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Water above spinodal

Liquid spinodal the bedrock of the water thermodynamics was for a long time discussed side by side with liquidliquid critical point elusively hidden beyond homogeneous nucleation limit inspiring numerous hypotheses aspiring to explain water anomalies. We combined these two concepts in their purest form creating a new equation of state capable of describing a real liquid. The equation expanding from the spinodal describes accurately the thermodynamic properties up to 100 MPa. We were able to predict spinodal in presence of liquid-liquid critical point and demonstrate that there is no such a feature as re-entering spinodal in water. It was predicted only because of the curved density surface caused by the existence of the second critical point. It practically excludes all other scenario except for the second critical point point at positive pressure. But critical point alone is not enough to capture the delicate shape of water; cooperativity the preference of nearby molecules to joint similar configurations proves to be important in the new equation. It designates water as a regular solution with phase separation driven not only by excess energy but also entropy. The heat capacity was also studied extensively making clear that disagreement between the two major experimental data sets available today is too profound. It could be explained only by additional underlying structural change. Because it is not present we can identify correct data.


I. INTRODUCTION
History of the scientific voyage to discover and then prove the origin of the water anomalies manifests difficulty and responsibly preceding the conclusive judgment shall be passed. We believe that this paper presents an important step forward by conclusively proving that all the proposed scenarios behind water anomalies can be explained by one phenomenon. But building upon decades of progress we have to put our work into the context.
Water was always a mysterious substance. Any organisms living in water very much appreciate its expansion upon freezing as well as the maximum density at four digress of Celsius enormously increasing chance of survival, on the other hand, melting under pressure reduces the friction for anyone skating above the frozen resistant realm. The list can go on up to astonishing 64 anomalies 1 some of them defining the life as we know it.
But the systematic effort to discover the origin of the anomalies is quite a new adventure. Early experiments discovered rapidly increasing heat capacity 2 and compressibility 3 of water with supercooling down to the point below which no bulk experiment can be performed because of rapid freezing 4 (the homogeneous nucleation limit) but did not reach a peak or a point of divergence. After a decade of the effort, Speedy came out with a very elegant way to analyze a different divergence this time a liquid spinodal 5 . He discovered that the spinodal seems to turn back towards positive pressure, jet another possible anomaly. That indicates a connection between these two divergences and give birth to the first a) Author to whom correspondence should be addressed: duskamich@gmail.com hypothesis of re-entering limit of stability. Very recent measurements of the compressibility and the correlation length of evaporating micro-droplets 6 (the results are still under heavy discussion mainly because of validity of temperature measurement 7,8 ) and the speed of sound of water in the double metastable region 9 (supercooled and stretched simultaneously) detected the Widom line 10 , which is believed to indicate the liquid-liquid critical point existence 11 . But that would be the step too far too early on our journey.
Then the new research tool changed everything. Molecular dynamics study discovered phase transition between high-density and low-density water ending in the second critical point 12 . The second hypothesis of liquidliquid critical point emerged. It took time to clarify if the low-density water is merely ice like fluctuation 13,14 or there is an energy barrier between the new face of water and ice. The former prevailed rigorous analysis 15 . This hypothesis has two special cases: the critical point hidden in zero temperature it is a singularity free scenario 16 and hiding the critical point bellow the liquid spinodal forms a critical point free scenario 17 . It means that not only existence of the critical point is crucial but also its position.
Several cleverly designed physical experiments found a way to prevent freezing. Fragile-to-strong dynamic transition of water confined in nanoporous material 18 and heat capacity of water solutions not canceling out hydrogen bond network 19 both substantiated the critical point scenario. But none of those results can be directly associated with the property of balk water. What was made clear instead: a straightforward localization of the critical point and confirmation of one of those scenarios is highly unlikely 20 ; that is where the difficulty is rooted and from that the responsibly arises to extrapolate to-wards unknown by most compelling way.
