Download Escaping the no man`s land: recent experiments on metastable

Escaping the no man’s land: recent experiments on
metastable liquid water
Frédéric Caupin
Institut Lumière Matière, UMR5306 Université Lyon 1-CNRS, Université de Lyon and
Institut 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 hitherto 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
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liquid below the equilibrium melting temperature, and to measure the properties 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
1 [email protected]
Preprint submitted to Journal of Non-Crystalline Solids
September 15, 2014
6
P
-1
-1
C (kJ K kg )
7
5
4
220
230
240
250
Temperature (K)
260
270
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 228 K. 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 (239.74 K). In a famous plot (Fig. 4 of Ref [2]), Speedy
10
and Angell noticed that many properties of stable and supercooled water (such
as αP , or the isothermal compressibility κT , the heat capacity at constant pressure CP , etc. . . ) could be fitted by power laws which extrapolated to a common
temperature of divergence of 228 K at atmospheric pressure, slightly below the
lower temperature limit of experiments. The following question therefore arises:
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if measurements could be performed at lower temperatures, would there be an
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
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ice, favored by certain impurities or surfaces. If precautions are taken to avoid
these, ice will still nucleate spontaneously from thermal fluctuations in the liquid. This occurs at the homogeneous nucleation temperature Th , which thus
gives a lower bound to the temperature at which measurements can be performed on liquid water cooled from ambient temperature. The exact value of
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Th depends on the experimental volume and cooling rate, and therefore varies
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
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temperatures than what can be obtained by directly cooling the room temperature liquid. Unfortunately this approach is also hindered by re-crystallization
of the amorphous ice, around 150 K [7] (or up to 190 K 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
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150 to 235 K at ambient pressure) is called the “no man’s land”, a name coined
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
40
proposed to explain the increasing anomalies of water when it is supercooled.
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 polymorphism). The rest of the discussion comes back to pure bulk water and describes
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the most recent experimental advances. Section 4 introduces new methods that
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
50
We should mention that we will not discuss the very interesting possibility
to study the ultraviscous liquid(s) that could be obtained by heating the amorphous ices just before crystallization occurs [10]. We also refer the reader to introductions, reviews or books where more details on several topics can be found:
metastable liquids and nucleation in general, with large parts about water [11],
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supercooled and glassy water [12, 13], thermodynamics of supercooled water [5],
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
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The focus of this review is experimental, but we need to introduce the competing 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 228 K,
Speedy looked for a source of instability. Any liquid can be brought below its
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saturated vapor pressure, but will eventually become unstable with respect to
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 thermodynamic explanation: if the line of density maxima (LDM) intersects the spin-
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odal curve in the pressure-temperature plane (see Fig. 2, top), the latter must
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 pressure. 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
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metastable liquid-vapor equilibrium, which can happen only at a liquid-vapor
critical point: such a singularity seems unlikely. However, two interesting questions 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
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another phase than the vapor)?
In contrast, molecular dynamics simulations of a water-like potential [23]
found a monotonic temperature dependence of the liquid-vapor spinodal pressure. Thermodynamic consistency was preserved, as the LDM reached a maximum temperature at negative pressure, and avoided the spinodal (Fig. 2, bot-
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tom). The simulations found another source for water anomalies. In the second
critical point scenario [23], a first-order transition separates two liquids with different structures in the supercooled region. This liquid-liquid transition (LLT)
terminates at a liquid-liquid critical point (LLCP) (237 ± 4 K and 167 ± 24 MPa
for the ST2 potential [24]), responsible for the large increase in many water
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properties. However, they diverge only at the critical point. Below the critical
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 crystallization, challenging [25, 26] or reasserting [27, 28, 29, 30, 31, 32, 33] the existence
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of a LLCP. It is beyond the scope of this review to detail this debate. However,
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 accessible 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
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loci of extrema in the response functions of water. They would be associated
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
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LLCP thus predicted lies at much lower pressure than in simulations (227 K,
13 MPa) [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 spinodal. In that case, upon cooling, there is no extremum in response function,
5
0 Pressure
liq-sol
C
liq-vap
T
Temperature
LDM
0 Pressure
spinodal
liq-sol
liq-liq
C
liq-vap
C'
T
Temperature
spinodal
LDM
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 liquidsolid 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).
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but instead a LLT, or if the high density liquid remains metastable with respect
to the low density liquid, a divergence on the high density liquid spinodal. The
singularity-free interpretation [39] offers a thermodynamically consistent picture, 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
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is noteworthy to mention that a cell model involving hydrogen bond cooperativity [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:
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• finding a genuine LLT;
• 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.
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3. Squeeze or hide: tricks to avoid crystallization
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 temperature 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
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the nucleation rate.
3.1. Confinement
For a detailed review on the effects of confinement on freezing and melting, 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
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liquid-crystal equilibrium temperature is depressed from its bulk value Teq by
an amount:
∆Teq =
4 σLC Teq
,
ρc L d
7
(1)
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
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modified to account for microscopic details, but the trend is correctly captured.
For water-ice, σLC is in the range 25−44 mJ m−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
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agrees with the observed melting temperatures for pore diameters from 3 to
4.4 nm (from 215.0 to 239.7 K). 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-
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like phase at very low temperatures. A recent review of all their work is available [43]. Let us simply mention here that, using a variety of techniques (neutron scattering, nuclear magnetic resonance, Fourier transform infrared spectroscopy. . . ), several features compatible with the second critical point scenario
were reported, such as: dynamic crossover [44], density minimum [45, 46] and
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density hysteresis [47]. These works have provoked a passionate debate about
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
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corresponds to at most 5 layers of water across. Two are in direct contact with
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.
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We would also like to mention that the exact pressure at which experiments
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 hy8
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 macro170
scopic Laplace law, the pressure jump is 2γLV /R ≃ −140/R in MPa if R is in
nm. In extremely narrow pores the validity of the macroscopic law is questionable. 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
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wonder if it should rather be with a liquid at negative pressure [55]. This idea
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 −100 MPa.
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3.2. Antifreeze
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-
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ical working concentrations, the cryoscopic constant K for water is somewhat
different from the infinite dilution limit, but the proportionality still holds with
K = 1.853 K kg mol−1 . Interestingly, the homogeneous crystallization temperature 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
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for many compounds, and the values found to be surprisingly grouped around
2. There are variations, correlated with the self-diffusion coefficient of the solute [57].
Murata and Tanaka have chosen glycerol as the solute [58]. Using an impressive set of experimental techniques, they concluded that a transition be-
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tween two different liquids with equal glycerol concentration occurred, without
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
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solutions of sugar and polyol molecules were studied: only with glycerol and
1,2,4-butanetriol a LLT was reported without concurrent demixing. Experiments 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
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reach a clear conclusion for pure water.
