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Exploring Potential Energy Landscapes: Thermodynamics and Dynamics in Supercooled Liquids and Glasses

This paper reviews the thermodynamic formalism in the Potential Energy Landscape (PEL) approach, comparing it with numerical simulations and discussing the development of a PEL Equation of State (EOS). It explores the dynamics and importance of understanding the slowing down phenomenon and the vanishing entropy difference in supercooled liquids. The study delves into the correlation function in the PEL, specific heat, time-dependent specific heat, and the distribution of local minima. It also covers the Gaussian distribution and T-dependence of energy in different models of liquids. The paper examines the Equations of State based on PEL properties, connections between thermodynamics and dynamics, and the aging phenomenon. Additionally, it discusses the fluctuation-dissipation relation and the physical aging of glass-forming liquids. The Soft Sphere Model and the concept of fictive parameters in liquid-to-liquid transitions are explored in detail.

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Exploring Potential Energy Landscapes: Thermodynamics and Dynamics in Supercooled Liquids and Glasses

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  1. Bangalore, June 2004 Potential Energy Landscape Description of Supercooled Liquids and Glasses

  2. Why do we case ? Thermodynamics and Dynamics Review of thermodynamic formalism in the PEL approach Comparison with numerical simulations Development of an PEL EOS Extention to non-equilibrium case (one or more fictive parameters ?) Outline

  3. Why do we care ? Dynamics 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).

  4. Why do we care:Thermodynamics Why do we care Thermodyanmics A vanishing of the entropy difference at a finite T ?

  5. Separation of time scales f(t) 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. fbasin i(T)= eIS+ kBTSln [hwj i/kBT] + fanharmonic i(T) fbasin i(T)= -kBT ln[Zi(T)] normal modes j Z(T)= S Zi(T) allbasins i

  9. Stillinger formalism

  10. 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>)]

  11. 1-d Cos(x) Landscape

  12. Didattic - Correlation Function in IS

  13. Specific Heat

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

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

  16. <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).

  17. minimization

  18. BKS Silica Eis nel tempo

  19. Evaluete the DOS diagonalization

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

  21. Vibrational Free Energy kBTSjln [hwj(eIS)/kBT] LW-OTP SPC/E S ln[wi(eIS)]=a+b eIS

  22. Pitfalls

  23. f anharmonic anharmonic eIS independent anharmonicity Weak eIS dependent anharmonicity

  24. Einstein Crystal

  25. Caso r2 per n-2n

  26. 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

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

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

  29. P(eIS,T)

  30. BMLJ Sconf BMLJ Configurational Entropy

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

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

  33. 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]

  34. Developing an EOS based on PES properties

  35. 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)

  36. Eis e S conf for silica… Esempio di forte

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

  38. V ~ (s/r)-n Soft Spheres with different softness

  39. Conclusion I The V-dependence of the statistical properties of the PEL can be quantified for models of liquids Accurate EOS can be constructed from these information Interesting features of the liquid state (TMD line) can be correlated to features of the PEL statistical properties Connections between Dynamics and Thermodynamics

  40. Simple (numerical) Aging Experiment

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

  42. Evolution of eIS in aging (BMLJ) W. Kob et al Europhys. Letters 49, 590 (2000). One can hardly do better than equilibrium !!

  43. 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

  44. A look to the meaning of Teff

  45. How to ask a system its Tin t

  46. 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).

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

  48. 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 a fictive T !

  49. 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

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

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