What makes water special is a possibility to form a hydrogen bond, a strong electromagnetic interaction between water molecules created by locality of hydrogen electron in the molecule itself. The hydrogen bond is very orientation-dependent, forming wide tetrahedral network responsible for lowering water density 21,22 (we virtually separate such water molecules and call it tetrahedral water for simplicity). Several methods demonstrated HB influence on shape of phase diagram and the ability to replicate the scenarios: a lattice-gas model incorporating HB microscopic behavior into a liquid phase 23,24 and HB implemented into Van der Waals equation of state 25,26 . HB strength and cooperativity has profound consequences 27 . If tuned properly, it can reproduce all the scenarios mentioned above. But none of them can describe thermodynamic properties of water with sufficient accuracy so these methods are not suitable to provide definite prove.
Increase of accuracy was successful with another approach, this time fully thermodynamic (no knowledge of the microscopic behavior is required) a two-state method. It reproduces water behavior very accurately 28,29 . The International Association for the Properties of Water and Steam adopted a two-state equation of state as the guideline for supercooled ordinary water 30 . Dispute the concept of seemingly simple mixture or reaction between two states of water it is representing a first approximation of the bond lattice 31 representing complex behavior of HB. It can generate all the scenarios, except for the reentering limit of stability 32 . Besides the critical point scenario prevalently implemented and tested with this method the singularity free scenario was also rigorously examined by Tanaka 33 , but later density measurements of Mishima 34 made a strong argument against that scenario, because it didn't confirm a behavior consistent with vicinity of the Widom line but with the liquid-liquid transition instead.
The main disadvantage is that a polynomial function is implemented in the two-state equations to represent the approximation of the high-density state of water (to simplify the description we will call it water with broken HB). Therefore unlike in method above the core part of the two-state equation is without explicit physical foundation. In the most advanced version, the limit of stability was added to the polynomial function to represent liquid spinodal 35,36 . We are proposing a different approach, based on the following considerations. If the line of compressibility maxima is the most likely reality, then the first-ever formulated scenario (the re-entering limit of stability) cant be true. But it doesnt mean that the re-entering feature is irrelevant, quite contrary! Its discovery is based on elegant extrapolation of density from the limit of stability up to elevated pressure based on simplest expansion possible but jet very accurate. So curvature of density surface creating the appearance of the re-entrance is very real and the theory capable of explaining the reason behind that is also best suited to explain the nature of water anomalies.
We used exactly the same equation as proposed by Speedy for the extrapolation, created its integrated form to be able to describe all measurable thermodynamic properties and implemented that into the two-state equation. It will be clear later, such a model constituting only from components with clear physical meaning is capable to discover several important issues. In such an arrangement this new equation of state is able of interpreting all the remaining scenarios and it's only up to measurements to judge which one is the most compelling.

II. EQUATION OF STATE
To choose the proper form of the model a nature of the two states interaction needs to be examined. Molecular dynamics studies indicate that mixing of the states is symmetric 29,37 . The HB is apparently not strong enough to form a polymer like super-molecule (then a stoichiometry would create asymmetry). So the preference to surround molecule with molecules interacting the same way is left on the cooperativity manifesting itself in a noneideal mixing contribution to the model.
Then the equation of state (EOS) in terms of Gibbs free energy G (in reduced temperatureT = T T VLCP and pressurep = p p VLCP ; where VLCP indicates conditions at vapor liquid critical point) is defined as non-ideal mixture of tetrahedral water G B with fraction of its molecules x and water with broken hydrogen bonds G A

G= G
where R is the specific gas constant, and ω = ω 0 1 + ω 1p + ω 2T + ω 3Tp is the cooperativity controlling preference of the species for its neighbors and if the two spices do not like each other separation of the two spices begun in a LLCP. The ω 0 ensure that critical point forms in selected temperature effectively serving as critical temperature parameter of the model. Introducing the cooperative dependent on pressure and temperature as well, we assume that water is a regular solution where phase separation is driven not just by energy but also by excess entropy.
A time required for reorganizing molecule of water (the structural relaxation time) is significantly faster than the time required for nucleating and growing a crystal, even far away from equilibrium 13 . Therefore we can apply the reaction equilibrium condition to find a metastable value of x in all range of pressures and temperatures.
