1. Liquid-Liquid Phase Transitions and Water-Like Anomalies in Liquids Erik Lascaris Final oral examination 9 July 2014 1

2. Outline Anomalies in water and simple models Liquid-liquid phase transition in water Liquid-liquid phase transition in silica Conclusions 2

3. Outline Anomalies in water and simple models Liquid-liquid phase transition in water Liquid-liquid phase transition in silica Conclusions 3

4. Water has many anomalies (compared to other liquids) Famous website by Martin Chaplin http://www1.lsbu.ac.uk/water/anmlies.html now lists 70 anomalies! (on 9 July 2014) Today we focus on 3 of them 4

5. 1. Density anomaly 5 At 1 atm Temperature of Maximum Density (TMD) is 4 °C As we increase T, density increases!

6. 2. Diffusion anomaly 6 In water, self-diffusion increases as density and pressure increase (at low T)

7. 3. Melting line with negative slope 7 Applying pressure can melt ice! Most liquids: details

8. Water has many anomalies (compared to other liquids) Where do these come from? What is their origin? 8 Let’s try a simple model!

9. Hard core + linear ramp potential Monatomic particles (spheres) Pairwise interaction: Particles can partially overlap 9 Hard core Linear ramp

10. PT diagram 10 Melting line with negative slope

11. PT diagram 11 Density anomaly increase T  increase density

12. PT diagram 12 Diffusion anomaly increase P  increase diffusivity

13. PT diagram 13 How can we explain these anomalies? How can we explain these anomalies?

14. Two length scales Particles have (1) no overlap or (2) partial overlap 14 Low density state (low T, low P) High density state (high T, high P)

15. Two length scales Particles have (1) no overlap or (2) partial overlap Increase T  more overlap  density increase 15 High T Low T

16. Two length scales Particles have (1) no overlap or (2) partial overlap Increase T  more overlap  density increase Increase P  more overlap  diffusion increase 16 High pressure Low pressure

17. Two length scales Particles have (1) no overlap or (2) partial overlap Increase T  more overlap  density increase Increase P  more overlap  diffusion increase 17 Low pressure High pressure

18. Two length scales Particles have (1) no overlap or (2) partial overlap Increase T  more overlap  density increase Increase P  more overlap  diffusion increase 18 High pressure Low pressure

19. Two length scales Particles have (1) no overlap or (2) partial overlap Increase T  more overlap  density increase Increase P  more overlap  diffusion increase 19 When T or P too high: Normal liquid again!

20. Melting line Clapeyron relation for slope melting line dP/dT: Change in entropy always positive:  S > 0 Usually crystal more dense than liquid:  V > 0 However, two length scales leads to  V < 0 20

21. Changing the model 21 To obtain a better understanding: try different potentials Hard Core + Linear RampLinear Ramp

22. PT diagram 22 Hard Core + Linear Ramp Linear Ramp

23. Changing the model 23 To obtain a better understanding: try different models Cut Ramp

24. PT diagrams of Cut Ramp potential 24 Several anomalies: Density anomaly Melting line with negative slope Diffusion anomaly All within same pressure range!

25. PT diagrams of Cut Ramp potential 25 How can we explain this? Look at Radial Distribution Function!

26. Radial Distribution Function (RDF) 26 RDF = probability for atom to find a neighbor a distance r away (normalized to ideal gas) ideal gascrystal

