Escaping the no man’s land: recent experiments on ...ilm-perso.univ-lyon1.fr/~fcaupin/fichiersPDF/Caupin_JNCS_2015.pdf · metastable liquids and nucleation in general, with large

Escaping the no man’s land: recent experiments onmetastable liquid water

Frederic Caupin

Institut Lumiere Matiere, UMR5306 Universite Lyon 1-CNRS, Universite de Lyon andInstitut Universitaire de France, 69622 Villeurbanne cedex, France

Abstract

The properties of supercooled water have been the subject of intense studies for

decades. One of the main goals was to follow the evolution of water anomalies,

already present in the stable liquid, as far as possible in the metastable phase.

All anomalies were found to become more pronounced, but their origin has hith-

erto remained hidden because of crystallization into ice. We review the recent

experimental developments in the field, with a focus on the techniques used

to reach a larger metastability, or to extend the investigations to the negative

pressure region of the phase diagram, where the liquid is also metastable with

respect to its vapor.

Keywords: water anomalies, supercooled water, negative pressure, Widom line

1. Introduction

When cooled from room temperature, water reaches a well known density

maximum near 4◦C at ambient pressure. It corresponds to a change in sign of

the isobaric expansion coefficient αP . It is experimentally possible to cool the

liquid below the equilibrium melting temperature, and to measure the proper-5

ties in this metastable state. Density is found to decrease faster and faster upon

cooling [1], which corresponds to αP becoming more and more negative. No sign

of slowing down in the increase in magnitude of αP is found down to the lowest

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Preprint submitted to Journal of Non-Crystalline Solids September 15, 2014

4

5

6

7

220 230 240 250 260 270

CP (

kJ

K-1 k

g-1)

Temperature (K)

Figure 1: (Color online) Isobaric heat capacity CP of water as a function of temperature.

The experimental data are from Refs. [3] (blue squares) and [4] (red circles); see Ref. [5] for a

review of these and other data. Despite a small discrepancy, they both show an appreciable

anomalous increase on cooling. The black solid curve is a power-law fit to Ref. [3] data with a

diverging temperature of 228K. The green dashed curve is a schematic drawing, to illustrate

the case in which CP passes through a maximum at a lower temperature than any that could

be studied to date.

measured temperature (239.74K). In a famous plot (Fig. 4 of Ref [2]), Speedy

and Angell noticed that many properties of stable and supercooled water (such10

as αP , or the isothermal compressibility κT , the heat capacity at constant pres-

sure CP , etc. . . ) could be fitted by power laws which extrapolated to a common

temperature of divergence of 228K at atmospheric pressure, slightly below the

lower temperature limit of experiments. The following question therefore arises:

if measurements could be performed at lower temperatures, would there be an15

actual divergence, or would an extremum be reached (Fig. 1)?

This outstanding question has been addressed by many authors, but has

remained elusive because of crystallization. In general, only a small degree of

2

supercooling can be achieved in water, because of heterogeneous nucleation of

ice, favored by certain impurities or surfaces. If precautions are taken to avoid20

these, ice will still nucleate spontaneously from thermal fluctuations in the liq-

uid. This occurs at the homogeneous nucleation temperature Th, which thus

gives a lower bound to the temperature at which measurements can be per-

formed on liquid water cooled from ambient temperature. The exact value of

Th depends on the experimental volume and cooling rate, and therefore varies25

between experiments. Nevertheless, a number of studies placed Th around 235

K at ambient pressure [6]. One might try approaching the problem from the

low-temperature end. Indeed, there are ways to prepare water in an amorphous

solid state [7]. Warming this glassy water might provide a liquid phase at lower

temperatures than what can be obtained by directly cooling the room temper-30

ature liquid. Unfortunately this approach is also hindered by re-crystallization

of the amorphous ice, around 150K [7] (or up to 190K for films under 150 nm

thick submitted to ultrafast heating rates [8]) at ambient pressure. The region

between the crystallization of amorphous ice and the limit of supercooling (from

150 to 235K at ambient pressure) is called the “no man’s land”, a name coined35

by Mishima and Stanley [9].

The purpose of this paper is to review the recent experiments which have

tried to “escape the no man’s land”, to solve the enduring mystery about water

anomalies. Section 2 gives a brief overview of the different theoretical scenarios

proposed to explain the increasing anomalies of water when it is supercooled.40

Section 3 presents two water proxies used to perform measurements below Th of

pure, bulk water. They provide some evidence in favor of one scenario, which

postulates that water can exist in two distinct liquid states (liquid polymor-

phism). The rest of the discussion comes back to pure bulk water and describes

the most recent experimental advances. Section 4 introduces new methods that45

have allowed to reach unprecedented supercooling (thus in effect lowering Th),

and gives an updated map of the no man’s land. Finally, Section 5 surveys

the progress in a rather untouched region of the phase diagram, at negative

pressure, where the key to water anomalies might become accessible.

