Dynamical crossover line in supercritical water

Dynamical crossover in water is studied by means of computer simulation. The crossover temperature is calculated from the behavior of velocity autocorrelation functions. The results are compared with experimental data. It is shown that the qualitative behavior of the dynamical crossover line is similar to the melting curve behavior. Importantly, the crossover line belongs to experimentally achievable (P, T) region which stimulates the experimental investigation in this field.

and dynamical properties of water was proposed. Since Frenkel line is the line of dynamical crossover in fluids some kind of relation between the Widom line and Frenkel line can exist.
As it was proposed in our earlier publications several methods to find the location of Frenkel line can be used [8][9][10] . The most convenient one is based on the lose of oscillations of vacfs. This criterion is used in the present work. Fig. 1(a,b) show the vacfs of oxygens for two densities: ρ = 1.0 and 1.3 g/cm 3 . One can see that the low temperature vacfs for these two densities look qualitatively different while at high temperatures they become very similar. In particular, as the temperature increases the oscillations of vacfs become less pronounced and finally disappear.
One more way to estimate the location of Frenkel line in P − T or ρ − T diagram is related to heat capacity of liquid [8][9][10] . In case of a monatomic fluid the magnitude of c V at the Frenkel line is 2 k B per particle (k B is Boltzmann constant). In case of water the heat capacity c V undergoes strong decay upon isochoric heating. Next to the melting line the heat capacity per molecule is about 9 k B or 3 k B per atom while at high temperatures it becomes as small as 1.5 k B per atom. If all degrees of freedom of the molecules are excited then the heat capacity per molecule at Frenkel line should be 6 k B per molecule or 2 k B per atom. The location of Frenkel line by c V criterion was evaluated from experimental data. The data were taken from NIST database 23 .
Frenkel line of water obtained from vacf criterion is shown in Fig. 2(a,b). From Fig. 2(a) one can see that Frenkel line starts at the boiling curve at temperature = ≈ .
where T c is the critical temperature. In our previous publications it was shown that the same ratio T F /T c takes place in Lennard-Jones fluid and liquid iron.
One observes extremely fast grow of the Frenkel line temperature in a narrow interval of densities ρ = (1.2-1.22)g/cm 3 . It is related to extremely slow disappearance of vacf oscillations in this region.  Importantly, the relation between the temperature at the Frenkel line T F and the melting temperature T m changes upon increasing the pressure. At low pressures (before the rapid increase) the ratio T F /T m is close to 2. At pressures ≈ P kbar 100 this ratio increases up to 5. On further rise of pressure it reaches the value of 9 at ≈ P kbar 900 . It means that in the range of pressures considered in the present work the Frenkel line bends up with respect to the melting line.
In our previous publications it was proposed that in the limit of high pressures Frenkel line should be parallel to the melting line in double logarithmic coordinates. In case of water we are not aware of any measurements of the melting curve above 1000 kbar. We expect that the Frenkel line and the melting curve will be parallel in the high pressure limit, but one needs to extend the melting curve to higher pressures in order to check it. Although this conclusion may be violated by transition of water into superionic phase under high pressure 24 . This phenomena can not be taken into account in frames of purely classical model used in the present work.
Very recently Frenkel line of water calculated by vacf criterion for a different model (TIP4P/2005) was reported 13 . This line is shown in Fig. 2(b) for the sake of comparison. One can see that this line is systematically higher then our line. However, the lines are very close to each other and this small discrepancy can be attributed to the different models under investigation. The authors of 13 studied the   Fig. 3. NN of water along the melting line was reported in refs 26,27. Although our results belong to an isotherm while the results of these publications are related to the melting line, the NNs are calculated in similar pressure interval. That is why we show them in the same plot for comparison. The difference between this work and the literature data should be referred not only to different lines in P − T plane but also to different models studied and different methods of calculation. One can see that up to pressure of approximately 50 kbar the coordination number rapidly increases while above this threshold it holds approximately constant. One can conclude, that at small pressures the local structure is very sensitive to the pressure change while at higher ones the local structure is very stable which is similar to the case of simple liquid. Therefore one can say that water becomes "simpler" upon increasing pressure.
We expect that similar rapid increase of Frenkel line in the density range ρ = . − . / g cm 1 2 1 22 3 can be observed in TIP4P/2005 as well. However, the results reported in the work 13 end up at the density 1.2 g/cm 3 . So one needs to extend them in order to check this assumption.
We report a computational study of dynamical crossover in water. The temperature of crossover (the Frenkel line temperature T F ) is calculated from the behavior of velocity autocorrelation functions. The results are compared to the experimental ones obtained from isochoric heat capacities. It is shown that qualitative behavior of Frenkel line is similar to the behavior of the melting curve. However, the ratio T F /T m increases with increasing of pressure. It means that the lines diverge in the range of pressures considered in the present work. This divergence can be related to the change of the local structure of water upon increasing the pressure.
Importantly, the temperatures of dynamical crossover appear to be rather low at moderate pressures. At pressures P < 30 kbar T F does not exceed 1000 K. It means that (P, T) parameters of Frenkel line can be achieved experimentally. Such experimental works would be important not only for deeper understanding of dynamical behavior of liquids but also could serve for supercritical technology. Importantly, the properties of fluids at the Frenkel line are close to the optimal ones for technological applications 28 . Water as well as carbon dioxide are among the most important and widely used supercritical fluids 29 . Supercritical water is used for many different applications, such as green solvent, as reaction medium for different chemical processes, for production of biofuel, oxidation of hazardous materials which is important for dangerous waste disposal. Applications of supercritical water include separation, extraction and purification of different substances and many others 30 .
The principal property of supercritical water providing its widespread application is its solving power. In ref. 13 the solubility of different solutes in carbon dioxide and its relation to the Frenkel line was discussed. It was shown that the solubility maxima are close to the Frenkel line. However, the data for solubility maxima of different solute in supercritical water are not available at the moment. Basing on the discussion of 13 and the results of the present paper one can propose that the optimal solving power of water should belong to the interval T = 700-1000 K and P = 1-3 GPa. Currently, the most widespread usage of supercritical water belongs to the temperatures interval 650-1000 K and to the pressures up to 0.5 kbar. However, these (P, T) conditions were found empirically and do not have any solid theoretical ground. Moreover, up to now the supercritical technology advances the theoretical foundations in the field. This work as well as ref. 13 allow to predict the best (P, T) conditions for supercritical water application which makes these publications the pioneering works in developing the theoretical basis of supercritical technologies.