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
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new phase exceeds a critical size, it will grow very fast and replace a large
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.
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Nucleation is also a kinetic process. The physics of the transformation determines the nucleation rate per unit volume and time, Γ, as a function of metastability (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,
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an experiment performed on a small sample during a short time will reach larger
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
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been obtained. For instance, Kanno et al. [60] reported Th values in the range 0−
300 MPa, with a minimum of 181 K at 200 MPa; the data up to 140 MPa 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 235 K for 300 and 6 µm diameter
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droplets, respectively, in line with the exponential temperature dependence of
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
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Taborek’s experiment [6], reviews many other nucleation rate measurements,
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
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have been used (see Ref. [66] for a review). They can be generated by adiabatic
expansion of a gas saturated with water vapor, leading to supersaturation conditions 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
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and temperature drop. The technique was introduced in the field of ice nucleation 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 spectroscopy to detect ice nucleation. With the supersonic nozzle, the droplets are
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much smaller than in emulsion studies, with diameters between 6.4 and 11.6 nm.
The liquid is therefore subject to a non-negligible Laplace pressure, from 54 to
29 MPa, respectively. Onset of ice was detected at 202 K and 215 K, respectively. 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
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water [68].
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
150
LLCP???
supercooled
stable
Pressure (MPa)
100
50
new border
0
P<0
-50
doubly
metastable
-100
-150
150
200
250
Temperature (K)
300
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 extrapolation 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 9 µm in diameter) can
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be probed with a femtosecond long pulse from an ultrabright X-ray laser at
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 227 K. Under these extreme conditions, water is liquid only for about
a microsecond. Its structure factor, measured down to 229 K, showed that
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the splitting of its first maximum into two peaks, already known at ambient
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
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unprecedented supercooling.
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
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dynamics simulations [65] is included in Fig. 3. Simulations with the potential
TIP4P/2005 [70] were chosen, because it is considered to be currently the potential that reproduces the most accurately a series of experimental properties
of water [71]. Looking at Fig. 3 thus suggests that a line of isothermal compressibility might exist in the region that has just become accessible. However,
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as with any water potential, simulations should be taken with caution, and the
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
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involved.
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
290
in pushing its border to lower temperatures (see Fig. 3), thus reducing the
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
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Instead of heading straight down along the slope of steepest cooling, we may
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
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negative pressure, that is a force literally trying to tear the liquid apart. The
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
305
(cavitation). Section 5.1 discusses the current limiting cavitation pressures that
can be reached. Knowing where to stop, liquid properties may be measured before nucleation occurs (Section 5.2), and even in the liquid that is both stretched
and supercooled (Section 5.3).
5.1. The cavitation limit
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Because water has a strong cohesion, as demonstrated for instance by its
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-
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imental values. For the most careful experiments, the agreement is very good
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
320
Pcav ≃ −30 MPa at room temperature. This lies far from the theoretical prediction based on classical nucleation theory (CNT). CNT involves the bulk surface
CNT
tension of water and predicts a cavitation pressure Pcav
≃ −140 MPa [73].
This value is actually reached by one single method, which was used by different groups. This is the microscopic Berthelot tube method, which consists
325
in cooling a micrometer size water droplet trapped in a quartz crystal, so that
the liquid, which sticks to the hydrophilic walls, follows an isochore (constant
density path). The group of Angell pioneered this technique, observing cavitation at a maximum tension of −140 MPa at 315 K [74], a value confirmed
by subsequent studies [75, 76, 77]. The reason for the large gap between the
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two sets of cavitation pressures remains unknown, but points towards an ubiquitous 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
335
techniques were used to create 120 µm long channels in silica, with a rectangular
cross section (4 µm wide 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
340
pressure (calculated from the Laplace equation) ranges from −1 MPa for the
largest pores, to −7 MPa 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.
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An usual way to remove pre-existing bubbles (that may be trapped on container walls for instance) is to pressurize the sample to large positive pressure, before performing the cavitation experiment [85]. Recently, this protocol
15
Cavitation pressure (MPa)
-15
-20
-25
-30
-35
0
10
20
30
Temperature (°C)
40
50
Figure 4: (Color online) Cavitation pressure as a function of temperature for different experiments. 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
350
positive applied pressures Pstat in the range 0 − 30 MPa, the cavitation pressure was found to exhibit a linear variation: averaging over all temperatures
Pcav = −3.46 − 0.92Pstat (with pressures in MPa). This gives a largest negative value Pcav = −31 MPa, which perfectly confirms our previous work on
acoustic cavitation. A detailed comparison (at each measured temperature) is
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shown in Fig. 4. In contrast to the technique using a standing wave in a resonator, we used bursts of focused ultrasound. A first, indirect estimate of the
pressure [86] led to Pcav from −26 MPa at 273.15 K to −17 MPa at 353.15 K.
Using a fiber-optic probe hydrophone, these values were later revised [17]: from
−34 MPa at 274 K to −25 MPa at 320 K. We emphasize that no change in Pcav
360
was observed, even after pressurizing the cell to Pstat = 20 MPa. This shows the
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 1.1 MHz wave were
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focused in water, tissue-mimicking materials, and real tissues. In water at room
temperature, Pcav = 27.4 ± 1.3 MPa, in good agreement with our previous work
(Fig. 4).
Ando and colleagues use the reflection at a liquid-vapor interface of a laserinduced shock-wave propagating in water, to generate a pulse of mechanical
370
tension. The pressure is calculated using Euler flow simulations. In a first
study [87], the cavitation threshold was reported to be Pcav = −60 ± 5 MPa,
exceeding all previously reported dynamic measurements. However, in a subsequent study[83], using two laser beams to create the superposition of two
reflected tension pulses, this value was revised to −20.1 ± 3.4 MPa, slightly less
375
negative than other acoustic methods (Fig. 4). The second study included some
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.