3

We should mention that we will not discuss the very interesting possibility50

to study the ultraviscous liquid(s) that could be obtained by heating the amor-

phous ices just before crystallization occurs [10]. We also refer the reader to in-

troductions, reviews or books where more details on several topics can be found:

metastable liquids and nucleation in general, with large parts about water [11],

supercooled and glassy water [12, 13], thermodynamics of supercooled water [5],55

crystallization in water [14], vitrification of water [15], amorphous ices [7], liquid

polymorphism [16], water at negative pressure [17, 18], metastable water with

emphasis on negative pressure [19], cavitation in water [20].

2. Theoretical background and controversy

The focus of this review is experimental, but we need to introduce the com-60

peting theoretical explanations that have been proposed. We will give a minimal

overview; see Refs. [13, 16, 21] for more detailed reviews.

Because of the apparent divergence of many properties extrapolated to 228K,

Speedy looked for a source of instability. Any liquid can be brought below its

saturated vapor pressure, but will eventually become unstable with respect to65

the vapor at the spinodal pressure. Speedy noticed that the equation of state

(EoS) of water at positive pressure extrapolated to a spinodal pressure with a

non-monotonic temperature dependence [22]. Interestingly, he found a thermo-

dynamic explanation: if the line of density maxima (LDM) intersects the spin-

odal curve in the pressure-temperature plane (see Fig. 2, top), the latter must70

reach a minimum pressure. In his stability limit conjecture [22], Speedy further

proposed that an instability line of the liquid would also exist at positive pres-

sure. It was later argued by Debenedetti [12] that a liquid-vapor spinodal could

not retrace all the way to positive pressure, because it would have to cross the

metastable liquid-vapor equilibrium, which can happen only at a liquid-vapor75

critical point: such a singularity seems unlikely. However, two interesting ques-

tions remain: (i) even if the liquid-vapor spinodal does not retrace to positive

pressure, does it reach a minimum pressure or not?; (ii) would there exist at

4

positive pressure a line where the supercooled liquid becomes unstable (towards

another phase than the vapor)?80

In contrast, molecular dynamics simulations of a water-like potential [23]

found a monotonic temperature dependence of the liquid-vapor spinodal pres-

sure. Thermodynamic consistency was preserved, as the LDM reached a maxi-

mum temperature at negative pressure, and avoided the spinodal (Fig. 2, bot-

tom). The simulations found another source for water anomalies. In the second85

critical point scenario [23], a first-order transition separates two liquids with dif-

ferent structures in the supercooled region. This liquid-liquid transition (LLT)

terminates at a liquid-liquid critical point (LLCP) (237±4K and 167±24MPa

for the ST2 potential [24]), responsible for the large increase in many water

properties. However, they diverge only at the critical point. Below the critical90

pressure, they will go through an extremum. Such a LLT was proposed for

many other water-like potentials. However, there has been recently an intense

debate about the stability of one of the two liquids with respect to crystalliza-

tion, challenging [25, 26] or reasserting [27, 28, 29, 30, 31, 32, 33] the existence

of a LLCP. It is beyond the scope of this review to detail this debate. However,95

we would like to emphasize the point of view of the experimentalist. Based on

a long history of attempts, the LLT, if it exists, would lie in a region acces-

sible to computers but apparently not to experiments. Therefore, even if the

LLT or LLCP were virtual, what would matter more is the existence or not of

loci of extrema in the response functions of water. They would be associated100

with the locus of extrema in the order parameter of the LLT, called the Widom

line [34, 35]. We also note that, assuming the existence of a LLCP, one can try

to locate it without using molecular dynamics simulations, but rather trying to

build an EoS consistent with experimental data measured on real water: the

LLCP thus predicted lies at much lower pressure than in simulations (227K,105

13MPa) [36].

We should mention that other scenarios have been proposed. The critical

point-free scenario [15, 37, 38] places the LLCP beyond the liquid-vapor spin-

odal. In that case, upon cooling, there is no extremum in response function,

5

Pressure

liq-vap

LDMTemperature

C

T

liq-sol

0

spinodal

Pressure

C' liq-vap

LDM

Temperature

C

T

liq-liqliq-sol

0

spinodal

Figure 2: (Color online) Sketch of the phase diagram of water for two of the scenarios proposed

to explain the anomalies of water. The blue curves show the equilibrium curves for the liquid-

solid and the liquid-vapor (with critical point C) transitions. The green short-dashed curve is

the line of density maxima (LDM), and the red long-dashed curve the liquid-vapor spinodal.

In the stability-limit conjecture (top) [22], the LDM intersects the spinodal which reaches

a minimum pressure; a line of instability exists in the supercooled liquid on which several

response functions of water diverge. In the second critical point scenario (bottom) [23], the

LDM avoids the spinodal. Water anomalies are due to a second critical point C′ terminating

a first-order liquid-liquid transition (purple curve).

