Supercooled and glassy water: Metastable liquid(s), amorphous solid(s), and a noman’s land
 Philip H. Handle^{a},^{b},^{1},^{2},
 Thomas Loerting^{b},^{1},^{2}, and
 Francesco Sciortino^{a},^{1},^{2}
 ^{a}Department of Physics, Sapienza University of Rome, I00185 Roma, Italy;
 ^{b}Institute of Physical Chemistry, University of Innsbruck, A6020 Innsbruck, Austria

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved October 3, 2017 (received for review September 1, 2017)
Abstract
We review the recent research on supercooled and glassy water, focusing on the possible origins of its complex behavior. We stress the central role played by the strong directionality of the water–water interaction and by the competition between local energy, local entropy, and local density. In this context we discuss the phenomenon of polyamorphism (i.e., the existence of more than one disordered solid state), emphasizing both the role of the preparation protocols and the transformation between the different disordered ices. Finally, we present the ongoing debate on the possibility of linking polyamorphism with a liquid–liquid transition that could take place in the noman’s land, the temperature–pressure window in which homogeneous nucleation prevents the investigation of water in its metastable liquid form.
Water, the molecule of life, is literally everywhere. It covers the Earth’s surface in the form of lakes, rivers, oceans, ice caps, and glaciers. It fills vast underground caverns and it resides in our atmosphere as vapor or in clouds. However, water is not restricted to Earth. It is found on the planets of our solar system and their moons and on asteroids and comets (1), possibly in its glass (i.e., amorphous ice) form (2). Water as an amorphous solid is also abundant in the interstellar medium (3), where it may play a fundamental role in the formation of complex organic molecules such as amino acids and sugars (4). In light of this ubiquity, it is not surprising that the scientific literature concerned with water and water solutions is immense.
In this paper we focus explicitly on bulk water in metastable conditions, that is, water below its equilibrium melting temperature,
both as a supercooled liquid and as an amorphous solid. We attempt to provide insights on the unconventional behavior of water
in its liquid and glass form. Indeed, despite water being the most abundant liquid, paradoxically it is scientifically considered
anomalous due to its rather complex behavior. For instance, the isobaric heat capacity <mml:math><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:math>
CP, the isothermal compressibility <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mrow></mml:math>
KT, and the viscosity <mml:math><mml:mi>η</mml:mi></mml:math>
η all show a nonmonotonic temperature <mml:math><mml:mi>T</mml:mi></mml:math>
T and/or pressure <mml:math><mml:mi>P</mml:mi></mml:math>
P dependence, which is amplified upon supercooling. The incomplete understanding of the origin of the numerous anomalies motivates
the scientific interest in liquid water and even more in its metastable liquid state. Similarly, the disordered solid form
of water also has its peculiarities. It displays polyamorphism (amorphous polymorphism), a term created in analogy to crystal
polymorphism, indicating the existence of more than one disordered amorphous form.
Since several highquality reviews focusing on supercooled and glass water have appeared in recent years (5????????–14), rather than being comprehensive we focus on a limited number of experimental and theoretical investigations. Specifically,
we attempt to elucidate the present understanding of bulk (as opposed to confined) water in the supercooled liquid state above
the homogeneous nucleation temperature <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:mrow></mml:math>
Th and in the amorphous state below <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx, the temperature at which the glass crystallizes on heating. These two temperatures define the intermediate window in the
<mml:math><mml:mi>T</mml:mi></mml:math>
T–<mml:math><mml:mi>P</mml:mi></mml:math>
P plane in which only crystalline ice is observed. This window, which presentday research attempts to shrink by working on
smaller and smaller samples and better and better aged glasses, is conventionally called noman’s land (15). We also discuss in detail the phenomenon of polyamorphism unraveled by Mishima et al. (16, 17) and Mishima (18) which has been subject to intense debate ever since. Foremost, we aim at briefly mentioning the key findings and pointing
to the open questions that split the field, hopefully providing a representation of some of the different ideas proposed to
interpret the available experimental results and the associated theoretical frameworks. It is indeed fair to state that even
today there is no consensus about the origin of the anomalous behavior of liquid water and about the origin and existence
of thermodynamically distinct disordered polyamorphs.
Liquid Water
The Physics of Water.
What makes water special compared with other liquids is the strong directional character of the intermolecular interaction
potential. The dominant contribution, called the hydrogen bond (HB), has a strength intermediate between the stronger covalent
bond and the weaker dipoleinduced interaction. In addition, the HB strength is significantly larger than the thermal energy
at room <mml:math><mml:mi>T</mml:mi></mml:math>
T. An HB requires a hydrogen atom pointing toward a closeby oxygen atom. The strength of the HB is maximized when the hydrogen
atom is collinear with the acceptor and donor oxygen and progressively weakens on increasing the HOO angle.
The strength of the HB and its directionality are key elements in controlling liquid water’s thermodynamic and dynamic behavior.
If the HB interaction did not exist, water would behave just like all other H_{2}X triatomic molecules, with X representing any chalcogen, and it would be gaseous at room <mml:math><mml:mi>T</mml:mi></mml:math>
T (19). If the interaction between water molecules was isotropic and described by an attraction comparable to the HB strength,
the gas–liquid critical <mml:math><mml:mi>T</mml:mi></mml:math>
T would be located at about one order of magnitude higher temperature than the real experimental value (<mml:math><mml:mrow><mml:mn>647</mml:mn></mml:mrow></mml:math>
647 K). What makes water a liquid at ambient <mml:math><mml:mi>T</mml:mi></mml:math>
T is the directionality of the HBs and the limited number of linear HBs that a molecule can form (at most four). Such limited
valence significantly lowers the critical <mml:math><mml:mi>T</mml:mi></mml:math>
T compared with the isotropic case (20). This originates a liquid phase at room <mml:math><mml:mi>T</mml:mi></mml:math>
T in which the number of nearest neighbors is around four, significantly lower than the typical value observed in simple liquids,
namely 12. Since on cooling percolation precedes the gas–liquid critical point (CP), the liquid phase can be described as
a percolating network of HBs (21), constantly restructuring itself on a picosecond time scale.