A. Water as a simple fluid Speedy's EOS 5 assumes that pressure expands above the limit of stability alongside an isotherm as a Taylor series of difference between density and density at spinodal. If only a second term of the expansion applies then the volume of the state A iŝ wherep S is the pressure at the spinodal as a function of the temperature,V S is the volume at spinodal and B is the second derivative of pressure with respect to density at constant temperature at spinodal. The B is zero at VLCP by definition. It means that the equation has limited validity at elevated temperature. This equation was proven to represent density up to 370 K and from 0 to 100 MPa very well and the addition of another valid term into the expansion is not increasing quality of fit considerably 5 .
State B came from the difference between Gibbs free energy of state B and A as where parameters a 0 , a 2 , a 3 ... represents first approximation of difference between G, V, S, and so on.
In this stage, knowing the cooperativity and the G B − G A it is possible to determine all three parameters of the Speedy's EOS: location of spinodal in phase diagramp S , the volume at spinodalV S and the second derivative B for any temperature. The two-state model is compensating the curvature of density for the existence of the second critical point and Speedys EOS fitts data to approximately localize spinodal and its properties. From that perspective, it is hard to see the temperature-dependent functions as free parameters of the model. They are first approximation of the real spinodal. There are no additional polynomial terms creating artificial distortion between the two-state model and extrapolation towards the spinodal predicting it as accurately as possible.
But to develop full thermodynamic description we use polynomials to represent the temperature dependent parameters of the sponodal Then we developed the integrated form for the Speedy's EOS by determining entropy of state A (it was more convenient then deriving Gibbs free energy itself) from the Maxwell relation ∂S ∂V T = ∂p ∂T V where the left side comes from Eqs. 3. where dT andŜ S is entropy at spinodal jet to be determined.

B. Optimization of EoS parameters
There are three major thermodynamic properties used for EOS development: density, speed of sound and isobaric heat capacity. As a first step, only the parameters of the two-state model were optimized to fit density at positive pressure accurately be the Speedys EOS and to predict spinodal. Then a spline representation of the predicted spinodal enables to evaluate accuracy of speed of the sound. This is relatively fast optimization serving as global optimization tool to find proper two state model. At the second step the parameters of the twostate model and the analytical representation of state A were optimized simultaneously by genetic optimization to fit density at positive pressure. Subsequently a derivative of entropy of state A at spinodalŜ ′ S = dŜS dT could be induced based on heat capacity at atmospheric pressure. With a selection of proper analytical representation, we could include heat capacity and speed of sound into optimization to further improve the whole model. There is an important issue to resolve: two major data sets for heat capacity in supercooled region ( Angell et al. 38 vs. Tombari et al. 39 and Voronov et al. 40 ) disagree significantly on where they start to diverge (Data of Archer and Carter 41 considered to diverge at the lowest temperature are actually not so different from the Angellet al. 38 to have a significant influence on the EOS.) For theŜ ′ S we choose simple cubic polynomial to impose a restriction on the continuous character of the property. It was possible to optimize model only for data of Tombari et al. 39 and Voronov et al. 40 (Table I) (EOS-VaT model) but not for Angellet al. 38 or Archer and Carter 41 where T S ′ S needs to turn down sharply, see Fig. 1. Such rapid change of speed in a grow of entropy T S ′ S happening independently on pressure, it means for both high and low-density water at the same temperature, is highly unlikely. It would mean change beyond hydrogen bonding. But from recent experiments in waterrich hydrazinium trifluoroacetate solution by Woutersen et al. 19 the only sharp change of heat capacity is because of the change in hydrogen bond structure indicating the liquid-liquid transition proximity. To show thermodynamic consequences of such a behavior we accepted existence of such a turn and use polynomial interpolation as an analytic representation of theTŜ ′ S T for Angell et al. 38 (EOS-A model). The author will provide if asked the parameters of this model. There is no significant difference between the models except for theTŜ ′ S T therefore all the five temperature dependent functions presented in Fig. 1 are for the EOS-VaT model (the solid line) with only dashed line for thê TŜ ′ S T of the EOS-A model. The pressure at spinodal p S T smoothly connects to Speed's limit of stability at higher temperatures (see Fig. 1 (a)) suggesting a vanishing of tetrahedral water. The volume at spinodalV S T ( Fig. 1 (b)) is monotonic function with gradually decreasing decline at supercooled region. The second derivative of pressure with respect to density at constant temperature at spinodal B T (Fig. 1 (c)) starts to increase around 280 K despite the theoretical condition to be zero in VLCP. It can mark a region where higher terms should start to play a role. But because of the focus of the paper on supercooled region we will not apply them. The derivative of entropy of state A at spinodalTŜ ′ S draw a sharp distinction between the heat capacity of the two data sets. It is clear that such a model with clear physical meaning of all parameters presents a distinct advantage, and enables to spot any unrealistic behavior.

