THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE

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1 CHAPTER 1 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE Frédéric Caupin 1 and Abraham D. Stroock 2 1 Institut Lumière Matière, Université Claude Bernard Lyon 1 and Institut Universitaire de France, Lyon, France 2 School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York INTRODUCTION Liquid water exhibits numerous anomalies, such as its expansion upon cooling below 4 C. One of the explanations put forward involves polymorphism: two different liquid phases could exist at low temperature, in the supercooled region; this proposal has focused significant experimental and theoretical interest on the supercooled regime. We study another metastable state of water, where the liquid is at negative pressure, as explained in Section 1.2. The properties of water in this region of the phase diagram are largely unknown, although they could shed light on the debate about supercooled water, as argued in Section 1.3. Recent experiments have questioned the exact value of the limit of metastability of water at negative pressure. In Section 1.4, we present some techniques used to explore this exotic state. In Section 1.5 we discuss studies of the stability limit of water under tension based on these techniques. In Section 1.6, we consider other properties of water in this regime. Finally, we Please enter \offprintinfo{(title, Edition)}{(Author)} at the beginning of your document. 1

2 2 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE conclude in Section 1.7 with the identification of several outstanding questions that should be addressed in the future. 1.2 WHAT IS NEGATIVE PRESSURE? It is sometimes believed that the pressure is always a positive quantity. This constraint holds for a gas, but not for a liquid. In the dense phase, the molecules are close to each other and experience large attractive forces responsible for the cohesion of the liquid. They may thus resist an externally applied mechanical tension, resulting in a negative internal pressure. The generality of this possibility was clarified by the van der Waals equation of state [31] (Fig. 1.1): at low temperatures, the pressure along the liquid branch of a van der Waals isotherm can pass through zero pressure (Fig. 1.1.A); the liquid is a thermodynamically metastable state that only becomes completely unstable with respect to the vapor at the minimum of the isotherm, where it crosses the spinodal (Fig. 1.1.B). The metastable state of a liquid under tension is a form of superheat. As with a superheated liquid, the emergence of the stable, vapor state occurs via a nucleated process with an activation barrier. While thermodynamically unstable, the liquid can remain intact for long periods of time until this activation barrier is overcome: the metastable Pressure stable liquid unstable metastable liquid (a) metastable vapor liquid-vapor equilibrium Volume P>0 P<0 stable vapor binodal Temperature stable vapor metastable critical point critical point spinodal unstable Density (b) binodal metastable stable liquid stable liquid Pressure binodal metastable P>0 P<0 spinodal unstable Temperature (c) Figure 1.1 Schematic cuts of the phase diagram of a pure substance, showing the stable, metastable, and unstable regions: (a) a low temperature isotherm with a metastable liquid branch reaching negative pressure, (b) temperature-density cut, and (c) pressure-temperature cut.

3 THE PHASE DIAGRAM OF WATER 3 state is kinetically stable. We note that even the liquid state of noble gases can support tension at sufficiently low temperatures [20, 21, 101]. The tensile properties of liquid water are not just a curiosity of the physical chemist. Indeed, the long standing Cohesion-Tension theory of transpiration motion of water from the root to the leaves of plants states that the flow of water in the capillaries is driven by a reduction of pressure in the leaves and that this pressure frequently drops below zero (for example in trees of greater than 10 m in height) [8, 34, 70, 28]. Experimental measurements indicate that water in certain plants can reach pressures down to approximately 10 MPa [79]. The suckers on the tentacles of octopi offer another example of nature s use of water at negative pressure: experimental measurements showed that the pressure in the liquid beneath a sucker could reach MPa [85]. Negative pressures also play a role in technological contexts. For example, large flow velocities or fluctuations in pressure in turbulent flows around propellers can bring the liquid to negative pressures transiently and cause cavitation; the subsequent collapse of bubbles formed by this process can damage the blades [5, 37]. This mechanical action associated with the growth and collapse of bubbles in an oscillatory pressure field is exploited in the process of sonication that is used for cleaning surfaces [60] and has been explored as a means of controlling chemical reactions [95]. The ability to reliably manipulate liquids at negative pressure could allow for improved technologies for heat transfer and water management [103]. 1.3 THE PHASE DIAGRAM OF WATER While water shares the generic features of van der Waals liquids in Fig. 1.1, it also presents numerous anomalous properties and the structure of its phase diagram remains an active topic of research [32]. Of particular interest over the past decades have been the properties of liquid water in the metastable state of supercooling: upon cooling below the line of fusion, Angell and others have shown that numerous thermodynamic and dynamic properties of liquid water differ in their evolution with temperature from that expected of simple liquids. Extrapolations of measurements on supercooled water even suggested a possible divergence of many properties at a common temperature of 228 K at ambient pressure [88]. Several explanations have been proposed to account for the anomalies of water: : (i) the stability limit conjecture [89], (ii) the metastable liquid-liquid critical point hypothesis [74], and (iii) the singularity free scenario [77]. An excellent review of the properties of supercooled water and the explanations proposed can be found in Ref. [32]. We just give here a brief overview. (i) Water exhibits a line of density maxima (LDM). Speedy has shown that, if the LDM meets the spinodal line, thermodynamics imposes that the latter reaches a minimum [89]. Speedy made a step further in proposing that it would retrace up to positive pressure, and be at the origin of the suggested

