upon cooling. Single ultrashort laser pulses were used to stimulate bubble nucleation in the metastable liquid,16 a precondition for subsequent measurements of the liquid–vapour homogenisation temperature Th (L + V → L), and thus for an accurate determination of the water density in the inclusions. (The reader who is not familiar with this technique may consult Fig. 1 in ref. 17.) Besides prograde homogenisation temperatures Th (upon heating) we also report retrograde liquid–vapour homogenisation temperatures Thr measured upon cooling, ice nucleation temperatures Ticenr both in the presence and absence of the vapour bubble (L + V → S and L → S), final ice melting temperatures Ticem, most of them measured in the absence of a vapour bubble (L + S → L), and some measurements of spontaneous, retrograde and prograde vapour bubble nucleation Tvapnr and Tvapnp (L → L + V), respectively. The IAPWS-95 formulation of Wagner and Pruss11 was used in this study to calculate water densities and simultaneously serves as a reference to compare with our data. We note that IAPWS-95 assumes a negative slope of the TMD line throughout the metastable region, which results in a similar trend of the spinodal as proposed in Speedy's “stability-limit conjecture”.5
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Fig. 2 Assemblage of synthetic fluid inclusions in quartz formed during hydrothermal experiments along a healed crack. |
A detailed description of the preparation and synthesis procedures used to produce the fluid inclusions in this study can be found in Section S2.1 in the ESI.†
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Fig. 3 Simplified scheme of the experimental setup used for this study. See the text for details. DM: dichroic mirror. |
(i) The heating/freezing stage (Linkam THMSG 600) is mounted on the microscope and allows for microthermometric measurements in the temperature range between −180 and 600 °C. Temperature calibrations of the stage were performed using synthetic H2O and H2O–CO2 fluid inclusion standards for the triple point of water (0.0 °C), the triple point of CO2 (−56.6 °C), and the critical point of CO2 (31.4 °C20). Based on these calibrations, we estimate the accuracy of the temperature measurements at ±0.1 °C around 0 °C, at ±0.2 °C below −20 °C and above 40 °C and at ±0.3 °C above 100 °C. The precision (reproducibility) determined from replicate measurements is ±0.05 °C. Further details are provided in Section S2.2 (ESI†).
(ii) An amplified femtosecond (fs) laser system (Coherent) was used to stimulate vapour bubble nucleation in the metastable liquid state of the inclusions by means of single ultrashort laser pulses. For a detailed description of the setup we refer to the ESI† and to Krüger et al.16 The experimental setup allows us to stimulate vapour bubble nucleation in selected fluid inclusions at different temperatures under microscopic observation. Subsequent microthermometric measurements can be performed without moving the sample.
(iii) Confocal Raman spectroscopy was used to identify the ice phase in the inclusions in order to determine ice nucleation temperatures. In pure water inclusions the phase transition is not visible from microscopic observations unless a vapour bubble is present that becomes strongly compressed upon ice nucleation. The fiber-coupled Raman setup consists of a 500 mW diode-pumped solid-state single mode laser with 532 nm wavelength and 1 MHz bandwidth (Torus, Laser Quantum), two dichroic mirrors, a spectrometer (Horiba iHR550M CORE 3) and a back-illuminated CCD camera (Horiba SYNAPSE BIVS); for Raman excitation 60 mW was used. Again, a detailed description of the Raman setup is given in the ESI.† The setup allows us to measure Raman spectra at different temperatures under simultaneous observation of the focus position of the excitation beam relative to the fluid inclusions.
The temperatures of the different phase transitions were measured at very low heating and cooling rates respectively to ensure that the sample was in thermal equilibrium with the heating and cooling block of the stage. When approaching the expected phase transition, i.e., for the final 1 or 2 centigrade the temperature was changed in 0.1 centigrade steps with intervals of 10 to 20 seconds in between. The different phase transitions are illustrated in Fig. 4 in a schematic representation of the p–T diagram of water. The phase diagram displays the liquid–vapour (L + V) and liquid–solid (L + S) equilibrium curves, the bubble–nucleation (cavitation) curve (Tvapnr and Tvapnp), the ice nucleation curve (Ticenr) and three liquid-isochores passing through a pressure minimum.
After Th measurements the inclusions were cooled to room temperature and vapour bubble nucleation was induced again by means of a femtosecond laser pulse. Subsequently, the inclusions were further cooled to measure the retrograde homogenisation temperature Thr following the same procedure as described for the Th measurements. But unlike prograde homogenisation, retrograde homogenisation can only be observed in high-density inclusions, in which Th
r is higher than Ticenr, the temperature of ice nucleation. Fig. 4 shows that the isochores of such high-density inclusions intersect the liquid–vapour equilibrium curve (L + V) twice, at Th and Th
r, respectively. The reproducibility of the measured Th and Th
r values was routinely checked by duplicate measurements alternately measuring prograde and retrograde homogenisation.