The directionality of the interaction is responsible for the peculiar correlation between local energy, density, and entropy.
The establishment of four linear HBs (a state of low energy) is possible only for welldefined orientations of the water molecule
(low entropy and density). The formation of such a state is driven by the decrease in energy but it is contrasted by the decrease
in entropy and the increase in the mechanical work (<mml:math><mml:mrow><mml:mi>P</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:math>
PV, <mml:math><mml:mi>V</mml:mi></mml:math>
V being volume) term. Similarly, the formation of locally denser arrangements, associated with the presence of additional molecules
in the first coordination shell, generates distorted HBs and states with higher local energy and entropy. The formation of
these more dense local environments is driven by entropy and <mml:math><mml:mrow><mml:mi>P</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:math>
PV and contrasted by the energy loss. Both of these structural motifs are found in liquid water. The most prominent examples
are environments characterized by four and three linear HBs. In the first case the local arrangement is to a good approximation
tetrahedral, while in the second case a fifth neighbor, locally distorting the HB pattern, is present in the first coordination
shell (22).
Water’s Anomalies.
The competition between different local environments has been identified as the source of the anomalies of liquid water, both
in its stable and metastable states (21). While in a simple liquid the role of <mml:math><mml:mi>T</mml:mi></mml:math>
T and <mml:math><mml:mi>P</mml:mi></mml:math>
P is limited to a smooth change in local properties, in water a change in these control variables modifies the equilibrium
between the different local environments. Generically, decreasing <mml:math><mml:mi>T</mml:mi></mml:math>
T favors fully bonded local configurations, and increasing <mml:math><mml:mi>P</mml:mi></mml:math>
P favors more distorted ones. Such equilibrium between different classes, which is not found in simple liquids, is responsible
for the existence of a <mml:math><mml:mi>T</mml:mi></mml:math>
T at which the density reaches a maximum value at constant pressure <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>M</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>
TMD (5). When <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>M</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>
T=TMD the increase in <mml:math><mml:mi>V</mml:mi></mml:math>
V associated with the formation of more bonded local configurations compensates the decrease in <mml:math><mml:mi>V</mml:mi></mml:math>
V associated to the reduction of the thermal vibrations. The latter is the mechanism responsible for contraction on cooling
in all materials. Equilibrium between different local environments also explains other anomalies, such as the speeding up
of the dynamics on increasing <mml:math><mml:mi>P</mml:mi></mml:math>
P (molecules with distorted bonds have a larger mobility).
The picture of water as composed of two or more classes of locally structurally distinct molecules is a recurrent theme in
water science research, often contrasted with an alternative picture based on a continuum of geometric arrangements. Already
the first computer simulations of water (28) provided evidence that all local properties are characterized by broad unimodal distributions with a <mml:math><mml:mi>T</mml:mi></mml:math>
T and <mml:math><mml:mi>P</mml:mi></mml:math>
P dependence of the average values, apparently ruling out the possibility of a meaningful definition of distinct local environments.
Only recently, progress in data analysis made it possible to show that classes of structurally different local environments
can robustly be defined if appropriate order parameters are evaluated (29, 30). These new analyses, applied to novel and old simulation data, show that water molecules differing in their local environment
can be treated as species in chemical equilibrium. Interestingly, the ability to resolve these classes requires information
on the degree on translational order of the secondnearestneighbor shell (31), sophisticated Markov state models (32), and, in some cases, the suppression (33) of the vibrational (translational and librational) component via the inherent structure methodology (34).
Are molecules with tetrahedral local order progressively clustering in space on cooling? Does this clustering—which could originate from the highly directional interactions (geometric correlation) and from the redistribution of the electronic density on bonding (polarizable and quantum effects)—become so intense as to induce a complete separation of these differently bonded molecules and possibly distinct disordered liquid phases? The answer to these questions is central in presentday research.
Supercooled Water.
Answering the above questions is even more relevant in the description of supercooled water, that is, water below its melting
temperature <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>
Tm. In this region, where liquid water becomes thermodynamically less stable than ice, it is still possible to perform measurements
in the liquid state for times significantly longer than its equilibration time but still shorter than the crystallization
time. Eventually, however, nucleation of ice prevails. For our purposes we can simply state that the homogeneous nucleation
temperature <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:mrow></mml:math>
Th sets the lower limit of existence of the metastable liquid phase. The <mml:math><mml:mi>T</mml:mi></mml:math>
T–<mml:math><mml:mi>P</mml:mi></mml:math>
P region in which nucleation prevents the observation of metastable liquid states has been named “noman’s land” (15). The border to the noman’s land is illdefined, since the location of <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:mrow></mml:math>
Th depends on the experimental time and on the sample size. Recent experiments (35) show that metastable liquid water can transiently exist on the millisecond time scale down to <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>227</mml:mn></mml:mrow></mml:math>
T=227 K, setting the present upper limit of the noman’s land. The very same study also indicates that tetrahedral ordering increases
in supercooled water. At timescales shorter than microseconds crystallization can even be avoided entirely (see Preparation Routes).
Possible Explanations.