6

but instead a LLT, or if the high density liquid remains metastable with respect110

to the low density liquid, a divergence on the high density liquid spinodal. The

singularity-free interpretation [39] offers a thermodynamically consistent pic-

ture, where there is no LLT, but no divergences either. The response functions

go through an extremum upon cooling, as a direct consequence of the LDM. It

is noteworthy to mention that a cell model involving hydrogen bond coopera-115

tivity [21] can realize each of the four above mentioned scenarios by tuning the

values of its parameters.

To conclude this section, we give a (non-exhaustive) list of experimental

signatures that would help to decide between the available scenarios:

• finding a genuine LLT;120

• finding an extremum in one of the responses functions that has a seemingly

diverging behavior in the supercooled region;

• measuring the EoS of water at negative pressure to elucidate the shape of

its LDM and of its liquid-vapor spinodal curve.

3. Squeeze or hide: tricks to avoid crystallization125

A trick of some sort has to be played to be able to perform measurements

on water molecules in a liquid phase below the homogeneous nucleation tem-

perature Th of ice in the bulk. Two leads have been followed: (i) confining

water in narrow pores, and (ii) mixing water with a substance that decreases

the nucleation rate.130

3.1. Confinement

For a detailed review on the effects of confinement on freezing and melt-

ing, see Ref. [40]. In a cylindrical pore of diameter d filled with a liquid that

perfectly wets the pore walls, the Gibbs-Thomson relation teaches us that the

liquid-crystal equilibrium temperature is depressed from its bulk value Teq by135

an amount:

∆Teq =4σLC Teq

ρc Ld, (1)

7

where σLC is the energy per unit area of the liquid-crystal interface, ρc is the

mass per unit volume of crystal, and L is the latent heat of melting per unit

mass. This direct application of macroscopic thermodynamic laws should be

modified to account for microscopic details, but the trend is correctly captured.140

For water-ice, σLC is in the range 25−44mJm−2 [41], so that ∆Teq ≃ 100/d with

d in nm. Remarkably, experiments performed on well defined pores in MCM-41

silica [42] have shown that the Gibbs-Thomson relation is satisfied for nanopores:

after a layer of nonfreezing water on the pore wall has been accounted for, it

agrees with the observed melting temperatures for pore diameters from 3 to145

4.4 nm (from 215.0 to 239.7K). However, for 2.5 nm pores, differential scanning

calorimetry does not detect any first-order transition.

Taking advantage of this effect, Chen, Mallamace and co-workers have used

water confined in very narrow nanopores to obtain measurements on a liquid-

like phase at very low temperatures. A recent review of all their work is avail-150

able [43]. Let us simply mention here that, using a variety of techniques (neu-

tron scattering, nuclear magnetic resonance, Fourier transform infrared spec-

troscopy. . . ), several features compatible with the second critical point scenario

were reported, such as: dynamic crossover [44], density minimum [45, 46] and

density hysteresis [47]. These works have provoked a passionate debate about155

the data interpretation [48, 49, 50, 51, 52, 53]. Without taking position about

the validity of the results, we would like to emphasize that their relation to

bulk water properties is not straightforward. Indeed, to avoid crystallization,

pores with a diameter of 1.4 − 1.5 nm have been used [44, 45, 46, 47]. This

corresponds to at most 5 layers of water across. Two are in direct contact with160

the pore walls, and the others are likely affected by its presence. Recently, a

careful optical Kerr effect spectroscopy experiment on partially hydrated Vycor

glass [54] has shown that the first and second layers from the walls have dynamic

properties differing from bulk water.

We would also like to mention that the exact pressure at which experiments165

in pores are performed is not really known. Helium is used as a pressurizing

medium [47], and the applied helium pressure is reported. However, for an hy-

8

drophilic pore, the liquid-vapor interface has a curvature such that the pressure

in the confined liquid is less than the external pressure. If one applies the macro-

scopic Laplace law, the pressure jump is 2γLV/R ≃ −140/R in MPa if R is in170

nm. In extremely narrow pores the validity of the macroscopic law is question-

able. The very notion of pressure as a scalar quantity should be replaced by an

anisotropic quantity related to the strong interaction with the pore walls. Still,

assuming that confined water might be compared with the bulk phase, one may

wonder if it should rather be with a liquid at negative pressure [55]. This idea175

has been recently developed by Soper [56]: combining neutron scattering from

and simulations of ordered cylindrical pores in silica with 2.5 nm diameter, he

concluded that the density of water was not homogeneous, and the density of the

core liquid less than that of the bulk, suggesting a pressure around −100MPa.

3.2. Antifreeze180

Another way to avoid crystallization is to add a solute to water. At very low

solute concentration c, this is well known to depress the melting point Tm by

an amount proportional to c (∆Tm = −Kc) but independent of the nature of

the solute. This is a colligative property, known as the cryoscopic law. At typ-

ical working concentrations, the cryoscopic constant K for water is somewhat185

different from the infinite dilution limit, but the proportionality still holds with

K = 1.853Kkgmol−1. Interestingly, the homogeneous crystallization temper-

ature of the solution is also depressed by an amount proportional to c, which

may be written ∆Th = λ∆Tm = −λKc. The parameter λ has been measured

for many compounds, and the values found to be surprisingly grouped around190

2. There are variations, correlated with the self-diffusion coefficient of the so-

lute [57].

