Thermodynamic Response Functions and Stokes-Einstein Breakdown in Superheated Water under Gigapascal Pressure

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Among the anomalous properties of water is the seeming divergence of the thermodynamic response functions and dynamic properties below the homogenous nucleation temperature (~232 K). Furthermore, water exhibits an increasingly decoupling of the viscosity and diffusion, upon cooling, resulting in the breakdown of the Stokes-Einstein relationship (SER). At high temperatures and pressures, however, water behaves more like a “simple” liquid. Nonetheless, experiments at 400 K and GPa pressures (Bove et al . (2011) Phys. Rev. Lett. , 111:185901) showed that although the diffusion decreases monotonically with the pressure, opposite to pressurized supercooled water, a decoupling of the viscosity and diffusion, larger than that found in supercooled water at normal pressure, is observed. Here, we studied the thermodynamic response functions and breakdown of the SER along the 400 K isotherm up to 3 GPa, through molecular dynamics. Seven water models were investigated. A monotonic increase of the density (~50 %) and decrease of the isothermal compressibility (~90 %) and thermal expansion (~65 %) is found. Our results also show that compressed hot water has various resemblances to cool water at normal pressure, with pressure inducing the formation of a new second coordination sphere and a monotonic decrease of the diffusion and viscosity coefficients. Whereas all water models provide a good account of the viscosity, the magnitude of the violation of the SER at high pressures (> ~1 GPa) is significantly smaller than that found through experiments. Thus, violation of the SER in simulations is comparable to that observed for liquid supercooled water, indicating possible limitations of the water models to account for the local structure and self-diffusion of superheated water above ~1 GPa. Superheated water Thermodynamic Response Function Gigapascal Pressure Stokes-Einstein Breakdown Molecular Dynamics Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. Introduction Liquid water exhibits numerous thermodynamic and dynamic anomalies, most of which intensify upon cooling below the freezing point [ 1 – 3 ]. Among the most prominent anomalies of liquid water at normal pressure are the increase of the magnitude of its response functions upon cooling [ 2 , 4 ]. These anomalies, related to an increase of the entropy and volume fluctuations at low temperatures, opposite to simple liquids, prevent yet, a comprehensive understanding of liquid water, including its temperature and pressure dependence [ 2 , 4 , 5 ]. Several explanations were put forward to provide a coherent picture of water’s anomalous behaviour. These include the ‘‘stability limit conjecture’’ (SLC)[ 6 ], the singularity-free hypothesis[ 7 ] or the liquid-liquid critical point (LLCP)[ 8 ]. The LLCP picture, the most widely investigated, foresees the co-existence of high-density and low-density liquid (HDL/LDL) forms of water below a second critical point located already below the homogeneous nucleation temperature ( T h ~232 K). The existence of HDL and LDL forms of liquid water would then relate to the abovementioned anomalies, including the temperature of maximum density ( ρ ) at 4º C, the minimum of the isothermal compressibility ( K T ) at 46 ºC, and a negative thermal expansion coefficient ( α P ) below 4º C[ 2 ]. Furthermore, in addition to the seeming divergence of these thermodynamic response functions, the LLCP would explain a similar apparent divergence of transport properties such as the viscosity, in deep supercooled water. The LLCP has been observed in molecular simulations of several water models[ 8 – 11 ] but since its estimated locus ( T ~ 170K-190 K; P ~ 0.17–0.19 GPa)[ 10 ] is already within the so-called “no man’s land” (101 K − 231 K), this poses serious experimental limitations. Several experiments attempted to overcome this kinetic limitation by investigating nanoconfined water[ 12 ] and water droplets[ 13 , 14 ], for which T h is below that of bulk water. However, no experimental study could yet refute or confirm the existence of a LLCP in liquid water. Among the dynamic anomalies of liquid water are the decrease of the viscosity, η , upon compression up to ~ 1 kbar, below ~ 10 ºC (see Fig. 4 of ref. [ 2 ]), and the increase of the self-diffusion, D , at low pressures, below room temperature[ 2 , 15 ]. Further, an increasingly decoupling of diffusion and viscosity, upon cooling below 350 K, at normal pressure, is observed, resulting in a breakdown of the Stokes-Einstein relationship (SER) [ 16 – 24 ]. The breakdown of the SER at low temperatures occurs because the viscosity increases faster than 1/ D . Thus, such a violation is magnified when the self-diffusion of low tetrahedrality populations is considered, since the latter exhibit a less pronounced decrease of the self-diffusion coefficient with the temperature[ 24 ]. Recently, three of us showed that the breakdown of SER can be quantitatively explained using the translational jump-diffusion (TJD) approach [ 22 , 25 – 30 ]. The TJD splits the total diffusion coefficient on a jump-diffusion term, emanating from the translational jump movements, and a residual diffusion term, the remaining part of the diffusion contributed by small step displacements. Interestingly, the residual diffusion remains almost strongly coupled to the viscosity in supercooled water. Lowering the temperature, pressure, or both, facilitates a dramatic growth of the contribution of jump-diffusion in supercooled water leading to a gradual collapse of the SER. Unlike supercooled water, superheated water behaves more like a simple liquid and the LLCP hypothesis should not have to be invoked to interpret its thermodynamic and structural properties. Nevertheless, the dynamics of superheated water at GPa pressure has been shown[ 31 ] to exhibit much larger deviations to the SER than those observed for supercooled water, at least, down to 240 K where both viscosity[ 16 ] and diffusion[ 32 , 33 ] data is available. Bove et al .[ 31 ], measured, through quasielastic incoherent neutron scattering, the translational and rotational diffusion coefficients of water along the 400 K isotherm up to 3 GPa, which is the melting point of ice VII. The translational diffusion D was found to strongly decrease with pressure. although the slowdown rate diminishes above 1 GPa. However, this reduction of D is slower by a factor of nearly two than the increase of viscosity η of water at the same pressures. Therefore, the product Dη increases with the pressure, violating the SER. This breakdown of the SER is particularly strong above 1 GPa and its relationship with the putative LLCP is not known. Herein, we performed molecular dynamics (MD) simulations of liquid water with several water models along the 400 K isotherm up to 3 GPa. We considered seven water models, namely, SPC/E[ 34 ] (rigid, nonpolarizable, three-site and three point-charge model), TIP3P-FB[ 35 ] (three-charge site model), TIP4P-FB[ 35 ] (four-charge site model), TIP5P[ 36 ] (five-charge site model), SWM4-NDP[ 37 ] (polarizable water model), TIP4P-Ew[ 38 ] (rigid nonpolarizable model with four sites and three-point charges), and TIP4P/2005[ 39 ] (rigid nonpolarizable model with four sites and three-point charges). The main goal of this work is twofold: to investigate whether the abovementioned water models can quantitatively predict the SER breakdown at high-temperatures and high-pressures and assess putative structural and thermodynamic anomalies in superheated water, that might accompany the SER violation. 2. Methods 2.1 Molecular Dynamics MD simulations were performed for 1500 water molecules in a cubic box with periodic boundary conditions, for the distinct water models (SPC/E, SWM4-NDP, TIP3P/FB, TIP4P/FB, TIP5P, TIP4P-Ew, and TIP4P/2005). The initial configuration was generated with the Packmol software[ 40 ], and the MD simulations were carried out with the GROMACS program[ 41 ]. The systems were simulated at 400 K and twelve different pressures: 0.01, 0.1, 0.2, 0.3, 0.5, 0.8, 1.2, 1.5, 1.9, 2.4, 2.7, and 3.0 GPa. Following steepest descent energy minimization and a 100 ps MD in the NVT ensemble, the systems were further equilibrated for 10 ns in the NPT ensemble. The trajectories were then propagated for 40 ns in the NPT ensemble. Electrostatic interactions were computed via the particle-mesh Ewald (PME) method[ 42 ]. A cut-off of 1 nm was used for non-bonded van der Waals and the PME real space electrostatic interactions. The temperature was controlled via the Nosé-Hoover thermostat [ 43 , 44 ] with a coupling constant of 0.5 ps. The pressure was controlled isotropically with the Parrinello-Rahman barostat [ 45 ] with a coupling constant of 1 ps. The Verlet- Leapfrog algorithm was used to solve the equation of motion with a time-step of 2 fs. 2.2. Dynamic Properties The diffusion coefficient was estimated from the mean square displacement through the Einstein relation[ 40 ], $$D=\frac{1}{6}\mathop {\lim }\limits_{{t \to \infty }} \frac{d}{{dt}}\left\langle {\sum\limits_{{i=1}}^{N} {{{\left| {{r_i}(t) - {r_i}(0)} \right|}^2}} } \right\rangle$$ 