The extrapolation of the thermodynamic response functions into the noman’s land is at the heart of possible thermodynamic
scenarios proposed to interpret water’s anomalies. The steep rise of <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mrow></mml:math>
KT and <mml:math><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:math>
CP (36, 37) on cooling (after the subtraction of a significant normal component) has been linked to a possible powerlaw divergence
around <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>≈</mml:mo><mml:mn>228</mml:mn></mml:mrow></mml:math>
T≈228 K and interpreted as originating from a thermodynamic instability.
An elegant explanation of such divergence has been proposed by Speedy (38). In standard liquids, as predicted by the van der Waals theory, the gas–liquid spinodal (the line emanating from the gas–liquid
CP defining the meanfield limit of stability of the liquid phase) is monotonic in the <mml:math><mml:mi>P</mml:mi></mml:math>
P–<mml:math><mml:mi>T</mml:mi></mml:math>
T plane approaching <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>
T=0 K at the maximum tensile (<mml:math><mml:mrow><mml:mi>P</mml:mi><mml:mo><</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>
P<0) strength value (39). Speedy (38), however, showed that thermodynamic constraints impose a reentrant behavior on the spinodal in the <mml:math><mml:mi>P</mml:mi></mml:math>
P–<mml:math><mml:mi>T</mml:mi></mml:math>
T plane if the liquid has a <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>M</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>
TMD line intersecting the spinodal line. If the spinodal traces back to positive <mml:math><mml:mi>P</mml:mi></mml:math>
P, the liquid becomes unstable to gaslike fluctuations both on heating and on cooling (40). Despite the difficulty of imagining a liquid that would vaporize on cooling, Speedy’s scenario is thermodynamically consistent
and it has been recently observed in onecomponent models of patchy colloids (24, 25). In these models the reentrance is provided by closedloop phase diagrams (Fig. 1 C and D), where the spinodal starts in the upper CP and terminates at the lower one. According to Debenedetti (5), Speedy’s scenario should not be considered as a possible explanation of water’s anomalies since there is no experimental
evidence of a reentrance of the metastable gas–liquid coexistence line (binodal).
Another possibility is to assume no divergence of the extrapolations of <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mrow></mml:math>
KT and <mml:math><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:math>
CP into the noman’s land but rather a maximum. Such a scenario is often called singularityfree scenario (Fig. 1 A and B). This scenario is consistent with intuition (and experimental and numerical observations) that supercooled water approaches
the structure of a fully bonded random tetrahedral network on cooling. The crossover from a weakly tetrahedral structure
at ambient <mml:math><mml:mi>T</mml:mi></mml:math>
T to a fully developed tetrahedral structure at very low <mml:math><mml:mi>T</mml:mi></mml:math>
T suggests indeed a sigmoidal <mml:math><mml:mi>T</mml:mi></mml:math>
T dependence of the local properties (e.g., energy and volume) and a maximum in their <mml:math><mml:mi>T</mml:mi></mml:math>
T derivatives. A maximum in the specific heat is typical of systems with quantized energy levels (Schottky anomaly) and commonly
observed when particles form a limited number of strong bonds, such that the energy provides a proxy to the number of bonded
pairs.
The <mml:math><mml:mi>T</mml:mi></mml:math>
Ts at which the response functions assume the maximum value, when evaluated at different pressures, define loci in the <mml:math><mml:mi>P</mml:mi></mml:math>
P–<mml:math><mml:mi>T</mml:mi></mml:math>
T plane along which fluctuations are maximized. In the LLCP scenario (26, 41) it is assumed that lines of <mml:math><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:math>
CP and <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mrow></mml:math>
KT extrema converge to a common line [Widom line (42)] that emanates from a CP at positive pressures. This novel CP is commonly indicated as the second CP, the gas–liquid CP
being the first. This scenario (Fig. 1 E and F), discovered in computer simulations of the ST2 water model (26, 41), implies that indeed a spatial correlation exists between molecules in tetrahedral and in distorted local environments,
resulting in a segregation of the two classes of molecules over macroscopic distances. This indicates the existence of a real
thermodynamic phase separation between two disordered liquid phases, a low (LDL) and a highdensity liquid (HDL). The two
liquids differ not only in density but also in local order, a feature which has suggested that water thermodynamics are controlled
by two order parameters (9).
Recently, extensive studies (43, 44) of the ST2 model have cleared up the field from erroneous results (45), thereby providing definitive evidence of an LL transition in this model. The universality of the LL transition in tetrahedral
network fluids has been discussed in studies based on primitive models of patchy colloidal particles (46). It was shown that an LL transition is a generic feature in this class of particles (47) and that a simple extension of the ST2 model for water displays an LLCP that progressively moves to a temperature where
the liquid is more stable than ice <mml:math><mml:msub><mml:mi mathvariant="normal">I</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math>
Ih (44). Finally, we stress that the singularityfree scenario is equivalent to an LLCP scenario in which the critical temperature
is located at zero <mml:math><mml:mi>T</mml:mi></mml:math>
T (9, 48).
Support for the LLCP scenario in water, in addition to the numerical studies of several water (30, 49????–54) and waterlike (55) potentials, comes from the work of Holten et al. (56). They presented an equation of state, built under the assumption of the presence of a critical component in the free energy,
compatible with all available experimental thermodynamic data. Their extended model estimate for the LLCP locus is <mml:math><mml:mrow><mml:mn>0.057</mml:mn></mml:mrow></mml:math>
0.057 GPa and <mml:math><mml:mrow><mml:mn>214</mml:mn></mml:mrow></mml:math>
214 K (56). Meanfield free energies based on the idea of a chemical equilibrium between two main water classes, supported by the previously
mentioned order parameter analysis of computer simulations data, have also been shown to be compatible with the LLCP hypothesis
(29, 30, 54).