380
The group of Stroock mimicks the mechanism by which trees pump the sap
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],
385
the membrane is a hydrogel, in which are embedded spherical cavities (in the
10 µm range) filled with water. As already noticed [84, 18], Pcav ≃ −22 MPa at
293 K, 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
390
to 54 MPa did not change Pcav [88], demonstrating that trapped pre-existing
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 ± 16 MPa. The
395
main focus of the work was the bubble dynamics: fast oscillations were observed
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 −14 MPa with a microelectromechanical
pressure sensor [91], and the interplay between poroelastic mass transport and
400
cavitation in the drying of ink-bottle porous media [92].
From this updated review, it is confirmed that only the microscopic Berthelot tube method (based on water filled inclusions in quartz) is able to reach
the theoretical predictions for Pcav , around −140 MPa. Many other methods,
including recent ones, cluster around −30 MPa (Fig. 4). The explanation of this
405
discrepancy is still unknown. The most reasonable hypothesis remains that an
ubiquitous impurity triggering cavitation at −30 MPa is present in all experiments 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 ,
CNT
down to Pcav
[74, 76]. It can be speculated that only a few inclusions are free
18
410
from the impurities favoring cavitation. Note that if the cavitation pressure is
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.
415
5.2. Properties of stretched water
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
420
the EoS of the metastable liquid. Indeed, extrapolations of the positive pressure
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.
425
The first work is due to Meyer [93] who used a Berthelot-Bourdon tube to
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 −20.3 MPa where it reaches 8.3◦ C [94, 95], and the metastable
430
melting curve down to −24 MPa [96]. Davitt et al. [97] used the acoustic method
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
435
found that the EoS is compatible with the standard extrapolation of the positive
pressure data [98, 99]. Their finding was recently confirmed down to −14 MPa
with a microelectromechanical pressure sensor [91]. However, the qualitative
differences between theoretical predictions mentioned above become manifest
only at larger negative pressure.
19
440
Because of the cavitation pressures measured up to now (see Section 5.1),
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,
445
the pressure must reach a largest negative value exactly at the crossing point
with the LDM. This, combined with the temperature variation of the liquidvapor 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
450
randomly in the range 40 to 47◦ C and occasionally not at all”, because the
energy barrier was just around the value for a finite cavitation probability. In
our recent work [77], an inclusion in which cavitation occurred around 325 K
was selected, and a total of 154 cavitation experiments were performed at three
different cooling rates, to obtain the statistical distribution of cavitation tem-
455
peratures. This gives the energy barrier as a function of temperature, which
was found to exhibit a curvature, from which a minimum was predicted to be
reached at Tmin = 321.4 ± 4.3 K. From the extrapolation of the positive pressure
EoS, Tmin = 317.6 K is calculated, in agreement with the experimental value.
This means that the LDM cannot depart too far from its extrapolation down
460
to the density studied, ρ = 922.8 kg m−3 .
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
465
it was not used in a systematic way to compare with an extrapolated EoS.
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 kg m−3 . At
these densities, the pressure at room temperature is in the −100 MPa range (see
Fig. 3). This time, a departure from the extrapolation of the positive pressure
470
EoS was found, with the extrapolated sound velocity being always lower than
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 re475
gion where liquid water is simultaneously metastable compared to the vapor
and to the ice. Thermodynamically, its borders are the metastable continuations of the liquid-vapor and the liquid-solid equilibrium lines beyond the triple
point. Note that these lines have been experimentally located in the pressuretemperature plane, down to 251 K for the former [100] and down to −24 MPa
480
for the latter [96].
Penetration in the doubly metastable region was reported only a couple of
times. First by Hayward [101], who reached a modest −0.02 MPa around 268 K
with a tension manometer. He humourously compared the water molecules to
a “bunch of schizophrenics”, “desperately anxious both to freeze and to boil”.
485
Using a modified version of Meyer’s Berthelot-Bourdon gauge, Henderson and
Speedy navigated along a curve going from −19.5 MPa at 273.15 K to −8 MPa
at 255.15 K [95]. The frontier was pushed further with our acoustic method: at
273.25 K we found Pcav = −26 MPa [86], which was revised to −34 MPa based
on fiber optic probe hydrophone calibration [17]. As before, the only known
490
method to reach larger tensions is to use water inclusions in quartz. It was
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 measurement was performed. The previous Brillouin study on water inclusions [75] did
495
investigate one of these survivors, but unfortunately measurements stopped at
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
500
spectra at low temperature because of viscosity, which made them impossible
21
to be analyzed properly below 258 K. 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 discrepancy became larger at lower temperatures. This proves that water was not
505
approaching the instability limit of this extrapolated EoS, located at 259 K for
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 273 K, which corresponds to a maximum in adiabatic
compressibility. Finally, at the lowest temperatures, the sound velocity along
510
an isotherm was found to exhibit a non monotonic behaviour. This led us to propose, 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 compressibility along isobars (see Fig. 3), one of the proxies used in simulations to detect
515
the Widom line that would be associated with a LLT and its LLCP. Although a
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
520
After decades of scrutiny from experimentalists, it may seem surprising that
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
525
not straightforward. Or they can continue to seek routes to greater supercooling.
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
530
experimental times.
Acknowledgments
Funding by the ERC under the European Community’s FP7 Grant Agreement 240113, and by the Agence Nationale de la Recherche ANR Grant 09BLAN-0404-01 are acknowledged.
535
References
[1] D. E. Hare, C. M. Sorensen, The density of supercooled water. II. bulk samples cooled to the homogeneou
The
Journal
of
Chemical
Physics
87
(8)
(1987)
4840–4845.
doi:doi:10.1063/1.453710.
URL http://jcp.aip.org/resource/1/jcpsa6/v87/i8/p4840_s1
540
[2] R. J. Speedy, C. A. Angell, Isothermal compressibility of supercooled water and evidence for a thermodyn
J. Chem. Phys. 65 (3) (1976) 851–858. doi:10.1063/1.433153.
URL http://link.aip.org/link/?JCP/65/851/1
[3] C.
A.
Angell,
W.
J.
Sichina,
M.
Oguni,
Heat capacity of water at extremes of supercooling and superheating,
545
The
Journal
of
Physical
Chemistry
86
(6)
(1982)
998–1002.
doi:10.1021/j100395a032.
URL http://pubs.acs.org/doi/abs/10.1021/j100395a032
[4] D. G. Archer, R. W. Carter, Thermodynamic properties of the NaCl + h 2 o system. 4. heat capacities o
The Journal of Physical Chemistry B 104 (35) (2000) 8563–8584.
550
doi:10.1021/jp0003914.