1 where the indicate an ensemble average of the mean square displacement (MSD), r i ( t ) and r i (0) are the positions of the i th oxygen atom at time t and the origin, respectively, and D is the diffusion coefficient. The shear viscosity, η, was calculated through integration of the stress (or pressure) tensor time correlation function ( \({P_{\alpha \beta }}\) )[ 46 , 47 ], $$\eta =\frac{V}{{10{k_B}T}}\int\limits_{0}^{\infty } {\left\langle {\sum\limits_{{\alpha \beta }}^{9} {{P_{\alpha \beta }}(0){P_{\alpha \beta }}(t)} } \right\rangle dt}$$ 2 where \({P_{\alpha \beta }}\) is the symmetrized traceless portion of the stress tensor, \({\sigma _{\alpha \beta }}\) , given by[ 48 ][ 42 ], $${P_{\alpha \beta }}=\frac{1}{2}\left( {{\sigma _{\alpha \beta }}+{\sigma _{\beta \alpha }}} \right) - \frac{1}{3}{\delta _{\alpha \beta }}\left( {\sum\limits_{\alpha } {{\sigma _{\alpha \alpha }}} } \right)$$ 3 \({\delta _{\alpha \beta }}\) is the Kronecker delta and there are six, out of the nine, distinct \({P_{\alpha \beta }}\) elements. This expression improves the statistics over the original Green-Kubo formula which involves the average over only the three different off-diagonal elements, \({\sigma _{\alpha \beta }}={\sigma _{\beta \alpha }}\) , of the stress tensor[ 49 ][ 40 ], $$\eta =\frac{V}{{3{k_B}T}}\int\limits_{0}^{\infty } {\left\langle {\sum\limits_{{\alpha \beta }}^{3} {{\sigma _{\alpha \beta }}(0){\sigma _{\alpha \beta }}(t)} } \right\rangle } dt$$ 4 also used here for comparison purposes; no significant differences were found between the viscosity calculated with Eqs. ( 2 ) and ( 4 ). For the definition of the stress tensor elements for periodic systems treated with the Ewald or the particle-mesh Ewald methods see refs [ 50 – 52 ]. 3. Results And Discussion 3.1. Thermodynamic Properties Various thermodynamic response functions, known to exhibit an abnormal behavior upon cooling, were calculated for the distinct models, to assess their pressure dependence in superheated water. These included the density ρ , the isothermal compressibility \({K}_{T}\) , and the thermal expansion coefficient \({\alpha }_{P}\) . The density ( \(\rho =m/V\) ) is the amount of mass ( m ) of a substance per unit volume ( V ) and is sensitive to temperature and pressure, especially for fluid systems. For simple liquids, a pressure increase, or temperature decrease, results in a volume reduction and, therefore, an increased density. The density of water, although increasing with pressure, decreases upon cooling below 4 ºC; hence, a volume increase is accompanied by an entropy decrease, a paradoxical behaviour not found in simple liquids. The isothermal compressibility, \({K_T}= - {(\partial \ln V/\partial P)_T}\) , provides a measure of the change of volume as a response to a change in pressure, at constant temperature. The thermal expansion coefficient, \({\alpha _P}={(\partial \ln V/\partial T)_P}\) , provides a measure of the change of volume of a fluid or solid as a response to a change in temperature, at constant pressure. These properties were calculated using the respective fluctuation formulas as discussed by Allen and Tildesley[ 53 ]. The pressure dependence of the above properties for the distinct water models is displayed in Fig. 1 and compared with the available experimental data[ 54 ]. The density of water increases monotonically (as expected) with the pressure increase (Fig. 1 a), consistent with the experimental trend. Moreover, the results are similar for the distinct models, except for SWM4-NDP and TIP5P, which predict, respectively, a lower and higher density at high pressures, compared to the experimental data. For the remaining five water models the density is in very good agreement with the experiments. For most models, ρ is almost coincident up to ~ 1 GPa. Thus, the density only diverges slightly, among the distinct water models, at high pressures (> 1 GPa), and an increase ~ 50% is found between the lowest and highest pressures studied, for most models. The isothermal compressibility, \({K}_{T}\) (Fig. 1 b), and thermal expansion coefficient, \({\alpha }_{P}\) (Fig. 1 c), decay monotonically with the pressure for all water models. The largest difference among the water models is found for TIP5P, which estimates significantly larger K T and α P , especially at low pressures. Except for TIP5P and SWM4-NDP the simulated \({\alpha }_{P}\) matches well the experimental data at low pressures, whereas at high pressures deviations are also observed for TIP4P-Ew and TIP4P-fb. The TIP3P-fb, TIP4P/2005, and SPC/E models outperform the remaining ones for the entire pressure range. For most models a decrease of K T and α P around 90% and 65%, respectively, is found. 3.2 Diffusion and Viscosity The self-diffusion coefficient was estimated from the MSD (see Eq. ( 1 )). The MSD comprises three characteristic domains: a sort-time ballistic regime ( \(\text{M}\text{S}\text{D}\propto {t}^{2}\) ), an intermediate-time sub-diffusive regime ( \(\text{M}\text{S}\text{D}\propto {t}^{\alpha }; 0 < {\alpha } < 1\) ), and a long-time linear diffusive regime (MSD \(\propto t\) ), from which the diffusion can be estimated (see Eq. ( 1 )). The self-diffusion and viscosity coefficients are displayed in Figs. 2 a and 2 b, respectively. The self-diffusion coefficient at pressures below ~ 2.5 GPa is consistent with the experimental data[ 31 ], except for the TIP5P water model. At higher pressures (> 2.5 GPa), a faster decay of the self-diffusion coefficient, compared with the experiments, can be seen. Opposite to D , the experimental viscosity coefficient[ 31 , 55 , 56 ] does not exhibit such a marked downtrend variation at high pressures; “experimental” viscosity data are the same used by Bove et al .[ 31 ] and has been interpolated from experimental data at 373 K and 437 K[ 56 ]. Again, except for TIP5P the simulation viscosity is in good agreement with the experimental data for every water model (see Fig. 2 b). The faster decay of D at high pressures, and the viscosity accord, compared with the experiments, indicates that model water should obey the SER more closely than real water. The SER, \(D\eta /T=c\) , where the constant \(c={k_B}/C\pi {r_s}\) ( \({k_B}\) is the Boltzmann constant, C is a constant, and \({r_s}\) is the hydrodynamic radius of the “molecule”), predicts a constant Dη product at constant temperature. Figure 2 c displays Dη , normalized by Dη at 0.01 GPa, as a function of pressure for all water models. As can be seen the simulated Dη values slightly increase with the pressure, indicating an increasing diffusion − viscosity decoupling and, therefore, a breakdown of the SER. However, the magnitude of this violation is significantly lower than that found from the product of the experimental D and η . The experimental Dη increases slowly up to 1 GPa, after which a sharp increase can be seen up to ~ 2.5 GPa. Above 2.5 GPa an even steeper increase is observed, denoting a major decoupling between the diffusion and viscosity. Thus, although the water models predict the decoupling phenomenon, this is moderate compared to that found for real water for pressures above ~ 1 GPa. This possibly reveals the limitations of the water models to accurately capture the diffusion-viscosity decoupling of pressurized hot water. The reason for this behavior is mostly related with the prediction of a faster decay of the diffusion coefficient at high pressures, compared to real water. The latter, in turn, could be associated with a poor description of the structure of water at high temperatures and pressures. We now turn attention to the analysis of the structure of water. 3.3 Structure Bove et al. attributed the SER breakdown to the invariance of the number of hydrogen bonds (H-bonds) of water under pressure, associated with a rigid first coordination sphere, also based on MD simulations with the TIP4P/2005 water model[ 31 ]. Herein, we also calculated the average number of H-bonds per water molecule. A geometric H-bond definition was adopted: r ODOA < 3.5 Å, r OAHD < 2.45 Å, and the angle H D O D O A < 30°; where “D” and “A” denote the H-bond donor and acceptor, respectively. Figure 3 shows that the average number of geometric H-bonds ( n HB ) increases monotonically with the pressure. A relatively rapid increase is observed at low pressures whereas a moderate increase occurs at high pressures. This is consistent with the decrease of the self-diffusion coefficient. The fact that n HB does not plateau out explains the rapid decrease of the simulated self-diffusion coefficients at high pressures, compared to real water, resulting in a moderate violation of the SER. The n HB for all water models (except TIP5P) are nearly coincindent, implying only minor differences among these water models with respect to H-bond formation. The TIP5P water model, however, predicts a much lower number of H-bonds compared to the other models. However, a larger growth upon compression is observed. Thus, the increase in the number of H-bond in moving from 0.001 to 3 GPa is about 30% for TIP5P, while for the other water models is ~ 12%. This is consistent with the more steep decline of the self-diffusion coefficient of TIP5P water with