4 4 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE 100 Pressure (MPa) Temperature (K) Figure 1.2 Phase diagram of water proposed by Speedy [89] plotted using the IAPWS EoS [102, 99]. The blue curves are equilibrium lines, the red curve is the liquidvapor spinodal, and the green curve the LDM. The green circles show the experimental determination of the LDM at negative pressure [43]. When the spinodal and the LDM meet, the spinodal pressure reaches a minimum, and (for this EoS) retraces to positive pressure at low temperature. However, note that this would imply an improbable crossing between the spinodal and the metastable liquid-vapor equilibrium (see text for details). divergences in supercooled water (Fig. 1.2). Debenedetti has argued that this was not possible because it was forbidden for the liquid-vapor spinodal to intersect the metastable continuation of the liquid-vapor equilibrium line: indeed, this can happen only at a critical point (for a more detailed argument, we refer the reader to Refs. [90, 33]). However, two possibilities remain for a spinodal showing a minimum: (i) the liquid-vapor spinodal could retrace, while always remaining at negative pressure; but in this case, one would need another explanation for water anomalies at positive pressure, such as a liquidsolid spinodal, a behavior observed in a lattice model [78]. (ii) in the case of a liquid-liquid transition with a critical point lying beyond the liquid-vapor spinodal (critical point free scenario, see below), the liquid-vapor spinodal could be pre-empted by the spinodal of the transition between high density to low density liquid water [4]; this spinodal is allowed to cross the metastable liquid-vapor equilibrium line. (ii) Another scenario, avoiding the need for a retracing spinodal, was first proposed by Poole et al. based on molecular dynamics simulations. A first order transition line in supercooled water could exist, separating two liquids with different structures, a phenomenon called liquid polymorphism. The transition would terminate at a metastable critical point (Fig. 1.3B). From this critical point emanates the locus of maxima of the correlation length,

5 THE PHASE DIAGRAM OF WATER 5 Figure 1.3 Different phase diagrams of water generated by changing parameters in the cell model of Stokely et al. [93]. (A) Singularity free (SF). (B) Liquid-liquid critical point (LLCP). (C) Liquid-liquid critical point at negative pressure. (D) Critical point free with reentrant stability limit (CPF/SL). HDL and LDL refer to high and low density liquids, respectively. The L-L Widom line is the locus of maxima in the correlation length emanating from the LLCP. Reprinted with permission from Ref. [93]. called the Widom line. Close to this line, several thermodynamic quantities exhibit a peak, which would explain the anomalies of supercooled water. Until recently, all molecular dynamics simulations with various potentials for water confirmed the existence of a liquid-liquid transition terminating at a second critical point (see Ref. [1] and references therein). However, these results were recently challenged by Limmer and Chandler, who argue that the second liquid does not exist, being unstable with respect to crystallization [56]. The controversy is ongoing, with recent papers reaching the opposite conclusion to Limmer and Chandler [80, 53]. (iii) In the words of Debenedetti, the singularity-free scenario [77] proposes that, the experimentally observed increases in waters response functions upon supercooling are explained as the thermodynamically inevitable consequences of the existence of density anomalies [32]. A first attempt to unify these different pictures was made by Poole et al. [73] who used an extension of the van der Waals equation incorporating the effects of hydrogen bonds. Two of the scenarios could be obtained, depending on the input for the hydrogen bond energy. A similar approach was recently reported by Stokely et al. [93]. With a microscopic cell model, they were able

6 6 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE to switch from one scenario to the other by changing the values of the model parameters (Fig. 1.3). For all values, they find a liquid-liquid transition. In Fig. 1.3.A, its critical temperature is brought down to zero, and the phase diagram corresponds to the singularity free scenario (iii). In Fig. 1.3.B the metastable liquid-liquid critical point hypothesis (ii) is predicted. Fig. 1.3.C gives a variant where the critical point lies at negative pressure. We note that this case has been predicted in some simulations[97, 17, 18]. Finally, Fig. 1.3.D shows the case where the critical point falls beyond the liquid-vapor spinodal. This would be similar to the re-entrant spinodal scenario (i), but with the liquid-vapor spinodal becoming a liquid-liquid spinodal at low temperature, as mentioned above. At negative pressure, these scenarios differ with respect to the shapes of the LDM and of the liquid-vapor spinodal curve P s (T ): scenario (i) predicts a monotonic LDM and a minimum of P s as a function of temperature, whereas scenarios (ii) and (iii) predict a turning point in the LDM and a monotonic spinodal. In an experiment, it is difficult to reach the spinodal: rather, the liquid will break before by nucleation of vapor bubbles (cavitation). Usually, impurities favor heterogeneous nucleation, and lead to irreproducible results. But for a pristine system, nucleation will occur homogeneously, at a welldefined pressure threshold P cav (T ), which is an intrinsic properties of the liquid. Figure 1.4 shows some calculations of P cav (T ). The simplest approach is called classical nucleation theory (CNT) [112, 36]. CNT considers the nucleation of a spherical bubble with an infinitely sharp interface between liquid and vapor; P cav is then directly related to the bulk surface tension of the liquid. At low temperature, the surface tension is large, and CNT may lead to values of P cav beyond the spinodal limit, which is unphysical. This situation is shown in Fig. 1.4 for two different equations of state (EoSs) which illustrate two of the above scenarios: (i) with a retracing spinodal (Fig. 1.4.A), and (ii) with a monotonic spinodal (Fig. 1.4.B). An improvement on CNT is provided by density functional theory (DFT), which explicitly incorporates the EoS and hence the spinodal, and accounts for a diffuse liquid-vapor interface. Figure 1.4 shows that the shape of the homogeneous nucleation curve P cav (T ) predicted by DFT is qualitatively similar to that of P s (T ) (green and red curves) [19]. Thus, it appears that measuring the EoS or the LDM at negative pressure, or the temperature variation of P cav, could help to distinguish between the proposed scenarios (i)-(iii) and, in turn, shed light on the proposed liquidliquid transition in water. 1.4 EXPERIMENTAL METHODS TO GENERATE TENSION A detailed review on cavitation in water is available [22]. We have tried to bring part of it up to date, focusing on the methods, summarized in Fig. 1.5,