In high-density inclusions featuring retrograde liquid–vapour homogenisation, ice nucleation occurs at positive pressures, while in low-density metastable liquid inclusions nucleation of ice takes place at negative pressures. Since these low-density inclusions do not exhibit retrograde homogenisation, we could also use them to measure Ticenr at saturation vapour pressure. To do so, we stimulated vapour bubble nucleation in the metastable liquid and cooled the inclusions along the liquid–vapour equilibrium curve (see Fig. 4). Upon ice nucleation the vapour bubble disappeared again due to the volume expansion of ice relative to liquid water (L + V → S; Fig. 6c).
The nucleation of ice results in an abrupt pressure increase in the inclusions. The pressure in the ice Ih state can be calculated using the equation of state by Feistel and Wagner,22 but since pressure in the inclusions is not measurable, we refrain from exact calculations. To assess the magnitude of the pressure, a rough estimate using Table 11 in ref. 22 is sufficient: in the inclusions with the lowest density of 922 kg m−3 ice nucleates at −30.5 °C (see Fig. 9) and the resulting pressure remains below 0.1 MPa (because the density of inclusions is the same as the density of ice at nearly zero pressure at the given temperature). However, in inclusions with ϱ > 942 kg m−3 the pressure becomes larger than 200 MPa after ice nucleation, exceeding thereby the range of validity of the ice Ih equation of state. Such high internal pressures can cause irreversible volume changes of the inclusions. Therefore, we routinely re-measured prograde homogenisation temperatures after Ticenr and subsequent Ticem measurements to check for a potential decrease of the water densities. In the case of irreversible volume changes, the initially measured Ticenr values did not reproduce in duplicate measurements but were systematically higher.
The inclusions with the lowest water densities analysed in this study (Th > 140 °C) allowed us to measure the ice melting temperature under three-phase conditions (see Fig. 7b). For this purpose, we used the femtosecond laser pulses to stimulate vapour bubble nucleation during melting at −0.1 °C in the presence of liquid water and ice (L + S → L + S + V). In the inclusion shown in Fig. 7b, the final ice melting was measured between 0.0 and +0.1 °C (L + V + S → L + V) as expected for pure water inclusions.
In the first place, we used the IAPWS-95 formulation to calculate the density of the liquid phase at saturation pressure, i.e., along the liquid–vapour equilibrium curve. At this point, we note that in addition to the IAPWS-95 formulation, Wagner and Pruss also reported a polynomial function that can be used alternatively to calculate liquid-densities along the liquid–vapour curve (eqn (2.6), Section 2.3.111). However, since this polynomial is an empirical fit to experimental data above 0 °C, it covers only the stable part of the liquid–vapour curve and should not be extrapolated into the supercooled metastable region. A comparison of the two modes of calculation revealed increasing divergence of the liquid densities below −20 °C, with lower densities resulting from IAPWS-95.
The IAPWS-95 formulation is considered to exhibit ‘reasonable’ behaviour when extrapolated into the metastable liquid region (Section 7.3.211). It allows us to extrapolate the liquid-isochores into the metastable region and to calculate the liquid-spinodal. Similar to the ‘stability limit conjecture’,5 IAPWS-95 assumes a negative slope of the TMD line in the metastable region, and, as a consequence, the spinodal turns up towards positive pressures at low temperatures.
Although IAPWS-95 does not provide any information about the solid states of water, Wagner and Pruss gave a polynomial function describing the p–T trend of the ice melting curve (Ih) (eqn (2.16), Section 2.4.111) derived from Wagner et al.23 The curve is based on a fit to experimental data, including a few measurements of Henderson and Speedy,24 providing ice melting temperatures in the metastable region down to −22.8 MPa. Due to the lack of experimental data at higher tensile stress, we extrapolated the ice melting curve further into the metastable region for comparison with our new ice melting data.