Murata and Tanaka have chosen glycerol as the solute [58]. Using an im-

pressive set of experimental techniques, they concluded that a transition be-

tween two different liquids with equal glycerol concentration occurred, without195

macroscopic phase separation. The fact that the composition remains constant

is key to the results, to distinguish the phenomenon from phase separation in

9

a binary mixture. The simultaneous appearance of cubic ice requires special

attention to reach a conclusion [58]. In a subsequent study [59], 14 aqueous

solutions of sugar and polyol molecules were studied: only with glycerol and200

1,2,4-butanetriol a LLT was reported without concurrent demixing. Experi-

ments at a mole fraction of glycerol less than 0.135 were not possible because of

homogeneous crystallization. This means that the no man’s land also extends

in the temperature-concentration plane (see Fig. 4 of Ref. [58]), preventing to

reach a clear conclusion for pure water.205

4. Small and fast: the quest for deeper supercooling

In this Section we discuss experiments on pure, bulk water, that attempt to

push the temperature Th of ice nucleation to lower and lower values.

Nucleation occuring in a metastable liquid is a catastrophic event: once the

new phase exceeds a critical size, it will grow very fast and replace a large210

part or the whole liquid phase. In addition, nucleation is very sensitive to

impurities, which can lower the energy barrier for the phase change, leading to

heterogeneous nucleation at a lower metastability. For these reasons, it is very

advantageous to divide the experimental sample into many independent units.

Nucleation is also a kinetic process. The physics of the transformation deter-215

mines the nucleation rate per unit volume and time, Γ, as a function of metasta-

bility (e.g. degree of supercooling below the melting point, T−Tm). This means

that the metastability that can be reached in a given experiment depends on its

typical volume V and duration τ : nucleation occurs when ΓV τ ≃ 1. Therefore,

an experiment performed on a small sample during a short time will reach larger220

Γ, and hence larger metastability and lower Th.

To achieve the best conditions for nucleation studies, a technique of choice

is therefore to disperse droplets in an emulsified sample. In this way, many

determination of the homogeneous crystallization temperature Th in water have

been obtained. For instance, Kanno et al. [60] reported Th values in the range 0−225

300MPa, with a minimum of 181K at 200MPa; the data up to 140MPa is shown

10

on Fig. 3. Using a calorimetric technique, Taborek [61] was able to measure

the nucleation rate at ambient pressure. He investigated the dependence on

droplet size, reaching a minimum Th = 239 and 235K for 300 and 6µm diameter

droplets, respectively, in line with the exponential temperature dependence of230

the nucleation rate per unit volume and time (see Fig. 2 of Ref. [61]). He also

found an effect of the surfactant used to prepare the emulsion, which shows that

heterogeneous nucleation might occur at the droplet surface; he also provides

data for heavy water. A recent paper, that presents a microfluidic version of

Taborek’s experiment [6], reviews many other nucleation rate measurements,235

and shows that their scatter is mainly determined by uncertainty in the sample

temperature.

To cool down at an even faster rate (and thus reach a shorter experiment

duration τ , and a lower Th), droplets suspended in vacuum or in a carrier gas

have been used (see Ref. [66] for a review). They can be generated by adiabatic240

expansion of a gas saturated with water vapor, leading to supersaturation con-

ditions that eventually produce a cloud of liquid droplets, which will freeze upon

further cooling. Another geometry involves a supersonic nozzle. A mixture of

water vapor and carrier gas flows in a diverging nozzle, leading to fast pressure

and temperature drop. The technique was introduced in the field of ice nucle-245

ation studies by Huang and Bartell [67]. Recently, Manka et al. [62] combined

it with pressure measurements along the nozzle, small angle X-ray scattering

to characterize the aerosol distribution, and Fourier transform infrared spec-

troscopy to detect ice nucleation. With the supersonic nozzle, the droplets are

much smaller than in emulsion studies, with diameters between 6.4 and 11.6 nm.250

The liquid is therefore subject to a non-negligible Laplace pressure, from 54 to

29MPa, respectively. Onset of ice was detected at 202K and 215K, respec-

tively. The small droplet size also makes it possible that surface crystallization

plays a non negligible role [62, 68]. The work was recently extended to heavy

water [68].255

To work with micrometer size droplets, the group of Nilsson uses a gas

dynamic virtual nozzle [69]. A liquid water jet is produced, that breaks into

11

new border

LLCP???