URL http://pubs.acs.org/doi/abs/10.1021/jp0003914
[5] V. Holten,
C. E. Bertrand,
M. A. Anisimov,
Thermodynamics of supercooled water,
The
J. V. Sengers,
Journal
of
Chemical
Physics 136 (9) (2012) 094507. doi:doi:10.1063/1.3690497.
555
URL http://jcp.aip.org/resource/1/jcpsa6/v136/i9/p094507_s1?isAuthorized=no
23
[6] B.
Riechers,
F.
Wittbracht,
A.
Htten,
T.
Koop,
The homogeneous ice nucleation rate of water droplets produced in a microfluidic device and the role of
Physical
Chemistry
Chemical
Physics
15
(16)
(2013)
5873.
doi:10.1039/c3cp42437e.
560
URL http://xlink.rsc.org/?DOI=c3cp42437e
[7] N. Giovambattista, K. Amann-Winkel, T. Loerting, Amorphous ices, in:
H. E. Stanley (Ed.), Liquid Polymorphism, John Wiley & Sons, Inc.,
2013, pp. 139–173.
URL http://onlinelibrary.wiley.com/doi/10.1002/9781118540350.ch7/summary
565
[8] A.
Sepulveda,
E.
Leon-Gutierrez,
M.
Gonzalez-Silveira,
C. Rodriguez-Tinoco, M. T. Clavaguera-Mora, J. Rodriguez-Viejo,
Glass transition in ultrathin films of amorphous solid water, The Journal
of Chemical Physics 137 (24) (2012) 244506. doi:10.1063/1.4771964.
URL http://scitation.aip.org/content/aip/journal/jcp/137/24/10.1063/1.4771964
570
[9] O. Mishima, H. E. Stanley, The relationship between liquid, supercooled and glassy water,
Nature 396 (6709) (1998) 329–335.
URL http://www.nims.jp/water/Publications/MS1998nature-b.pdf
[10] K. Amann-Winkel, C. Gainaru, P. H. Handle, M. Seidl, H. Nelson,
R. Bohmer, T. Loerting, Water’s second glass transition, Proceedings
575
of the National Academy of Sciences 110 (44) (2013) 17720–17725.
doi:10.1073/pnas.1311718110.
URL http://www.pnas.org/cgi/doi/10.1073/pnas.1311718110
[11] P. G. Debenedetti, Metastable liquids, Princeton University Press, 1996.
[12] P. G. Debenedetti, H. E. Stanley, Supercooled and glassy water, Phys.
580
Today 56 (6) (2003) 40–46. doi:10.1063/1.1595053.
URL http://link.aip.org/link/PHTOAD/v56/i6/p40/
[13] P. G. Debenedetti, Supercooled and glassy water, Journal of Physics:
24
Condensed Matter 15 (45) (2003) R1669–R1726.
URL http://www.iop.org/EJ/abstract/0953-8984/15/45/R01
585
[14] S. Balibar,
uid phase,
F. Caupin,
Nucleation of crystals from their liq-
Comptes Rendus Physique 7 (9-10) (2006) 988–999.
doi:10.1016/j.crhy.2006.10.024.
[15] C. A. Angell, Insights into phases of liquid water from study of its unusual glass-forming properties,
Science 319 (5863) (2008) 582–587. doi:10.1126/science.1131939.
590
URL http://www.sciencemag.org/cgi/doi/10.1126/science.1131939
[16] H. E. Stanley (Ed.), Advances in Chemical Physics, Liquid Polymorphism,
John Wiley & Sons, 2013.
[17] F. Caupin, A. Arvengas, K. Davitt, M. E. M. Azouzi, K. I.
Shmulovich,
595
C.
Ramboz,
D.
A.
Sessoms,
A.
D.
Stroock,
Exploring water and other liquids at negative pressure,
nal
of
Physics:
Condensed
Matter
24
(28)
Jour(2012)
284110.
doi:10.1088/0953-8984/24/28/284110.
URL http://iopscience.iop.org/0953-8984/24/28/284110
[18] F. Caupin, A. D. Stroock, The stability limit and other open questions
600
on water at negative pressure, in: H. E. Stanley, S. Rice (Eds.), Liquid
polymorphism, no. 152 in Advances in Chemical Physics, Wiley, New
York, 2013.
[19] S. L. Meadley, C. A. Angell, Water and its relatives: the stable, supercooled and particularly the stretche
URL http://arxiv-web3.library.cornell.edu/abs/1404.4031?context=cond-mat
605
[20] F. Caupin, E. Herbert, Cavitation in water: a review, Comptes Rendus
Physique 7 (9-10) (2006) 1000–1017. doi:10.1016/j.crhy.2006.10.015.
[21] K.
Stokely,
M.
G.
Mazza,
H.
E.
Stanley,
G.
Franzese,
Effect of hydrogen bond cooperativity on the behavior of water,
Pro-
ceedings of the National Academy of Sciences 107 (4) (2010) 1301–1306.
25
610
doi:10.1073/pnas.0912756107.
URL http://www.pnas.org/cgi/doi/10.1073/pnas.0912756107
[22] R. J. Speedy, Stability-limit conjecture. an interpretation of the
properties
of
water,
J.
Phys.
Chem.
86
(6)
(1982)
982–991.
doi:10.1021/j100395a030.
615
[23] P.
H.
Poole,
F.
Sciortino,
U.
Essmann,
H.
E.
Stanley,
Phase behaviour of metastable water, Nature 360 (6402) (1992) 324–328.
doi:10.1038/360324a0.
URL http://dx.doi.org/10.1038/360324a0
[24] Y.
620
Liu,
A.
Z.
Panagiotopoulos,
P.
G.
Low-temperature fluid-phase behavior of ST2 water,
Debenedetti,
The
Journal
of
Chemical Physics 131 (10) (2009) 104508. doi:10.1063/1.3229892.
URL http://link.aip.org/link/JCPSA6/v131/i10/p104508/s1&Agg=doi
[25] D. T. Limmer, D. Chandler, The putative liquid-liquid transition is a liquid-solid transition in atomistic m
The
625
Journal
of
Chemical
Physics
135
(13)
(2011)
134503.
doi:10.1063/1.3643333.
URL http://link.aip.org/link/JCPSA6/v135/i13/p134503/
[26] D. T. Limmer, D. Chandler, The putative liquid-liquid transition is a liquid-solid transition in atomistic m
The
Journal
of
Chemical
Physics
138
(21)
(2013)
214504.
doi:10.1063/1.4807479.