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Fig. 9
ϱ–T diagram illustrating the results of the fluid inclusion measurements. Ticenr: ice nucleation in the absence of the vapour phase (L → S). Ticem: final ice melting in the absence of the vapour phase (L + S → L). Th∞ and Th![]() |
Inclusion |
T
h![]() |
T
h![]() ![]() |
T icenr (L → S) ±0.2 °C | T icem (L + S → L) ±0.1 °C | r(@20 °C) ±0.15 μm | T h∞ (°C) |
T
h![]() |
V (μm3) | ϱ bulk(Th∞) (kg m−3) | ϱ(Ticenr) (kg m−3) | ϱ(Ticem) (kg m−3) |
---|---|---|---|---|---|---|---|---|---|---|---|
13-2-3 | 28.7 ± 0.1 | −12.4 ± 0.1 | −43.3 | 0.4 | 0.61 | 30.18+0.46−0.32 | −13.63−0.39+0.29 | 450−260+400 | 995.55−0.14+0.10 | 997.75−0.13+0.09 | 995.23−0.12+0.10 |
13-2-2 | 29.8 ± 0.1 | −13.2 ± 0.1 | −43.3 | 0.4 | 0.46 | 31.64+0.72−0.46 | −14.70−0.60+0.38 | 160−110+230 | 995.10−0.23+0.15 | 997.34−0.21+0.13 | 994.53−0.18+0.11 |
13-3-3 | 36.4 ± 0.2 | −16.8 ± 0.1 | −43.0 | 0.6 | 0.70 | 37.67+0.45−0.37 | −17.80−0.29+0.23 | 330−170+260 | 993.05−0.16+0.14 | 995.44−0.15+0.12 | 993.37−0.15+0.12 |
13-3-4 | 45.0 ± 0.2 | −21.5 ± 0.2 | −42.4 | 1.0 | 0.55 | 46.42+0.58−0.43 | −22.55−0.37+0.37 | 90−60+100 | 989.57−0.25+0.19 | 992.22−0.23+0.17 | 989.86−0.23+0.17 |
2-2-6 | 49.2 ± 0.2 | −23.2 ± 0.2 | −42.1 | 1.1 | 0.99 | 50.09+0.31−0.29 | −23.84−0.28+0.26 | 430−160+250 | 987.96−0.14+0.13 | 990.70−0.13+0.12 | 985.90−0.12+0.11 |
14-1-4 | 51.4 ± 0.2 | −24.2 ± 0.2 | −42.3 | 1.2 | 0.51 | 52.85+0.62−0.46 | −25.22−0.49+0.37 | 70−30+70 | 986.68−0.29+0.22 | 989.51−0.28+0.20 | 987.87−0.27+0.20 |
14-1-3 | 61.8 ± 0.2 | −28.6 ± 0.2 | −41.1 | 1.8 | 0.61 | 63.00+0.47−0.39 | −29.39−0.37+0.32 | 60−30+60 | 981.59−0.25+0.21 | 984.68−0.26+0.18 | 981.45−0.24+0.20 |
2-1-3 | 71.8 ± 0.2 | −31.9 ± 0.2 | −40.4 | 2.1 | 0.91 | 72.64+0.32−0.29 | −32.42−0.27+0.25 | 160−70+90 | 976.21−0.19+0.17 | 979.55−0.18+0.16 | 978.51−0.18+0.16 |
2-1-3c | 74.5 ± 0.2 | −32.9 ± 0.2 | −40.1 | 2.1 | 0.91 | 75.34+0.31−0.30 | −33.42−0.27+0.25 | 150−60+90 | 974.61−0.18+0.18 | 978.02−0.18+0.17 | 977.00−0.17+0.17 |
8-1-4 | 79.9 ± 0.2 | −34.9 ± 0.2 | −39.7 | 2.3 | 1.60 | 80.44+0.24−0.23 | −35.21−0.22+0.22 | 690−180+230 | 971.49−0.15+0.15 | 975.06−0.16+0.12 | 973.87−0.14+0.14 |
2-1-4c | 81.6 ± 0.2 | −35.1 ± 0.2 | −39.8 | 2.2 | 1.20 | 82.26+0.27−0.26 | −35.48−0.24+0.23 | 280−90+120 | 970.35−0.17+0.16 | 973.94−0.16+0.16 | 972.94−0.16+0.16 |
8-1-1 | 88.8 ± 0.2 | −36.8 ± 0.2 | −38.5 | 2.8 | 1.05 | 89.52+0.28−0.27 | −37.19−0.24+0.23 | 160−60+80 | 965.62−0.19+0.18 | 969.37−0.17+0.19 | 968.24−0.18+0.17 |
8-1-7 | 91.7 ± 0.2 | Not observed | −37.7 | 3.0 | 2.04 | 92.42+0.28−0.27 | — | 150−60+70 | 963.66−0.19+0.18 | 967.49−0.19+0.18 | 966.55−0.19+0.17 |
8-2-7 | 94.8 ± 0.2 | n.o. | −37.3 | 3.1 | 1.35 | 95.38+0.25−0.25 | — | 310−90+100 | 961.61−0.18+0.17 | 965.51−0.17+0.17 | 963.50−0.17+0.17 |
10-5-6 | 111.5 ± 0.3 | n.o. | −36.1 | 4.4 | 1.14 | 112.14+0.36−0.36 | — | 140−50+60 | 949.31−0.28+0.28 | 953.62−0.27+0.27 | 952.74−0.27+0.27 |