7 EXPERIMENTAL METHODS TO GENERATE TENSION Pressure (MPa) A Temperature (K) Pressure (MPa) B Temperature (K) Figure 1.4 Cavitation pressure as a function of temperature for two scenarios for water: re-entrant spinodal scenario (A) and liquid-liquid critical point scenario (B). These scenarios predict a different temperature behavior for the liquid-vapor spinodal (red curve), either with a minimum (A, based on extrapolation of positive pressure data [19]), or monotonic (B, based on molecular dynamics simulations with the TIP5P potential [111]). The blue curve shows the prediction of CNT based on the bulk surface tension of water; it becomes unphysical when it goes beyond the liquid-vapor spinodal. The green curve is the DFT prediction [19] which correctly remains above the spinodal, and reflects its temperature dependence.

8 8 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE A liquid B vapor: T, P vap /P sat < 1 piezo liquid (pure): T, P liq C D Figure 1.5 Sketches of several methods used to put a liquid under mechanical tension. (A) Acoustic method. A hemispherical piezoelectric transducer emits focused ultrasound bursts (yellow arrows) into a bulk liquid [44]. (B) Metastable vapor-liquid equilibrium. A nanoporous membrane or gel mediates the equilibrium of a bulk volume of liquid and it sub-saturated vapor[103]. (C) Berthelot tube. A rigid container partially filled with a liquid in equilibrium with its vapor is heated until the liquid expands to fill the entire volume. Upon cooling, the liquid follows a isochore and its pressure decreases [11, 114]. (D) Centrifugal method. A tube formed with two symmetrical bends at each end (a z-tube) is spun around its mid-point such that the pressure in the liquid drops due to the centripetal acceleration acting on the column of liquid [13]. which are the most promising to gain knowledge on the properties of water at negative pressure Acoustic cavitation As an acoustic wave is a succession of compression and tension, it can stretch a liquid to negative pressure, provided the amplitude is large enough. However, early experiments reached only moderate tensions, presumably because of heterogeneous nucleation. The first studies which reported high tensions,

9 EXPERIMENTAL METHODS TO GENERATE TENSION 9 down to -20 MPa [38, 40], used a focused wave, which avoided heterogeneous nucleation on the container walls. They also obtained larger tensions by using cleaner water with a reduced concentration of dissolved gas. Both used hydrophones to estimate the pressure, but this demands calibration. Greenspan and Tschiegg also used a static pressure method: they measured the variation of the driving voltage of the acoustic emitter with the ambient pressure, P stat in the liquid (in the absence of any sound wave)[40]. Greenspan and Tschiegg found a linear variation for P stat between 0 and 0.8 MPa. Extrapolating linearly to zero voltage gives an estimate of the cavitation pressure. The acoustic method which has been extensively studied in Paris uses a hemispherical piezoelectric transducer (Fig. 1.5.A) to focus bursts of a 1 MHz sound wave [44]. The high frequency allows to stretch a small volume of liquid (typically (100 µm) 3 ), during a short time (typically 100 ns). The nucleated bubbles can be detected optically, by the audible sound emitted, or by the echo they reflect back to the transducer. The three methods provide perfectly consistent results with respect to one another [45]. The third detection method is easier to use because the transducer itself serves as a detector: its voltage changes when the echo reaches its surface. The cavitation pressure has been calibrated by three methods. The first uses a commercial piezoelectric needle hydrophone placed at the focus. This method has a limited accuracy (13% uncertainty in the hydrophone gain), and moreover, as the needle is fragile, it can be used only at low amplitude, so that an extrapolation to the cavitation threshold is required [44]. The second method is the static pressure method mentioned above, which was extended to P stat between 0 and 10 MPa. Again, a linear relation was found, which could be extrapolated to zero voltage to give an estimate of P cav [44]. The two first methods agree with each other, but both involve long extrapolations. The third and most accurate method uses a fiber optic probe hydrophone (FOPH)[92, 6]. The reflectivity of the interface between an optical fiber and water gives an absolute measurement of the liquid density. When the fiber is placed at the focus of the wave, the density is modulated, and its minimum value can be obtained up to amplitudes close to the cavitation threshold [30]. The last method gives a slightly more negative pressure than the others, which is likely due to non-linearity in the focusing of the wave. With all of these methods of calibration, P cav was found to be a monotonically increasing function of temperature from -34 MPa at 0 C to -5 MPa at 200 C; this magnitude is in stark disagreement with predictions (Fig. 1.4). We return to this topic in Section 1.5. Note that the acoustic method was also used to investigate other liquids [7]. It gives P cav in good agreement with CNT for liquids with a surface tension lower than water (heptane and ethanol); for dimethylsulfoxide, with an intermediate surface tension, the method gives a stability limit that disagrees with CNT, but to a lesser degree than for water. Recently, the Paris group has shown that other analyses can be performed with the acoustic method. One interesting feature of the method is its high reproducibility, which allows the use of repeated bursts with the same exper-