630
URL http://scitation.aip.org/content/aip/journal/jcp/138/21/10.1063/1.4807479
[27] P.
H.
Poole,
S.
R.
Becker,
F.
Sciortino,
F.
W.
Starr,
Dynamical behavior near a liquidliquid phase transition in simulations of supercooled water,
J. Phys. Chem. B 115 (48) (2011) 14176–14183. doi:10.1021/jp204889m.
URL http://pubs.acs.org/doi/abs/10.1021/jp204889m
635
[28] F.
Sciortino,
I.
Saika-Voivod,
P.
H.
Poole,
Study of the ST2 model of water close to the liquidliquid critical point,
Physical
Chemistry
Chemical
26
Physics
13
(44)
(2011)
19759.
doi:10.1039/c1cp22316j.
URL http://xlink.rsc.org/?DOI=c1cp22316j
640
[29] Y. Liu, J. C. Palmer, A. Z. Panagiotopoulos, P. G. Debenedetti,
Liquid-liquid transition in ST2 water, The Journal of chemical physics
137 (21) (2012) 214505.
URL http://scitation.aip.org/content/aip/journal/jcp/137/21/10.1063/1.4769126
[30] T.
645
A.
Kesselring,
Buldyrev,
H.
J.
G.
Franzese,
Herrmann,
S.
H.
E.
V.
Stanley,
Nanoscale dynamics of phase flipping in water near its hypothesized liquid-liquid critical point,
Scientific Reports 2. doi:10.1038/srep00474.
URL http://www.nature.com/srep/2012/120629/srep00474/full/srep00474.html
[31] P.
650
H.
Poole,
R.
K.
Bowles,
I.
Saika-Voivod,
F.
Sciortino,
Free energy surface of ST2 water near the liquid-liquid phase transition,
The
Journal
of
Chemical
Physics
138
(3)
(2013)
034505.
doi:10.1063/1.4775738.
URL http://scitation.aip.org/content/aip/journal/jcp/138/3/10.1063/1.4775738
[32] T.
655
S.
A.
V.
Kesselring,
Buldyrev,
E.
H.
J.
Lascaris,
G.
Herrmann,
H.
Franzese,
E.
Stanley,
Finite-size scaling investigation of the liquid-liquid critical point in ST2 water and its stability with resp
The
Journal
of
Chemical
Physics
138
(24)
(2013)
244506.
doi:10.1063/1.4808355.
URL http://scitation.aip.org/content/aip/journal/jcp/138/24/10.1063/1.4808355
660
[33] J.
A.
C.
Palmer,
Z.
F.
Martelli,
Panagiotopoulos,
Y.
P.
Liu,
G.
R.
Car,
Debenedetti,
Metastable liquidliquid transition in a molecular model of water,
Na-
ture 510 (7505) (2014) 385–388. doi:10.1038/nature13405.
URL http://www.nature.com/doifinder/10.1038/nature13405
665
[34] P.
H.
Poole,
I.
Saika-Voivod,
Density minimum and liquidliquid phase transition,
27
F.
Sciortino,
Journal
of
Physics:
Condensed
Matter
17
(43)
(2005)
L431–L437.
doi:10.1088/0953-8984/17/43/L01.
URL http://stacks.iop.org/0953-8984/17/i=43/a=L01?key=crossref.1c7586d6ba50981758718957
670
[35] J. Luo,
L. Xu,
E. Lascaris,
H. E. Stanley,
S. V. Buldyrev,
Behavior of the widom line in critical phenomena, Physical Review Letters 112 (13). doi:10.1103/PhysRevLett.112.135701.
URL http://link.aps.org/doi/10.1103/PhysRevLett.112.135701
[36] V. Holten, M. A. Anisimov, Entropy-driven liquidliquid separation in supercooled water,
675
Scientific Reports 2. doi:10.1038/srep00713.
URL http://www.nature.com/doifinder/10.1038/srep00713
[37] P.
Poole,
F.
Sciortino,
T.
Grande,
H.
Stanley,
C.
Angell,
Effect of hydrogen bonds on the thermodynamic behavior of liquid water,
Phys.
680
Rev.
Lett.
73
(12)
(1994)
1632–1635.
doi:10.1103/PhysRevLett.73.1632.
URL http://prl.aps.org/abstract/PRL/v73/i12/p1632_1
[38] Q.
J.
Zheng,
J.
Shao,
G.
Green,
J.
H.
Wolf,
Kieffer,
C.
P.
H.
Austen
Poole,
Angell,
Limiting tensions for liquids and glasses from laboratory and MD studies,
685
in: A. R. Imre, H. J. Maris, P. R. Williams (Eds.), Liquids Under Negative
Pressure, Vol. 84 of NATO Science Series, Springer Netherlands, 2002,
pp. 33–46.
URL http://www.springerlink.com/content/t4r2725487423504/abstract/
[39] S. Sastry, P. G. Debenedetti, F. Sciortino, H. E. Stanley, Singularity-
690
free interpretation of the thermodynamics of supercooled water, Physical
Review E 53 (6) (1996) 6144–6154.
[40] C.
K.
Alba-Simionesco,
E.
Gubbins,
R.
B.
Coasne,
G.
Radhakrishnan,
Dosseh,
M.
Effects of confinement on freezing and melting,
28
G.
Dudziak,
Sliwinska-Bartkowiak,
Journal
of
695
Physics:
Condensed
Matter
18
(6)
(2006)
R15–R68.
doi:10.1088/0953-8984/18/6/R01.
URL http://stacks.iop.org/0953-8984/18/i=6/a=R01?key=crossref.f5c1d69a7ce3d8d5ec6733131
[41] L.
Granasy,
T.
Pusztai,
P.
F.
James,
Interfacial properties deduced from nucleation experiments: A cahnhilliard analysis,
700
The
Journal
of
Chemical
Physics
117
(13)
(2002)
6157.
doi:10.1063/1.1502652.
URL http://link.aip.org/link/JCPSA6/v117/i13/p6157/s1&Agg=doi
[42] S.
Jhnert,
A.
705
F.
Schreiber,
Vaca
M.
Chvez,
Schnhoff,
G.
E.
Schaumann,
G.
H.
Findenegg,
Melting and freezing of water in cylindrical silica nanopores,
Physical
Chemistry
Chemical
Physics
10
(39)
(2008)
6039.
doi:10.1039/b809438c.
