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5th Liquid Matter Conference

5th Liquid Matter Conference. Konstanz, 14-18 September 2002. Potential Energy Landscape Equation of State. Emilia La Nave, Francesco Sciortino, Piero Tartaglia (Roma). Stefano Mossa (Boston/Paris). Outline. Brief introduction to the inherent-structure (IS) formalism (Stillinger&Weber)

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5th Liquid Matter Conference

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  1. 5th Liquid Matter Conference Konstanz, 14-18 September 2002 Potential Energy Landscape Equation of State Emilia La Nave, Francesco Sciortino, Piero Tartaglia (Roma) Stefano Mossa (Boston/Paris)

  2. Outline • Brief introduction to the inherent-structure (IS) formalism (Stillinger&Weber) • Statistical Properties of the Potential Energy Landscape (PEL). PEL Equation of State (PEL-EOS) • Aging in the IS framework. Comparison with numerical simulation of aging systems.

  3. Potential Energy Landscape Statistical description of the number [W(eIS)deIS], depth [eIS] and volume [log(w)] of the PEL-basins e IS P IS w

  4. 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>)]

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

  6. F(T)=-kBT ln[W(<eIS>)]+fbasin(<eIS>,T) From simulations….. • <eIS>(T) (steepest descent minimization) • fbasin(eIS,T) (harmonic and anharmonic contributions) • F(T) (thermodynamic integration from ideal gas) In this talk….. Data for two rigid-molecule models: LW-OTP, SPC/E-H20

  7. fbasin(eIS,T)= eIS+kBTSln [hwj(eIS)/kBT] +fanharmonic(T) Basin Free Energy normal modes LW-OTP SPC/E S ln[wi(eIS)]=a+b eIS normal modes

  8. 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 normal modes Predictions: <eIS(T)>=E0-bs 2 - s 2/kT Sconf(T)=aN-(<eIS (T)>-E0)2/2s 2

  9. T-dependence of <eIS> SPC/E LW-OTP

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

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

  12. 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]

  13. Comparing the PEL-EOS with Simulation Results (LW-OTP)

  14. SPC/E WaterP(T,V)=Pconst(V)+PT(V) T + P1/T(V)/T

  15. Conclusion I The V-dependence of the statistical properties of the PEL has been quantified for two models of molecular 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

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

  17. From Equilibrium to OOE…. P(T,V)= Pconf(T,V)+ Pvib(T,V) If we know which equilibrium basin the system is exploring… • eIS(V,Tf),V,T • log(w) • Pvib • eIS(V,Tf),V • Pconf eIS acts as a fictive T !

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

  19. Liquid-to-Liquid T-jump at constant V P-jump at constant T

  20. Conclusion II • The hypothesis that the system samples in aging the same basins explored in equilibrium allows to develop an EOS for OOE-liquids which depends on one additional parameter • Short aging times, small perturbations are consistent with such hypothesis. Work is requested to evaluate the limit of validity. • The parameter can be chosen as fictive T, fictive P or depth of the explored basin eIS

  21. 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 Pconf 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

  22. References and Acknowledgements We acknowledge important discussions, comments, criticisms from P. Debenedetti, S. Sastry, R. Speedy, A. Angell, T. Keyes, G. Ruocco and collaborators Francesco Sciortino 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). Emilia La Nave, Stefano Mossa and Francesco Sciortino Potential Energy Landscape Equation of State Phys. Rev. Lett., 88, 225701 (2002). Stefano Mossa, Emilia La Nave, Francesco Sciortino and Piero Tartaglia, Aging and Energy Landscape: Application to Liquids and Glasses., cond-mat/0205071

  23. Entering the supercooled region

  24. Same basins in Equilibrium and Aging ?

  25. 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

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

  27. T-dependence of <eIS> (LW-OTP)

  28. P=-∂F/∂V Reconstructing P(T,V) F(V,T)=-TSconf(T,V)+<eIS(T,V)>+fvib(T,V) P(T,V)= Pconf(T,V) + Pvib(T,V)

  29. Numerical TestsCompressing at constant T Pf Pi T time

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