III. EXPERIMENTAL DATA THE TOUCHSTONE
It is important to point out that proposed EOS aims to be valid up to 100 MPa but we will compare it to data up to 200 MPa to ensure proper extrapolation ability. The thermodynamic data selected for the comparison presents the state of the art, a description of the particular data sets can be found in caption of figures. We have already shown that our model predicts normal-like spinal without the re-entrant feature. Now we will present how well can the model explain the curvature of the density surface Fig. 2. At the positive pressure, we can relay on density measurement of bulk and emulsified water. For the negative pressure there is the speed of sound data measured at density and temperature and analyzed to determine pressure by Pallares et al. (2016) 44 . So the measured density surface of ordinary water is quite a complete picture extending far below the temperature of freezing to 200 K and down to -110 MPa as well. But for the meaningful model, there are other conditions to follow. The proximity of density of state B to the density of tetrahedral ice at atmospheric pressure was required through-out the optimization to match the observation in modeled water by Palmer et al. 15 . For continuous pathway to amorphous ice, it is also important to form a proper shape of the liquid-liquid transition which proved to be a nontrivial problem 36 . We were successful with highdensity water extrapolating towards high-density amorphous ice. But because of the cubic term in the G B − G A added to increase the flexibility of the LLTL and the Widom line, there is instability of state B below 200 K (not visible from the Fig. 2) preventing proper behavior for the small region below LLTL. Disagreement with density at low temperature and pressure above 100 MPa is also a consequence of simplified the G B − G A it could be improved by higher density at LLTL (the approach implemented by 36 ). But then it would require to sharply slow the speed with which density increases at LLT to meet the high-density amorphous ice. Additional terms could eliminate that but further complicate the model. We consider the results as a reasonable compromise especially that major focus is toward the spinodal.
Phase diagram Fig. 3 pictures the complex shape of the LLTL and the Widom line even better. LLTL is hidden behind the homogeneous nucleation line as predicted but with a decrease of temperature, it is not turning fast enough to meet amorphous ice transition at 200 MPa. On the other hand, the Widom line instead of monotonously decreasing with increasing temperature slows and creates S-shape which is crucial to interpret following data properly. Locus of density and compressibility maxima measured by a group of Frédéric Caupin 9,44 are represented correctly even though their data was not used for optimization. But there is remaining slight disagreement between the position of compressibility maximum determined by Kim et al. 6 in micro-droplets as much as 4.4 K. But it is hard to reconcile the trend of those data with data of Kanno and Angell (1979) 3 as pointed out by 36 and can be seen from Fig. 4, the precise position of this compressibility maximum may not be a dogma. Such flexibility of the LLTL and the Widom line is crucial and can be seen best from lack of it. It was found in a recent study that without the cubic term of the G B − G A it is impossible to reconcile those two crucial data sets properly. Adding the one term to the G B − G A is clearly reasonable.
After forming the basic shape of density second part the two-state model the cooperativity became open for scrutiny. Once the criticality is introduced higher terms are responsible for changing the fraction of conversion x shaping significantly density surface. Having the EOS without any possibility to create additional curvature it is up to the two-state model to explain it. We conclude that the introduction of cooperativity ω dependent not only on pressure 28 but also on temperature is important to capture the delicate shape of water, adding two extra parameters to the model. But such a result should not be surprising after so much evidence about importance of cooperativity of HB mentioned above.
The location of the liquid-liquid critical point although  an important parameter of the model is a mere guess because of the two-state model being the mean-field approximation of phase transition. We localized it in 221.0 K and 53.5 MPa, which means that the critical pressure is little less than a minimum of 60 MPa expected by Ni et al. 53 from a recent study of water diffusion. Most probably because of the less flexible LLTL and Widom line discussed above. A consequence of the lower critical pressure is apparent from a slightly higher compressibility maximum at zero pressure Fig. 4.
