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Potential Energy Landscape Description of Supercooled Liquids and Glasses

Potential Energy Landscape Description of Supercooled Liquids and Glasses. INIZIO LEZIONE. Riferimenti. http://mc2tar.phys.uniroma1.it/~fs/didattica/dottorato/. D. Wales  Energy Landscapes Cambridge University Press

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Potential Energy Landscape Description of Supercooled Liquids and Glasses

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  1. Potential Energy Landscape Description of Supercooled Liquids and Glasses

  2. INIZIO LEZIONE Riferimenti 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 glassesJ. Stat. Mech. 050515, 2005 Articoli Gruppo Roma (molti dei quali sul landscape)   http://glass.phys.uniroma1.it/sciortino/publications.htm

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

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

  5. Gr e sq Local Order Indicators Radial Distribution Function - Structure Factor Conditional probability of finding a particle center at distance r (in a spherical shell of volume 4p r2 dr) given that there is another one at the origin

  6. S(q) Static Structure Factor

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

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

  9. Sself fq

  10. Evolution of (g(r) with T

  11. r2

  12. fqt

  13. diff

  14. Summary of T-dependence of tau alpha

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

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

  17. 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)

  18. Citazioni goldstein, stillinger

  19. 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

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

  21. Q(T,V)= S Qi(T,V) Formalismo di Stillinger-Weber ‘ allbasins i Non-crystalline

  22. Stillinger formalism

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

  24. 1-d Cos(x) Landscape

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

  26. <eIS>(T,V) (steepest descent minimization) fbasin(eIS,T,V) (harmonic and anharmonic contributions) F(T,V) (thermodynamic integration from ideal gas) From simulations….. F(T,V)=-kBT ln[W(<eIS>)]+fbasin(<eIS>,T,V)

  27. minimization

  28. BKS Silica Si02 Eis nel tempo

  29. Specific Heat Slow Dyn. High T

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

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

  32. Evaluete the DOS Basin Shape diagonalization

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

  34. Vibrational Free Energy kBTSjln [hwj(eIS)/kBT] LW-OTP SPC/E S ln[wi(eIS)]=a+b eIS +c eIS2

  35. Pitfalls

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

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

  38. Thermodynamic Integration Thermodynamic integration

  39. Frenkel-Ladd (Einstein Crystal)

  40. Application to Vibrational Free Energy

  41. n-2n

  42. BMLJ Sconf BMLJ Configurational Entropy

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

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

  45. Fine Seconda Parte

  46. Recall f vib

  47. The Random Energy Model for eIS Gaussian Landscape Hypothesis: e-(eIS -E0)2/2s 2 W(eIS)deIS=eaN -----------------deIS 2ps2 Sconf(eIS)/N=a-(eIS-E0)2/2s 2

  48. Partitin function

  49. Prediction 1 Predictions of Gaussian Landscape

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

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