10-5-4 | 113.5 ± 0.3 | n.o. | −35.9 | 4.5 | 1.18 | 114.11+0.37−0.35 | — | 150−50+60 | 947.78−0.29+0.27 | 952.13−0.28+0.27 | 951.26−0.27+0.26 |
4-2-6 | 115.2 ± 0.3 | n.o. | −36.1 | 4.6 | 1.07 | 115.86+0.38−0.36 | — | 100−40+50 | 946.41−0.30+0.28 | 950.81−0.29+0.28 | 949.94−0.29+0.27 |
4-2-4 | 120.9 ± 0.3 | n.o. | −35.2 | 5.1 | 1.19 | 121.50+0.36−0.35 | — | 130−50+60 | 941.46−0.30+0.28 | 946.42−0.29+0.28 | 945.58−0.29+0.27 |
4-1-3 | 126.7 ± 0.3 | n.o. | −34.1 | 5.3 | 1.12 | 127.32+0.37−0.35 | — | 100−30+50 | 937.09−0.31+0.30 | 941.77−0.30+0.29 | 940.94−0.30+0.29 |
7-7-10 | 127.4 ± 0.3 | n.o. | −34.4 | 5.4 | 1.20 | 127.99+0.36−0.35 | — | 130−40+50 | 936.53−0.30+0.29 | 941.21−0.30+0.28 | 940.40−0.29+0.28 |
4-1-7 | 132.1 ± 0.3 | n.o. | −33.1 | 5.7 | 1.80 | 132.53+0.33−0.32 | — | 400−90+100 | 932.67−0.28+0.28 | 937.44−0.27+0.27 | 936.65−0.27+0.27 |
4-1-6 | 135.2 ± 0.3 | n.o. | −32.3 | 6.0 | 1.11 | 135.82+0.37−0.36 | — | 90−30+40 | 929.83−0.32+0.31 | 934.66−0.31+0.30 | 933.89−0.31+0.30 |
4-1-1 | 137.8 ± 0.3 | n.o. | −32.1 | 6.1 | 1.82 | 138.22+0.33−0.32 | — | 380−90+100 | 927.71−0.29+0.29 | 932.60−0.28+0.28 | 931.84−0.28+0.28 |
4-1-4 | 141.9 ± 0.3 | n.o. | −31.4 | 6.4 | 1.52 | 142.38+0.34−0.33 | — | 210−60+70 | 924.00−0.31+0.30 | 928.98−0.30+0.29 | 928.23−0.30+0.29 |
6-2-4 | 148.0 ± 0.3 | n.o. | −30.7 | 6.7 | 1.86 | 148.41+0.33−0.32 | — | 360−80+100 | 918.49−0.30+0.30 | 923.60−0.30+0.29 | 922.88−0.29+0.30 |
6-2-1 | 149.7 ± 0.3 | n.o. | −30.5 | 6.8 | 1.55 | 150.17+0.34−0.33 | — | 201−53+65 | 916.85−0.32+0.31 | 922.00−0.31+0.30 | 921.28−0.31+0.30 |
Inclusion | T icenr (L + V → S) ±0.2 °C | T h∞ (°C) | V (μm3) | r(Ticenr) ±0.15 μm |
---|---|---|---|---|
7-2-1 | −37.8 | 145.75+0.33−0.33 | 250−60+80 | 1.36 |
7-6-1 | −38.1 | 145.70+0.32−0.32 | 420−90+100 | 1.64 |
7-6-6 | −38.3 | 146.10+0.37−0.35 | 80−30+40 | 0.94 |
7-6-5 | −38.3 | 146.14+0.35−0.34 | 120−40+50 | 1.06 |
7-6-3 | −38.1 | 146.21+0.33−0.32 | 360−80+100 | 1.52 |
7-6-9 | −38.5 | 146.45+0.33−0.33 | 250−60+80 | 1.39 |
7-6-7 | −38.5 | 147.12+0.38−0.36 | 70−30+30 | 0.90 |
7-2-2 | −37.9 | 147.14+0.33−0.32 | 280−70+80 | 1.42 |
7-6-4 | −38.4 | 147.14+0.34−0.33 | 270−70+80 | 1.40 |
6-2-4 | −38.4 | 148.31+0.33−0.32 | 360−80+100 | 1.59 |
6-2-7 | −38.7 | 148.47+0.36−0.35 | 100−30+40 | 1.02 |
6-2-5 | −39.0 | 148.62+0.37−0.36 | 70−20+30 | 0.90 |
6-2-6 | −39.1 | 148.87+0.36−0.35 | 100−30+40 | 1.03 |
6-3-8 | −39.5 | 149.71+0.35−0.34 | 140−40+50 | 1.12 |
6-3-9 | −39.6 | 150.01+0.35−0.34 | 140−40+50 | 1.14 |
6-2-3 | −38.7 | 150.21+0.37−0.36 | 70−30+30 | 0.91 |
6-2-2 | −38.8 | 151.04+0.38−0.36 | 60−20+30 | 0.87 |
6-3-13 | −39.6 | 150.99+0.35−0.34 | 120−40+50 | 1.10 |
6-2-1 | −38.9 | 150.92+0.35−0.34 | 130−40+50 | 1.13 |
4-2-4 | −39.2 | 121.50+0.36−0.35 | 130−50+60 | 0.86 |
4-2-6 | −39.1 | 115.86+0.38−0.36 | 100−40+50 | 0.72 |
8-2-7 | −39.2 | 95.38+0.25−0.25 | 310−90+100 | 0.55 |
10-5-6 | −38.2 | 112.14+0.36−0.36 | 140−50+60 | 0.71 |
Inclusion | T vapnr (L → L + V) ±0.2 °C | T vapnp (L → L + V) ±0.1 °C | r(20 °C) ±0.15 μm | T h∞ (°C) | V (μm3) | ϱ bulk(Th∞) (kg m−3) | ϱ(Tvapnr) (kg m−3) | ϱ(Tvapnp) (kg m−3) |