URL http://xlink.rsc.org/?DOI=b809438c
[43] F.
710
Mallamace,
C.
Corsaro,
S.-H.
Chen,
H.
Transport and dynamics in supercooled confined water,
E.
Stanley,
in:
H.
E.
Stanley (Ed.), Liquid Polymorphism, John Wiley & Sons, Inc., 2013, pp.
203–262.
URL http://onlinelibrary.wiley.com/doi/10.1002/9781118540350.ch10/summary
[44] L.
715
Liu,
S.-H.
Chen,
A.
Faraone,
C.-W.
Yen,
C.-Y.
Mou,
Pressure dependence of fragile-to-strong transition and a possible second critical point in supercooled co
Phys.
Rev.
Lett.
95
(11)
(2005)
117802.
doi:10.1103/PhysRevLett.95.117802.
URL http://link.aps.org/doi/10.1103/PhysRevLett.95.117802
[45] D. Liu, Y. Zhang, C.-C. Chen, C.-Y. Mou, P. H. Poole, S.-H. Chen,
720
Observation of the density minimum in deeply supercooled confined water,
Proceedings of the National Academy of Sciences 104 (23) (2007) 9570–
9574. doi:10.1073/pnas.0701352104.
URL http://www.pnas.org/content/104/23/9570.abstract
29
[46] F. Mallamace, C. Branca, M. Broccio, C. Corsaro, C.-Y. Mou, S.-H. Chen,
725
The anomalous behavior of the density of water in the range {30 K < T < 373 K},
Proceedings of the National Academy of Sciences 104 (47) (2007) 18387–
18391. doi:10.1073/pnas.0706504104.
URL http://www.pnas.org/content/104/47/18387.abstract
[47] Y.
730
Zhang,
Liu,
C.-Y.
A.
Mou,
Faraone,
J.
B.
W.
A.
Leo,
S.
Kamitakahara,
K.-H.
Chang,
Chen,
S.-H.
Density hysteresis of heavy water confined in a nanoporous silica matrix,
Proceedings of the National Academy of Sciences 108 (30) (2011) 12206–
12211.
URL http://www.pnas.org/content/108/30/12206.short
735
[48] S.
Cerveny,
J.
Colmenero,
A.
Alegra,
Comment on pressure dependence of fragile-to-strong transition and a possible second critical point in s
Physical Review Letters 97 (18). doi:10.1103/PhysRevLett.97.189802.
URL http://link.aps.org/doi/10.1103/PhysRevLett.97.189802
[49] J. Swenson, Comment on pressure dependence of fragile-to-strong transition and a possible second critic
740
Physical Review Letters 97 (18). doi:10.1103/PhysRevLett.97.189801.
URL http://link.aps.org/doi/10.1103/PhysRevLett.97.189801
[50] S.-H. Chen, L. Liu, A. Faraone, Chen, liu, and faraone reply, Physical
Review Letters 97 (18) (2006) 189803.
URL http://prl.aps.org/abstract/PRL/v97/i18/e189803
745
[51] A. K. Soper, Density minimum in supercooled confined water, Proceedings of the National Academy of Sciences 108 (47) (2011) E1192–E1192.
URL http://www.pnas.org/content/108/47/E1192.short
[52] Y.
Liu,
750
Zhang,
C.-Y.
A.
Mou,
Faraone,
J.
B.
W.
Leao,
A.
S.
Kamitakahara,
K.-H.
Chang,
Chen,
S.-H.
Reply to soper: Density measurement of confined water with neutron scattering,
30
Proceedings of the National Academy of Sciences 108 (47) (2011) E1193–
E1194. doi:10.1073/pnas.1113408108.
URL http://www.pnas.org/cgi/doi/10.1073/pnas.1113408108
[53] A. K. Soper, Density profile of water confined in cylindrical pores in MCM-41 silica,
755
Journal of Physics:
Condensed Matter 24 (6) (2012) 064107.
doi:10.1088/0953-8984/24/6/064107.
URL http://iopscience.iop.org/0953-8984/24/6/064107
[54] A. Taschin,
P. Bartolini,
A. Marcelli,
R. Righini,
R. Torre,
A comparative study on bulk and nanoconfined water by time-resolved optical kerr effect spectroscopy,
760
Faraday Discuss. 167 (0) (2014) 293–308. doi:10.1039/C3FD00060E.
URL http://pubs.rsc.org/en/content/articlelanding/2013/fd/c3fd00060e
[55] F. Caupin, Question to s.-h. chen after his talk at the sixth international
discussion meeting on relaxations in complex systems, roma (Sep. 2009).
[56] A. K. Soper, Radical re-appraisal of water structure in hydrophilic confinement,
765
Chemical
Physics
Letters
590
(2013)
1–15.
doi:10.1016/j.cplett.2013.10.075.
URL http://linkinghub.elsevier.com/retrieve/pii/S0009261413013559
[57] N. Kimizuka, T. Suzuki, Supercooling behavior in aqueous solutions,
The Journal of Physical Chemistry B 111 (9) (2007) 2268–2273.
770
doi:10.1021/jp065379b.
URL http://pubs.acs.org/doi/abs/10.1021/jp065379b
[58] K.-I. Murata, H. Tanaka, Liquidliquid transition without macroscopic phase separation in a waterglycero
Nature Materials 11 (5) (2012) 436–443. doi:10.1038/nmat3271.
URL http://www.nature.com/doifinder/10.1038/nmat3271
775
[59] K.-i. Murata, H. Tanaka, General nature of liquidliquid transition in aqueous organic solutions,
Nature Communications 4. doi:10.1038/ncomms3844.
URL http://www.nature.com/doifinder/10.1038/ncomms3844
31
[60] H.
Kanno,
R.
J.
Speedy,
C.
A.
Angell,
Supercooling of water to -92\textsuperscript{o}{C} under pressure, Sci780
ence 189 (4206) (1975) 880–881. doi:10.1126/science.189.4206.880.
URL http://www.sciencemag.org/content/189/4206/880
[61] P. Taborek, Nucleation in emulsified supercooled water, Phys. Rev. B
32 (9) (1985) 5902–5906. doi:10.1103/PhysRevB.32.5902.