10 10 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE imental conditions: temperature, ambient pressure and pressure amplitude. The observed cavitation is random, and the cavitation probability can be defined as the fraction of bursts leading to cavitation. When the sound amplitude is increased, the cavitation probability goes smoothly from 0 to 1, with a characteristic S-shape which is expected for a thermally activated nucleation mechanism [44]. This detailed information on the cavitation statistics contrasts with many other experiments, where either only the largest negative pressure obtained in one run or sample is reported, or the series of cavitation pressure does not obey any well-defined statistical law. The slope of the probability versus pressure curve gives access to the size of the critical bubble through the nucleation theorem [71]. It is around 10 nm 3 at room temperature [30]. Finally, the acoustic method can be used to generate an EoS at negative pressure. A difficulty lies in the short time during which the tension is generated (typically 100 ns for a 1 MHz wave). Yet, Davitt et al. have were able to perform a time-resolved Brillouin experiment by averaging over repeated pulses [29]. The light scattered from the water at the acoustic focus was collected, and its spectral features analyzed during a time window of around 100 ns synchronized with the sound wave. The shift of the Brillouin lines thus gives the sound velocity of water, while it is submitted to negative pressure. As the density of the liquid could also be measured with the FOPH under the same conditions, this yields an experimental EoS of water at negative pressure. Davitt et al. could measure it down to -26 MPa, and found good agreement with standard extrapolations of positive pressure data [29]. Unfortunately, no measurements could be performed at larger tensions, because of the increasing cavitation probability Metastable vapor-liquid equilbrium A liquid and its vapor can coexist at pressures that deviate from the binodal. This phenomenon is familiar in the context of capillarity: if a curved meniscus separates a liquid and its vapor, the Young-Laplace law predicts a difference in pressure between the two phases: P liq P vap = 2γ cos θ r, (1.1) where γ is the surface tension, θ is the contact angle, and r is the radius of curvature of the interface that separates the phases. For a positively curved meniscus (r > 0), as for a droplet of liquid, the liquid pressure is larger than that of the surrounding vapor; for a negatively curved meniscus (r < 0), as for a liquid confined in a pore in a wettable solid, the liquid pressure is lower. If the fluid is allowed to equilibrate between the two phases, the pressures must also satisfy the balance of the chemical potential µ vap (P vap, T ) = µ liq (P liq, T ),

11 EXPERIMENTAL METHODS TO GENERATE TENSION 11 which yields Pvap Pliq µ 0 + v vap (P, T ) dp = µ 0 + v liq (P, T ) dp, (1.2) P 0 P 0 where P 0 is the binodal pressure, P vap and P liq are the unknown pressures, and v vap and v liq are the molar volumes. If we consider the vapor to be an ideal gas and the liquid to be incompressible, Eq. 1.2 becomes: P liq P 0 = RT v liq ln ( Pvap P 0 ), (1.3) where R is the ideal gas constant. Taken together, Eqs. 1.1 and 1.3 form the basis for the Kelvin-Laplace equation and define a relationship between P vap and P liq for a given radius of curvature of the meniscus separating the phases. Of interest for this review, Eq. 1.3 predicts that a small reduction in the relative humidity of the vapor (RH = P vap /P 0 ) can induce a large reduction in the pressure of the liquid: for example, for RH = 0.9, P liq P 0 14 MPa at T = 20 C. Therefore, one could use the RH of a vapor to generate tension in a liquid. Figure 1.6 presents the variation of P liq with RH, calculated with Eq. 1.2 using the IAPWS EoS [102, 99]. Furthermore, with an appropriate membrane, one could allow this equilibrium to occur between a macroscopic volume of liquid and its sub-saturated vapor, as shown schematically in Fig. 1.5.B. An appropriate membrane for this purpose must stabilize the liquid phase, present sufficient mechanical rigidity to withstand the pressure difference between the liquid and vapor, and allow for transfer of the fluid between the phases. The microstructure of two such membranes is shown in the expanded views in Fig. 1.5.B: a porous medium with wettable, rigid walls and a gel in which the liquid exists as a molecular mixture with a cross-linked polymer. For a bulk liquid placed in equilibrium with a subsaturated vapor (RH < 1), the pressure will be below the binodal pressure and the liquid will be thermodynamically metastable; we thus refer to this method as metastable vapor-liquid equilibrium (MVLE). Early examples of this approach were pursued by the plant scientists who proposed the Cohesion-Tension theory for transpiration in vascular plants [70]. In 1896, Askenasy reported on an apparatus in which a porous, ceramic cup was attached to a vertical glass tube filled with water at the top and mercury at the bottom. He showed that evaporation from the porous membrane led to a rise of the column of mercury to 820 mm before cavitation occurred, with the water at a slightly negative pressure [8] (see also [72]). Machin was the first to pursue this approach for the study of liquids under tension. He formed a sealed tube with a macroscopic internal volume from nanoporous Vycor glass [59]. With Vycor membranes, Machin placed butane at an estimated pressure of -4 MPa, but his experiments with water failed due to cracking of the glass. Wheeler and Stroock [103] have employed a different class of materials, hydrogels, to separate the phases. Various acrylate-based