---|---|---|---|---|---|---|---|---|
7-6-2 | 53.7 | 29.6 | 1.86 | 145.21+0.33−0.32 | 370−80+100 | 921.43−0.30+0.30 | 924.50−0.29+0.29 | 925.17−0.29+0.29 |
7-2-1 | 55.9 | 22.7 | 1.64 | 145.75+0.33−0.33 | 250−60+80 | 920.94−0.31+0.30 | 923.96−0.30+0.29 | 924.74−0.31+0.49 |
7-6-1 | 59.7 | 22.8 | 1.94 | 145.70+0.32−0.32 | 420−90+100 | 920.99−0.30+0.29 | 923.89−0.29+0.28 | 924.91−0.29+0.29 |
7-6-10 | 53.8 | 29.6 | 1.82 | 145.72+0.33−0.32 | 350−80+100 | 920.97−0.30+0.30 | 924.05−0.29+0.29 | 924.72−0.29+0.29 |
7-6-6 | 64.7 | 17.2 | 1.13 | 146.10+0.37−0.35 | 80−30+40 | 920.62−0.34+0.32 | 923.39−0.33+0.31 | 924.69−0.33+0.31 |
7-6-5 | 65.5 | 17.1 | 1.28 | 146.14+0.35−0.34 | 120−40+50 | 920.58−0.32+0.31 | 923.32−0.31+0.30 | 924.66−0.31+0.30 |
7-6-3 | 63.3 | 17.3 | 1.85 | 146.21+0.33−0.32 | 360−80+100 | 920.52−0.30+0.30 | 923.33−0.29+0.29 | 924.59−0.29+0.29 |
7-6-9 | 62.2 | 23.8 | 1.63 | 146.45+0.33−0.33 | 250−60+80 | 920.30−0.31+0.30 | 923.15−0.30+0.29 | 924.21−0.30+0.29 |
7-6-8 | 52.9 | 26.2 | 2.15 | 146.57+0.32−0.32 | 560−110+130 | 920.19−0.29+0.29 | 923.32−0.28+0.28 | 924.05−0.29+0.28 |
7-6-7 | 62.9 | 18.1 | 1.06 | 147.12+0.38−0.36 | 70−30+30 | 919.68−0.35+0.33 | 922.53−0.33+0.32 | 923.76−0.34+0.32 |
7-2-2 | 61.8 | 24.3 | 1.70 | 147.14+0.33−0.32 | 280−70+80 | 919.66−0.31+0.30 | 922.55−0.30+0.29 | 923.59−0.30+0.29 |
7-6-4 | 68.8 | 16.7 | 1.68 | 147.14+0.34−0.33 | 270−70+80 | 919.66−0.31+0.30 | 922.33−0.30+0.29 | 923.77−0.30+0.29 |
6-2-4 | 64.4 | n.o. | 1.88 | 148.31+0.33−0.32 | 360−80+100 | 918.58−0.30+0.30 | 921.43−0.29+0.29 | — |
6-2-7 | 64.8 | 20.1 | 1.20 | 148.47+0.36−0.35 | 100−30+40 | 918.43−0.33+0.32 | 921.27−0.32+0.31 | 922.50−0.32+0.31 |
6-2-5 | 54.4 | 29.1 | 1.07 | 148.62+0.37−0.36 | 70−20+30 | 918.29−0.35+0.33 | 921.44−0.34+0.32 | 922.14−0.34+0.32 |
6-2-6 | 59.1 | 29.5 | 1.20 | 148.87+0.36−0.35 | 100−30+40 | 918.06−0.33+0.32 | 921.08−0.32+0.31 | 921.91−0.32+0.31 |
6-3-8 | 69.8 | 19.5 | 1.38 | 149.71+0.35−0.34 | 140−40+50 | 917.28−0.32+0.31 | 920.00−0.31+0.30 | 921.40−0.31+0.31 |
6-3-9 | 65.9 | 18.2 | 1.38 | 150.01+0.35−0.34 | 140−40+50 | 917.00−0.32+0.32 | 919.85−0.31+0.30 | 921.16−0.31+0.31 |
6-2-3 | 65.2 | 22.6 | 1.09 | 150.21+0.37−0.36 | 70−30+30 | 916.81−0.35+0.33 | 919.69−0.34+0.32 | 920.87−0.34+0.32 |
6-2-2 | 72.8 | 16.8 | 1.01 | 151.04+0.38−0.36 | 60−20+30 | 916.03−0.36+0.34 | 918.70−0.35+0.33 | 920.25−0.35+0.33 |
6-3-13 | 71.8 | n.o. | 1.29 | 150.99+0.35−0.34 | 120−40+50 | 916.41−0.33+0.32 | 919.10−0.32+0.31 | — |
6-2-1 | 72.8 | 12.6 | 1.35 | 150.92+0.35−0.34 | 130−40+50 | 916.15−0.33+0.32 | 918.81−0.32+0.31 | 920.46−0.32+0.31 |
ϱ(Ticenr) = −5.8481T + 743.53 |