the pressure. These results show that while H-bonds are gradually broken upon compression in supercooled water, a pressure increase in superheated water increases geometric H-bonding. A nearly constant average number of geometric H-bonds is found in pressurized polarizable model water at room temperature[ 57 ]. Further insight into the structure of water can be gauged from water’s partial radial distribution functions (rdfs). The rdf measures the probability of finding a particle at a distance r from some reference particle, revealing the local number density variation along the distance from the reference particle. The oxygen-oxygen, g OO ( r ), and oxygen-hydrogen, g OH ( r ), partial rdfs, are displayed in Figs. 4 and 5 , respectively. Figure 4 shows that as pressure increases up to ~ 0.5 GPa, the second coordination sphere vanishes, with a new one starting to form at a higher distance upon further compression. This is accompanied by a broadening of the first coordination sphere. As the second coordination sphere becomes more and more prominent upon compression, it also starts shifting to shorter distances. A compression of the first coordination sphere concerning the position of the first minimum can also be seen above 0.5 GPa. The depth of the first minimum, in turn, starts to increase (i.e., deeper minimum) significantly above 0.5 GPa denoting a lower diffusion coefficient as water molecules are found less frequently crossing between the first and second coordination spheres. The g OH ( r ) (see Fig. 5 ) shows a first peak around 0.18 nm, corresponding to the nearest H atoms of neighbouring water molecules, which can form H-bonds with the reference O atom, and a second peak at around 0.32 nm. The first and second peaks undergo a shift to lower distances upon compression. However, while the height of the first peak slightly decreases that of the second increases. Moreover, the depth of the first minimum decreases (i.e., less deep) while that of the second exhibits the opposite behaviour. The concomitant decrease of the height of the first peak and depth of the first minimum for all water models, explain the near constancy of the number of H neighbours across the low- and high-pressure ranges (see Fig. 6 ). The emergence of a clear third peak is also visible at high pressures, again revealing some resemblances with the structure of liquid water upon cooling. The first shell coordination number (CN) of water (O-O and O-H), calculated by integrating the radial distribution functions are shown in Fig. 6 . Clearly there is a sharp increase of CN as the pressure increases to 0.5 GPa, which is caused by the sudden shift to longer distances of the first minimum in the rdf. However, the O-H coordination number remains almost insensitive to the pressure, for the reasons previously noted. We have also calculated the tetrahedral order parameter q for the TIP4P/2005 model at all pressures using the following equation[ 58 , 59 ] . $$q=1-\frac{3}{8}\sum _{j=1}^{3}\sum _{k=j+1}^{4}{\left({cos}{\theta }_{jk}+\frac{1}{3}\right)}^{2}$$ 5 Figure 7 exhibits p ( q ), the distributions of q , at six representative pressures. At 0.01 GPa the tetrahedral order distribution peaks at around q ~ 0.48 and exhibits a shoulder around q ~ 0.7. With increasing pressure, the peak height increases, and the shoulder gradually vanishes such that the shoulder of the distribution is almost invisible at the highest pressure P = 3 GPa. Figure 7 b plots the average tetrahedral order parameter as a function of pressure. The tetrahedrality remains almost constant up to 0.5 GPa followed by a steep decrease upon further compression. Therefore, although the number of H-bonds increases with increasing pressure, the tetrahedrality decreases, indicating the formation of more strained and possibly higher energy H-bonds. These structural changes should impact the diffusion-viscosity decoupling of highly pressurized water at 400 K, possibly inducing a faster decline of the diffusion coefficient, compared to the experiments. 4. Conclusions Liquid water remains one of the most studied liquids both because of its importance to multiple chemical processes and its peculiar behavior, especially in supercooled metastable conditions. Opposite to supercooled water, superheated water behaves more like a simple liquid. Nonetheless, experimental neutron diffraction experiments revealed a large decoupling of the viscosity and diffusion upon compression. The possible connection between this decoupling and the putative second critical point of liquid water is unknown. Herein, to gain insight into the behaviour of superheated model water, molecular dynamics simulations at 400 K up to 3 GPa were performed with seven water models: SPC/E, TIP3P-FB, TIP4P-FB, TIP5P, SWM4-NDP, TIP4P-Ew, and TIP4P/2005. Our results show that the TIP5P water model predicts the most disparate results among the models investigated, seemingly associated with a lower number of H-bonds. This model predicts a higher isothermal compressibility and thermal expansion coefficient as well as a lower viscosity and a higher self-diffusion coefficient, especially at low pressures. Upon compression the thermodynamic response functions and transport coefficients of TIP5P water become less dissimilar from the remaining water models. This is linked with a steeper increase of the number of the H-bonds with pressure, making TIP5P water’s H-bond network more similar to that of the remaining models, at high pressures. Although the number of H-bonds increases, the tetrahedrality of the H-bond network decreases with increasing pressure indicating a more strained network. The distinct water models estimate a more moderate violation of the SER, than that found by Bove et al .[ 31 ]. Thus, a diffusion-viscosity decoupling similar to that found in real supercooled water, down to 242 K, is observed. The reason is that water models predict a more pronounced decrease of the self-diffusion coefficient at high pressures, than experiments. This suggests that water models commonly used to study liquid water, including metastable supercooled water, cannot accurately describe the structural and dynamic responses to compression, of superheated water. This problem should be revisited using other water models and/or ab initio MD to gain further insight into the structural source of the strong diffusion-viscosity decoupling observed in real superheated water. Declarations Acknowledgements This paper is dedicated to Prof. Pratim K. Chattaraj, Professor, IIT Kharagpur, who has been the source of inspiration for us. We respectfully acknowledge his seminal contribution in the broad area of theoretical chemistry. Shivam, Archita, and Vikas acknowledge IIT Patna for their fellowships. N. G. acknowledges the work support by UIDB/04046/2020 and UIDP/04046/2020 centre grants from FCT, Portugal (to BioISI), by the Portuguese National Distributed Computing Infrastructure (http://www.incd.pt). S. D. acknowledges computational facility from IIT Patna. Author contributions S.D. contributed to the methodology, analysis, and writing original draft. A. M. contributed to the analysis, and writing original draft, V.D. contributed to the methodology and analysis, N.G. contributed to the conceptualization, methodology, analysis, and writing original draft. 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Doi: 10.1103/PhysRevB.49.755 N Galamba, CA Nieto de Castro, JF Ely (2004) The Journal of Physical Chemistry B 108: 3658. Doi: 10.1021/jp036234x MP Allen, DJ Tildesley (2017) Computer simulation of liquids. Oxford university press, S Wiryana, LJ Slutsky, JM Brown (1998) Earth and Planetary Science Letters 163: 123. EH Abramson, JM Brown (2004) Geochimica et Cosmochimica Acta 68: 1827. EH Abramson (2007) Physical Review E 76: 051203. N Galamba (2013) The Journal of Physical Chemistry B 117: 589. Doi: 10.1021/jp309312q PL Chau, AJ Hardwick (1998) Molecular Physics 93: 511. Doi: 10.1080/002689798169195 JR Errington, PG Debenedetti (2001) Nature 409: 318. Doi: 10.1038/35053024 Additional Declarations No competing interests reported. Supplementary Files TOCGraphic.png Cite Share Download PDF Status: Published Journal Publication published 20 Apr, 2023 Read the published version in Theoretical Chemistry Accounts → Version 1 posted Editorial decision: Major revision 06 Apr, 2023 Reviews received at journal 06 Apr, 2023 Reviewers agreed at journal 05 Apr, 2023 Reviewers invited by journal 02 Jan, 2023 Editor assigned by journal 28 Dec, 2022 Submission checks completed at journal 28 Dec, 2022 First submitted to journal 27 Dec, 2022 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-2419984","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":163309387,"identity":"72ae40e0-4a8a-419f-af86-9435e206476d","order_by":0,"name":"Shivam Dueby","email":"","orcid":"","institution":"Indian Institute of Technology Patna","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Shivam","middleName":"","lastName":"Dueby","suffix":""},{"id":163309389,"identity":"6ce1f4c4-a537-417c-9ff3-e72990f06416","order_by":1,"name":"Archita Maiti","email":"","orcid":"","institution":"Indian Institute of Technology