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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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## References

1. Speedy, R. J. & Angell, C. A. Isothermal compressibility of supercooled water and evidence for a thermodynamic singularity at −45 °C. J. Chem. Phys. 65, 851–858 (1976)

2. Angell, C. A., Oguni, M. & Sichina, W. J. Heat capacity of water at extremes of supercooling and superheating. J. Phys. Chem. 86, 998–1002 (1982)

3. Hare, D. E. & Sorensen, C. M. Densities of supercooled H2O and D2O in 25 μm glass capillaries. J. Chem. Phys. 84, 5085–5089 (1986)

4. Mishima, O. & Stanley, H. E. Decompression-induced melting of ice IV and the liquid–liquid transition in water. Nature 392, 164–168 (1998)

5. Soper, A. K. & Ricci, M. A. Structures of high-density and low-density water. Phys. Rev. Lett. 84, 2881–2884 (2000)

6. Winkel, K., Elsaesser, M. S., Mayer, E. & Loerting, T. Water polyamorphism: reversibility and (dis)continuity. J. Chem. Phys. 128, 044510 (2008)

7. Huang, C. et al. The inhomogeneous structure of water at ambient conditions. Proc. Natl Acad. Sci. USA 106, 15214–15218 (2009)

8. Clark, G. N. I., Hura, G. L., Teixeira, J., Soper, A. K. & Head-Gordon, T. Small-angle scattering and the structure of ambient liquid water. Proc. Natl Acad. Sci. USA 107, 14003–14007 (2010)

9. Poole, P. H., Sciortino, F., Grande, T., Stanley, H. E. & Angell, C. A. Effect of hydrogen-bonds on the thermodynamic behavior of liquid water. Phys. Rev. Lett. 73, 1632–1635 (1994)

10. Tanaka, H. Simple view of waterlike anomalies of atomic liquids with directional bonding. Phys. Rev. B 66, 064202 (2002)

11. Holten, V. & Anisimov, M. A. Entropy-driven liquid–liquid separation in supercooled water. Sci. Rep. 2, 713 (2012)

12. Poole, P. H., Sciortino, F., Essmann, U. & Stanley, H. E. Phase behavior of metastable water. Nature 360, 324–328 (1992)

13. Moore, E. B. & Molinero, V. Structural transformation in supercooled water controls the crystallization rate of ice. Nature 479, 506–508 (2011)

14. Speedy, R. J. Stability-limit conjecture. An interpretation of the properties of water. J. Phys. Chem. 86, 982–991 (1982)

15. Sastry, S., Debenedetti, P. G., Sciortino, F. & Stanley, H. E. Singularity-free interpretation of the thermodynamics of supercooled water. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53, 6144–6154 (1996)

16. Amann-Winkel, K. et al. Water’s second glass transition. Proc. Natl Acad. Sci. USA 110, 17720–17725 (2013)

17. Liu, Y., Palmer, J. C., Panagiotopoulos, A. Z. & Debenedetti, P. G. Liquid–liquid transition in ST2 water. J. Chem. Phys. 137, 214505 (2012)

18. Poole, P. H., Bowles, R. K., Saika-Voivod, I. & Sciortino, F. Free energy surface of ST2 water near the liquid–liquid phase transition. J. Chem. Phys. 138, 034505 (2013)

19. Palmer, J. C., Car, R. & Debenedetti, P. G. The liquid–liquid transition in supercooled ST2 water: a comparison between umbrella sampling and well-tempered metadynamics. Faraday Discuss. 167, 77–94 (2013)

20. Li, Y. P., Li, J. C. & Wang, F. Liquid–liquid transition in supercooled water suggested by microsecond simulations. Proc. Natl Acad. Sci. USA 110, 12209–12212 (2013)

21. Overduin, S. D. & Patey, G. N. An analysis of fluctuations in supercooled TIP4P/2005 water. J. Chem. Phys. 138, 184502 (2013)

22. Limmer, D. T. & Chandler, D. The putative liquid–liquid transition is a liquid-solid transition in atomistic models of water. J. Chem. Phys. 135, 134503 (2011)

23. Limmer, D. T. & Chandler, D. The putative liquid–liquid transition is a liquid–solid transition in atomistic models of water. II. J. Chem. Phys. 138, 214504 (2013)

24. Stillinger, F. H. & Rahman, A. Improved simulation of liquid water by molecular dynamics. J. Chem. Phys. 60, 1545–1557 (1974)

25. Lee, J. Y. & Kosterlitz, J. M. New numerical method to study phase transitions. Phys. Rev. Lett. 65, 137–140 (1990)

26. Moore, E. B. & Molinero, V. Ice crystallization in water’s ‘no-man’s land’. J. Chem. Phys. 132, 244504 (2010)

27. Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784–805 (1983)

28. Shiratani, E. & Sasai, M. Molecular scale precursor of the liquid–liquid phase transition of water. J. Chem. Phys. 108, 3264–3276 (1998)

29. Matsumoto, M., Saito, S. & Ohmine, I. Molecular dynamics simulation of the ice nucleation and growth process leading to water freezing. Nature 416, 409–413 (2002)

