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
XT0T )( CT
Thermodynamic transition
gT KT
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)
aE
),( tTPa
Energy of basin a
Probability of being in basin a at t
),(),( tTPEtTE aa
a
0
0000
),(),(),(
TT
tTEtTEttTC
)0,(TC
),( TC
： Quenched
： Annealed
a
)10,10,10( 642coolCt
)10( 2heatCt
☆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, 044505 (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
CTT
CTT
State realized in the presence of a suitable constraint
}{ iRConfigurational 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
..}){,,,(ln}){,,,( EKRNVTQTkRNVTF iBi
Configurational Partition Function for a constrained system
])(exp[})({ 2 i
iiii RrCRr
})({ ii Rr Choice for the gate function
① Within topologically identical Voronoi polyhedra:
mathematically well-defined, but hard to calculate
② Gaussian fields: practical
Niiii drdrRrrVN
RNVTQ 1})({})]({exp[!
1}){,,,(
Simultaneously and cooperatively rearranging regions
SRR: Difference between two adjacent basins
CRR: Atoms involved in the transition state
108
523.0
N
Density functional theoryfielddensity : )(r potential Grand: )]([ r
][)]}{,([ lrandomi RcTT
Glass formation
Y. Singh et al PRL(1985), C. Kaur & S. P.Das PRL(2001)
)0(
}){,(])(exp[)( 2i
iiC RRrr
})]{,([})({ ii RR
as a function of }{ iR
Free energy landscape
))()()(()(2
1
))(()(
log)(][)]([
212121 ll
ll
l
cdd
dd
rrrrrr
rrr
rrr
: Direct correlation function)(rc
Percus-Yevick approximation
Ramakrishnan-Yussouff free energy functional
108
523.0
N
Forced relaxation in FCC
basin1 basin2
No of atoms in the core ： 32555.0 362
Ｓｔｒｉｎｇ ｍｏｔｉｏｎ and CRR
No of atoms in the core ： 18501.0 362
basin 1 basin 2
Ｓｔｒｉｎｇ ｍｏｔｉｏｎ and CRR
basin1 basin2
No of atoms in the core ： 10501.0 32
Ｓｔｒｉｎｇ ｍｏｔｉｏｎ and CRR
Density dependence of the size of CRR
CRRN # of atoms in the core below which no relaxation occurs
Principal component analysisfor molecular simulations
Representative point in configurational space.
( )x t
N
ttyty
3
1T
0
0
)()(
lll VV
A
ttxtx T)()(
A
N
l llii tyVtx3
1 , )()(),()( T txVty ll
Mode projection onto 3D-real space
Fast Slow Total dynamics
Slow dynamics
Fast dynamics
600 K
N
Ll llii tyVtx3
1 ,Slow )()(
L
l llii tyVtx1 ,
Fast )()(
N
l llii tyVtx3
1 , )()(
FEL in Principal component analysis
FEL :
---The observed rate of yl
in a simulation.
( ) log ( )l l B l lf y k T p y
Probability distribution for yl
total,obs,obs /)()( lllll ZyZyp
)(,obs ll yZ
)(,obstotal
,obs llll yZdyZ
Dynamics on FEL
yl / λl1/2
400 K (>Tg) 200 K (<Tg)
2D contour maps of FEL’s. 10 1, / 2 3 10 [ ]eff l s
y l+1 / λ
l+11/
2
y l+1 / λ
l+11/
2
yl / λl1/2
Waiting time distribution for slow relaxation
g
dgCgp0
])(exp[)()( Prob. of activation free energy
2)( tt
)(
)()(1
)(*
gcg
gcgcc
TsT
TsTTTs
S
TkTs
Waiting time distribution
gneww 0
)(/* TsSn c :Size of CRR by Adam and Gibbs
SRR
CRR
Unifying concept
0t1 0T 0)( 0 Tsc
2t xT10 gTt 0D
2)(
)(
0
0
TT
TT
TsT
TsT
g
X
gcg
XcX
)/(
12
0TTT
T
gg
x
Characteristic Temperature Equation
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
Unified understanding by the FEL -Crystallization
T
Liquid
}{ iR
mT
Crystal
TCT
Super cooled Liquid
slow relaxation fast relaxation
Glass
gT
t
Ideal Glass
Trapped in a basin
KT
Liquid
}{ iR
Unified understanding by the FEL -Vitrification
ConclusionPhenomenological 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