URL http://link.aps.org/abstract/PRB/v32/p5902
785
[62] A. Manka, H. Pathak, S. Tanimura, J. Wlk, R. Strey, B. E. Wyslouzil,
Freezing water in no-man’s land, Physical Chemistry Chemical Physics
14 (13) (2012) 4505. doi:10.1039/c2cp23116f.
URL http://xlink.rsc.org/?DOI=c2cp23116f
[63] H. Laksmono, T. A. McQueen, J. A. Sellberg, C. Huang, N. D. Loh, R. G.
790
Sierra, D. Starodub, D. Norlund, M. Beye, D. P. Deponte, A. Martin,
A. Barty, J. Feldkamp, S. Boutet, G. J. Williams, M. J. Bogan, A. Nilsson,
Probing homogenous ice nucleation within supercooled bulk water droplet in ”no man’s land” with an u
2013, pp. 984–986. doi:10.1063/1.4803439.
URL http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4803439
795
[64] J. A. Sellberg, Temperature-dependent x-ray scattering of liquid water,
Licentiate thesis, Stockholm University (May 2012).
[65] G.
Pallares,
Aragones,
M.
J.
L.
El
Mekki
F.
Azouzi,
Abascal,
C.
M.
A.
Gonzalez,
Valeriani,
F.
Anomalies in bulk supercooled water at negative pressure,
800
J.
L.
Caupin,
Proceedings
of the National Academy of Sciencesdoi:10.1073/pnas.1323366111.
URL http://www.pnas.org/cgi/doi/10.1073/pnas.1323366111
[66] M. P. Anisimov, E. G. Fominykh, S. V. Akimov, P. K. Hopke,
Vaporgas/liquid nucleation experiments: A review of the challenges,
Journal
805
of
Aerosol
Science
40
(9)
(2009)
733–746.
doi:10.1016/j.jaerosci.2009.06.002.
URL http://linkinghub.elsevier.com/retrieve/pii/S0021850209001074
32
[67] J. Huang, L. S. Bartell, Kinetics of homogeneous nucleation in the freezing of large water clusters,
The Journal of Physical Chemistry 99 (12) (1995) 3924–3931.
URL http://pubs.acs.org/doi/pdf/10.1021/j100012a010
810
[68] A.
Bhabhe,
H.
Pathak,
B.
E.
Wyslouzil,
Freezing of heavy water (d 2 o) nanodroplets, The Journal of Physical Chemistry A 117 (26) (2013) 5472–5482. doi:10.1021/jp400070v.
URL http://pubs.acs.org/doi/abs/10.1021/jp400070v
[69] J. A. Sellberg, C. Huang, T. A. McQueen, N. D. Loh, H. Laksmono,
815
D. Schlesinger,
R. G. Sierra,
D. Nordlund,
C. Y. Hampton,
D. Starodub, D. P. DePonte, M. Beye, C. Chen, A. V. Martin,
A. Barty, K. T. Wikfeldt, T. M. Weiss, C. Caronna, J. Feldkamp, L. B. Skinner, M. M. Seibert, M. Messerschmidt, G. J.
Williams, S. Boutet, L. G. M. Pettersson, M. J. Bogan, A. Nilsson,
820
Ultrafast x-ray probing of water structure below the homogeneous ice nucleation temperature,
Nature 510 (7505) (2014) 381–384. doi:10.1038/nature13266.
URL http://www.nature.com/nature/journal/v510/n7505/full/nature13266.html
[70] J. L. F. Abascal, C. Vega, A general purpose model for the condensed phases of water: TIP4p/2005,
The
825
Journal
of
Chemical
Physics
123
(23)
(2005)
234505.
doi:10.1063/1.2121687.
URL http://link.aip.org/link/JCPSA6/v123/i23/p234505/
[71] C. Vega, J. L. F. Abascal, Simulating water with rigid non-polarizable models: a general perspective,
Physical Chemistry Chemical Physics 13 (44) (2011) 19663–19688.
doi:10.1039/c1cp22168j.
830
URL http://xlink.rsc.org/?DOI=c1cp22168j
[72] M. T. Tyree, M. H. Zimmermann, Xylem Structure and the Ascent of Sap,
2nd Edition, no. XIV in Springer Series in Wood Science, Springer-Verlag,
Berlin, Heidelberg, New-York, 2002.
URL http://www.springer.com/life+sci/forestry/book/978-3-540-43354-5
33
835
[73] F.
Caupin,
Liquid-vapor
diagram of water,
interface,
cavitation,
and
the
phase
Physical Review E 71 (5) (2005) 051605.
doi:10.1103/PhysRevE.71.051605.
[74] Q.
Zheng,
D.
J.
Durben,
G.
H.
Wolf,
C.
A.
Angell,
Liquids at large negative pressures: water at the homogeneous nucleation limit,
840
Science 254 (5033) (1991) 829–832. doi:10.1126/science.254.5033.829.
URL http://www.sciencemag.org/cgi/content/abstract/254/5033/829
[75] A.
D.
Alvarenga,
M.
Grimsditch,
R.
Elastic properties of water under negative pressures,
J.
Bodnar,
J. Chem. Phys.
98 (11) (1993) 8392–8396. doi:10.1063/1.464497.
845
URL http://link.aip.org/link/?JCP/98/8392/1
[76] K. I. Shmulovich, L. Mercury, R. Thiry, C. Ramboz, M. El Mekki,
Experimental superheating of water and aqueous solutions,
Geochimica
et
Cosmochimica
Acta
73
(9)
(2009)
2457–2470.
doi:10.1016/j.gca.2009.02.006.
850
URL http://www.sciencedirect.com/science/article/B6V66-4VNH449-B/2/4eea2ba80cbb333602e
[77] M. El Mekki Azouzi,
C. Ramboz,
J.-F. Lenain,
A coherent picture of water at extreme negative pressure,
F. Caupin,
Nat
Phys
9 (1) (2013) 38–41. doi:10.1038/nphys2475.
URL http://www.nature.com/nphys/journal/v9/n1/full/nphys2475.html
855
[78] K. Davitt, A. Arvengas, F. Caupin, Water at the cavitation limit: Density
of the metastable liquid and size of the critical bubble, Europhys. Lett.
90 (1) (2010) 16002. doi:10.1209/0295-5075/90/16002.
[79] C.
Duan,
R.
Karnik,
M.-C.
Lu,
A.
Majumdar,
Evaporation-induced cavitation in nanofluidic channels,
860
PNASdoi:10.1073/pnas.1014075109.
URL http://www.pnas.org/content/early/2012/02/14/1014075109
34
[80] A.
