Free Energy Free Energy LandscapeLandscape

-Evolution of the -Evolution of the TDM-TDM-

Takashi Odagaki

Kyushu University

IV WNEP Round Table Discussion September 22, 2006

Trapping diffusion Model

What do we have to What do we have to understand?understand?

ThermodynamicsThermodynamics What is the transition?What is the transition? Cooling rate dependence?Cooling rate dependence? TTgg? T? TKK??

Slow process

Fast process DynamicsDynamics Slow dynamics? TSlow dynamics? T00?? Fast dynamics?Fast dynamics? TTgg? T? Txx??

How do we proceed?How do we proceed?

Phenomenological UnderstandingPhenomenological Understanding of Experimentsof Experiments

Fundamental TheoryFundamental Theory

New ParadigmNew Paradigm

2nd order Phase Transition

Glass transition

Configuration

Fre

e en

ergy gTT

gTT

Degree of quenching (t obs)=

basins of #

visitedbasins of #1

Annealed

quenched

C

gTT

disordered

ordered

C

CTT

doq

T

T

CTT

CTT F waittgTAt

Slow & fast relaxations

Trapping diffusion model

• Single particle description of the FEL picture

2)( tt

)(

)()(

gcg

gcgc

TsT

TsTTTs

Waiting time distribution

0t1 0T 0)( 0 Tsc

2t xT10 gTt 0D

• Findings

• Gaussian – non-Gaussian Transition at

•

• Characteristic Temperature Equation1

0 )(2 TTTT ggx

gT

0TTK

Characteristic Temperature Equation

V B Kokshenev & P D Borges, JCP 122, 114510 (2005)

g

C

T

T

0/TTg

g

C

T

T

0/TTg

• 20 basins:Einstein oscillators

slow

fast

Specific heat

Probability of being in basin a at t

aE

),( tTPa

Energy of basin a

),(),( tTPEtTE aa

a

i

ii TT

tTEtTEttTC

),(),(

),( 00

a

anneale

d

quen

che

d

•Findings

• Annealed to quenched transition

• Cooling rate dependence

• Characteristic behaviors of ac specific heat

ConclusionConclusion

The FEL picture is the only frame The FEL picture is the only frame work that provides unified work that provides unified understanding of the glass transition.understanding of the glass transition.

The FEL can be constructed by The FEL can be constructed by thethe

density functional approach.density functional approach.

RemarRemarkk

The 1st order ac specific heat

D

eb

ye

10-2 10-1 100 10110-1

100

101

p

ea

kw

0

(T-T0)/Tg

914.00 )(~ TT

gT

10-2 10-1 100

2

4

6

8

(T-T0)/Tg

)]/(022.0exp[04.1~ 0TTTg

gT

)75.0/( 0 TgT

Real part

0/Wpeak

Imaginary part

10-2

100

102

0

1T/T g=0.85

T/T g=0.85

T/T g=1.0

T/T g=1.0

T/T g=1.25

T/T g=1.25

(C 1()-C quench )/(C aneeal -C quench )~

/w 0

Return

10−2 100 102

−2

−1

0

1

/w0

(C"2()−C2(∞ )) (C2(0)−C2(∞ ))/~ ~ ~ ~

T/Tg=0.825T/Tg=0.875T/Tg=1.0T/Tg=1.25T/Tg=2.5

10−2 100 102

−2

−1

0

1

/w0

(C'2()−C2(∞ )) (C2(0)−C2(∞ ))/~ ~ ~ ~

T/Tg=0.825T/Tg=0.875T/Tg=1.0T/Tg=1.25T/Tg=2.5

The 2nd order ac specific heat Real part Imaginary part

10−2 10−1 100 101

10−2

100

=0.97

(T−T0)/Tg

Min of real part

Max of imaginary partMin of imaginary part

=1.05=1.01

Tg

)(~ 0TT

Return

Free energy landscape

)()( tuRtr iii

])(exp[})({ 2 i

iiii RrCRr For practical calculation

dRrrHN

RNVTZ iiii })({})]({exp[!

1}){,,,(

}){,,,(ln}){,,,( iBi RNVTZTkRNVT

ttt 0for )(trR ii where

Separation of microscopic and structural-relaxation time scales

))()()(()(2

1

))(()(

log)(][)]([

212121 ll

ll

l

cdd

dd

rrrrrr

rrr

rrr

: Direct correlation function

)(rc

Ramakrishnan-Yussouff free energy functional for hard spheres

])(exp[)( 2 i

iC Rrr

Simultaneously and cooperatively rearranging regions

SRR: Difference between two adjacent basins

CRR: Atoms involved in the transition state

108

523.0

N

return

No of atoms in the core ： 32555.0 362

String motion and CRRReturn