To scrutinies our model even further we analyze the speed of sound. The model represents data very precisely at positive pressure up to 100 MPa, see Fig. 5 but gradually disagree with data at negative pressure. With the model as it is, the only possibility to increase overall accuracy at negative pressure is to decrease accuracy at the positive pressure revealing that there is a difference in speed of growth of density satisfying data at positive and negative pressure respectively. We decided to stick to the correct interpretation of positive pressure and use the negative pressure data as an indication of extrapolation abilities. There is another issue worthy of mentioning: EOS predicts the lower speed of sound at the negative pressure with an increasing difference at higher temperatures (it starts above 300 K) probably because of increasing influence from the missing higher terms of the Speedys EOS as discussed above. But the prediction of its minimum is quite accurate. The model represents the isobaric heat capacity at atmospheric pressure accurately as required by definition of the model. Fig. 6 (b) is revealing a consequences of the sharp decline ofTŜ ′ S necessary to fit data of Angellet al. 38 . The decline if not compensated by an increase of heat capacity caused by the proximity of the Widom line leads to a decrease of heat capacity itself at a pressure slightly above atmospheric pressure. It can be even better seen from isochoric heat capacity Fig. 7 (b). Such a decrease of heat capacity is not present in a model water 37 . But on the other hand the EOS-VaT model (see the Fig. 7 (a)) is very similar to the model water further confirming that heat capacity data of Tombari et al. 39 and Voronov et al. 40 are describing the speed of the divergence better. The Fig. 6 (a) is also disclosing very good agreement for measurements at elevated pressure.

IV. CONCLUSION
Density surface (as a function of pressure and temperature) with its curious shape inspired investigation of vast regions below freezing point and even in negative pressure and sparked several hypotheses aiming to find the rational behind its pattern. We presented first prove that the seemingly reentering spinodal, the first-ever formulated hypothesis, is a consequence of the liquid-liquid critical point existence at positive pressure. We were able to do that by combining the two-state model with the Speedys EOS in exactly the same way as it was used to discover the re-entering spinodal. Proposed equation of state despite its deliberate simplicity can describe all available thermodynamic data with a reasonable accuracy. Possibility that the critical point is hidden in zero temperature, the singularity free scenario, seems to be clearly against available data. It means that despite lack of direct proof of the existence of the second critical point there is no other plausible explanation left. The elusive liquid-liquid critical point of water is the only explanation we can offer to the nature of substance shaping all living in so profound way.
The main advantage of the proposed EOS is the ability to predict spinodal in the presence of the second critical point. The spinodal is the result of not input into the fitting procedure. Another advantage is its simplicity and clarity of all constituency of the model allowing to spot which parts of the model are playing an important role. The cooperativity seems to be more important than anticipated in previous two-state models, denoting water as a regular solution with phase separation driven not only by excess energy but also entropy. We observed that the most compelling way to reconcile the latest data of maximum compressibility 6,9 is to introduce higher flexibility to the curve forming the LLTL and the Widom line with an appearance of inflection point before crossing the spinodal. We also found out that the difference between heat capacity measured by Voronov et al. 40 and Tombari et al. 39 , and by Angellet al. 38 are so profound that there is no rational basis for such a strong change in heat capacity with respect to temperature. Measurements of Voronov et al. 40 and Tombari et al. 39 are according to our equation the most probable representation of heat capacity in the metastable region.

ACKNOWLEDGMENTS
We acknowledge Mikhail A. Anisimov for suggesting this research project and for providing new experimental data on the heat capacity of supercooled water, F. Caupin providing new data on the speed of sound at negative pressure and fruitful discussions with J.V. Singers and J. Hruby. This research originated from a project Towards an IAPWS Guideline for the Thermodynamic Properties of Supercooled Heavy Water (preliminary results were made public 61 ) under support from Young Scientist IAPWS Fellowship. Continuation of the research was supported by the Institute of Thermodynamics, The Czech Academy of Sciences (the institutional support RVO:61388998).