From the experimental side, indications of the LLCP can be found in the density dependence of the structure factor <mml:math><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
S(q) and on the analogies between the <mml:math><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
S(q) measured at low and high pressure and the <mml:math><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
S(q) measured in the low and highdensity amorphous ices (discussed below). Ref. 57 reports the observation of a continuous transformation with increasing <mml:math><mml:mi>P</mml:mi></mml:math>
P from a lowdensity form of water with an open, hydrogenbonded tetrahedral structure to a highdensity form of water with
nontetrahedral OOO angles and a collapsed second coordination shell. Recent timeresolved optical Kerr effect investigations
of the vibrational dynamics and relaxation processes in supercooled bulk water (58) show evidence of the existence of two main local configurations with an increasing weight of the open tetrahedral structure
at low <mml:math><mml:mi>T</mml:mi></mml:math>
T. Consistent interpretations have also been formulated by Sellberg et al. (35) on the basis of Xray emission and absorption experiments. Strong experimental support in favor of an LL transition has
been presented by Mishima and Stanley (59) and Mishima (60). They measured the metastable melting lines of ices III, IV, and V and found discontinuities in the case of ices IV and
V and a continuous line in the case of ice III. The discontinuity indicates that the properties of the liquid in equilibrium
with the crystal change abruptly, an indirect evidence of the crossing of the line of LL phase transition. The continuous
line for ice III indicates that in this case the LL transition line is not crossed. The presence of a possible LLCP is also
hinted at by the temperature dependence of the <mml:math><mml:mi>V</mml:mi></mml:math>
V measured in emulsified water (61), the <mml:math><mml:mi>T</mml:mi></mml:math>
T and <mml:math><mml:mi>P</mml:mi></mml:math>
P dependence of the bulk viscosity (62), and experimental studies of water solutions (63?–65). In summary, available experimental data suggest the locus of the possible LLCP in the area <mml:math><mml:mrow><mml:mn>214</mml:mn><mml:mo>?</mml:mo><mml:mn>232</mml:mn></mml:mrow></mml:math>
214?232 K and <mml:math><mml:mrow><mml:mn>0.02</mml:mn><mml:mo>?</mml:mo><mml:mn>0.10</mml:mn></mml:mrow></mml:math>
0.02?0.10 GPa (56, 59, 61).
We note in passing that while the observation of a compressibility maximum at positive <mml:math><mml:mi>P</mml:mi></mml:math>
P is possibly preempted by crystal nucleation such observation could become accessible at negative <mml:math><mml:mi>P</mml:mi></mml:math>
P. Very recent experiments in microscopic inclusions of water in quartz building on the seminal work of ref. 66 suggest that this is indeed the case (67, 68).
Amorphous Ice
Polyamorphism.
We now turn our attention to the arrested disordered states of water observed below the noman’s land. One of the most intriguing
findings in the physics of amorphous water is its polyamorphism, sometimes also called amorphous polymorphism. The term polymorphism
indicates the existence of several distinct crystalline phases for the same component. In case of water, 17 ice polymorphs
are known to date (69). Adding the term amorphous, however, complicates our affairs. In principle, polyamorphism should indicate the existence
of several distinct amorphous phases for the same component. This, however, brings the issue whether or not there are distinct
“amorphous phases.” Amorphous solids are outofequilibrium systems and their properties depend strongly on preparation history.
Their properties also change with time and this relaxation proceeds at different rate at different <mml:math><mml:mi>T</mml:mi></mml:math>
T and <mml:math><mml:mi>P</mml:mi></mml:math>
P. Consequently, the same phase may appear as a distinct one depending on the state of relaxation. At a certain <mml:math><mml:mi>T</mml:mi></mml:math>
T, the structural relaxation may either be immeasurably slow, take place on experimentally accessible time scales, or be interrupted
by crystallization. If it takes place on accessible time scales and crystallization is slower than observation time the amorphous
material passes through numerous relaxation states and ultimately reaches a fully relaxed state. In this metastable equilibrium
there is still excess enthalpy and entropy compared with the stable crystalline phase. Thus, a very important measure to judge
whether or not amorphous materials may be called “phases” rather than “states” is the separation of time scales between structural
relaxation and crystallization.
Taking these considerations into account, we operationally define polyamorphism as the existence of more than one amorphous phase in metastable equilibrium (i.e., nonaging and noncrystallizing). Different amorphous phases have to differ in terms of their properties. Furthermore, just like in the case of distinct crystalline phases, phase boundaries to stable or metastable phases have to exist. We note that phase boundaries between phases with the same symmetry can end in a CP. The usually different symmetry of coexisting crystal phases is indeed the reason why crystal–crystal coexistence lines do not end in a CP. A CP can, however, be the end point of an amorphous–amorphous coexistence line since amorphous phases are isotropic.
Preparation Routes.
In the case of water we operationally discriminate three amorphous ices according to their respective density: low (LDA), high (HDA), and veryhighdensity amorphous ice (VHDA). Fig. 2 provides a schematic representation of common paths experimentally used to prepare samples of the three types.
