Duesseldorf February 5 2004

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

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

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

6. Citazioni goldstein, stillinger

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

8. Z(T)= S Zi(T) allbasins i fbasin i(T)= -kBT ln[Zi(T)] fbasin(eIS,T)= eIS+ kBTSln [hwj(eIS)/kBT] + fanharmonic(T) normal modes j

9. Thermodynamics in the IS formalism Stillinger-Weber F(T)=-kBT ln[W(<eIS>)]+fbasin(<eIS>,T) with Basin depth and shape fbasin(eIS,T)= eIS+fvib(eIS,T) and Number of explored basins Sconf(T)=kBln[W(<eIS>)]

10. Distribution of local minima (eIS) Real Space Configuration Space + Vibrations (evib) rN evib eIS

11. <eIS>(T) (steepest descent minimization) fbasin(eIS,T) (harmonic and anharmonic contributions) F(T) (thermodynamic integration from ideal gas) From simulations….. F(T)=-kBT ln[W(<eIS>)]+fbasin(<eIS>,T) E. La Nave et al., Numerical Evaluation of the Statistical Properties of a Potential Energy Landscape, J. Phys.: Condens. Matter 15, S1085 (2003).

12. BKS Silica Eis nel tempo

13. Basin Free Energy kBTSln [hwj(eIS)/kBT] LW-OTP SPC/E S ln[wi(eIS)]=a+b eIS …if b=0 …..

14. The Random Energy Model for eIS Hypothesis: e-(eIS -E0)2/2s 2 W(eIS)deIS=eaN -----------------deIS 2ps2 S ln[wi(eIS)]=a+b eIS Predictions: <eIS(T)>=E0-bs 2 - s 2/kT Sconf(T)=aN-(<eIS (T)>-E0)2/2s 2

15. Gaussian Distribution ? eIS=SeiIS E0=<eNIS>=Ne1IS s2= s2N=N s21

16. T-dependence of <eIS> SPC/E LW-OTP T-1 dependence observed in the studied T-range Support for the Gaussian Approximation

17. BMLJ Sconf BMLJ Configurational Entropy

18. The V-dependence of a, s2, E0 e-(eIS -E0)2/2s 2 W(eIS)deIS=eaN -----------------deIS 2ps2

19. Landscape Equation of State P=-∂F/∂V|T F(V,T)=-TSconf(T,V)+<eIS(T,V)>+fvib(T,V) In Gaussian (and harmonic) approximation P(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T Pconst(V)= - d/dV [E0-bs2] PT(V) =R d/dV [a-a-bE0+b2s2/2] P1/T(V) = d/dV [s2/2R]

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

21. SPC/E P(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T FS, E. La Nave, and P. Tartaglia, PRL. 91, 155701 (2003)

22. AG per Silica Correlating Thermodynamics and Dynamics: Adam-Gibbs Relation t~exp(A/TSconf) BKS Silica Ivan Saika-Voivod et al, Nature 412, 514 (2001).

23. 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. Simple (numerical) Aging Experiment

25. Aging in the PEL-IS framework Ti Tf Tf Starting Configuration (Ti) Short after the T-change (Ti->Tf) Long time

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

27. Which T in aging ? F(T, Tf)=-TfSconf (eIS)+fbasin(eIS,T) Relation first derived by S. Franz and M. A. Virasoro, J. Phys. A 33 (2000) 891, in the context of disordered spin systems

28. How to ask a system its Tin t

29. 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. Soft sphere F(V, T, Tf)=-TfSconf (eIS)+fbasin(eIS,T) Support from the Soft Sphere Model

31. P(T,V)= Pconf(T,V)+ Pvib(T,V) From Equilibrium to OOE…. If we know which equilibrium basin the system is exploring… eIS, V, T .. We can correlate the state of the aging system with an equilibrium state and predict the pressure (OOE-EOS) eIS acts as effective T !

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

33. Numerical TestsHeating a glass at constant P T P time

34. Numerical TestsCompressing at constant T Pf Pi T time

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

36. 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 eIS

37. 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. 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. Entering the supercooled region

40. Same basins in Equilibrium and Aging ?

41. T-dependence of Sconf (SPC/E)