The ice nucleation line intersects the liquid–vapour equilibrium curve at −38.2 °C, providing an estimate of the expected ice nucleation temperature Ticenr at saturation pressure (L + V → S). In comparison to this, the real Ticenr measurements performed in the presence of the vapour bubble (not shown in Fig. 9) yielded slightly lower values, ranging between −37.8 and −39.6 °C (see Fig. S5, ESI†). Although these measurements suggest a potential effect of the vapour bubble on ice nucleation temperatures, we found only a weak correlation between Ticenr and the bubble radius r at Ticenr (see Fig. S6, ESI†). Replicate measurements of individual inclusions reproduced, with few exceptions, within ±0.5 °C. Again, Table 2 and Fig. S5 and S6 (ESI†) report the lowest Ticenr (L + V → S) values measured in each inclusion.
At −36.2 °C, Fig. 9 and Fig. S4 (ESI†) additionally show an intersection of the ice nucleation line with the liquid-spinodal predicted by IAPWS-95 (dashed curve). As a result, ice nucleation temperatures measured in low-density inclusions (ϱ(Ticenr) < 955 kg m−3) are plotted on the left hand side of the spinodal i.e., within the IAPWS-95 defined region of the unstable liquid. Therefore, IAPWS-95 cannot provide any pressures at Ticenr for these low-density inclusions, which is another reason for presenting our data in a ϱ–T diagram. It should be noted that we did not observe any indication for a change in the state of liquid water above Ticenr. Moreover, we found that the application of femtosecond laser pulses in the doubly metastable region, even close to Ticenr, resulted in vapour bubble nucleation but never stimulated ice nucleation. We therefore conclude that ice nucleation in fluid inclusions is not triggered by bubble nucleation as it was observed by Barrow et al.34 in Berthelot tube experiments.
In addition, Fig. 9 also displays some Tvapnr data from previous studies.10,12,13,29,31 For the calculation of the corresponding ϱ(Tvapnr) from these studies we treated Thobs equal to Th∞ since they only report Th
obs values. In fact in inclusions with ϱ < 930 kg m−3, the relative difference in ϱbulk resulting from the difference between Th
obs and Th∞ is less than 0.05%, and thus, is negligible. In contrast to our own data, the Tvapnr values reported in ref. 10, 29 and Table 1 of ref. 13 show a much larger scatter by several tens of degrees for different inclusions of similar densities. We therefore display only the lowest of these values in Fig. 9. The locus of the nucleation data points in the ϱ–T plane appears to form a bubble nucleation (cavitation) curve that exhibits a maximum at around 40 °C (see Discussion in Section 4.4).
Fig. 10 demonstrates that the Th∞–Thr∞ data obtained in this study are in excellent agreement with IAPWS-95 (also see Fig. S7, ESI†), and thus, confirm the results of the IAPWS-5 formulation with respect to saturation liquid densities in the supercooled region. This finding suggests that the IAPWS-95 isochores for densities larger than 970 kg m−3, as drawn through the metastable region from Th
r∞ to Th∞, through the temperature of maximum density, are essentially correct (the pseudo-isochore p(T|ϱ = 970 kg m−3) is included in Fig. 12b). However, our data discussed in the following paragraphs indicate that the validity of the IAPWS-95 extrapolation becomes questionable when diving deeper into the stretched metastable region.