Three paths lead to LDA. Two of these are condensation of gaseous water without crystallization (70) and cooling of the liquid without crystallization. The latter procedure, called vitrification, is the standard procedure
of producing a glass. Water, however, is a bad glass former, crystallizing very easily. Thus, hyperquenching at <mml:math><mml:mrow><mml:mo>></mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:math>
>106 K s^{?1} is required to avoid detectable crystallization (71). A third path is via transformation of HDA (e.g., by heating or decompressing HDA) (17, 72). After proper annealing all three routes lead to samples with the same density, structure, and calorimetric behavior, such
that all of them are often considered to be the same glass, namely LDA (8). Also for HDA several distinct preparation routes are possible, including quenching of pressurized emulsions of pure water
(73) (here cooling rates of <mml:math><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math>
103 K min^{?1} are sufficient to avoid crystallization) and pressurizing hexagonal ice at <mml:math><mml:mrow><mml:mn>77</mml:mn></mml:mrow></mml:math>
77 K beyond the low <mml:math><mml:mi>T</mml:mi></mml:math>
T metastable extension of the liquid–ice coexistence (16). Nowadays, this material is referred to as unannealed HDA (uHDA) to stress its unrelaxed character, whereas its relaxed
variant is called expanded HDA (eHDA) (72, 74). HDA can also be prepared from the other two amorphous ices, namely by compressing LDA or by decompressing VHDA (Fig. 2C). VHDA can be prepared by pressurizing hexagonal ice at temperatures between about <mml:math><mml:mrow><mml:mn>130</mml:mn></mml:mrow></mml:math>
130 and <mml:math><mml:mrow><mml:mn>150</mml:mn></mml:mrow></mml:math>
150 K (8) or by heating either ice <mml:math><mml:msub><mml:mi mathvariant="normal">I</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math>
Ih (75) or HDA at high <mml:math><mml:mi>P</mml:mi></mml:math>
P (76).
Structural Properties.
In LDA each water molecule is surrounded by four next neighbors (77), a pattern consistent with ice <mml:math><mml:msub><mml:mi mathvariant="normal">I</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math>
Ih (78). This indicates LDA is a representation of a fully bonded ideal random tetrahedral network. The vibrational character of
LDA is further reminiscent of the one of highly ordered substances, missing the excess density of states characteristic of
most other glasses (6). In this respect LDA comes close to the experimental realization of an ideal glass state (6). HDA is characterized by a structure that significantly differs from LDA. At ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P each molecule has four hydrogen partners on average, but a fifth molecule has moved toward the first shell due to a contraction
of the second shell (77). This contraction is even more extreme at <mml:math><mml:mn>2</mml:mn></mml:math>
2 GPa where the total coordination number goes up to roughly nine (79). Please note that structurally uHDA and eHDA are very similar. They, however, differ in that they have lots (uHDA) or almost
no (eHDA) nanocrystalline domains embedded in the amorphous matrix (80). In the VHDA structure at ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P even two molecules from the second shell are pushed toward the first shell (81). Hence, the structures of HDA and VHDA significantly differ at ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P and low <mml:math><mml:mi>T</mml:mi></mml:math>
T. At high pressures, however, the structural differences between VHDA and HDA become less pronounced, because VHDA is much
less compressible than HDA. Klotz et al. (79) have argued that the structure of HDA near <mml:math><mml:mrow><mml:mn>0.7</mml:mn></mml:mrow></mml:math>
0.7 GPa resembles the structure of VHDA at ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P.
We note that the crossover from the four linear HBs in LDA to the distorted HBs where one (HDA) or two (VHDA) interstitial molecules are found between the first and second coordination shell resembles the pressure dependence of the HB pattern in the liquid state.
Amorphous–Amorphous Transitions.
The possibility to associate the LDA and HDA structures to the two distinct liquids predicted by the LLCP scenario has prompted
a significant experimental effort in the study of amorphous ices, specifically on the stability of and the relations among
the different disordered solid forms. It becomes indeed very relevant to assess if the different highdensity forms are members
of the same family and if the high and lowdensity forms are thermodynamically distinct phases (i.e., whether the glasses
are proxies of the two thermodynamically distinct liquids). When tackling those questions, the abovenamed difficulties due
to the peculiarities of amorphous solids arise. For HDA a strong history dependence was reported for the temperature of transformation
to LDA at ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P (82?–84) and for <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx at higher <mml:math><mml:mi>P</mml:mi></mml:math>
P (80). All this makes it particularly important to carefully define the preparation routes when comparing different experiments.
For the properly relaxed amorphous state the transformation temperatures reach a limiting value.
HDA can be transformed to LDA by heating at low <mml:math><mml:mi>P</mml:mi></mml:math>
P. This transformation is accompanied by a <mml:math><mml:mrow><mml:mo>≈</mml:mo><mml:mn>25</mml:mn></mml:mrow></mml:math>
≈25% change in density occurring in a jumplike fashion. When first observed, this transformation immediately prompted an interpretation
as a firstorder transition (17), despite the outofequilibrium nature of the entire process. This isobaric transition is irreversible. If, however, an
isothermal path at sufficiently high <mml:math><mml:mi>T</mml:mi></mml:math>
T is followed one can switch back and forth between HDA and LDA (Fig. 2C). The transitions on compression of LDA or on decompression of HDA are very sharp and they appear at different <mml:math><mml:mi>P</mml:mi></mml:math>
P, that is, there is hysteresis (18). This hysteresis can be traced up to the crystallization at <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx (18) and it corroborates the firstorderlike nature of the transition. Further evidence for a discontinuous nature of the LDA<mml:math><mml:mo>?</mml:mo></mml:math>
? HDA transitions was provided by other methods (83, 85). Earlier scattering experiments claiming the possibility of a continuous transition from HDA to LDA (86) can presumably be explained by structural relaxation of HDA preceding the firstorderlike transition. Gromnitskaya et al.
(87) showed that for wellrelaxed samples the initial relaxation stage is absent, simplifying the nature of the transition (88). In our opinion, the experimental evidence, therefore, clearly speaks in favor of LDA and HDA being two distinct amorphous
phases, converting through a firstorderlike transition. Being a solid–solid transition it is likely influenced by mechanical
stress, which should disappear for the HDL<mml:math><mml:mo>?</mml:mo></mml:math>
?LDL transition (89).