-150

-100

-50

0

50

100

150

150 200 250 300

Pre

ssu

re (

MP

a)

Temperature (K)

stable

supercooled

P<0doublymetastable

Figure 3: (Color online) Phase diagram of water. The colored areas show the different possible

states of liquid water. The delimitating dashed lines were obtained by polynomial extrapola-

tion of the positive pressure data. The pink region is the no man’s land, between the line of

crystallization of amorphous ice (left black curve) [9] and the of homogeneous nucleation of ice

in the supercooled liquid. The latter was formerly located by an experiment on emulsions [60],

but has been pushed to the left by recent experiments: the line connecting purple triangles

shows where ice starts to nucleate in the nanodroplets of Ref. [62], and the circle gives the

lowest temperature estimated for the microdroplets of Refs. [63, 64]. The red and green thick

curves show the path followed in the Brillouin experiment of Ref. [65] (see Section 5), with the

pressure estimated from simulations with the TIP4P/2005 potential. These simulations also

give the locus of maxima of the isothermal compressibility (brown diamonds), which suggests

a divergence at the location of the white plus sign that might be a critical point terminating

a liquid-liquid transition. (adapted from Ref. [65]).

12

micrometer size droplets. The droplets evaporate in vacuum at a high cooling

rate. The originality is that each small drop (down to 9µm in diameter) can

be probed with a femtosecond long pulse from an ultrabright X-ray laser at260

Linac Coherent Light Source, and 200 nm ice crystals detected on a single-shot

basis. Supercooled water was detected down to a temperature estimated to

reach 227K. Under these extreme conditions, water is liquid only for about

a microsecond. Its structure factor, measured down to 229K, showed that

the splitting of its first maximum into two peaks, already known at ambient265

temperature, increases continuously upon cooling. Although the temperature

calibration is a delicate matter (because the evaporation model needs as input

some quantities that are presently unknown at this low temperature, such as

water heat capacity, see Fig. 1), this is a promising route to probe water at an

unprecedented supercooling.270

To sum up, Fig. 3 shows an updated map of the phase diagram of water and

of its no man’s land. The recent works on evaporating droplets [62, 68, 69] have

pushed Th downward compared to the work on emulsions. Will this be enough

to find the origin of water anomalies? For comparison, a prediction of molecular

dynamics simulations [65] is included in Fig. 3. Simulations with the potential275

TIP4P/2005 [70] were chosen, because it is considered to be currently the po-

tential that reproduces the most accurately a series of experimental properties

of water [71]. Looking at Fig. 3 thus suggests that a line of isothermal com-

pressibility might exist in the region that has just become accessible. However,

as with any water potential, simulations should be taken with caution, and the280

features they predict might well suffer from a shift in temperature and pressure

(or even not exist in real water!). Anyhow, one of the purposes of Fig. 3 is

to motivate more measurements of water at extreme supercooling. Note that

this will require special techniques, compatible with the very short timescales

involved.285

I would like to conclude this section with a terminology issue. The most

recent works [62, 68, 63] claim that they enter the “no man’s land”. As the

boundary of the no man’s land is a matter of experimental definition, this is not

13

correct. What can be said is that the experimental efforts have been successful

in pushing its border to lower temperatures (see Fig. 3), thus reducing the290

interval spanned by the no man’s land. But the very concept is akin to Tantalus’

punishment: when one tries to enter no man’s land, its location recedes, eluding

our grasp.

5. An unexplored territory: water at negative pressure

Instead of heading straight down along the slope of steepest cooling, we may295

take a detour through a less trodden path, at negative pressures. Water, like

any liquid, can be stretched to a metastable state at densities lower than its

density at equilibrium with vapor. This is a manifestation of the cohesive forces

between the liquid molecules. Their mutual attraction can even counterbalance

negative pressure, that is a force literally trying to tear the liquid apart. The300

existence of negative pressures (mechanical tension) may sound surprising, but

this state occurs routinely in nature, in the sap of trees [72].

The knowledge of the properties of water at negative pressure is still in its

infancy. The main reason is the difficulty to avoid nucleation of vapor bubbles

(cavitation). Section 5.1 discusses the current limiting cavitation pressures that305

can be reached. Knowing where to stop, liquid properties may be measured be-

fore nucleation occurs (Section 5.2), and even in the liquid that is both stretched

and supercooled (Section 5.3).

5.1. The cavitation limit

Because water has a strong cohesion, as demonstrated for instance by its310

high surface tension, one expects it may reach a large degree of metastability.

Its maximum value was the topic of our previous review [20], partly updated

in [18]. We can give here only a brief summary. First, care must be taken to

avoid heterogeneous nucleation, which accounts for a large scatter among exper-

imental values. For the most careful experiments, the agreement is very good315

in the high temperature region, at positive pressure, commonly denominated

14

superheated water. However, at lower temperature, where cavitation occurs in

the stretched liquid, there is a surprising splitting of the cavitation pressure Pcav

into two sets. A first set of very different methods leads to a consistent value of

Pcav ≃ −30MPa at room temperature. This lies far from the theoretical predic-320

tion based on classical nucleation theory (CNT). CNT involves the bulk surface

tension of water and predicts a cavitation pressure PCNTcav ≃ −140MPa [73].