27. RDF of Linear Ramp (0% cut) 27 Low pressure (no anomalies) RDF at T = 0.040

28. RDF of Linear Ramp (0% cut) 28 Low pressure (no anomalies) RDF at T = 0.040

29. RDF of Linear Ramp (0% cut) 29 Low pressure (no anomalies) RDF at T = 0.040

30. RDF of Linear Ramp (0% cut) 30 Medium pressure (inside anomaly region) RDF at T = 0.040

31. RDF of Linear Ramp (0% cut) 31 High pressure (no anomalies) RDF at T = 0.040

32. RDF of 50% Cut Ramp 32 Low pressure (no anomalies) RDF at T = 0.040

33. RDF of 50% Cut Ramp 33 Medium pressure (no anomalies) RDF at T = 0.040

34. RDF of 50% Cut Ramp 34 High pressure (no anomalies) RDF at T = 0.040

35. Two “competing” length scales 35 Particles are either near r=0 or r=1 but rarely on the ramp! Within anomaly region, some particles are on the ramp 0% cut: anomalies 50% cut: no anomalies HCLR

36. Anomalies: conclusions For anomalies to occur, we require: Need two length scales in potential – Liquid has two preferred liquid states Length scales need to be “competing” – Anomalies occur when liquid is in between states 36 Water has anomalies  does it have two length scales?

37. Two length scales in water! 37 High-Density Amorphous ice (HDA) Katrin Amann-Winkel (2013) Loerting Group, Universität Innsbruck Low-Density Amorphous ice (LDA) When cooled super fast, it’s possible to create two types of “glassy” water! Spontaneous crystallization

38. Liquid-liquid phase transition in water Two liquids below nucleation temperature: – Low density liquid (LDL) – High density liquid (HDL) Separated by liquid-liquid phase transition (LLPT) line Line ends in a liquid-liquid critical point (LLCP) Hypothesis (Poole/Sciortino/Essmann/Stanley, Nat. 1992)

39. Outline Anomalies in water and simple models Liquid-liquid phase transition in water (using ST2 model) Liquid-liquid phase transition in silica Conclusions 39

40. ST2 water model 40 + + – – Stillinger & Rahman, J. Chem. Phys. 60, 1545 (1974)

41. Isochores in NVT ensemble 41 Adapted from: Poole et al., JPCM 17, L431 (2005) 0.80 g/cm 3 NVT: Simulations with constant Number of molecules, constant Volume, and constant Temperature Isochores are found by connecting data points of the same density

42. Isochores in NVT ensemble 42 Adapted from: Poole et al., JPCM 17, L431 (2005) 0.80 g/cm 3 0.81 g/cm 3 NVT: Simulations with constant Number of molecules, constant Volume, and constant Temperature Isochores are found by connecting data points of the same density

43. Isochores in NVT ensemble 43 0.80 g/cm 3 0.81 g/cm 3 0.82 g/cm 3 Adapted from: Poole et al., JPCM 17, L431 (2005) NVT: Simulations with constant Number of molecules, constant Volume, and constant Temperature Isochores are found by connecting data points of the same density

44. Isochores in NVT ensemble 44 HDL LDL NVT: Simulations with constant Number of molecules, constant Volume, and constant Temperature Isochores are found by connecting data points of the same density Adapted from: Poole et al., JPCM 17, L431 (2005) 0.80 g/cm 3 Isochores cross at the Liquid-Liquid Critical Point (LLCP)

45. Measuring location (T, P) of the LLCP It’s hard to locate LLCP accurately using only crossing isochores… Alternative method: Fitting order parameter to 3D Ising model! Requires NPT (constant Pressure) simulations 45 Will be explained on next few slides!

46. “Phase flipping” in NPT ensemble 46 LDL HDL LLCP Density as a function of time:

47. Density distribution Phase flipping  histogram of density 47 LDL HDL

48. Energy distribution Phase flipping  histogram of energy 48 LDL HDL

49. Order parameter distribution 2D histogram of energy & density 49 LDL HDL

50. Order parameter distribution 2D histogram of energy & density Order parameter: M =  + 27.6 E Order parameter: M =  + 27.6 E 50 LDL HDL

51. Order parameter distribution Order parameter M can be fit to 3D Ising model 51 Order parameter: M =  + 27.6 E Order parameter: M =  + 27.6 E LDL HDL LDLHDL