While for LDA and HDA experiments show a hysteresis, enabling us to approximately map the binodal, this has so far not been
achieved for HDA and VHDA. A binodal between HDA and VHDA, if it exists, is very hard to trace. At low <mml:math><mml:mi>T</mml:mi></mml:math>
T the observation of the transformation is impeded due to slow kinetics and at higher <mml:math><mml:mi>T</mml:mi></mml:math>
T due to the small density difference between HDA and VHDA. Despite the difficulty, a relatively sharp density change corresponding
to the HDA<mml:math><mml:mo>→</mml:mo></mml:math>
→VHDA transformation was observed at <mml:math><mml:mrow><mml:mn>125</mml:mn></mml:mrow></mml:math>
125 K (90, 91). This is accompanied by a sudden change of compressibility (92), Raman spectra (92), and relaxation dynamics (93). However, also in this case sample history plays an important role: The sharpness of the density jump was found to decrease
with compression rate (90) or to disappear both at lower <mml:math><mml:mi>T</mml:mi></mml:math>
T (90) and when starting from annealed samples (91). Furthermore, it has been argued that the compressibility and Raman spectra change smoothly (94). The VHDA<mml:math><mml:mo>→</mml:mo></mml:math>
→HDA transition can also be observed at <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>≥</mml:mo><mml:mn>125</mml:mn></mml:mrow></mml:math>
T≥125 K but is kinetically hindered at lower <mml:math><mml:mi>T</mml:mi></mml:math>
T (72, 95). Even though the evidence is ambiguous, we favor the interpretation of a continuous nature of the HDA<mml:math><mml:mo>?</mml:mo></mml:math>
?VHDA transitions at <mml:math><mml:mrow><mml:mn>125</mml:mn></mml:mrow></mml:math>
125 K. Similar conclusions were drawn from isobaric experiments (93, 96, 97). Given the same symmetry of HDA and VHDA, a lowlying CP could exist, possibly located around <mml:math><mml:mrow><mml:mn>0.8</mml:mn></mml:mrow></mml:math>
0.8 GPa. In any case, confirming or refuting this hypothesis needs future work. Despite these open questions it is fair to state
that HDA and VHDA behave as distinct materials if studied at low <mml:math><mml:mi>T</mml:mi></mml:math>
T. We further note that in aqueous LiCl solution the HDA<mml:math><mml:mo>?</mml:mo></mml:math>
? VHDA transition fulfills all criteria of a firstorder transition (98). In other studies of salty HDA, however, this firstorder transition was not found (99, 100).
Phase Diagram of Noncrystalline Water.
To provide some guidance Fig. 3 summarizes the <mml:math><mml:mi>P</mml:mi></mml:math>
P–<mml:math><mml:mi>T</mml:mi></mml:math>
T behavior of LDA, HDA, and VHDA. We focus specifically on their domain of existence, that is, the <mml:math><mml:mi>P</mml:mi></mml:math>
P–<mml:math><mml:mi>T</mml:mi></mml:math>
T window where these glasses can be observed, bounded by the temperatures of crystallization <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx and polyamorphic transition. In this context we also review the calorimetric glasstoliquid transition as characterized
by its onset temperature <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg.
LDA (Fig. 3A) exists at relative low <mml:math><mml:mi>P</mml:mi></mml:math>
P and it crosses over to HDA at higher <mml:math><mml:mi>P</mml:mi></mml:math>
P. Calorimetric data show the glass transition onset around <mml:math><mml:mrow><mml:mn>136</mml:mn></mml:mrow></mml:math>
136 K at ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P (101), followed by crystallization. This leaves a small <mml:math><mml:mi>T</mml:mi></mml:math>
T window where an ultraviscous liquid could exist on the low <mml:math><mml:mi>T</mml:mi></mml:math>
T side of the noman’s land. Numerical results suggest that <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg(LDA) should decrease with increasing pressure (103). As a result, the window in which LDL can be studied should widen on increasing <mml:math><mml:mi>P</mml:mi></mml:math>
P.
HDA (Fig. 3B) converts to VHDA when heated at high <mml:math><mml:mi>P</mml:mi></mml:math>
P and hence its calorimetric <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg and <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx can only be properly characterized at low <mml:math><mml:mi>P</mml:mi></mml:math>
P. At ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P HDA can be heated high enough before the conversion to LDA (or LDL), such that a calorimetric glass transition onset around
<mml:math><mml:mrow><mml:mn>116</mml:mn></mml:mrow></mml:math>
116 K can be observed (101). This is in fact only the case for eHDA, but not for uHDA. For HDA noncalorimetric experiments suggest that <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg(HDA) increases significantly with pressure (107). Since <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx increases at a similar rate, the <mml:math><mml:mi>T</mml:mi></mml:math>
T window where HDL could be accessible to experiments remains approximately at the same size upon pressure variation.
VHDA (Fig. 3C) converts slowly to HDA at low pressure and hence its calorimetric <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg and <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx can only be properly characterized at high <mml:math><mml:mi>P</mml:mi></mml:math>
P. On heating at high <mml:math><mml:mi>P</mml:mi></mml:math>
P, calorimetry data show the onset of a glass transition around <mml:math><mml:mrow><mml:mn>140</mml:mn></mml:mrow></mml:math>
140 K (102). On further heating VHDA crystallizes to nearly pure ice XII (108). Noncalorimetric experiments suggest (93, 105) that <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg(VHDA) is approximately constant with increased pressure. Since <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx increases the window for a possible observation of the metastable liquid increases with pressure.
Relations Between Glasses and Supercooled Water.