12 12 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE liquid pressure (MPa) T = K 0.9 nm 0.4 r = 3.6 nm 1.8 nm 1.2 nm rela ve humidity 1.0 Figure 1.6 Prediction of pressure in liquid water in metastable equilibrium with its vapor at sub-saturated relative humidities (RH). Equation 1.2 was solved using the IAPWS EoS for v liq (P, T ) [102, 99]. The labels of the red arrows present simple estimates of the radii of pores in a wettable material that would allow the pore liquid to reach the pressures indicated (-40, -80, -120, and -160 MPa), based on equation 1.1. hydrogels with high mass fraction of solids allowed for bulk volumes of liquid water to come to metastable equilibrium with vapors of RH down to 0.86, corresponding to an estimated pressure of -22 MPa at 20 C [104]. This stability threshold was reproducible across hundreds of experiments. Further, this threshold was identical within experimental uncertainty for membranes formed of materials with macroscopic contact angles with water ranging from 13 to 45 o and for samples with and without pre-pressurization out to positive pressures of 54 MPa [104]. These observations suggest that, if impurities or pre-existing nuclei define this threshold, they are ubiquitous, calibrated and independent of the macroscopically observable affinity of the membrane material. The group of Marmottant recently exploited this system to investigate the dynamics of the growth of cavitation bubbles as a function of confinement and mass transfer [100]. Generating negative pressures via MVLE has the advantage that a static, macroscopic volume of liquid can be studied in the metastable state. The approach described allows one to control two thermodynamic state variables: the temperature and the chemical potential (via the RH of the vapor Eqs. 1.2 or 1.3). The pressure or density must then be extracted with an extrapolated EoS (Fig. 1.6) unless a distinct measurement is performed simultaneously, for example, of pressure or speed of sound. This method also opens a route toward studying the dynamics of liquids under tension. For example, if two membranes coupled via a liquid-filled capillary are exposed to vapors of distinct,

13 EXPERIMENTAL METHODS TO GENERATE TENSION 13 sub-saturated RHs, then a net flow of liquid under tension can be generated in the capillary; such a system represents a synthetic mimic of the basic structure of the a vascular plant, as shown by Wheeler and Stroock [103]. Such a system could form the basis for performing viscometry at negative pressures. A liability of this method is the potential for contamination of the liquid or the presence of sites for heterogeneous nucleation due to contact between the liquid water and the boundaries that define the liquid-filled volume. We note briefly, that one can try to use the method of MVLE to interpret observations made on liquids confined within the pore space of wettable, porous solids, rather than considering a bulk liquid volume coupled to the vapor via this pore liquid. Machin suggested that one could potentially extract information about liquids under tension by measuring the adsorptiondesorption isotherms of fluids in nanoporous solids. He showed he could fit the desorption branches of these isotherms with an extrapolation of Speedy s EoS for water [59]. We caution though against the association of the properties of confined water with those of the bulk. Firstly, we note that to achieve large tensions (large degrees of metastability), the pores must be small (diameter in the nanometer range based on Eq. 1 see Fig. 1.6). At such dimensions, the influence of the walls of the pores on the thermodynamics of the confined liquid can become non-negligible: while the chemical potential and temperature of the pore liquid can be set by the RH and temperature of the external environment, one cannot assume that the relationship between these variables and pressure or density (the EoS) is the same as that for the bulk. Secondly, we note that simulation [16, 66, 87] indicate that highly confined pore liquids can exist in spatially heterogeneous states (for example, with the density varying with the distance from the walls), such that well-defined state variables cannot be assigned Berthelot tube In this section we review a method used over more than 150 years, and named after its inventor, Marcellin Berthelot [11]. The Berthelot method consists in the following (Fig. 1.5.C). A vessel is filled with liquid and sealed with a remaining gas bubble. The vessel is then warmed up until the bubble dissolves completely; from the dissolution temperature T d, the liquid density is deduced. The vessel is then cooled down, the liquid sticks to its walls and the pressure decreases, down to negative pressure if the temperature is low enough. At some temperature T cav, cavitation occurs and the liquid goes back to equilibrium with its vapor. The maximum tension reached by Berthelot was estimated to be 5 MPa at 18 C [11]. The exact value depends on the estimate of the volume change of the liquid upon cavitation, complicated by deformation of the containing vessel; see Ref. [22] for details. More importantly, to convert the volume change into a pressure requires the use of an EoS, in a range where it has not been measured, so that it involves uncertain extrapolations.

14 14 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE A successful variation on the Berthelot tube uses water inclusions in a quartz crystal. Water trapped in small pockets (in the µm range) inside crystals can be found in nature. Roedder [76] used such microscopic inclusions to prepare ice crystals and liquid water in metastable equilibrium. He started with liquid and vapor in the inclusion. Upon freezing, the vapor disappeared because of the greater volume of ice. When the inclusion was melted again, if the vapor did not nucleate, a negative pressure developed as the system followed the metastable melting line. A maximum ice-liquid equilibrium temperature of 6.5 C was observed; by an extrapolation of the melting line measured at positive pressure, Roedder estimated the corresponding pressure to be at least 90 MPa. Angell and his group used the Berthelot method with synthetic inclusions [114]. A quartz crystal is quench-fractured, and then heated with water in an autoclave at high pressure and temperature. The fractures eventually heal, trapping small pockets of water in the crystal. They found that all inclusions in a given sample had the same T d and hence the same density. They observed two distinct cavitation behaviors: when T d > 250 C (autoclaving temperature higher than 400 C), T cav was the same within ±2 C for all inclusions in a given sample. At low enough density, P cav was positive, and the results agreed well with the superheating experiments [9, 84]. On the other hand, when T d < 250 C (high density inclusions), T cav was scattered. For fluorite and calcite, T cav was always scattered, and the estimated P cav was less negative that in quartz. Angell and his group attribute the scatter to heterogeneous nucleation, and its source to possibly surfactant molecules cluster destroyed by annealing at the higher temperatures. The maximum tension was obtained in one sample with high density inclusions (0.91 g ml 1 and T d = 160 C); Angell and his group report that some [inclusions] could be cooled to 40 C without cavitation, and one was observed in repeated runs to nucleate randomly in the range 40 to 47 C and occasionally not at all [114]: they estimate that nucleation occurred at P cav 140 MPa. The fact that no inclusion that survived cooling to 40 C ever nucleated bubbles during cooling to lower temperatures was interpreted as an evidence that the isochore crosses the metastable LDM, thus retracing to less negative pressure at low temperature. This interpretation fits Speedy s scenario, at least in the sense that the LDM keeps a negative slope deep into the negative pressure region in the P T plane. Alvarenga et al. later pointed out the possibility that some inclusions in quartz deform significantly during cooling and thus invalidate the assumption of a near-isochoric path made to estimate P cav [2]. They were able to show a volume change in a platelet-like inclusion, but for roughly spherical inclusions the constant volume assumption appeared to be appropriate. It should be emphasized that in the work by Angell and his group, the inclusions in which they estimated P cav 140 MPa were not of well-rounded form, like those on which the reliable and reproducible high temperature data were obtained [115]. However, the method has been recently