Patna","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Archita","middleName":"","lastName":"Maiti","suffix":""},{"id":163309390,"identity":"cecccff0-0c45-40a7-9bed-4b439167a916","order_by":2,"name":"Vikas Dubey","email":"","orcid":"","institution":"Indian Institute of Technology Patna","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Vikas","middleName":"","lastName":"Dubey","suffix":""},{"id":163309392,"identity":"33dd2f24-ce95-4dfb-812d-1dd37f326679","order_by":3,"name":"Nuno Galamba","email":"","orcid":"","institution":"University of Lisbon","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Nuno","middleName":"","lastName":"Galamba","suffix":""},{"id":163309395,"identity":"8525ce9d-3481-4942-958c-13665e377641","order_by":4,"name":"Snehasis Daschakraborty","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA00lEQVRIiWNgGAWjYHACNiCWYOBnZmyA8JmJ1SLZzNjYQIoWBgaDAwwwawgA+dnNxx78+GNhb3ycuf1xAYOdPAM77wG8WgzuHEs37G2TSNx2mLGxeQZDsmEDM18Cfi0SOWYSvA0SCWYgLTwMzAkMzDwG+B02I/+b5J8/EvbGzWAt9YS1MNzIYZPmYZNg3MAM1nKYsBaDG2lm0rJAv8wAOmz2DIPjhm2EHZb8TPLNnzp7/v7jDz4XVFTL8/OfIeAwZMDMYACLJuK1jIJRMApGwSjAAgB8JzlKQ91/ZwAAAABJRU5ErkJggg==","orcid":"","institution":"Indian Institute of Technology Patna","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Snehasis","middleName":"","lastName":"Daschakraborty","suffix":""}],"badges":[],"createdAt":"2022-12-27 20:14:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2419984/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2419984/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00214-023-02991-0","type":"published","date":"2023-04-20T20:31:48+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":31106055,"identity":"098d32c5-7ae5-4d52-a380-8b2cf18d1bb8","added_by":"auto","created_at":"2023-01-04 17:15:47","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":36881,"visible":true,"origin":"","legend":"\u003cp\u003eSee mage above for figure legend.\u003c/p\u003e","description":"","filename":"Figure1.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/dc0d84b04b2d040660ae9fd8.png"},{"id":31105155,"identity":"a779f035-4779-4b44-bb2e-7798e523939b","added_by":"auto","created_at":"2023-01-04 16:59:47","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":33351,"visible":true,"origin":"","legend":"\u003cp\u003eSee mage above for figure legend.\u003c/p\u003e","description":"","filename":"Figure2.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/5b6dd048d782ed18959912c9.png"},{"id":31106054,"identity":"02a750a7-bf36-4314-bfd0-40304d3b9b58","added_by":"auto","created_at":"2023-01-04 17:15:47","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":20018,"visible":true,"origin":"","legend":"\u003cp\u003eAverage number of hydrogen bonds/molecule at distinct pressures for the different water models\u003c/p\u003e","description":"","filename":"Figure3.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/820fe3b1e1b3d2b478146756.png"},{"id":31105159,"identity":"ce1e4f9d-54b9-4c4b-a547-d6cd77f274dd","added_by":"auto","created_at":"2023-01-04 16:59:47","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":32468,"visible":true,"origin":"","legend":"\u003cp\u003eRadial distribution function (OW-OW), \u003cem\u003eg\u003c/em\u003e(\u003cem\u003er\u003c/em\u003e), for the distinct water models at 400 K and six representative pressures (0.01, 0.2, 0.5, 1.2, 1.9, and 2.7 GPa).\u003c/p\u003e","description":"","filename":"Figure4.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/7d6b287bf63b234e0aa38662.png"},{"id":31105693,"identity":"cd690412-0f83-48c6-90e7-882dd126f366","added_by":"auto","created_at":"2023-01-04 17:07:47","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":30699,"visible":true,"origin":"","legend":"\u003cp\u003eRadial distribution function (OW-HW), \u003cem\u003eg\u003c/em\u003e(\u003cem\u003er\u003c/em\u003e), for the distinct water models at 400 K and six representative pressures (0.01, 0.2, 0.5, 1.2, 1.9, and 2.7 GPa).\u003c/p\u003e","description":"","filename":"Figure5.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/c0c4ae1b020ff495c6782f6a.png"},{"id":31105160,"identity":"11ee2f86-c3a6-4e16-916e-b5609ac5d494","added_by":"auto","created_at":"2023-01-04 16:59:47","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":19724,"visible":true,"origin":"","legend":"\u003cp\u003eCoordination Number (CN) (a) (OW-OW), and (b) (OW-HW), for the distinct water models at 400 K and six representative pressures (0.01, 0.2, 0.5, 1.2, 1.9, and 2.7 GPa).\u003c/p\u003e","description":"","filename":"Figure6.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/46f1da86b861822404031e7c.png"},{"id":31105694,"identity":"0dffd234-c6d8-4801-a35b-4bd86a2ff3fe","added_by":"auto","created_at":"2023-01-04 17:07:47","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":17536,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Distributions of tetrahedral order parameter \u003cem\u003ep\u003c/em\u003e(\u003cem\u003eq\u003c/em\u003e) for TIP4P/2005 water models and (b) average value of tetrahedral order parameter \u0026lt;\u003cem\u003eq\u003c/em\u003e\u0026gt; as a function of pressure.\u003c/p\u003e","description":"","filename":"Figure7.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/8dabac10919e2f8e4807784a.png"},{"id":44726135,"identity":"55646d47-9146-4fb8-80c9-c6bf97fbdbbd","added_by":"auto","created_at":"2023-10-16 20:45:26","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":444728,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/b37b5192-8fcb-4062-a408-53503449a3ec.pdf"},{"id":31105690,"identity":"0df7ad5d-4557-4256-bab7-515dab133df8","added_by":"auto","created_at":"2023-01-04 17:07:47","extension":"png","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":50043,"visible":true,"origin":"","legend":"","description":"","filename":"TOCGraphic.png","url":"https://assets-eu.researchsquare.com/files/rs-2419984/v1/8a338ff89bc13b83956f1e64.png"}],"financialInterests":"No competing interests reported.","formattedTitle":"Thermodynamic Response Functions and Stokes-Einstein Breakdown in Superheated Water under Gigapascal Pressure","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eLiquid water exhibits numerous thermodynamic and dynamic anomalies, most of which intensify upon cooling below the freezing point [\u003cspan additionalcitationids=\"CR2\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Among the most prominent anomalies of liquid water at normal pressure are the increase of the magnitude of its response functions upon cooling [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. These anomalies, related to an increase of the entropy and volume fluctuations at low temperatures, opposite to simple liquids, prevent yet, a comprehensive understanding of liquid water, including its temperature and pressure dependence [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eSeveral explanations were put forward to provide a coherent picture of water\u0026rsquo;s anomalous behaviour. These include the \u0026lsquo;\u0026lsquo;stability limit conjecture\u0026rsquo;\u0026rsquo; (SLC)[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], the singularity-free hypothesis[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] or the liquid-liquid critical point (LLCP)[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. The LLCP picture, the most widely investigated, foresees the co-existence of high-density and low-density liquid (HDL/LDL) forms of water below a second critical point located already below the homogeneous nucleation temperature (\u003cem\u003eT\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e ~232 K). The existence of HDL and LDL forms of liquid water would then relate to the abovementioned anomalies, including the temperature of maximum density (\u003cem\u003eρ\u003c/em\u003e) at 4\u0026ordm; C, the minimum of the isothermal compressibility (\u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sub\u003e) at 46 \u0026ordm;C, and a negative thermal expansion coefficient (\u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003eP\u003c/em\u003e\u003c/sub\u003e) below 4\u0026ordm; C[\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Furthermore, in addition to the seeming divergence of these thermodynamic response functions, the LLCP would explain a similar apparent divergence of transport properties such as the viscosity, in deep supercooled water.\u003c/p\u003e \u003cp\u003eThe LLCP has been observed in molecular simulations of several water models[\u003cspan additionalcitationids=\"CR9 CR10\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] but since its estimated locus ( \u003cem\u003eT\u003c/em\u003e\u0026thinsp;~\u0026thinsp;170K-190 K; \u003cem\u003eP\u003c/em\u003e\u0026thinsp;~\u0026thinsp;0.17\u0026ndash;0.19 GPa)[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] is already within the so-called \u0026ldquo;no man\u0026rsquo;s land\u0026rdquo; (101 K \u0026minus;\u0026thinsp;231 K), this poses serious experimental limitations. Several experiments attempted to overcome this kinetic limitation by investigating nanoconfined water[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] and water droplets[\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], for which \u003cem\u003eT\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e is below that of bulk water. However, no experimental study could yet refute or confirm the existence of a LLCP in liquid water.