This value is actually reached by one single method, which was used by dif-

ferent groups. This is the microscopic Berthelot tube method, which consists

in cooling a micrometer size water droplet trapped in a quartz crystal, so that325

the liquid, which sticks to the hydrophilic walls, follows an isochore (constant

density path). The group of Angell pioneered this technique, observing cav-

itation at a maximum tension of −140MPa at 315K [74], a value confirmed

by subsequent studies [75, 76, 77]. The reason for the large gap between the

two sets of cavitation pressures remains unknown, but points towards an ubiqui-330

tous impurity that would destabilize the metastable liquid in most experimental

techniques, or more surprisingly stabilize the inclusions of water in quartz [78].

Here we would like to update the experimental survey with a few recent

references that were not included in our previous reviews. Nanofabrication

techniques were used to create 120µm long channels in silica, with a rectangular335

cross section (4µmwide and from 20 to 120 nm high) [79]. The channels, initially

filled with water, dry by evaporation, creating negative pressure in the liquid

because of the Laplace pressure jump across the liquid-vapor menisci. The

channels empty by cavitation, instead of recession of the menisci. The cavitation

pressure (calculated from the Laplace equation) ranges from −1MPa for the340

largest pores, to −7MPa for the narrowest. These values are still far from the

above ones, because nucleation was triggered by an unstability of the meniscus

that produced a bubble near the pore entrance, which then moved to the center

and expanded.

An usual way to remove pre-existing bubbles (that may be trapped on con-345

tainer walls for instance) is to pressurize the sample to large positive pres-

sure, before performing the cavitation experiment [85]. Recently, this protocol

15

-35

-30

-25

-20

-15

0 10 20 30 40 50

Cav

itat

ion

pre

ssure

(M

Pa)

Temperature (°C)

Figure 4: (Color online) Cavitation pressure as a function of temperature for different ex-

periments. Recent acoustic-based experiments are compared to our previous work (filled red

circles) [80]: acoustic resonator (empty blue squares) [81], short focused burst (upward green

triangle) [82] (from which we have taken the largest negative pressure reached), and reflected

shock waves (downward purple triangle) [83]. The result using artificial trees (black crossed

square) [84], with the pressure calculated from the controlled vapor activity, is also shown.

For a comparison with older data, see Fig. 4 of Ref. [17].

16

was again tested for acoustic cavitation: negative pressures are generated in a

standing wave created by the resonance of the water-filled container [81]. For

positive applied pressures Pstat in the range 0 − 30MPa, the cavitation pres-350

sure was found to exhibit a linear variation: averaging over all temperatures

Pcav = −3.46 − 0.92Pstat (with pressures in MPa). This gives a largest neg-

ative value Pcav = −31MPa, which perfectly confirms our previous work on

acoustic cavitation. A detailed comparison (at each measured temperature) is

shown in Fig. 4. In contrast to the technique using a standing wave in a res-355

onator, we used bursts of focused ultrasound. A first, indirect estimate of the

pressure [86] led to Pcav from −26MPa at 273.15K to −17MPa at 353.15K.

Using a fiber-optic probe hydrophone, these values were later revised [17]: from

−34MPa at 274K to −25MPa at 320K. We emphasize that no change in Pcav

was observed, even after pressurizing the cell to Pstat = 20MPa. This shows the360

advantage of using a focused wave, which avoids the effect of bubbles trapped

at the container wall, because negative pressures occur only in a small region

inside the bulk liquid during a short time. This approach was followed in an

ultrasound-therapy oriented study [82]. Short pulses of a 1.1MHz wave were

focused in water, tissue-mimicking materials, and real tissues. In water at room365

temperature, Pcav = 27.4± 1.3MPa, in good agreement with our previous work

(Fig. 4).

Ando and colleagues use the reflection at a liquid-vapor interface of a laser-

induced shock-wave propagating in water, to generate a pulse of mechanical

tension. The pressure is calculated using Euler flow simulations. In a first370

study [87], the cavitation threshold was reported to be Pcav = −60 ± 5MPa,

exceeding all previously reported dynamic measurements. However, in a sub-

sequent study[83], using two laser beams to create the superposition of two

reflected tension pulses, this value was revised to −20.1± 3.4MPa, slightly less

negative than other acoustic methods (Fig. 4). The second study included some375

statistics on cavitation, to report the threshold at a cavitation probability of

0.5. The first study, which only reported an individual threshold at a cavitation

probability of 1, thus seems to have overestimated the limit of metastability.

17

The technique of artificial trees is an interesting alternative to acoustics.