ϱ(Ticem) = −0.01279T4 + 0.1368T3 − 0.06136T2 − 9.637T + 1000 |
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Fig. 11 ϱ–T diagram illustrating the deviation of our experimental ice melting data (dots) from the extrapolated reference curve23 (dotted line). Solid line: weighted fit to our data. |
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Fig. 12 (a) Enlarged detail of the ϱ–T phase diagram of water displaying the cavitation (bubble nucleation) curve ϱcav(T) (thick grey curve) derived from experimental data of this study and previous studies as well as the final ice melting data (this study). The cavitation curve exhibits a maximum at around 40 °C. Besides the thermodynamic spinodal (dashed line, IAPWS-95), the diagram features two kinetic spinodals for δT = 0 and +0.1 nm40 (lower and upper grey dash-dotted curves, respectively). The two TMD lines (IAPWS-95 and Pallares et al.15) have negative slopes. As a guide to the eye the diagram displays two pseudo-isochores (solid, nearly horizontal grey lines) at Th∞ of 143 and 151 °C, respectively. (b) p–T representation of the water phase diagram derived from (a). The pseudo-isochores (solid grey curves) exhibit a pressure minimum, while the cavitation curve pcav(T) (thick grey line) is only gently curved. The two kinetic spinodals for δT = 0 and +0.1 nm40 (lower and upper grey dash-dotted curves, respectively) are illustrated as well. |
Data points that we regarded as less reliable (see Section 3.1) were weighted low for curve fitting. The deviations from the reference increase with increasing Ticem, i.e., with decreasing density ϱ(Ticem) and are significantly larger than the uncertainty of the temperature measurements. At the present stage of research, it is hard to tell if these deviations mainly indicate the limits of the extrapolation or if effects associated with the confinement of the ice–liquid system in a microscopic inclusion need to be considered.
One effect that definitely plays a role is elastic deformation of the inclusion walls under large tensile stress, which results in a volume decrease and thus an increase of the water density. To account for the difference between our data and the extrapolated curve we would have to assume elastic volume changes of up to 1.8% which seem too much considering the maximum tensile stress of −130 MPa at Ticem in the lowest density inclusions (according to IAPWS-95). However, Burnley and Davis36 have demonstrated that elastic volume changes of fluid inclusions depend not only on the magnitude of differential stress but also on the size and the shape of the inclusions, the thickness of the inclusion walls and the orientation of the inclusions within an anisotropic crystal.
The effect of surface tension at the ice–liquid interface might also have an impact on ice melting temperatures. Actually, one would expect that the surface tension affects ice melting in a similar way as it affects the liquid–vapour homogenisation temperature. (recall Section 2.4.2. and ref. 17). However, for high density inclusions with Ticem near zero our data approach the results of Henderson and Speedy24 obtained in a macroscopic Berthelot tube. This may indicate that the surface tension at the ice–water interface is sufficiently small (estimates vary from 0.04 mN m−1 by van Oss et al.37 to about 30 mN m−1 by Hardy,38 compared to 75 mN m−1 for water), and the size of the inclusions is sufficiently large to neglect the effect of surface tension. On the other hand, interfaces with the host material may also play a role.
Setting aside, for now, the question of the validity of the melting curve extrapolation, we conclude the discussion of ice melting by pointing out the following observation: the lowest melting points nearly approach the IAPWS-95 spinodal, but nevertheless, the stretched ice melts into a homogeneous stretched liquid. This is yet another indication for the limitation of the IAPWS-95 extrapolation into the deeply stretched region. In the next section, we shall have a closer look at this finding.
A new finding of our study is the existence of prograde vapour bubble nucleation (cavitation) that occurs upon heating of metastable liquid inclusions above the ice melting temperature. The prograde Tvapnp(ϱ) appears to slightly increase with increasing density, in contrast to the decreasing trend of retrograde Tvapnr(ϱ). This finding suggests to associate the pro- and retrograde values with a single cavitation curve ϱcav(T) which is not monotonic, but rather curved, exhibiting a maximum at around 40 °C. In our study we never reached this maximum, lacking samples with bulk densities in a small interval between 924 and 921 kg m−3; therefore, the gap. The thick grey line in Fig. 12a represents a suggestion for the ϱcav(T) curve. The nucleation data for low density inclusions at high temperatures10 are taken into account in a simplified fashion by requiring the ϱcav(T) to end at the critical point.