15 EXPERIMENTAL METHODS TO GENERATE TENSION 15 reproduced and extended to several aqueous solutions [83], and the overall results confirmed. Clearly, a direct measurement of pressure or another thermodynamic function (e.g. speed of sound) is necessary to make the method reliable. A way to measure pressure within the tube is to shape the glass capillary serving as a Berthelot tube into a helix to make a Bourdon gauge. The change in internal pressure makes the helix coil or uncoil, and the rotation is measured with a mirror attached to it [65, 41, 43] or a capacitance distance meter [35]. These gauges were calibrated at positive pressure. Meyer studied water, ethanol and ether, obtaining P cav = 3.4 MPa at 24 C, 3.95 MPa at 23 C, and 7.3 MPa at 18 C, respectively [65]. He measured the pressure-volume relation, and found it to be linear, except for ether where he noticed a curvature. He also measured the LDM of water and found it to lie near the linear extrapolation of positive pressure measurements. Evans measured P cav in the range 3 to 5 MPa, between 35 C and 20 C [35]. Henderson and Speedy found cavitation at 16 MPa at 38 C [41]. They also extended the work of Meyer on the LDM down to 20.3 MPa at 8.3 C (open, green circles in Fig. 1.4) [43]. Finally one can use an electrical strain gauge pressure transducer. This was done in Trevena s group with stainless steel Berthelot tubes [24, 52]. Of particular note, Ohde s group, automated the experiments to repeat thousands of thermal cycles. They reached a minimum P cav of 18.5 MPa at 53 C [46] and reported persistent variability, run-to-run. All of these methods to measure P cav are expected to be quite accurate, because the extrapolation of the calibration relies only on the controlled and calibrated deformation of solids (glass or steel), not on an assumption about the properties of the stretched liquid. The large discrepancy in the magnitude of P cav relative to predictions (Fig. 1.4) suggests heterogeneous nucleation. Alvarenga et al. [2] introduced the use of Brillouin scattering to measure the sound velocity in stretched liquid water during cooling. They reported tensions beyond 100 MPa at 20 C. The pressure was calculated from the change in sound velocity before and after cavitation, assuming a linear relation based on positive pressure data. To conclude with mineral inclusions, we shall mention a work focusing on kinetic aspects, by measuring the statistics of lifetimes of one inclusion at fixed temperatures [96]. The largest negative pressure achieved in this work is 16.7 MPa at C, and the lifetimes followed a Poisson distribution. Berthelot method has the advantage of generating tension in a static, macroscopic volume of liquid; this characteristic facilitates mechanical and spectroscopic characterization. As we have noted, this method, when the volume of water is around (10 µm) 3 has also provided access to the largest range of tensions, apparently approaching the predicted stability limit of water. Outstanding opportunities for the Berthelot approach include its use with other spectroscopic methods such as Raman [39] or IR [64] to access dynamics in the metastable regime and its use to perform systematic studies of the doubly metastable regime of supercooling under tension.

16 16 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE Centrifuge method This method, first employed by Reynolds (cited in [108]), consists in rotating at high speed a tube containing water. Because of the centrifugal force, a negative pressure is developed on the rotation axis: P = P ρω2 r 2 where P 0 is the pressure outside the tube, ρ is the water density, and r is the distance between the center and the liquid-gas interface. The first studies achieved a minimum value for P cav of 0.49 MPa [108] and 0.57 MPa [98]. Briggs introduced an important advance in this technique with the development of the z-shaped capillary (Fig. 1.5.D) that provides auto-stabilization of the liquid filament. He proceeded to measure the most negative P cav with boiled distilled water in Pyrex capillaries ( mm inner diameter) [13]. Briggs also investigated the temperature variation of P cav : he found a minimum of 27.7 MPa at 10 C, with P cav = 2 MPa at 0 C and 22 MPa at 50 C. He aslo investigated other liquids [14, 15]. Later quartz tubes were used with water [94]; P cav was found to be highly variable between tubes and to vary with time in the same tube, reaching at best 17.5 MPa. Winnick attempted to use centrifugation to check the EoS [105]. During rotation of a tube containing water, he measured the angular velocity and the meniscus position; from this he tried to deduce the specific volume of the liquid, averaged over the pressure range developed along the liquid column (down to 10 MPa). However, the analysis has been criticized [58] and reconsidered [106, 50]. The average over the tube length makes it difficult to fully determine the P V curve. Finally, we note that the centrifuge method is used by plant physiologist: it was adapted by Cochard to measure xylem hydraulic conductance [26], and by Holbrook et al. to provide a definitive calibration of the widely used pressure bomb method of measuring water pressure in plants [47]. 1.5 LIMIT(S) OF METASTABILITY Comparison between the different methods Figure 1.7 compares the results obtained for the cavitation pressure with different methods, excluding those performed in quartz inclusions. Note that usually only the cavitation pressure reported is the most negative that could be observed with a given method. In contrast, the experiments using short acoustic bursts focused from a hemispherical transducer (Section 1.4.1) or MVLE (Section 1.4.2) give much more reproducible results, with cavitation occurring in a narrow range of pressure. Despite the variety of experimental approaches, close values of P cav are obtained. They are all far less negative than the theoretical predictions (see Section 1.3 and Fig. 1.4). The centrifuge method (Section 1.4.4) stands out from the others by the low temperature behavior of its results. A more than tenfold drop in tension is observed below 10 C. At first sight, this could be an argument in favor of

