T. Odagaki Department of Physics, Kyushu University

1. Free Energy Landscape Approach to Glass Transition T. Odagaki Department of Physics, Kyushu University T. Yoshidome, A. Koyama, A. Yoshimori and J. Matsui Japan-France Seminar, Paris September 30, 2005

2. Thermodynamic transition Dynamic transition Phenomenological understanding Free energy landscape Glass transition singularities

3. Energy of basin a Probability of being in basin a at t a ：Quenched ：Annealed Phenomenological understanding : Heat capacity T. Tao &T.O(PRE 2002),T.O et al (JCP 2002),T. Tao et al (JCP2005)

4. slow fast T. Tao, T. O and A. Yoshimori: JCP 122, 044505 (2005) • 20 basins:Einstein oscillators Annealed-to-quenched transition and cooling rate dependence ☆Annealed to quenched transition ☆Cooling rate dependence

5. Outline 1. Free Energy Landscape, CRR and SRR 2. Density Functional Theory and FEL 3. Principal Component Analysis and FEL 4．Unifying Concept for Glass Transition

6. Landau theory for phase transitions State realized in the presence of a suitable constraint

7. Configurational space State realized in the presence of a suitable constraint Free energy landscape picture

8. Definition of the free energy landscape • Many basins appear below some temperature • Support fast and slow relaxations • Quasi-thermodynamic transition Potential energy landscape does not have these properties.

9. Configurational Partition Function for a constrained system Choice for the gate function ①　Within topologically identical Voronoi polyhedra: mathematically well-defined, but hard to calculate ② Gaussian fields: practical Free energy landscape Basic Concept for the FEL

10. Simultaneously and cooperatively rearranging regions SRR: Difference between two adjacent basins CRR: Atoms involved in the transition state

11. Glass formation as a function of Y. Singh et al PRL(1985), C. Kaur & S. P.Das PRL(2001) Free energy landscape Density functional theory

12. : Direct correlation function Ramakrishnan-Yussouff free energy functional Percus-Yevick approximation

13. Forced relaxation in FCC

14. No of atoms in the core ：32 basin1 basin2 Ｓｔｒｉｎｇ　ｍｏｔｉｏｎ and CRR

15. No of atoms in the core ：18 basin 2 basin 1 Ｓｔｒｉｎｇ　ｍｏｔｉｏｎ and CRR

16. No of atoms in the core ：10 basin2 basin1 Ｓｔｒｉｎｇ　ｍｏｔｉｏｎ and CRR

17. Density dependence of the size of CRR # of atoms in the core below which no relaxation occurs

18. Representative point in configurational space. Principal component analysisfor molecular simulations

19. Mode projection onto 3D-real space Fast Slow Total dynamics Slow dynamics Fast dynamics 600 K

20. FEL in Principal component analysis Probability distribution for yl ---The observed rate of yl in a simulation. FEL :

21. Dynamics on FEL 400 K (>Tg) 200 K (<Tg) yl+1 / λl+11/2 yl+1 / λl+11/2 yl / λl1/2 yl / λl1/2 2D contour maps of FEL’s.

22. :Size of CRR by Adam and Gibbs CRR Prob. of activation free energy Waiting time distribution SRR Waiting time distribution for slow relaxation

23. Characteristic Temperature Equation Unifying concept

24. Characteristic Temperature Equation V B Kokshenev & P D Borges, JCP 122, 114510 (2005)

25. Crystal Liquid T Unified understanding by the FEL -Crystallization

26. Ideal Glass Glass Super cooled Liquid Liquid slow relaxation fast relaxation Trapped in a basin T Unified understanding by the FEL -Vitrification

27. Phenomenological understanding Construction of free energy landscape Separation of slow dynamics Conclusion ○Dynamics: Gaussian to non-Gaussian transition ○Thermodynamics: Annealed to quenched transition ○Density functional theory ○Clear definition of CRR and SRR ○Principal component analysis ○Dynamics in the FEL