Arvengas,
K.
Davitt,
F.
Caupin,
Fiber optic probe hydrophone for the study of acoustic cavitation in water,
Rev. Sci. Instrum. 82 (3) (2011) 034904. doi:10.1063/1.3557420.
865
URL http://link.aip.org/link/RSINAK/v82/i3/p034904/
[81] K. B. Bader, J. L. Raymond, J. Mobley, C. C. Church, D. F. Gaitan,
The effect of static pressure on the inertial cavitation threshold,
The
Journal of the Acoustical Society of America 132 (2) (2012) 728–737.
URL http://scitation.aip.org/content/asa/journal/jasa/132/2/10.1121/1.4733539
870
[82] A. D. Maxwell, C. A. Cain, T. L. Hall, J. B. Fowlkes, Z. Xu,
Probability of cavitation for single ultrasound pulses applied to tissues and tissue-mimicking materials,
Ultrasound
in
Medicine
&
Biology
39
(3)
(2013)
449–465.
doi:10.1016/j.ultrasmedbio.2012.09.004.
URL http://linkinghub.elsevier.com/retrieve/pii/S0301562912005698
875
[83] P. A. Quinto-Su, K. Ando, Nucleating bubble clouds with a pair of laser-induced shocks and bubbles,
Journal of Fluid Mechanics 733. doi:10.1017/jfm.2013.456.
URL http://www.journals.cambridge.org/abstract_S0022112013004564
[84] T. D. Wheeler, A. D. Stroock, The transpiration of water at negative pressures in a synthetic tree,
Nature 455 (7210) (2008) 208–212. doi:10.1038/nature07226.
880
URL http://www.nature.com/doifinder/10.1038/nature07226
[85] E. N. Harvey, D. Barnes, W. McElroy, A. Whiteley, D. Pease,
Removal of gas nuclei from liquids and surfaces, Journal of the American
Chemical Society 67 (1) (1945) 156–157.
URL http://pubs.acs.org/doi/abs/10.1021/ja01217a600
885
[86] E. Herbert, S. Balibar, F. Caupin, Cavitation pressure in water, Physical
Review E 74 (4) (2006) 041603. doi:10.1103/PhysRevE.74.041603.
[87] K. Ando, A.-Q. Liu, C.-D. Ohl, Homogeneous nucleation in water in microfluidic channels,
Physical Review Letters 109 (4). doi:10.1103/PhysRevLett.109.044501.
URL http://link.aps.org/doi/10.1103/PhysRevLett.109.044501
35
890
[88] T. D. Wheeler, A. D. Stroock, Stability limit of liquid water in metastable equilibrium with subsaturated
Langmuir 25 (13) (2009) 7609–7622. doi:10.1021/la9002725.
URL http://dx.doi.org/10.1021/la9002725
[89] O.
Vincent,
P.
Marmottant,
P.
A.
Quinto-Su,
C.-D.
Ohl,
Birth and growth of cavitation bubbles within water under tension confined in a simple synthetic tree,
895
Physical Review Letters 108 (18). doi:10.1103/PhysRevLett.108.184502.
URL http://link.aps.org/doi/10.1103/PhysRevLett.108.184502
[90] O. Vincent, P. Marmottant, S. R. Gonzalez-Avila, K. Ando, C.-D. Ohl,
The fast dynamics of cavitation bubbles within water confined in elastic solids,
Soft Matter 10 (10) (2014) 1455. doi:10.1039/c3sm52697f.
900
URL http://xlink.rsc.org/?DOI=c3sm52697f
[91] V. Pagay, M. Santiago, D. A. Sessoms, E. J. Huber, O. Vincent, A. Pharkya, T. N. Corso, A. N. Lakso, A. D. Stroock,
A microtensiometer capable of measuring water potentials below 10 MPa,
Lab Chip 14 (15) (2014) 2806–2817. doi:10.1039/C4LC00342J.
905
URL http://pubs.rsc.org.docelec.univ-lyon1.fr/en/content/articlelanding/2014/lc/c4lc003
[92] O. Vincent, D. A. Sessoms, E. J. Huber, J. Guioth, A. D. Stroock,
Drying by cavitation and poroelastic relaxations in extreme ink-bottle porous media,
arXiv:1402.6776 [cond-mat]arXiv: 1402.6776.
URL http://arxiv.org/abs/1402.6776
910
[93] J. Meyer, Zur kenntnis des negativen druckes in flssigkeiten, Abhandlungen der Deutschen BunsenGessellschaft 6 (1911) 1–53.
[94] S. Henderson, R. Speedy, A berthelot-bourdon tube method for studying
water under tension, Journal of Physics E: Scientific Instruments 13 (1980)
778.
915
[95] S. J. Henderson, R. J. Speedy, Temperature of maximum density in water at negative pressure,
The Journal of Physical Chemistry 91 (11) (1987) 3062–3068.
36
doi:10.1021/j100295a084.
URL http://dx.doi.org/10.1021/j100295a084
[96] S. J. Henderson, R. J. Speedy, Melting temperature of ice at positive and negative pressures,
920
The Journal of Physical Chemistry 91 (11) (1987) 3069–3072.
doi:doi:10.1021/j100295a085.
URL http://dx.doi.org/10.1021/j100295a085
[97] K. Davitt, E. Rolley, F. Caupin, A. Arvengas, S. Balibar, Equation of
state of water under negative pressure, The Journal of Chemical Physics
925
133 (2010) 174507.
[98] W. Wagner, The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substanc
Journal of Physical and Chemical Reference Data 31 (2002) 387–535.
doi:10.1063/1.1461829.
URL http://link.aip.org/link/?JPR/31/387/
930
[99] T.
I.
A.
for
the
Properties
of
Water
and
Steam,
Revised release on the IAPWS formulation 1995 for the thermodynamic properties of ordinary water su
(Sep. 2009).
URL http://www.iapws.org/relguide/IAPWS95-Rev.pdf
[100] G. F. Kraus, S. C. Greer, Vapor pressures of supercooled water and deuterium oxide,
935
The Journal of Physical Chemistry 88 (20) (1984) 4781–4785.
URL http://pubs.acs.org/doi/abs/10.1021/j150664a067
[101] A. T. Hayward, Negative pressure in liquids: Can it be harnessed to
serve man? a review of the history of negative pressure experiments and
discussion of current work and applications, American Scientist (1971)
940
434–443.
37