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Metastable liquid–liquid transition in a molecular model of water

Abstract

Liquid water’s isothermal compressibility1 and isobaric heat capacity2, and the magnitude of its thermal expansion coefficient3, increase sharply on cooling below the equilibrium freezing point. Many experimental4,5,6,7,8, theoretical9,10,11 and computational12,13 studies have sought to understand the molecular origin and implications of this anomalous behaviour. Of the different theoretical scenarios9,14,15 put forward, one posits the existence of a first-order phase transition that involves two forms of liquid water and terminates at a critical point located at deeply supercooled conditions9,12. Some experimental evidence is consistent with this hypothesis4,16, but no definitive proof of a liquid–liquid transition in water has been obtained to date: rapid ice crystallization has so far prevented decisive measurements on deeply supercooled water, although this challenge has been overcome recently16. Computer simulations are therefore crucial for exploring water’s structure and behaviour in this regime, and have shown13,17,18,19,20,21 that some water models exhibit liquid–liquid transitions and others do not. However, recent work22,23 has argued that the liquid–liquid transition has been mistakenly interpreted, and is in fact a liquid–crystal transition in all atomistic models of water. Here we show, by studying the liquid–liquid transition in the ST2 model of water24 with the use of six advanced sampling methods to compute the free-energy surface, that two metastable liquid phases and a stable crystal phase exist at the same deeply supercooled thermodynamic condition, and that the transition between the two liquids satisfies the thermodynamic criteria of a first-order transition25. We follow the rearrangement of water’s coordination shell and topological ring structure along a thermodynamically reversible path from the low-density liquid to cubic ice26. We also show that the system fluctuates freely between the two liquid phases rather than crystallizing. These findings provide unambiguous evidence for a liquid–liquid transition in the ST2 model of water, and point to the separation of time scales between crystallization and relaxation as being crucial for enabling it.

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Figure 1: Thermodynamic equilibrium between metastable liquid polymorphs.
Figure 2: Finite-size scaling and the liquid–liquid interface.
Figure 3: Free-energy surface from unconstrained simulations.
Figure 4: Structural and topological order in the metastable coexisting liquids and in cubic ice.

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Acknowledgements

Computations were performed at the Terascale Infrastructure for Groundbreaking Research in Engineering and Science (TIGRESS) facility at Princeton University. P.G.D. acknowledges support from the National Science Foundation (CHE 1213343), A.Z.P. acknowledges support from the US Department of Energy (DE-SC0002128), and R.C. acknowledges support from the US Department of Energy (DE-SC0008626).

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Authors and Affiliations

Authors

Contributions

J.C.P., R.C., A.Z.P. and P.G.D. planned the study. J.C.P., Y.L. and F.M. performed the simulations and numerical data analysis. J.C.P. and P.G.D. wrote the main paper and methods information. All authors discussed the results and commented on the manuscript at each stage.

Corresponding author

Correspondence to Pablo G. Debenedetti.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Reversible free-energy surfaces at 228.6 K and 2.4 kbar computed using different sampling techniques.

Surfaces on the top row were computed using (from left to right) umbrella sampling MC, well-tempered metadynamics and unconstrained MC; the bottom row shows results from hybrid MC, parallel tempering MC and Hamiltonian exchange MC simulations. The free-energy barrier separating the liquid basins is 4kBT for all of the surfaces shown. Contours are 1kBT apart and uncertainties are estimated to be less than 0.5kBT.

Extended Data Figure 2 Autocorrelation functions for different sampling techniques.

Autocorrelation functions for density (blue) and Q6 (red) computed in the LDL region using unconstrained MC (left), hybrid MC (centre) and Hamiltonian exchange MC (right). The correlation functions were calculated by averaging results from at least 12 independent simulations. Density and Q6 fluctuations decay on very similar timescales, despite exhibiting technique-dependent transient behaviour where these processes may be separated by more than one order of magnitude.

Extended Data Figure 3 Time evolution of the crystalline order parameter in two-phase MC simulations of the ice Ic–liquid interface at 2.6 kbar.

The MC simulations were initiated from configurations containing 512 and 670 ST2 water molecules in the ice Ic and liquid phases, respectively. The x and y dimensions of the simulation cells were fixed in accord with the lattice constant for ice Ic, which was determined at each temperature by performing a separate calculation for the bulk ice phase, while the z dimension was allowed to fluctuate so as to impose a constant pressure of 2.6 kbar perpendicular to the ice–liquid interface. Drift of Q6 towards higher or lower values indicates that the system is freezing or melting. The melting temperature of 273 ± 3 K at 2.6 kbar was estimated by averaging the lowest and highest temperatures, respectively, at which melting and freezing were observed.

Extended Data Figure 4 Reversible free-energy surface at 275 K and 2.7 kbar demonstrating ice Ic–liquid coexistence.

The liquid and ice Ic basins have equal depths with respect to the saddle point, indicating that the reported state condition is a point of coexistence. Such results confirm the estimates of the melting temperature for ice Ic at 2.6 kbar obtained from TI calculations using the EEOS and the two-phase MC simulations of the ice–liquid interface. Contours are 1kBT apart.

Extended Data Table 1 Sampling methods
Extended Data Table 2 Comparison of ice Ic–liquid free-energy differences obtained from thermodynamic integration and from results presented in the text for the ST2 model
Extended Data Table 3 Comparison of ice Ic–liquid free-energy differences obtained from thermodynamic integration and from results presented by Limmer and Chandler23 for the ST2 water model
Extended Data Table 4 Estimates of the melting temperature for ice Ic at 2.6 kbar for the ST2 water model

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Palmer, J., Martelli, F., Liu, Y. et al. Metastable liquid–liquid transition in a molecular model of water. Nature 510, 385–388 (2014). https://doi.org/10.1038/nature13405

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