17 LIMIT(S) OF METASTABILITY 17 0 Cavitation pressure (MPa) Berthelot Berthelot Berthelot centrifuge shock wave acoustic acoustic acoustic trees Temperature ( C) Figure 1.7 Comparison of the cavitation pressure of water as a function of temperature obtained with different techniques: a Berthelot-Bourdon tubes (solid triangle up - [41], solid triangle down - [43]); metal Berthelot tube with pressure transducer (open diamond - [46]); z-tube centrifuge ( - [13]); shock wave (box plus - [109]); acoustic (black bullet - [38], black solid square - [40], red bullet - [30]); and MVLE (green solid diamond - [103]). An arrow means that cavitation was not observed. Reproduced with permission from Ref. [23]. the reentrant spinodal scenario (Section 1.3). However, this abrupt change in the limit of metastability is not supported by the other techniques. Beyond the fundamental aspects, such a drop would be significant for the survival of plants in cold regions, because tensions higher than 2 MPa could not be withstood. This is why Cochard et al. [27] revisited the centrifuge method, but using a yew segment in place of the capillary. Even though less negative values of P cav could be reached at room temperature, they vary monotonically with temperature, reaching 7 MPa at 1 C, larger than Briggs result, which must have been subject to some artifact. Figure 1.8.A compares the most extensive sets of measurements of the stability limit by the acoustic method [44], which were extended up to 190 C [30], and Berthelot method in quartz inclusions [114]. The quartz inclusions appear to give a much more negative P cav. But one has to remember that the liquid density only is known (assuming that the inclusions keep a constant volume and remain sealed), and that the pressure is deduced with an extrapolated EoS. Therefore, a direct measurement of the liquid density at the nucleation threshold has been performed in the acoustic experiment [30], thanks to a

18 18 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE fiber optic probe hydrophone [6]. Figure 1.8.B presents a comparison of the acoustic and quartz inclusion measurements in terms of density. The red bullets are direct measurements acquired with the hydrophone and they compare well in trend and magnitude with pressure estimates that were converted to density with an EoS, but the major discrepancy with the quartz inclusions is persistent Origin of the discrepancy in the limits of metastability The discrepancy between acoustic and inclusion experiments remains to be explained. This section proposes some speculative ideas. Before proceeding, let us exclude two reasons sometimes invoked. One might think that the acoustic experiment quenches the liquid too rapidly compared to the inclusion method. However, the acoustic tension lasts around 50 ns, much longer than microscopic relaxation times (typically in the ps range [68]). There is a dependence of P cav on the experimental time, but it is logarithmic, and has the opposite effect than required to explain the discrepancy: P cav is more negative for a shorter time [44]. Furthermore, other experiments (e.g. with the centrifugal [13] or MVLE [103] methods) have time-scales comparable to the inclusion method, but still reach values of P cav close to the acoustic ones (Fig. 1.7). One might also wonder if water in the inclusions is somehow stabilized by a confinement effect from the silica walls. However, one expects that any interaction (for example, dispersion forces) that could cause this stabilization would have a range that is much smaller (< 10 nm) [51] than the inclusion size (a few microns). We are therefore left with an explanation involving impurities. There are two possibilities: (i) a stabilizing impurity that extends the achievable range of metastability is present only in some of the inclusions; or (ii) a destabilizing impurity that reduces the achievable range of metastability is present in all experiments, except in some inclusions. (i) Stabilizing impurities could be created from the quartz during the fabrication of inclusions, when water reaches supercritical conditions where it is known to dissolve silica. Silica nanoclusters could be formed. However their effect on cavitation is not clear. They might stabilize the neighboring hydrogen bond network, but this influence is expected to have only a short range, and thus stabilization of the whole sample would require a large cluster concentration, which might be detected by suitable methods. A recent IR spectroscopic study of a water inclusion in quartz [64] hints in this direction. (ii) Destabilizing impurities may be present in all experiments, but disappear during the fabrication of inclusions. This could happen thanks to the high pressure and temperatures reached: surfactant molecules cluster destroyed by annealing at the higher temperatures have been invoked in [114]. We emphasize that a nearly ubiquitous impurity is required to explain the reproducibility of our cavitation statistics from one water sample to another. For at least one impurity to be present in the volume involved in the acoustic experi-

19 LIMIT(S) OF METASTABILITY 19 Cavitation pressure (MPa) Temperature ( C) Density (kg m -3 ) Temperature ( C) Figure 1.8 Comparison of the cavitation pressure (top) and density (bottom) of water as a function of temperature obtained with acoustic method and quartz inclusions used as Berthelot tubes. The symbols represent: acoustic method with calibration by static pressure method (open diamonds - [44] and solid blue diamonds - [30]), acoustic method with fiber optic probe hydrophone (red bullets - [6]), quartz inclusions (open squares - [114]). In the lower panel, cavitation of an inclusion during melting is also included (black filled square - [113]) and red and grey arrows indicate isothermal and isochoric paths, respectively. Green lines are the binodals. Lower panel reproduced with permission from Ref. [30].