Methods
In the present work we study the behavior of water by means of molecular dynamics simulations. An SPC/E model of water is used 31 . The phase diagram of this model was reported in several publications. In 32 a comparison of solid part of the phase diagram of SPC/E, several variants of TIP4P model and experimental results is given. One can see that all models fail to reproduce the whole complexity of the experimental phase diagram, but manage to describe some parts of it. In particular, SPC/E model is good in describing boiling curve of water. In ref. 33  It was shown that the discrepancy of SPC/E model and ab-initio results is of the order of 15-20% for T = 1000 K and pressure up to about 100 kbar. However, at T = 2000 K and pressures up to approximately 90 kbar the agreement of ab-initio and SPC/E results is within 5% which should be considered as good agreement. One can guess that at T = 1000 K and so high pressure the results are affected by crystallization effects while at temperatures well above the melting line SPC/E model can be used to study the high pressure behavior of water.
A system of 4000 water molecules in a cubic box was simulated in molecular dynamics at constant volume, number of particles and temperature (canonical ensemble). The temperature was held constant by Nose-Hoover thermostat. The density was varied from ρ = . / g cm 0 8 min 3 up to ρ = . / g cm 2 0 max 3 and the temperatures from T min = 275 K up to = ⋅ T K 5 10 max 4 . Initially the system was equilibrated for 1ns with a time step dt = 1fs. After that it was simulated more 1ps with the same time step in order to calculate the thermodynamic properties. Finally, 10 5 steps with timestep dt = 0.1fs were made in order to well reproduce the decay of velocity autocorrelation function (vacf).
All simulations were performed using lammps simulation package 35 .