t. odagaki department of physics, kyushu university t. yoshidome, a. koyama, a. yoshimori and j....
TRANSCRIPT
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