\u003c/p\u003e \u003cp\u003eAmong the dynamic anomalies of liquid water are the decrease of the viscosity, \u003cem\u003eη\u003c/em\u003e, upon compression up to ~\u0026thinsp;1 kbar, below ~\u0026thinsp;10 \u0026ordm;C (see Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e of ref. [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]), and the increase of the self-diffusion, \u003cem\u003eD\u003c/em\u003e, at low pressures, below room temperature[\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. Further, an increasingly decoupling of diffusion and viscosity, upon cooling below 350 K, at normal pressure, is observed, resulting in a breakdown of the Stokes-Einstein relationship (SER) [\u003cspan additionalcitationids=\"CR17 CR18 CR19 CR20 CR21 CR22 CR23\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. The breakdown of the SER at low temperatures occurs because the viscosity increases faster than 1/\u003cem\u003eD\u003c/em\u003e. Thus, such a violation is magnified when the self-diffusion of low tetrahedrality populations is considered, since the latter exhibit a less pronounced decrease of the self-diffusion coefficient with the temperature[\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eRecently, three of us showed that the breakdown of SER can be quantitatively explained using the translational jump-diffusion (TJD) approach [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan additionalcitationids=\"CR26 CR27 CR28 CR29\" citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. The TJD splits the total diffusion coefficient on a jump-diffusion term, emanating from the translational jump movements, and a residual diffusion term, the remaining part of the diffusion contributed by small step displacements. Interestingly, the residual diffusion remains almost strongly coupled to the viscosity in supercooled water. Lowering the temperature, pressure, or both, facilitates a dramatic growth of the contribution of jump-diffusion in supercooled water leading to a gradual collapse of the SER.\u003c/p\u003e \u003cp\u003eUnlike supercooled water, superheated water behaves more like a simple liquid and the LLCP hypothesis should not have to be invoked to interpret its thermodynamic and structural properties. Nevertheless, the dynamics of superheated water at GPa pressure has been shown[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e] to exhibit much larger deviations to the SER than those observed for supercooled water, at least, down to 240 K where both viscosity[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] and diffusion[\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e] data is available. Bove \u003cem\u003eet al\u003c/em\u003e.[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], measured, through quasielastic incoherent neutron scattering, the translational and rotational diffusion coefficients of water along the 400 K isotherm up to 3 GPa, which is the melting point of ice VII. The translational diffusion \u003cem\u003eD\u003c/em\u003e was found to strongly decrease with pressure. although the slowdown rate diminishes above 1 GPa. However, this reduction of \u003cem\u003eD\u003c/em\u003e is slower by a factor of nearly two than the increase of viscosity \u003cem\u003eη\u003c/em\u003e of water at the same pressures. Therefore, the product \u003cem\u003eDη\u003c/em\u003e increases with the pressure, violating the SER. This breakdown of the SER is particularly strong above 1 GPa and its relationship with the putative LLCP is not known.\u003c/p\u003e \u003cp\u003eHerein, we performed molecular dynamics (MD) simulations of liquid water with several water models along the 400 K isotherm up to 3 GPa. We considered seven water models, namely, SPC/E[\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e] (rigid, nonpolarizable, three-site and three point-charge model), TIP3P-FB[\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] (three-charge site model), TIP4P-FB[\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] (four-charge site model), TIP5P[\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e] (five-charge site model), SWM4-NDP[\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e] (polarizable water model), TIP4P-Ew[\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e] (rigid nonpolarizable model with four sites and three-point charges), and TIP4P/2005[\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e] (rigid nonpolarizable model with four sites and three-point charges).\u003c/p\u003e \u003cp\u003eThe main goal of this work is twofold: to investigate whether the abovementioned water models can quantitatively predict the SER breakdown at high-temperatures and high-pressures and assess putative structural and thermodynamic anomalies in superheated water, that might accompany the SER violation.\u003c/p\u003e"},{"header":"2. Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Molecular Dynamics\u003c/h2\u003e \u003cp\u003eMD simulations were performed for 1500 water molecules in a cubic box with periodic boundary conditions, for the distinct water models (SPC/E, SWM4-NDP, TIP3P/FB, TIP4P/FB, TIP5P, TIP4P-Ew, and TIP4P/2005). The initial configuration was generated with the Packmol software[\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e], and the MD simulations were carried out with the GROMACS program[\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e]. The systems were simulated at 400 K and twelve different pressures: 0.01, 0.1, 0.2, 0.3, 0.5, 0.8, 1.2, 1.5, 1.9, 2.4, 2.7, and 3.0 GPa. Following steepest descent energy minimization and a 100 ps MD in the \u003cem\u003eNVT\u003c/em\u003e ensemble, the systems were further equilibrated for 10 ns in the \u003cem\u003eNPT\u003c/em\u003e ensemble. The trajectories were then propagated for 40 ns in the \u003cem\u003eNPT\u003c/em\u003e ensemble. Electrostatic interactions were computed via the particle-mesh Ewald (PME) method[\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. A cut-off of 1 nm was used for non-bonded van der Waals and the PME real space electrostatic interactions. The temperature was controlled via the Nos\u0026eacute;-Hoover thermostat [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e, \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e] with a coupling constant of 0.5 ps. The pressure was controlled isotropically with the Parrinello-Rahman barostat [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e] with a coupling constant of 1 ps. The Verlet- Leapfrog algorithm was used to solve the equation of motion with a time-step of 2 fs.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Dynamic Properties\u003c/h2\u003e \u003cp\u003eThe diffusion coefficient was estimated from the mean square displacement through the Einstein relation[\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e],\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$D=\\frac{1}{6}\\mathop {\\lim }\\limits_{{t \\to \\infty }} \\frac{d}{{dt}}\\left\\langle {\\sum\\limits_{{i=1}}^{N} {{{\\left| {{r_i}(t) - {r_i}(0)} \\right|}^2}} } \\right\\rangle$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere the \u0026lt;\u0026thinsp;\u0026gt;\u0026thinsp;indicate an ensemble average of the mean square displacement (MSD), \u003cem\u003er\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) and \u003cem\u003er\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(0) are the positions of the \u003cem\u003ei\u003c/em\u003eth oxygen atom at time \u003cem\u003et\u003c/em\u003e and the origin, respectively, and \u003cem\u003eD\u003c/em\u003e is the diffusion coefficient.\u003c/p\u003e \u003cp\u003eThe shear viscosity, η, was calculated through integration of the stress (or pressure) tensor time correlation function (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P_{\\alpha \\beta }}\\)\u003c/span\u003e\u003c/span\u003e)[\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e, \u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e],\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\eta =\\frac{V}{{10{k_B}T}}\\int\\limits_{0}^{\\infty } {\\left\\langle {\\sum\\limits_{{\\alpha \\beta }}^{9} {{P_{\\alpha \\beta }}(0){P_{\\alpha \\beta }}(t)} } \\right\\rangle dt}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P_{\\alpha \\beta }}\\)\u003c/span\u003e\u003c/span\u003e is the symmetrized traceless portion of the stress tensor, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma _{\\alpha \\beta }}\\)\u003c/span\u003e\u003c/span\u003e, given by[\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e][\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e],\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${P_{\\alpha \\beta }}=\\frac{1}{2}\\left( {{\\sigma _{\\alpha \\beta }}+{\\sigma _{\\beta \\alpha }}} \\right) - \\frac{1}{3}{\\delta _{\\alpha \\beta }}\\left( {\\sum\\limits_{\\alpha } {{\\sigma _{\\alpha \\alpha }}} } \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\delta _{\\alpha \\beta }}\\)\u003c/span\u003e \u003c/span\u003e is the Kronecker delta and there are six, out of the nine, distinct \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P_{\\alpha \\beta }}\\)\u003c/span\u003e\u003c/span\u003e elements. This expression improves the statistics over the original Green-Kubo formula which involves the average over only the three different off-diagonal elements, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma _{\\alpha \\beta }}={\\sigma _{\\beta \\alpha }}\\)\u003c/span\u003e\u003c/span\u003e, of the stress tensor[\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e][\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e],\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\eta =\\frac{V}{{3{k_B}T}}\\int\\limits_{0}^{\\infty } {\\left\\langle {\\sum\\limits_{{\\alpha \\beta }}^{3} {{\\sigma _{\\alpha \\beta }}(0){\\sigma _{\\alpha \\beta }}(t)} } \\right\\rangle } dt$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ealso used here for comparison purposes; no significant differences were found between the viscosity calculated with Eqs.