The formation of the cavitation curve is illustrated in Fig. 12b, where we translated all ϱ–T values into p–T using the IAPWS-95 formulation. The extrapolation of IAPWS-95 deep into the metastable region is questionable, but still the (pseudo-) isochores are qualitatively correct, exhibiting a pressure minimum close to the TMD line recently determined by Pallares et al.15 Thus, our data are consistent with the following picture: when moving along an pseudo-isochore p(T|ϱ), irrespective of the temperature increase or decrease, cavitation will occur at a temperature T where the pressure p(T|ϱ) approaches a certain critical value pcav, which depends on the temperature. In Fig. 12b the thick grey line is the proposed pcav(T) curve obtained by IAPWS-95 transformation of ϱcav(T) from Fig. 12a. While pseudo-isochores shown in Fig. 12b are curved, exhibiting a minimum at the TMD line, the cavitation curve pcav(T) is a monotonic, gently curved function of temperature. Because of the difference in curvature, an isochore p(T|ϱ) can intersect the cavitation curve pcav(T) twice, or not at all. Unfortunately, it is not clear from our data whether or not pcav(T) exhibits a pressure minimum at low temperatures. If so, the pressure minimum would be at a temperature below the intersection with the TMD line predicted by IAPWS-95.
In Fig. 12a and b we plot one of the existing theoretical proposals for the cavitation curve, namely the “kinetic spinodal” by Kiselev and Ely for comparison.40 The two kinetic spinodals were calculated based on IAPWS-95, but in addition took into account density fluctuations and the effect of the curvature of the liquid–vapour interface. The two grey dash-dotted curves, thus correspond to two values of the curvature parameter Tolman length (δT). Apparently, a kinetic spinodal with a Tolman length between 0 and 0.1 nm would coincide with our proposal at high temperatures, but at temperatures around the maximum of the ϱcav(T) curve the differences become large. This could be due to the failure of the extrapolation of IAPWS-95 deep into the metastable region. Notice that the form of the two kinetic spinodals resembles the form of the re-entrant IAPWS-95 spinodal. In fact, it has been proposed (Caupin,41 Caupin and Stroock42) to exploit the cavitation data to obtain information on the p–T trend of the spinodal. Looking at our cavitation data from this point of view, we confirm our conclusion from ice nucleation data in Section 3.2: the actual spinodal, if re-entrant at all, must have its pressure minimum at a much lower temperature than that predicted by IAPWS-95 and a steeper negative slope at low temperatures.
However, we feel that one should not over-interpret the cavitation data and their theories. Unfortunately, there is still an unanswered question about the origin of the large difference in cavitation pressures obtained in microscopic inclusions and macroscopic Berthelot tubes or by the acoustical technique.42 Thus, one should remain open to a substantial revision of the theories of cavitation in the stretched state of water. Perhaps one should take explicitly into account the temperature dependence of the strength of the hydrogen bond,43 which plays a role in the cohesion of water as well as in adhesion to the host material.
On the one hand, we confirm IAPWS-95 concerning saturation liquid-densities in the supercooled region. Thus, IAPWS-95 seems to be appropriate for inclusion densities larger than 970 kg m−3.
On the other hand, we disproof the low-temperature trend of the liquid-spinodal predicted by IAPWS-95, due to inconsistency with ice nucleation data in the doubly metastable region. This finding is further supported by our ice melting data, demonstrating that ice melts into a homogeneous liquid even for low densities with a melting point close to the predicted spinodal. Such inclusions exhibit prograde vapour bubble nucleation upon heating, in addition to the well-known retrograde bubble nucleation upon cooling. The resulting bubble nucleation (or cavitation) curve is not monotonic in the ϱ–T representation, but exhibits a maximum at around 40 °C. (Is this related with the compressibility minimum?) The cavitation pressure curve pcav(T) obtained therefrom using IAPWS-95, is monotonic and only slightly curved.
Our findings do not determine whether the spinodal is re-entrant or not. However, looking at the ϱ–T data in Fig. 9, which are not affected by the questionable EOS extrapolation, one closing remark seems to be appropriate: liquid water with a density of 921 kg m−3 remains in a homogeneous state during cooling down to a temperature of −30.5 °C, where it is transformed into ice whose density corresponds to zero pressure. Is this finding compatible with any of the proposed scenarios?
L | Liquid phase |
S | Solid phase |
V | Vapour phase |
Qtz | Quartz host |
TMD | Temperature of maximum density |
T | Temperature (°C) |
V | Volume (μm3) |
p | Pressure (MPa) |
ϱ | Density (kg m−3) |
obs | Observed (measured) value |
h | Liquid–vapour homogenisation |
n | Nucleation |
m | Melting |
∞ | Hypothetical value for infinitely large volumes |
p | Phase transition occurring upon heating |
r | Phase transition occurring upon cooling |
Footnotes |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6cp04250c |
‡ Current address: Institute of Environmental Physics, University of Heidelberg, Im Neuenheimer Feld 229, 69102 Heidelberg, Germany. |
§ Current address: Institute of Earth and Environmental Sciences, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476 Potsdam-Golm, Germany. |
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