The abrupt transition between HDA and LDA on heating at low <mml:math><mml:mi>P</mml:mi></mml:math>
P or decompressing at sufficiently high <mml:math><mml:mi>T</mml:mi></mml:math>
T is clearly consistent with the possibility that such a transition is the outofequilibrium echo of a true thermodynamic
coexistence curve between two (arrested) liquids of different density. The existence of more than one glass of water does
resonate with the existence of more than one liquid water form (41). One important precondition for the existence of two distinct liquids is that their relaxation times need to be clearly
shorter than their transformation times. It was shown very recently that indeed the dielectric relaxation times in HDA are
about three orders of magnitude shorter than the polyamorphic transformation, and similarly the relaxation times in LDA are
about three orders of magnitude shorter than crystallization times (109).
The observation of a calorimetric glass transition temperature for HDA (the socalled second glass transition) clearly distinct
from the one of LDA has been interpreted as evidence that water has two distinct phases above its glass transition temperatures
(101). These phases could possibly be identified as the two distinct ultraviscous liquids, LDL and HDL. This is strong evidence
for two distinct metastable liquid forms, one metastable only with respect to crystallization [associated to the <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg(LDA)] and the other metastable both with respect to crystallization and with respect to the lowdensity liquid form [associated
with the <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg(eHDA)]. Evidence in favor of two liquid phases has also been presented in ref. 83. During the experiment, eHDA is decompressed at <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>140</mml:mn></mml:mrow></mml:math>
T=140 K down to <mml:math><mml:mrow><mml:mn>70</mml:mn></mml:mrow></mml:math>
70 MPa. On this path, the sample first crosses the glass transition line where HDA turns into ultraviscous HDL. Then, it crosses
the transformation line from HDL to LDL (Fig. 3B). Immediate quenching to <mml:math><mml:mrow><mml:mn>77</mml:mn></mml:mrow></mml:math>
77 K at the transformation point freezes the sample. Subsequent visual examination and Xray diffraction experiments at ambient
<mml:math><mml:mi>P</mml:mi></mml:math>
P show that the sample presents spatially separated regions of LDA and HDA, suggesting that the conversion from HDL to LDL
has been interrupted by the quench (83). Very recent ambient pressure static and dynamic Xray experiments reveal a diffusive collective relaxation process in both
LDA and HDA above the glass transition temperature (88). This suggests an ultraviscous liquid nature for LDL and HDL. If this is indeed the case, then the bulk, firstorder HDL<mml:math><mml:mo>→</mml:mo></mml:math>
→LDL transition just below <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>
Tx at <mml:math><mml:mrow><mml:mn>70</mml:mn></mml:mrow></mml:math>
70 MPa was experimentally observed in ref. 83.
We stress, however, that even though the interpretation of the previously presented calorimetric data in terms of a glasstoliquid
transition is corroborated by several studies (93, 104?–106, 110, 111) the identification of the calorimetric glass transition with the onset of a liquid phase is still debated. Some researchers
believe the heat capacity increase to indicate the unfreezing of orientational (as opposed to translational) motion (112, 113). Alternatively, it has been argued that the rise in <mml:math><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:math>
CP is a sub<mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg effect (114)—a hypothesis which did not withstand careful scrutiny for LDA (115) and was taken back (116). However, it was revived and is now debated for the case of HDA (117?–119). Also, an order–disorder transition (120) or the presence of impurities (121) were quoted as a source of the heat capacity increase, but none of these explanations is able to explain the full phenomenology,
for example the ratedependent shifts of the calorimetric <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg onset. Although the scientific debate on the proper interpretation is ongoing, we do not see any stringent piece of evidence
that rules out the interpretation of a transition from glassy solids toward ultraviscous liquids.
Perspectives and Conclusions
In our opinion, three key controversial questions will drive future research on supercooled and glassy water: (i) the existence or absence of an LLCP in supercooled water, (ii) the number of different amorphous forms of water, and (iii) the nature of their glass transition(s).
Concerning i, we foresee strong effort in the direction of shrinking the noman’s land from both high and low temperatures, in an attempt
to enlarge the accessible window. Most likely the field will see some advance through the application of temperature jump
experiments, coupled with ultrafast probing techniques. This will allow measurements of bulk properties in the time interval
between (metastable) equilibration and crystallization. One instrumentally highly demanding avenue would be to couple the
hyperquenching technique with ultrafast Xray or laser techniques so that a water droplet can be probed in the microsecond
time scale, while it cools from room temperature to <mml:math><mml:mrow><mml:mn>77</mml:mn></mml:mrow></mml:math>
77 K. The detection of extrema in the response functions, especially in the isothermal compressibility, would be highly rewarding,
since it could signal the crossing of the Widom line. Large effort will possibly be devoted to fastquench experiments under
high <mml:math><mml:mi>P</mml:mi></mml:math>
P, to probe the region where the LLCP and the associated HDL–LDL spinodal are expected. The <mml:math><mml:mi>T</mml:mi></mml:math>
T jump can either be a heating pulse on amorphous ice or a cooling pulse on pressurized stable water. For the heating pulses
the improved annealing procedures to produce more stable HDA developed over the last few years will be instrumental to access
the ultraviscous (metastable) liquid state in between the glass transition and the crystallization temperature. We also expect
efforts in the direction of performing this type of sophisticated studies at negative pressures.