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and (\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e). For the definition of the stress tensor elements for periodic systems treated with the Ewald or the particle-mesh Ewald methods see refs [\u003cspan additionalcitationids=\"CR51\" citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results And Discussion","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Thermodynamic Properties\u003c/h2\u003e \u003cp\u003eVarious thermodynamic response functions, known to exhibit an abnormal behavior upon cooling, were calculated for the distinct models, to assess their pressure dependence in superheated water. These included the density \u003cem\u003eρ\u003c/em\u003e, the isothermal compressibility \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{T}\\)\u003c/span\u003e\u003c/span\u003e, and the thermal expansion coefficient \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{P}\\)\u003c/span\u003e\u003c/span\u003e. The density (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho =m/V\\)\u003c/span\u003e\u003c/span\u003e) is the amount of mass (\u003cem\u003em\u003c/em\u003e) of a substance per unit volume (\u003cem\u003eV\u003c/em\u003e) and is sensitive to temperature and pressure, especially for fluid systems. For simple liquids, a pressure increase, or temperature decrease, results in a volume reduction and, therefore, an increased density. The density of water, although increasing with pressure, decreases upon cooling below 4 \u0026ordm;C; hence, a volume increase is accompanied by an entropy decrease, a paradoxical behaviour not found in simple liquids. The isothermal compressibility,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_T}= - {(\\partial \\ln V/\\partial P)_T}\\)\u003c/span\u003e\u003c/span\u003e, provides a measure of the change of volume as a response to a change in pressure, at constant temperature. The thermal expansion coefficient, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha _P}={(\\partial \\ln V/\\partial T)_P}\\)\u003c/span\u003e\u003c/span\u003e, provides a measure of the change of volume of a fluid or solid as a response to a change in temperature, at constant pressure. These properties were calculated using the respective fluctuation formulas as discussed by Allen and Tildesley[\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe pressure dependence of the above properties for the distinct water models is displayed in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and compared with the available experimental data[\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e]. The density of water increases monotonically (as expected) with the pressure increase (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea), consistent with the experimental trend. Moreover, the results are similar for the distinct models, except for SWM4-NDP and TIP5P, which predict, respectively, a lower and higher density at high pressures, compared to the experimental data. For the remaining five water models the density is in very good agreement with the experiments. For most models, \u003cem\u003eρ\u003c/em\u003e is almost coincident up to ~\u0026thinsp;1 GPa. Thus, the density only diverges slightly, among the distinct water models, at high pressures (\u0026gt;\u0026thinsp;1 GPa), and an increase\u0026thinsp;~\u0026thinsp;50% is found between the lowest and highest pressures studied, for most models.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe isothermal compressibility, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{T}\\)\u003c/span\u003e\u003c/span\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb), and thermal expansion coefficient, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{P}\\)\u003c/span\u003e\u003c/span\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ec), decay monotonically with the pressure for all water models. The largest difference among the water models is found for TIP5P, which estimates significantly larger \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sub\u003e and α\u003csub\u003e\u003cem\u003eP\u003c/em\u003e\u003c/sub\u003e, especially at low pressures. Except for TIP5P and SWM4-NDP the simulated \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{P}\\)\u003c/span\u003e\u003c/span\u003e matches well the experimental data at low pressures, whereas at high pressures deviations are also observed for TIP4P-Ew and TIP4P-fb. The TIP3P-fb, TIP4P/2005, and SPC/E models outperform the remaining ones for the entire pressure range. For most models a decrease of \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003eP\u003c/em\u003e\u003c/sub\u003e around 90% and 65%, respectively, is found.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Diffusion and Viscosity\u003c/h2\u003e \u003cp\u003eThe self-diffusion coefficient was estimated from the MSD (see Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e)). The MSD comprises three characteristic domains: a sort-time ballistic regime (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{M}\\text{S}\\text{D}\\propto {t}^{2}\\)\u003c/span\u003e\u003c/span\u003e), an intermediate-time sub-diffusive regime (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{M}\\text{S}\\text{D}\\propto {t}^{\\alpha }; 0 \u0026lt; {\\alpha } \u0026lt; 1\\)\u003c/span\u003e\u003c/span\u003e), and a long-time linear diffusive regime (MSD\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\propto t\\)\u003c/span\u003e\u003c/span\u003e), from which the diffusion can be estimated (see Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e)).\u003c/p\u003e \u003cp\u003eThe self-diffusion and viscosity coefficients are displayed in Figs.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea and \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb, respectively. The self-diffusion coefficient at pressures below ~\u0026thinsp;2.5 GPa is consistent with the experimental data[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], except for the TIP5P water model. At higher pressures (\u0026gt;\u0026thinsp;2.5 GPa), a faster decay of the self-diffusion coefficient, compared with the experiments, can be seen. Opposite to \u003cem\u003eD\u003c/em\u003e, the experimental viscosity coefficient[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e, \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e, \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e] does not exhibit such a marked downtrend variation at high pressures; \u0026ldquo;experimental\u0026rdquo; viscosity data are the same used by Bove \u003cem\u003eet al\u003c/em\u003e.[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e] and has been interpolated from experimental data at 373 K and 437 K[\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e]. Again, except for TIP5P the simulation viscosity is in good agreement with the experimental data for every water model (see Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb). The faster decay of \u003cem\u003eD\u003c/em\u003e at high pressures, and the viscosity accord, compared with the experiments, indicates that model water should obey the SER more closely than real water.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe SER, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(D\\eta /T=c\\)\u003c/span\u003e\u003c/span\u003e, where the constant \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(c={k_B}/C\\pi {r_s}\\)\u003c/span\u003e\u003c/span\u003e (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k_B}\\)\u003c/span\u003e\u003c/span\u003e is the Boltzmann constant, \u003cem\u003eC\u003c/em\u003e is a constant, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({r_s}\\)\u003c/span\u003e\u003c/span\u003eis the hydrodynamic radius of the \u0026ldquo;molecule\u0026rdquo;), predicts a constant \u003cem\u003eDη\u003c/em\u003e product at constant temperature. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ec displays \u003cem\u003eDη\u003c/em\u003e, normalized by \u003cem\u003eDη\u003c/em\u003e at 0.01 GPa, as a function of pressure for all water models. As can be seen the simulated \u003cem\u003eDη\u003c/em\u003e values slightly increase with the pressure, indicating an increasing diffusion\u0026thinsp;\u0026minus;\u0026thinsp;viscosity decoupling and, therefore, a breakdown of the SER. However, the magnitude of this violation is significantly lower than that found from the product of the experimental \u003cem\u003eD\u003c/em\u003e and \u003cem\u003eη\u003c/em\u003e. The experimental \u003cem\u003eDη\u003c/em\u003e increases slowly up to 1 GPa, after which a sharp increase can be seen up to ~\u0026thinsp;2.5 GPa. Above 2.5 GPa an even steeper increase is observed, denoting a major decoupling between the diffusion and viscosity. Thus, although the water models predict the decoupling phenomenon, this is moderate compared to that found for real water for pressures above ~\u0026thinsp;1 GPa. This possibly reveals the limitations of the water models to accurately capture the diffusion-viscosity decoupling of pressurized hot water. The reason for this behavior is mostly related with the prediction of a faster decay of the diffusion coefficient at high pressures, compared to real water. The latter, in turn, could be associated with a poor description of the structure of water at high temperatures and pressures. We now turn attention to the analysis of the structure of water.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Structure\u003c/h2\u003e \u003cp\u003eBove \u003cem\u003eet al.\u003c/em\u003e attributed the SER breakdown to the invariance of the number of hydrogen bonds (H-bonds) of water under pressure, associated with a rigid first coordination sphere, also based on MD simulations with the TIP4P/2005 water model[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. Herein, we also calculated the average number of H-bonds per water molecule. A geometric H-bond definition was adopted: \u003cem\u003er\u003c/em\u003e\u003csub\u003eODOA\u003c/sub\u003e\u0026lt; 3.5 \u0026Aring;, \u003cem\u003er\u003c/em\u003e\u003csub\u003eOAHD\u003c/sub\u003e\u0026lt; 2.45 \u0026Aring;, and the angle H\u003csub\u003eD\u003c/sub\u003eO\u003csub\u003eD\u003c/sub\u003eO\u003csub\u003eA\u003c/sub\u003e\u0026lt; 30\u0026deg;; where \u0026ldquo;D\u0026rdquo; and \u0026ldquo;A\u0026rdquo; denote the H-bond donor and acceptor, respectively.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows that the average number of geometric H-bonds (\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003eHB\u003c/em\u003e\u003c/sub\u003e) increases monotonically with the pressure. A relatively rapid increase is observed at low pressures whereas a moderate increase occurs at high pressures. This is consistent with the decrease of the self-diffusion coefficient. The fact that \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003eHB\u003c/em\u003e\u003c/sub\u003e does not plateau out explains the rapid decrease of the simulated self-diffusion coefficients at high pressures, compared to real water, resulting in a moderate violation of the SER.\u003c/p\u003e \u003cp\u003eThe \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003eHB\u003c/em\u003e\u003c/sub\u003e for all water models (except TIP5P) are nearly coincindent, implying only minor differences among these water models with respect to H-bond formation. The TIP5P water model, however, predicts a much lower number of H-bonds compared to the other models. However, a larger growth upon compression is observed. Thus, the increase in the number of H-bond in moving from 0.001 to 3 GPa is about 30% for TIP5P, while for the other water models is ~\u0026thinsp;12%. This is consistent with the more steep decline of the self-diffusion coefficient of TIP5P water with the pressure.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThese results show that while H-bonds are gradually broken upon compression in supercooled water, a pressure increase in superheated water increases geometric H-bonding. A nearly constant average number of geometric H-bonds is found in pressurized polarizable model water at room temperature[\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFurther insight into the structure of water can be gauged from water\u0026rsquo;s partial radial distribution functions (rdfs). The rdf measures the probability of finding a particle at a distance \u003cem\u003er\u003c/em\u003e from some reference particle, revealing the local number density variation along the distance from the reference particle. The oxygen-oxygen, \u003cem\u003eg\u003c/em\u003e\u003csub\u003eOO\u003c/sub\u003e(\u003cem\u003er\u003c/em\u003e), and oxygen-hydrogen, \u003cem\u003eg\u003c/em\u003e\u003csub\u003eOH\u003c/sub\u003e(\u003cem\u003er\u003c/em\u003e), partial rdfs, are displayed in Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, respectively.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows that as pressure increases up to ~\u0026thinsp;0.5 GPa, the second coordination sphere vanishes, with a new one starting to form at a higher distance upon further compression. This is accompanied by a broadening of the first coordination sphere. As the second coordination sphere becomes more and more prominent upon compression, it also starts shifting to shorter distances. A compression of the first coordination sphere concerning the position of the first minimum can also be seen above 0.5 GPa. The depth of the first minimum, in turn, starts to increase (i.e., deeper minimum) significantly above 0.5 GPa denoting a lower diffusion coefficient as water molecules are found less frequently crossing between the first and second coordination spheres.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe \u003cem\u003eg\u003c/em\u003e\u003csub\u003eOH\u003c/sub\u003e(\u003cem\u003er\u003c/em\u003e) (see Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) shows a first peak around 0.18 nm, corresponding to the nearest H atoms of neighbouring water molecules, which can form H-bonds with the reference O atom, and a second peak at around 0.32 nm. The first and second peaks undergo a shift to lower distances upon compression. However, while the height of the first peak slightly decreases that of the second increases. Moreover, the depth of the first minimum decreases (i.e., less deep) while that of the second exhibits the opposite behaviour.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe concomitant decrease of the height of the first peak and depth of the first minimum for all water models, explain the near constancy of the number of H neighbours across the low- and high-pressure ranges (see Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e). The emergence of a clear third peak is also visible at high pressures, again revealing some resemblances with the structure of liquid water upon cooling. The first shell coordination number (CN) of water (O-O and O-H), calculated by integrating the radial distribution functions are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Clearly there is a sharp increase of CN as the pressure increases to 0.5 GPa, which is caused by the sudden shift to longer distances of the first minimum in the rdf. However, the O-H coordination number remains almost insensitive to the pressure, for the reasons previously noted.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWe have also calculated the tetrahedral order parameter \u003cem\u003eq\u003c/em\u003e for the TIP4P/2005 model at all pressures using the following equation[\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e, \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e] .\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$q=1-\\frac{3}{8}\\sum _{j=1}^{3}\\sum _{k=j+1}^{4}{\\left({cos}{\\theta }_{jk}+\\frac{1}{3}\\right)}^{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e exhibits \u003cem\u003ep\u003c/em\u003e(\u003cem\u003eq\u003c/em\u003e), the distributions of \u003cem\u003eq\u003c/em\u003e, at six representative pressures. At 0.01 GPa the tetrahedral order distribution peaks at around \u003cem\u003eq\u003c/em\u003e\u0026thinsp;~\u0026thinsp;0.48 and exhibits a shoulder around \u003cem\u003eq\u003c/em\u003e\u0026thinsp;~\u0026thinsp;0.7. With increasing pressure, the peak height increases, and the shoulder gradually vanishes such that the shoulder of the distribution is almost invisible at the highest pressure \u003cem\u003eP\u003c/em\u003e\u0026thinsp;=\u0026thinsp;3 GPa. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003eb plots the average tetrahedral order parameter\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eq\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;as a function of pressure. The tetrahedrality\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eq\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;remains almost constant up to 0.5 GPa followed by a steep decrease upon further compression. Therefore, although the number of H-bonds increases with increasing pressure, the tetrahedrality decreases, indicating the formation of more strained and possibly higher energy H-bonds. These structural changes should impact the diffusion-viscosity decoupling of highly pressurized water at 400 K, possibly inducing a faster decline of the diffusion coefficient, compared to the experiments.