30. Frenkel, D. & Smit, B. Understanding Molecular Simulation: From Algorithms to Applications 2nd edn (Academic, 2002)

31. Rossky, P. J., Doll, J. D. & Friedman, H. L. Brownian dynamics as smart Monte Carlo simulation. J. Chem. Phys. 69, 4628–4633 (1978)

32. Liu, Y., Panagiotopoulos, A. Z. & Debenedetti, P. G. Low-temperature fluid-phase behavior of ST2 water. J. Chem. Phys. 131, 104508 (2009)

33. Torrie, G. M. & Valleau, J. P. Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J. Comput. Phys. 23, 187–199 (1977)

34. Kumar, S., Bouzida, D., Swendsen, R. H., Kollman, P. A. & Rosenberg, J. M. The weighted histogram analysis method for free-energy calculations on biomolecules. 1. The method. J. Comput. Chem. 13, 1011–1021 (1992)

35. Hub, J. S., de Groot, B. L. & van der Spoel, D. g_wham: a free weighted histogram analysis implementation including robust error and autocorrelation estimates. J. Chem. Theory Comput. 6, 3713–3720 (2010)

36. Sugita, Y. & Okamoto, Y. Replica-exchange molecular dynamics method for protein folding. Chem. Phys. Lett. 314, 141–151 (1999)

37. Reinhardt, A., Doye, J. P. K., Noya, E. G. & Vega, C. Local order parameters for use in driving homogeneous ice nucleation with all-atom models of water. J. Chem. Phys. 137, 194504 (2012)

38. Bonomi, M., Barducci, A. & Parrinello, M. Reconstructing the equilibrium boltzmann distribution from well-tempered metadynamics. J. Comput. Chem. 30, 1615–1621 (2009)

39. Gee, J. & Shell, M. S. Two-dimensional replica exchange approach for peptide-peptide interactions. J. Chem. Phys. 134, 064112 (2011)

40. King, S. V. Ring configurations in a random network model of vitreous silica. Nature 213, 1112–1113 (1967)

41. Barducci, A., Bussi, G. & Parrinello, M. Well-tempered metadynamics: a smoothly converging and tunable free-energy method. Phys. Rev. Lett. 100, 020603 (2008)

42. Duane, S., Kennedy, A. D., Pendleton, B. J. & Roweth, D. Hybrid Monte Carlo. Phys. Lett. B 195, 216–222 (1987)

43. Kesselring, T. A. et al. Finite-size scaling investigation of the liquid–liquid critical point in ST2 water and its stability with respect to crystallization. J. Chem. Phys. 138, 244506 (2013)

44. Choukroun, M. & Grasset, O. Thermodynamic model for water and high-pressure ices up to 2.2 GPa and down to the metastable domain. J. Chem. Phys. 127, 124506 (2007)

45. Choukroun, M. & Grasset, O. Thermodynamic data and modeling of the water and ammonia-water phase diagrams up to 2.2 GPa for planetary geophysics. J. Chem. Phys. 133, 144502 (2010)

46. Shilling, J. E. et al. Measurements of the vapor pressure of cubic ice and their implications for atmospheric ice clouds. Geophys. Res. Lett. 33, L17801 (2006)

47. Mayer, E. & Hallbrucker, A. Cubic ice from liquid water. Nature 325, 601–602 (1987)

48. Yamamuro, O., Oguni, M., Matsuo, T. & Suga, H. Heat capacity and glass transition of pure and doped cubic ices. J. Phys. Chem. Solids 48, 935–942 (1987)

49. Handa, Y. P., Klug, D. D. & Whalley, E. Difference in energy between cubic and hexagonal ice. J. Chem. Phys. 84, 7009–7010 (1986)

50. Mcmillan, J. A. & Los, S. C. Vitreous ice: irreversible transformations during warm-up. Nature 206, 806–807 (1965)

51. Weber, T. A. & Stillinger, F. H. Pressure melting of ice. J. Chem. Phys. 80, 438–443 (1984)

52. Miller, T. F. et al. Symplectic quaternion scheme for biophysical molecular dynamics. J. Chem. Phys. 116, 8649–8659 (2002)

53. Miyamoto, S. & Kollman, P. A. SETTLE: an analytical version of the SHAKE and RATTLE algorithm for rigid water models. J. Comput. Chem. 13, 952–962 (1992)

54. Palmer, J. C. General discussion. Faraday Discuss. 167, 118–127 (2013)

55. Hunter, J. E. & Reinhardt, W. P. Finite-size scaling behavior of the free energy barrier between coexisting phases: determination of the critical temperature and interfacial tension of the Lennard–Jones fluid. J. Chem. Phys. 103, 8627–8637 (1995)

56. Vega, C. & de Miguel, E. Surface tension of the most popular models of water by using the test-area simulation method. J. Chem. Phys. 126, 154707 (2007)

57. Handel, R., Davidchack, R. L., Anwar, J. & Brukhno, A. Direct calculation of solid–liquid interfacial free energy for molecular systems: TIP4P ice–water interface. Phys. Rev. Lett. 100, 036104 (2008)

## 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).

## Author information

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.

## Ethics declarations

### Competing interests

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.

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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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• DOI: https://doi.org/10.1038/nature13405

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