On the theoretical side we cannot avoid mentioning it is quite unsatisfactory that rather similar water models apparently predict different behaviors in supercooled states. While it has been claimed that several of them do show an LLCP, it is fair to state that up to today crystalclear evidence of an LL transition has only been provided for the ST2 water potential. Recent methods devoted to study ST2 water (43) will surely be applied to other models. We do expect that these more sophisticated investigations will clarify under which conditions a model potential shows or does not show (44, 48, 122) an LLCP and how the presence of an LLCP affects the phenomenology of the disordered solid phases (123). We do also expect that future research will focus on the way model parameters affect the thermodynamic scenarios, the barriers to crystallization, and the model glassforming ability. There is evidence that small variations in the model parameters can convert from an LLCP to a reentrant spinodal scenario as well as from an LLCP to a singularityfree scenario (48, 122). It has also been recently demonstrated (44) that the driving force for crystallization sensitively depends on the bond flexibility and on the softness of the interparticle repulsion (a quantity controlling the ability to interpenetrate). The ultimate goal will definitively be an ab initio quantummechanical numerical investigation of supercooled water. Steps in this direction have already been taken. Simulation studies of models with and without an LLCP can also be exploited to assess how the presence of an LLCP affects the phenomenology of the disordered solid phases (123). We note that interesting suggestions on the different thermodynamic scenarios could also come from the investigation of colloidal particles with directional interactions (124?–126).
Concerning ii, it will be necessary to clarify whether LDA, HDA, and VHDA are the same phase, differing only due their state of relaxation,
or whether they are formally separated by thermodynamic transitions. While the majority of results point toward a firstorderlike
transition between LDA and HDA, much less clear evidence has been presented for an HDA–VHDA transition. The nature of the
latter transition can be clarified with presentday technology. Extremely long and slow experiments, requiring a high demand
of time and patience, are needed. Furthermore, there is a need to characterize the glass(es) obtained by quenching liquid
water at (several) high pressures, from <mml:math><mml:mrow><mml:mn>100</mml:mn></mml:mrow></mml:math>
100 MPa on. The present evidence is limited to Mishima’s study at <mml:math><mml:mrow><mml:mn>500</mml:mn></mml:mrow></mml:math>
500 MPa (73). Indeed, while LDA results from ambient pressure hyperquenching, there is no consensus on the materials obtained by quenching
at high <mml:math><mml:mi>P</mml:mi></mml:math>
P. Simulation studies seem to suggest that the disordered glass closest to the highdensity liquid is the VHDA form (127).
Concerning iii, it seems clear that the interpretation of the calorimetric glass transitions of the amorphous ices will continue to receive
particular attention. Indeed, under the hypothesis that LDA and HDA are the glass proxies of LDL and HDL evidence of two distinct
ultraviscous phases could corroborate the LLCP scenario. Today there is no clear consensus on the origin of the calorimetric
signal associated to the lowdensity form above <mml:math><mml:mrow><mml:mn>136</mml:mn></mml:mrow></mml:math>
136 K (the first glass transition of water) and similarly the highdensity form above <mml:math><mml:mrow><mml:mn>116</mml:mn></mml:mrow></mml:math>
116 K (the second glass transition). Clarification of this point is rather relevant: If the calorimetric signal can effectively
be associated to a glasstoliquid transition, then the reported evidence of two distinct <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tgs at ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P for LDA and HDA (101) would provide indisputable evidence of two distinct ultraviscous liquid phases. This would offer the possibility to explore
metastable LDL and doubly metastable ultraviscous HDL at ambient <mml:math><mml:mi>P</mml:mi></mml:math>
P. We cannot avoid noticing that most of the calorimetric studies have been performed at ambient pressure. Calorimetric studies
at higher pressure, in the realm in which HDA and VHDA are more stable, would be most welcome. Such instruments, not commercially
available, would be very valuable since it is expected [based on volumetric studies (93, 106)] that also under high <mml:math><mml:mi>P</mml:mi></mml:math>
P an ultraviscous liquid state should exist.
To resolve the conflict on the two opposite interpretations of the calorimetric glass transition (orientational glass versus
ultraviscous liquid) novel experiments and novel concepts will be required. This could encompass diffusion studies on tracer
molecules as well as mechanical probing of the liquid, for example rheological studies on the sample at 140 K, just after
the glass transition and just before its crystallization. Such studies, or studies based on the experimental methods recently
exploited to quantify diffusion in ultraviscous water (128, 129), will be most helpful in interpreting the increase in <mml:math><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:math>
CP observed both for HDA and LDA at their two distinct <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tgs. The same experiments speak in favor of ultraviscous liquid nature, at least for LDA above its <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:math>
Tg (128, 129).
Surely new input from the experimental and theoretical side is highly needed to clear the haze and to come closer to the ultimate goal of understanding water. Based on previous history, it is highly likely that diametric viewpoints will persist and, by competing, drive the field forward.
Acknowledgments
T.L. and F.S. thank all students and collaborators with whom they have been studying this fascinating topic. T.L. thanks in particular R. B?hmer (Technical University of Dortmund), D. T. Bowron (ISIS), and N. Giovambattista (City University of New York) for their longterm collaboration on the subject. We also thank L. Rovigatti and J. Russo for helpful comments on the manuscript. This work was supported by Austrian Science Fund Schr?dinger Fellowship J3811 N34 (to P.H.H.), STARTPreis Y391, and bilateral project I1392 (T.L.), European Research Council Starting Grant SULIWA (to T.L.), the Austrian Academy of Sciences (several DOC projects for coworkers of T.L.), and ETNCOLLDENSE Grant 642774 (to F.S.).
Footnotes

?^{1}P.H.H., T.L., and F.S. contributed equally to this work.
 ?^{2}To whom correspondence may be addressed. Email: philip.handle{at}uibk.ac.at, thomas.loerting{at}uibk.ac.at, or francesco.sciortino{at}uniroma1.it.

Author contributions: P.H.H., T.L., and F.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.
Published under the PNAS license.
References
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