The group of Stroock mimicks the mechanism by which trees pump the sap380

up their trunk. Water exchange through a porous membrane in the leaves

allows to reach a metastable equilibrium between an undersaturated vapor in

the atmosphere (relative humidity less than 100%), and a liquid at negative

pressure. The process is similar to osmotic effects. In the synthetic trees [84],

the membrane is a hydrogel, in which are embedded spherical cavities (in the385

10µm range) filled with water. As already noticed [84, 18], Pcav ≃ −22MPa at

293K, consistent with (although slightly less negative than) Pcav in the acoustic

experiments. Here Pcav was calculated based on the relative humidity of the

vapor with which the liquid was equilibrated. Interestingly, pressurization up

to 54MPa did not change Pcav [88], demonstrating that trapped pre-existing390

bubbles do not play a role in the synthetic trees. This method was re-used in

the group of Marmottant [89, 90]. Measuring the volume of a cavity just before

and just after cavitation, they could estimate the pressure based on the liquid

compressibility, although with a large uncertainty: Pcav = 30 ± 16MPa. The

main focus of the work was the bubble dynamics: fast oscillations were observed395

and accounted for with a model involving the droplet confinement and elasticity.

Recent developments in the group of Stroock include the measurement of the

equation of state of water down to −14MPa with a microelectromechanical

pressure sensor [91], and the interplay between poroelastic mass transport and

cavitation in the drying of ink-bottle porous media [92].400

From this updated review, it is confirmed that only the microscopic Berth-

elot tube method (based on water filled inclusions in quartz) is able to reach

the theoretical predictions for Pcav, around −140MPa. Many other methods,

including recent ones, cluster around −30MPa (Fig. 4). The explanation of this

discrepancy is still unknown. The most reasonable hypothesis remains that an405

ubiquitous impurity triggering cavitation at −30MPa is present in all experi-

ments but the one with inclusions[78]. More precisely, for a given quartz sample,

the inclusions contain liquid at the same density but exhibit a range of Pcav,

down to PCNTcav [74, 76]. It can be speculated that only a few inclusions are free

18

from the impurities favoring cavitation. Note that if the cavitation pressure is410

extremely sensitive to trace impurities, most thermodynamic properties are not

expected to be. Therefore any method to generate negative pressure may be

used to prepare the liquid and measure its properties. Inclusions in quartz just

provide the method that can, at present, access the largest tensions.

5.2. Properties of stretched water415

Now that we know down to which negative pressure each experiment can

be performed, we may attempt to measure properties of stretched water before

if breaks by cavitation. Although the first attempt was published one century

ago [93], there have not been many since. The first information of interest is

the EoS of the metastable liquid. Indeed, extrapolations of the positive pressure420

EoS [22] and molecular dynamics simulations (e.g. Refs. [23] and [34] for the ST2

potential) differ qualitatively in their predictions of the liquid-vapor spinodal or

the line of density maxima (see Section 2). An experimental test is therefore

needed.

The first work is due to Meyer [93] who used a Berthelot-Bourdon tube to425

measure the relation between pressure and density down to -3.4 MPa at 24◦C.

In the 1980’s, Henderson and Speedy took over the challenge. Using a modified

version of Meyer’s Berthelot-Bourdon gauge, they measured the line of density

maxima down to −20.3MPa where it reaches 8.3◦C [94, 95], and the metastable

melting curve down to −24MPa [96]. Davitt et al. [97] used the acoustic method430

to stretch water, and measured simultaneously two physical quantities of the

metastable liquid: the density with a fiber optic probe hydrophone [80], and

the sound velocity with a time-resolved Brillouin scattering experiment [97].

They were thus able to obtain the EoS down to -26 MPa at 23.3◦C. They

found that the EoS is compatible with the standard extrapolation of the positive435

pressure data [98, 99]. Their finding was recently confirmed down to −14MPa

with a microelectromechanical pressure sensor [91]. However, the qualitative

differences between theoretical predictions mentioned above become manifest

only at larger negative pressure.

19

Because of the cavitation pressures measured up to now (see Section 5.1),440

the only experimental method that makes this region accessible is that using

water inclusions in quartz as microscopic Berthelot tubes. Indirect information

on the line of density maxima was obtained recently by a statistical analysis of

cavitation in one single water inclusion [77]. As the inclusion follows an isochore,

the pressure must reach a largest negative value exactly at the crossing point445

with the LDM. This, combined with the temperature variation of the liquid-

vapor surface tension, results in a minimum in the energy barrier for cavitation

as a function of temperature. This effect was already noticed by the group

of Angell [74], where an inclusion “was observed in repeated runs to nucleate

randomly in the range 40 to 47◦C and occasionally not at all”, because the450

energy barrier was just around the value for a finite cavitation probability. In

our recent work [77], an inclusion in which cavitation occurred around 325K

was selected, and a total of 154 cavitation experiments were performed at three

different cooling rates, to obtain the statistical distribution of cavitation tem-

peratures. This gives the energy barrier as a function of temperature, which455

was found to exhibit a curvature, from which a minimum was predicted to be

reached at Tmin = 321.4±4.3K. From the extrapolation of the positive pressure

EoS, Tmin = 317.6K is calculated, in agreement with the experimental value.