52. Outline Anomalies in water and simple models Liquid-liquid phase transition in water Liquid-liquid phase transition in silica (using WAC model and BKS model) Conclusions 52

53. Two models of silica (SiO 2 ) WAC model Woodcock, Angell, and Cheeseman Introduced 1976 Ions: Si +4 and O –2 53 Simple 1:2 mixture of Si and O ions Each ion has a charge + electrostatics Repulsive potential between ions to prevent Si-O fusion BKS model van Beest, Kramer, and van Santen Introduced 1990 Ions: Si +2.4 and O –1.2

54. PT diagram of BKS 54 Saika-Voivod, Sciortino, and Poole Phys. Rev. E 63, 011202 (2000) Lascaris, Hematti, Buldyrev, Stanley, and Angell J. Chem. Phys. 140, 224502 (2014) LLCP? Crossing isochores? No LLCP at low T

55. PT diagram of WAC 55 Lascaris, Hematti, Buldyrev, Stanley, and Angell J. Chem. Phys. 140, 224502 (2014) LLCP? Crossing isochores? Maybe LLCP at low T! Saika-Voivod, Sciortino, and Poole Phys. Rev. E 63, 011202 (2000)

56. Response functions WAC 56 At critical point: all response functions should diverge! Maxima are at different (T,P)  no LLCP!

57. Why is WAC closer to LLCP than BKS? 57 Studies by Molinero et al. suggest that “tetrahedrality” of the liquid plays a role Liquids like water and SiO 2 form tetrahedral bonds Rigid bonds lead to fast crystallization into hexagonal crystal

58. Why is WAC closer to LLCP than BKS? 58 Studies by Molinero et al. suggest that “tetrahedrality” of the liquid plays a role Rigid bonds lead to fast crystallization into hexagonal crystal High-Density Liquid Compact Liquid with high diffusivity Floppy bonds lead to high-density liquid

59. Why is WAC closer to LLCP than BKS? 59 Studies by Molinero et al. suggest that “tetrahedrality” of the liquid plays a role Rigid bonds lead to fast crystallization into hexagonal crystal High-Density Liquid Compact Liquid with high diffusivity Liquid-liquid phase transition requires existence of two liquids: LDL and HDL Low-Density Liquid Expanded Local structure closer to crystal

60. Why is WAC closer to LLCP than BKS? 60 Studies by Molinero et al. suggest that “tetrahedrality” of the liquid plays a role Conclusion: bond stiffness needs to be just right Too stiff  crystallization Too flexible  only HDL Just right  LDL + HDL

61. Si-O-Si bond angle distribution 61 Distribution WAC is sharper: stiffDistribution BKS more broad: floppy Hard to compare: bond angle depends on temperature! details

62. Bond angle as spring system 62 Pretend Si-O-Si bond angle is controlled by a spring:

63. 63 Imaginary spring (spring constant k 2 ) WAC more stiff  closer to LLCP details Bond angle as spring system

64. Outline Anomalies in water and simple models Liquid-liquid phase transition in water Liquid-liquid phase transition in silica Conclusions 64

65. Conclusions (1) Simple models explain origin of anomalies: – Two length scales Low density structure (expanded) High density structure (collapsed) – Region with anomalies is where these structures energetically compete 65

66. Conclusions (2) Best way to locate liquid-liquid critical point (LLCP) is by fitting order parameter to 3D Ising We did not find LLCP in silica models WAC, BKS Liquid-liquid phase transitions might be related to bond angle stiffness – Modeling liquid as network of springs might help predicting if liquid has a LLCP 66

67. Also a big THANKS to my collaborators: – Gene Stanley (advisor) – Sergey Buldyrev – Austen Angell – Mahin Hemmati – Giancarlo Franzese – Tobias Kesselring – Hans Herrmann – Gianpietro Malescio 67 THANK YOU! Liquid-Liquid Phase Transitions and Water-Like Anomalies in Liquids