1. Potential Energy Landscape Description of Supercooled Liquids and Glasses

2. INIZIO LEZIONE http://mc2tar.phys.uniroma1.it/~fs/didattica/dottorato/ D. Wales Energy Landscapes Cambridge University Press F. Sciortino Potential energy landscape description of supercooled liquids and glasses J. Stat. Mech. 050515, 2005 Potential energy landscape description of supercooled liquids and glasses Articoli Gruppo Roma (molti dei quali sul landscape) http://glass.phys.uniroma1.it/sciortino/publications.htm Riferimenti

3. Outline Introduzione ai vetri ed ai liquidi sottorrafreddati Formalismo Termodinamico nel PEL Confronti con dati numerici Sviluppo di una PEL EOS Termodinamica di fuori equilibrio Sommario

4. Structural Glasses: Self-generated disorder Nomenclature Routes to Vitrification: Quench Crunch Chemical Vitrification Vapor Deposition Ion bombardment Crystal Amorphization Long Range Order Missing Short Range Order Present Introduzione -routes to crystal state

5. Local Order Indicators Radial Distribution Function - Structure Factor Conditional probability of finding a particle center at distance r (in a spherical shell of volume 4  r 2 dr) given that there is another one at the origin Gr e sq

6. S(q) Static Structure Factor

7. Generalization of S(q) to dynamics How a density fluctuation decays….. How a particle decorrelate over a distance of the order of q -1 S(q,t)

8. Two well known models for S self (q,t) (if x i is a gaussian random process - Kubo) Free Diffusion Motion in an harmonic potential, Two models for Sself

9. Sself fqfq

10. Evolution of (g(r) with T

11. r2

12. fqt

13. diff

14. Summary of T-dependence of tau alpha

15. Dynamics Strong-Fragile P.G. Debenedetti, and F.H. Stillinger, Nature 410, 259 (2001). A slowing down that cover more than 15 order of magnitudes

16. Excess Entropy Thermodyanmics A vanishing of the entropy difference at a finite T ?

17. van Megen and S.M. Underwood Phys. Rev. Lett. 70, 2766 (1993)  (t) log(t) Separation of time scales Supercooled Liquid Glass

18. Citazioni goldstein, stillinger

19. IS P e  Statistical description of the number, depth and shape of the PEL basins Potential Energy Landscape, a 3N dimensional surface The PEL does not depend on T The exploration of the PEL depends on T

20. De Broglie wavelength 1/k B T Starting thermodynamic definitions Pair-wise additive spherical potentials System of identical particles

21. all basins i Q(T,V)=  Q i (T,V) Non-crystalline ‘ Formalismo di Stillinger-Weber

22. Stillinger formalism

23. Thermodynamics in the IS formalism Stillinger- Weber F(T,V)=-k B T ln[  ( )]+f basin (,T,V) with f basin (e IS,T,V)= e IS +f vib (e IS,T,V) and S conf (T,V)=k B ln[  ( )] Basin depth and shape Number of explored basins

24. 1-d Cos(x) Landscape

25. Real Space rNrN Distribution of local minima (e IS ) Vibrations (e vib ) + e IS e vib Configuration Space ekek

26. F(T,V)=-k B T ln[  ( )]+f basin (,T,V) From simulations….. (T,V) (steepest descent minimization) f basin (e IS,T,V) (harmonic and anharmonic contributions) F(T,V) (thermodynamic integration from ideal gas)

27. minimization

28. Eis nel tempo BKS Silica Si0 2

29. Specific Heat High T Slow Dyn.

30. Time-Dependent Specific Heat in the IS formalism Specific Heat

31. BMLJ V T A Liquid Entropy (in B) CP Liquid Entropy B

32. diagonalization Evaluete the DOS Basin Shape

33. Harmonic Basin free energy Very often approximated with……

34. Vibrational Free Energy SPC/E LW-OTP  ln[  i (e IS )]=a+b e IS +c e IS 2 k B T  j  ln [h  j (e IS )/k B T]

35. Pitfalls

36. f anharmonic e IS independent anharmonicity Weak e IS dependent anharmonicity anharmonic

37. Differences of 0.1-0.2 can arise from different handling of the anharmonic entropy Example wih soft sphere V=  (  /r) n n=12 D(e IS )

38. Thermodynamic Integration Thermodynamic integration

39. Frenkel-Ladd (Einstein Crystal)

40. Application to Vibrational Free Energy

41. n-2n

42. BMLJ Configurational Entropy BMLJ Sconf

43. (SPC/E) T-dependence of S conf (SPC/E)

44. Excess Entropy Thermodyanmics A vanishing of the entropy difference at a finite T ?

45. Fine Seconda Parte

46. Recall f vib

47. The Random Energy Model for e IS Hypothesis:  e IS )de IS = e  N -----------------de IS e -(e IS -E 0 ) 2 /2  2 2222 S conf (e IS )/N=  - (e IS -E 0 ) 2 /2  2 Gaussian Landscape

48. Partitin function

49. Predictions of Gaussian Landscape Prediction 1

50. Predictions of Gaussian Landscape II Eis vs T, Scon vs T Ek Tk Prediction grafics

51. e IS =  e i IS E 0 = =N e 1 IS  2 =  2 N =N  2 1 Gaussian Distribution ?