This means that the LDM cannot depart too far from its extrapolation down

to the density studied, ρ = 922.8 kgm−3.460

The previous result is still indirect. Because of the type of samples, it seems

that direct measurements can only be performed by optical methods. Brillouin

scattering gives the sound velocity which can, for instance, be compared to the

extrapolation of available EoSs. A first measurement was performed [75], but

it was not used in a systematic way to compare with an extrapolated EoS.465

We decided to revisit this experiment [65]. We measured the sound velocity

in two samples, along isochores at ρ1 = 933.2 and ρ2 = 952.5 kgm−3. At

these densities, the pressure at room temperature is in the −100MPa range (see

Fig. 3). This time, a departure from the extrapolation of the positive pressure

EoS was found, with the extrapolated sound velocity being always lower than470

20

the measured one. But the low temperature behaviour was more surprising, and

is described in the next section.

5.3. Doubly metastable water

We are now entering an almost virgin territory: the doubly metastable re-

gion where liquid water is simultaneously metastable compared to the vapor475

and to the ice. Thermodynamically, its borders are the metastable continua-

tions of the liquid-vapor and the liquid-solid equilibrium lines beyond the triple

point. Note that these lines have been experimentally located in the pressure-

temperature plane, down to 251K for the former [100] and down to −24MPa

for the latter [96].480

Penetration in the doubly metastable region was reported only a couple of

times. First by Hayward [101], who reached a modest −0.02MPa around 268K

with a tension manometer. He humourously compared the water molecules to

a “bunch of schizophrenics”, “desperately anxious both to freeze and to boil”.

Using a modified version of Meyer’s Berthelot-Bourdon gauge, Henderson and485

Speedy navigated along a curve going from −19.5MPa at 273.15K to −8MPa

at 255.15K [95]. The frontier was pushed further with our acoustic method: at

273.25K we found Pcav = −26MPa [86], which was revised to −34MPa based

on fiber optic probe hydrophone calibration [17]. As before, the only known

method to reach larger tensions is to use water inclusions in quartz. It was490

already noticed by the group of Angell [74] that “no inclusion that survived

cooling to 40◦C ever nucleated bubbles during cooling to lower temperatures

(although ice probably nucleated without being observable)”, but no measure-

ment was performed. The previous Brillouin study on water inclusions [75] did

investigate one of these survivors, but unfortunately measurements stopped at495

0◦C. The two inclusions we selected (at densities ρ1 and ρ2, see Section 5.2 and

Fig. 3), had the ability to reach the doubly metastable region, because along

these isochores, the minimum energy barrier to cavitation remains high enough.

The only limitation to the measurements was the broadening of the Brillouin

spectra at low temperature because of viscosity, which made them impossible500

21

to be analyzed properly below 258K. Still we found interesting anomalies [65].

First, as mentioned in Section 5.2, the sound velocity was found to be higher

than the prediction of the extrapolation of the positive pressure EoS. The dis-

crepancy became larger at lower temperatures. This proves that water was not

approaching the instability limit of this extrapolated EoS, located at 259K for505

the density ρ1, where the sound velocity should have been nearly twice lower

than what we measured. Next, the sound velocity along the ρ1 isochore reached

a clear minimum near 273K, which corresponds to a maximum in adiabatic

compressibility. Finally, at the lowest temperatures, the sound velocity along

an isotherm was found to exhibit a non monotonic behaviour. This led us to pro-510

pose, based on comparison with molecular dynamics simulations of TIP4P/2005

performed in the group of Valeriani, that the isochores were actually crossing a

line of anomalies. This line is the locus of maxima in the isothermal compress-

ibility along isobars (see Fig. 3), one of the proxies used in simulations to detect

the Widom line that would be associated with a LLT and its LLCP. Although a515

line of maxima in isothermal compressibility does not require a LLCP (see the

singularity free scenario, Section 2), our results rule out scenarios according to

which water properties would diverge on a line of instability (see Section 2).

6. Conclusion

After decades of scrutiny from experimentalists, it may seem surprising that520

there is still something to measure or progress to be made on supercooled water.

Yet this is the case, but scientists have to face a dilemma. They can choose to

confine water or mix it with a solute to avoid crystallization. This opens new

regions of temperature to measurements, but the relation to pure bulk water is

not straightforward. Or they can continue to seek routes to greater supercooling.525

Recent publications show that they can be successful. However, the very short

lifetime of this highly supercooled state makes measurements challenging, but

worth working on. An alternative is to dive in the doubly metastable region,

which is presumably further away from the putative LLT, but allows for longer

22

experimental times.530

Acknowledgments

Funding by the ERC under the European Community’s FP7 Grant Agree-

ment 240113, and by the Agence Nationale de la Recherche ANR Grant 09-

BLAN-0404-01 are acknowledged.

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