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003eLiquid water remains one of the most studied liquids both because of its importance to multiple chemical processes and its peculiar behavior, especially in supercooled metastable conditions. Opposite to supercooled water, superheated water behaves more like a simple liquid. Nonetheless, experimental neutron diffraction experiments revealed a large decoupling of the viscosity and diffusion upon compression. The possible connection between this decoupling and the putative second critical point of liquid water is unknown. Herein, to gain insight into the behaviour of superheated model water, molecular dynamics simulations at 400 K up to 3 GPa were performed with seven water models: SPC/E, TIP3P-FB, TIP4P-FB, TIP5P, SWM4-NDP, TIP4P-Ew, and TIP4P/2005.\u003c/p\u003e \u003cp\u003eOur results show that the TIP5P water model predicts the most disparate results among the models investigated, seemingly associated with a lower number of H-bonds. This model predicts a higher isothermal compressibility and thermal expansion coefficient as well as a lower viscosity and a higher self-diffusion coefficient, especially at low pressures. Upon compression the thermodynamic response functions and transport coefficients of TIP5P water become less dissimilar from the remaining water models. This is linked with a steeper increase of the number of the H-bonds with pressure, making TIP5P water\u0026rsquo;s H-bond network more similar to that of the remaining models, at high pressures.\u003c/p\u003e \u003cp\u003eAlthough the number of H-bonds increases, the tetrahedrality of the H-bond network decreases with increasing pressure indicating a more strained network.\u003c/p\u003e \u003cp\u003eThe distinct water models estimate a more moderate violation of the SER, than that found by Bove \u003cem\u003eet al\u003c/em\u003e.[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. Thus, a diffusion-viscosity decoupling similar to that found in real supercooled water, down to 242 K, is observed. The reason is that water models predict a more pronounced decrease of the self-diffusion coefficient at high pressures, than experiments. This suggests that water models commonly used to study liquid water, including metastable supercooled water, cannot accurately describe the structural and dynamic responses to compression, of superheated water. This problem should be revisited using other water models and/or \u003cem\u003eab initio\u003c/em\u003e MD to gain further insight into the structural source of the strong diffusion-viscosity decoupling observed in real superheated water.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis paper is dedicated to Prof. Pratim K. Chattaraj, Professor, IIT Kharagpur, who has been the source of inspiration for us. We respectfully acknowledge his seminal contribution in the broad area of theoretical chemistry. Shivam, Archita, and Vikas acknowledge IIT Patna for their fellowships. N. G. acknowledges the work support by UIDB/04046/2020 and UIDP/04046/2020 centre grants from FCT, Portugal (to BioISI), by the Portuguese National Distributed Computing Infrastructure (http://www.incd.pt). S. D. acknowledges computational facility from IIT Patna.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e S.D. contributed to the methodology, analysis, and writing original draft. A. M. contributed to the analysis, and writing original draft, V.D. contributed to the methodology and analysis, N.G. contributed to the conceptualization, methodology, analysis, and writing original draft. S.D. contributed to the conceptualization, methodology, analysis, writing original draft, and supervision of the project.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e This study was funded by FCT (CEEC/2018) (Portugal) and SERB Early Career Award (File No. ECR/2017/002335) (India).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclarations Conflict of Interest\u0026nbsp;\u003c/strong\u003eThe authors declare no competing interests.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003e\u003cspan\u003eRJ Speedy, CA Angell (1976) The Journal of Chemical Physics 65: 851. Doi:\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1063/1.433153\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003ePG Debenedetti (2003) Journal of Physics: Condensed Matter 15: R1669.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eP Gallo, K Amann-Winkel, CA Angell, et al. 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Doi:\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1038/35053024\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"theoretical-chemistry-accounts","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"tcac","sideBox":"Learn more about [Theoretical Chemistry Accounts](http://link.springer.com/journal/214)","snPcode":"214","submissionUrl":"https://submission.nature.com/new-submission/214/3","title":"Theoretical Chemistry Accounts","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Superheated water, Thermodynamic Response Function, Gigapascal Pressure, Stokes-Einstein Breakdown, Molecular Dynamics","lastPublishedDoi":"10.21203/rs.3.rs-2419984/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2419984/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eLiquid water is the most intriguing liquid in nature, both because of its importance to every known form of life, and its numerous anomalous properties, largely magnified under supercooled conditions. Among the anomalous properties of water is the seeming divergence of the thermodynamic response functions and dynamic properties below the homogenous nucleation temperature (~232 K). Furthermore, water exhibits an increasingly decoupling of the viscosity and diffusion, upon cooling, resulting in the breakdown of the Stokes-Einstein relationship (SER). At high temperatures and pressures, however, water behaves more like a “simple” liquid. Nonetheless, experiments at 400 K and GPa pressures (Bove \u003cem\u003eet al\u003c/em\u003e. (2011) \u003cem\u003ePhys. Rev. Lett.\u003c/em\u003e, 111:185901) showed that although the diffusion decreases monotonically with the pressure, opposite to pressurized supercooled water, a decoupling of the viscosity and diffusion, larger than that found in supercooled water at normal pressure, is observed. Here, we studied the thermodynamic response functions and breakdown of the SER along the 400 K isotherm up to 3 GPa, through molecular dynamics. Seven water models were investigated. A monotonic increase of the density (~50 %) and decrease of the isothermal compressibility (~90 %) and thermal expansion (~65 %) is found. Our results also show that compressed hot water has various resemblances to cool water at normal pressure, with pressure inducing the formation of a new second coordination sphere and a monotonic decrease of the diffusion and viscosity coefficients. Whereas all water models provide a good account of the viscosity, the magnitude of the violation of the SER at high pressures (\u0026gt; ~1 GPa) is significantly smaller than that found through experiments. Thus, violation of the SER in simulations is comparable to that observed for liquid supercooled water, indicating possible limitations of the water models to account for the local structure and self-diffusion of superheated water above ~1 GPa.\u003c/p\u003e","manuscriptTitle":"Thermodynamic Response Functions and Stokes-Einstein Breakdown in Superheated Water under Gigapascal Pressure","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-01-04 16:59:42","doi":"10.21203/rs.3.rs-2419984/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major revision","date":"2023-04-06T15:19:18+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2023-04-06T12:42:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"5f1315f1-9315-45e8-88b1-6fae06a5a334","date":"2023-04-05T06:03:11+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2023-01-02T15:45:07+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2022-12-28T09:58:59+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2022-12-28T05:58:37+00:00","index":"","fulltext":""},{"type":"submitted","content":"Theoretical Chemistry Accounts","date":"2022-12-27T20:10:12+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"theoretical-chemistry-accounts","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"tcac","sideBox":"Learn more about [Theoretical Chemistry Accounts](http://link.springer.com/journal/214)","snPcode":"214","submissionUrl":"https://submission.nature.com/new-submission/214/3","title":"Theoretical Chemistry Accounts","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"84b3c1d7-2f5c-4a7d-8a07-237dca862d14","owner":[],"postedDate":"January 4th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2023-10-16T20:39:18+00:00","versionOfRecord":{"articleIdentity":"rs-2419984","link":"https://doi.org/10.1007/s00214-023-02991-0","journal":{"identity":"theoretical-chemistry-accounts","isVorOnly":false,"title":"Theoretical Chemistry Accounts"},"publishedOn":"2023-04-20 20:31:48","publishedOnDateReadable":"April 20th, 2023"},"versionCreatedAt":"2023-01-04 16:59:42","video":"","vorDoi":"10.1007/s00214-023-02991-0","vorDoiUrl":"https://doi.org/10.1007/s00214-023-02991-0","workflowStages":[]},"version":"v1","identity":"rs-2419984","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2419984","identity":"rs-2419984","version":["v1"]},"buildId":"WrCJVZZCHTDjtuVLN7oU0","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}