T. Odagaki Department of Physics, Kyushu University T. Yoshidome, A. Koyama, A. Yoshimori and J. Matsui Japan-France Seminar, Paris September 30, 2005

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

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

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

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 Outline

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

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

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.

Basic Concept for the FEL Free energy landscape 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

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

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

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

Forced relaxation in FCC

basin1basin2 No of atoms in the core ： 32 Ｓｔｒｉｎｇ ｍｏｔｉ ｏｎ and CRR

No of atoms in the core ： 18 basin 1 basin 2 Ｓｔｒｉｎｇ ｍｏｔｉ ｏｎ and CRR

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

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

Principal component analysis for molecular simulations Representative point in configurational space.

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

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

Dynamics on FEL y l / λ l 1/2 400 K (>T g ) 200 K (<T g ) 2D contour maps of FEL’s. y l+1 / λ l+1 1/2 y l / λ l 1/2

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

Unifying concept Characteristic Temperature Equation

V B Kokshenev & P D Borges, JCP 122, (2005)

Unified understanding by the FEL -Crystallization T Liquid Crystal

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

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