20 20 THE STABILITY LIMIT AND OTHER OPEN QUESTIONS ON WATER AT NEGATIVE PRESSURE ment at 2 MHz, the impurity concentration c must exceed mol L 1 [30]. On the other hand, in the most metastable inclusions, there should be no impurity in the inclusion volume of around (10 µm) 3, requiring c < mol L 1. The overlap between ranges is compatible with the hypothesis that the acoustic experiment is sensitive to impurities whereas the inclusion experiment is not, but the margin is narrow. Another possibility is that the impurities present in bulk water are deactivated by adsorption sites present on the inclusion walls. Assuming c = 10 7 mol L 1 in bulk water, a cubic inclusion with side a = 10 µm contains impurities, and 1 site per (100 nm) 2 would be needed. c = 10 7 mol L 1 was chosen because hydronium and hydroxide ions are spontaneously created in neutral water (ph=7) with this concentration by autoprotolysis. This speculation gains support from the fact that inclusions filled with a 1 mol L 1 NaOH solution give much less scattered results than with pure water, and on average more negative pressures [83]. Hydronium ions would therefore be good candidates for a ubiquitous impurity, spontaneously present in sufficient quantity in water Remaining issues with inclusions A difficulty with these impurity scenarios is that inclusion experiments exhibit large scatter in the cavitation thresholds for samples with similar densities (Fig. 1.8.B); therefore, there should be several types of impurities involved, which affect the activation barrier for nucleation, E b in quantitatively different ways. Furthermore, a fully consistent picture should also explain the surprising behavior of ice-melting in inclusions [113]; this observation was reported in Zhengs thesis and has been hitherto overlooked. After an inclusion had been frozen and filled with ice, the ice melted upon heating. Because of the higher density of the liquid relative to ice, the pressure of the mixture of ice and water decreased during melting, following the metastable liquid-ice equilibrium line, until cavitation occurred. Surprisingly, the largest P cav obtained with this method is 22.8 MPa [113] (Fig. 1.8B, lowest filled black square), in an inclusion that cavitated at MPa with the usual isochoric liquid cooling method [113]. 1 The only change is the presence of the liquid-ice interface. In case (i) of Section 1.5.2, it would force the stabilizing impurities to disappear or lose their efficiency, by being incorporated in ice or deactivated at its surface, but the mechanism remains unclear. In case (ii), ice would trigger heterogeneous nucleation. With CNT, this scenario could occur due to incomplete wetting of a substrate [22]. However, a reduction of P cav by a factor of 4.5, as observed in one of the inclusions, would require a contact angle of the liquid on ice of 113 o, far above the observed 1 o [54]. A microscopic inter- 1 We mentioned in Section that ice-liquid equilibrium was observed up to 6.5 C in natural fluorite inclusions [76]. However, this work involved aqueous solutions of unknown composition and contained no corresponding study of high temperature cavitation.

21 OTHER TOPICS IN THE STUDY OF LIQUIDS UNDER TENSION 21 pretation is needed: ice being less dense than the liquid, the interface might provide heterogeneous nucleation sites facilitating the formation of vapor Path-dependent nucleation An alternative, more exotic hypothesis to explain the discrepancy is that the nucleation mechanism depends on the thermodynamic path followed in the phase diagram as water is stretched. We have seen in Fig. 1.8.B that the discrepancy between the acoustic and inclusion experiments cannot be explained by such a mechanism, because our high temperature experiments cross the path followed by the inclusions. Nevertheless, this scenario might be invoked to explain the behavior of ice-melting in inclusions just mentioned. What could be the microscopic origin of such a path-dependent cavitation? Using CNT with the bulk surface tension of water yields P cav values close to those estimated for inclusions cooled along an isochore [114, 19]. During ice-melting in inclusions, nucleation could occur through an intermediate metastable state which would lower E b because of a lower interfacial tension: this is called the Ostwald step rule, often invoked in crystallization, and recently verified experimentally [25]. This intermediate state could be another liquid phase of water, such as the low density liquid phase discussed in Section 1.3, provided that the transition between the low and high density liquids lies at negative pressure. Furthermore, simulations of a model globular protein have shown how the vicinity of a fluid-fluid critical point in the phase diagram of the protein largely reduces E b for its crystallization [107]. When water in the inclusions is stretched during ice-melting, cavitation could thus be affected by the vicinity of a liquid-liquid critical point if it is situated at negative pressure. On the other hand, the isochoric path usually followed by the inclusions goes through a region of large tension at high temperature. Thus, it may avoid the critical region and yield P cav predicted with CNT. To check the possibility of this mechanism, simulations of the kinetics of nucleation along these various trajectories are clearly needed. 1.6 OTHER TOPICS IN THE STUDY OF LIQUIDS UNDER TENSION Equation of state of water at negative pressure Molecular dynamics simulations can easily be performed in the metastable region, because the duration of the simulation is short enough to make nucleation highly improbable, except very close to the spinodal limit. The simulated EoS is therefore accessible. On the other hand, only a few attempts have been made to gain experimental knowledge on the EoS of water at negative pressure, which could put to a test the interaction potential used in the simulations. As explained in Section IV.C., the first work is due to Meyer [65] who used a Berthelot-Bourdon tube to measure the relation between pressure and density down to -3.4 MPa at 24 C. Henderson and Speedy later measured the