1. Duesseldorf Feb 2004 Duesseldorf February 5 2004 Potential Energy Landscape Description of Supercooled Liquids and Glasses Luca Angelani, Stefano Mossa, Ivan Saika- Voivod, Emilia La Nave, Piero Tartaglia

2. Duesseldorf Feb 2004 Why do we care ? Thermodynamics and Dynamics Review of thermodynamic formalism in the PEL approach Comparison with numerical simulations Development of a PEL EOS Generalization to non-equilibrium (one or more effective parameters ?) Outline

3. Duesseldorf Feb 2004 Why do we care ? Dynamics P.G. Debenedetti, and F.H. Stillinger, Nature 410, 259 (2001). A slowing down that cover more than 15 order of magnitudes 1

4. Duesseldorf Feb 2004 Why do we care: Thermodynamics Why do we care Thermodyanmics Is the excess entropy vanishing at a finite T ? 1

5. Duesseldorf Feb 2004 van Megen and S.M. Underwood Phys. Rev. Lett. 70, 2766 (1993)  (t) log(t) Separation of time scales Supercooled Liquid Glass glass liquid

6. Duesseldorf Feb 2004 Citazioni goldstein, stillinger

7. Duesseldorf Feb 2004 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

8. Duesseldorf Feb 2004 f basin i (T)= -k B T ln[Z i (T)] all basins i f basin (e IS,T)= e IS + k B T   ln [h  j (e IS )/k B T] + f anharmonic (T) normal modes j Z(T)=  Z i (T)

9. Duesseldorf Feb 2004 Thermodynamics in the IS formalism Stillinger-Weber F(T)=-k B T ln[  ( )]+f basin (,T) with f basin (e IS,T)= e IS +f vib (e IS,T) and S conf (T)=k B ln[  ( )] Basin depth and shape Number of explored basins

10. Duesseldorf Feb 2004 Real Space rNrN Distribution of local minima (e IS ) Vibrations (e vib ) + e IS e vib Configuration Space

11. Duesseldorf Feb 2004 F(T)=-k B T ln[  ( )]+f basin (,T) From simulations….. (T) (steepest descent minimization) f basin (e IS,T) (harmonic and anharmonic contributions) F(T) (thermodynamic integration from ideal gas) E. La Nave et al., Numerical Evaluation of the Statistical Properties of a Potential Energy Landscape, J. Phys.: Condens. Matter 15, S1085 (2003).

12. Duesseldorf Feb 2004 Eis nel tempo BKS Silica

13. Duesseldorf Feb 2004 Basin Free Energy SPC/E LW-OTP  ln[  i (e IS )]=a+b e IS k B T   ln [h  j (e IS )/k B T] …if b=0 …..

14. Duesseldorf Feb 2004 The Random Energy Model for e IS Hypothesis: Predictions :  e IS )de IS = e  N -----------------de IS e -(e IS -E 0 ) 2 /2  2 2222  ln[  i (e IS )]=a+b e IS =E 0 -b  2 -  2 /kT S conf (T)=  N- ( -E 0 ) 2 /2  2

15. Duesseldorf Feb 2004 e IS =  e i IS E 0 = =N e 1 IS  2 =  2 N =N  2 1 Gaussian Distribution ?

16. Duesseldorf Feb 2004 T-dependence of SPC/ELW-OTP T -1 dependence observed in the studied T-range Support for the Gaussian Approximation

17. Duesseldorf Feb 2004 BMLJ Configurational Entropy BMLJ Sconf

18. Duesseldorf Feb 2004 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

19. Duesseldorf Feb 2004 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]

20. Duesseldorf Feb 2004 Developing an EOS based on PEL properties SPC/E water

21. Duesseldorf Feb 2004 SPC/E P(T,V)=P const (V)+P T (V) T + P 1/T (V)/T FS, E. La Nave, and P. Tartaglia, PRL. 91, 155701 (2003)

22. Duesseldorf Feb 2004 Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation BKS Silica Ivan Saika- Voivod et al, Nature 412, 514 (2001). AG per Silica  exp(A/TS conf )

23. Duesseldorf Feb 2004 Conclusion I The V-dependence of the statistical properties of the PEL can be quantified for models of liquids Accurate EOS can be constructed Peculiar features of the liquid state (TMD line) can be connected to features of the PEL statistical properties Relation between Dynamics and Thermodynamics can be explored

24. Duesseldorf Feb 2004 Simple (numerical) Aging Experiment

25. Duesseldorf Feb 2004 Aging in the PEL-IS framework Starting Configuration (T i ) Short after the T-change (T i ->T f ) Long time TiTi TfTf TfTf

26. Duesseldorf Feb 2004 Evolution of e IS in aging (BMLJ), following a T-jump W. Kob et al Europhys. Letters 49, 590 (2000). One can hardly do better than equilibrium !!

27. Duesseldorf Feb 2004 F(T, T f )=-T f S conf (e IS )+f basin (e IS,T) Relation first derived by S. Franz and M. A. Virasoro, J. Phys. A 33 (2000) 891, in the context of disordered spin systems Which T in aging ?

28. Duesseldorf Feb 2004 How to ask a system its Tin t

29. Duesseldorf Feb 2004 Fluctuation Dissipation Relation (Cugliandolo, Kurcian, Peliti, ….) FS and Piero Tartaglia Extension of the Fluctuation-Dissipation theorem to the physical aging of a model glass-forming liquid Phys. Rev. Lett. 86, 107 (2001).

30. Duesseldorf Feb 2004 Support from the Soft Sphere Model Soft sphere F(V, T, T f )=-T f S conf (e IS )+f basin (e IS,T)

31. Duesseldorf Feb 2004 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 effective T ! e IS, V, T.. We can correlate the state of the aging system with an equilibrium state and predict the pressure (OOE-EOS)

32. Duesseldorf Feb 2004 Numerical Tests Liquid-to-Liquid T-jump at constant V P-jump at constant T S. Mossa et al. EUR PHYS J B 30 351 (2002)

33. Duesseldorf Feb 2004 Numerical Tests Heating a glass at constant P T P time

34. Duesseldorf Feb 2004 Numerical Tests Compressing at constant T PfPf T time PiPi

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

36. Duesseldorf Feb 2004 Conclusion II  The hypothesis that the system samples in aging the same basins explored in equilibrium allows us to develop an EOS for OOE-liquids (with one additional parameter)  Small aging times, small perturbations are consistent with such hypothesis. Work is ongoing to evaluate the limit of validity.  The additional parameter can be chosen as effective T, P or depth of the explored basin e IS

37. Duesseldorf Feb 2004 Perspectives  An improved description of the statistical properties of the potential energy surface.  A deeper understanding of the concept 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

38. Duesseldorf Feb 2004 Acknowledgements We acknowledge important discussions, comments, collaborations, criticisms from… A. Angell, P. Debenedetti, T. Keyes, G. Ruocco, S. Sastry, R. Speedy … and their collaborators

39. Duesseldorf Feb 2004 Entering the supercooled region

40. Duesseldorf Feb 2004 Same basins in Equilibrium and Aging ?

41. Duesseldorf Feb 2004 (SPC/E) T-dependence of S conf (SPC/E)