52. T-dependence of SPC/ELW-OTP T -1 dependence observed in the studied T-range Support for the Gaussian Approximation

53. P(e IS,T)

54. BMLJ Configurational Entropy BMLJ Sconf

55. (SPC/E) T-dependence of S conf (SPC/E)

56. Come misuriamo Sigma2, alpha, E0, b Come misuriamo

57. The V-dependence of ,  2, E 0  e IS )de IS =e  N -----------------de IS e -(e IS -E 0 ) 2 /2  2 2222

58. Landscape Equation of State P=-∂F/∂V |T F(V,T)=-TS conf (T,V)+ +f vib (T,V) In Gaussian (and harmonic) approximation P(T,V)=P const (V)+P T (V) T + P 1/T (V)/T P const (V)= - d/dV [E 0 -b  2 ] P T (V) =R d/dV [  -a-bE 0 +b 2  2 /2] P 1/T (V) = d/dV [  2 /2R]

59. Developing an EOS based on PES properties

60. SPC/E P(T,V)=P const (V)+P T (V) T + P 1/T (V)/T

61. Non-Gaussian behavior in BKS Silica

62. Eis e S conf for silica… Esempio di forte Non-Gaussian Behavior in SiO 2 Non gaussian silica

63. Landscape of Strong Liquid SW if # of bonded particles <= N max HS if # of bonded particles > N max V(r) r Maximum Valency Angel’s work !

64. Viscosity and Diffusivity: Arrhenius  =1  C v small Stokes-Einstein Relation Other strong properties: percolating

65. Energy per Particle Ground State Energy Known ! It is possible to equilibrate at low T ! E(T) is known and hence free energy can be calculated exactly down to T=0

66. It is possible to calculate exactly the vibrational entropy of one single bonding pattern (basin free energy) Basin Free energy (Ladd and Frenkel)

67. sconf Non zero ground state entropy

68. Landscape of strong and fragile liquids Realistic Model Network Primitive Model for Network Fragile Liquid

69. Dinamics !

70. Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation BKS Silica Ivan Saika- Voivod et al, Nature 412, 514 (2001). AG per Silica

71. SPC/E water

72. V ~ (  /r) -n Soft Spheres with different softness De Michele et al

73. Summary The statistical properties of the PEL can be quantified with a proper analysis of simulation data Accurate EOS can be constructed from these information (but we may have to go beyond the Gaussian approximation) Interesting features of the liquid state (TMD line) can be correlated to features of the PEL statistical properties Connections between Dynamics and Thermodynamics need further studies !!

74. End of Thirth Lecture

75. Simple (numerical) Aging Experiment

76. Aging in the PEL-IS framework Starting Configuration (T i ) Short after the T-change (T i ->T f ) Long time TiTi TfTf TfTf Same Basins as eq.!

77. Evolution of e IS in aging (BMLJ) One can hardly do better than equilibrium !!

78. The “TAP” free energies……

79. F(T, T f )=-T f S conf (e IS )+f basin (e IS,T) S. Franz and M. A. Virasoro, J. Phys. A 33 (2000) 891, Which T in aging ? Equivalent form:

80. If basins have identical shape …..

81. bmlj

82. A look to the meaning of T eff

83. Heat flows….. (case of basins of identical shape )

84. How to ask a system its Tin t

85. Didattic - Correlation Function in IS

86. Fluctuation Dissipation Relation (Cugliandolo, Kurcian, Peliti, ….)

87. Support from the Soft Sphere Model Soft sphere F(V, T, T f )=-T f S conf (e IS )+f basin (e IS,T)

88. From Equilibrium to OOE…. P(T,V)= P conf (T,V)+ P vib (T,V) If we know which equilibrium basin the system is exploring … e IS acts as a fictive T ! e IS, V, T.. We can correlate the state of the aging system with an equilibrium state and predict the pressure (OOE-EOS)

89. Numerical Tests Liquid-to-Liquid T-jump at constant V P-jump at constant T

90. Numerical Tests Heating a glass at constant P T P time

91. Numerical Tests Compressing at constant T PfPf T time PiPi

92. Breakdowns ! (things to be understood)

93. Breaking of the out-of-equilibrium theory…. Kovacs (cross-over) effect S. Mossa and FS, PRL (2004)

94. Breakdown - e is -dos From Kovacs

95. P(e IS,t w )

96. BMLJ

97. Exploring New Regions of Phase Space ?

98. Summary II  The hypothesis that the system samples in aging the same basins explored in equilibrium allows to develop an EOS for OOE-liquids depending on one additional parameter  Small aging times, small perturbations are consistent with such hypothesis. Work must be done to evaluate the limit of validity.  The aditional parameter can be chosen as fictive T, fictive P or depth of the explored basin e IS

99. Perspectives  An improved description of the statistical properties of the potential energy surface.  Role of the statistical properties of the PEL in liquid phenomena  A deeper understanding of the concept of P conf and of EOS of a glass.  An estimate of the limit of validity of the assumption that a glass is a frozen liquid (number of parameters)  Connections between PEL properties and Dynamics

100. Acknowledgements I acknowledge important comments, criticisms, discussions with P. Debenedetti, S. Sastry, R. Speedy, A. Angell, T. Keyes, G. Ruocco, P. Poole and their collaborators I thank, among others, E. La Nave, I. Saika-Voivod, C. Donati, A. Scala, L. Angelani, C. De Michele, F. Starr N. Giovambattista, A. Moreno, G. Foffi with whom I had the pleasure to work on PEL ideas.