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A new control parameter for the glass transition of glycerol.

D. L’Hôte, R. Tourbot, F. Ladieu, P. Gadige

Service de Physique de l’Etat Condensé (CNRS, MIPPU/ URA 2464), DSM/IRAMIS/SPEC/SPHYNX

CEA Saclay, France

Main Funding:

Additionnal Funding:

-4 -2 0 2 4

0

2

4

6

Tg(E

st)-

Tg(0

) , [

mK

]

Est , [MV/m]

Glycerol (Tg=187K at E

st=0)

The most emblematic claim of this work :

· Small effect: discovered through a nonlinear technique

The most interesting is not dTg but what we learn when varying P, Est.

Previously, the unique way to change Tg was the

Pressure P

Est is a new control parameter in Glycerol.

Outline:I) Motivations for nonlinear experiments

II) Our specially designed experiment III) Results on Glycerol

• Order of magnitude and comparison to the Box model • Relation to Ncorr

• Tg shift

IV) Some naives questions/ideas.

Summary and Perspectives.

How to combine the existence of correlations with the absence of order ?

12

10

8

6

4

2

0

-2

-4

Tg / T

Lo

g (

visc

osi

ty) t a

Angell, Science 267 (1995)

Tg / T0 0.2 0.4 0.6 0.8 1.0

Ea when T

T

Ea

e~~

Correlations when T

No (static) cristalline orderS

(q)

[a.u

.]

Polybutadiene, Tg 180K

What happens around Tg ???Relaxation time ta

tα

tα~ 10-12sTTmTg

LiquidSupercooled

liquid

Glass

ta=100s

(Crystal)

Dynamical Heterogeneities in supercooled liquids• Ncorr = average number of dynamically correlated molecules :

Hurley, Harowell, PRE, 52, 1694, (1995)

… directly observed in granular matter or in numerical simulations.

Example : numerical simulations on soft spheres :

3corrN

…Experimentally, the heterogeneous nature of the dynamics has been established through various breakthroughs:

• NMR experiments

• Local measurements

Tracht et al. PRL81, 2727 (98), J. Magn. Res. 140 460 (99),…

E. Vidal Russell and N.E. Israeloff , Nature 408, 695

(2000).

« clusters » of 30-90

monomers

When T: Ncorr would , which would explain why ta increases so much

Hole burning experiments

Dynamical Heterogeneities and NHB.

Non Res Hole Burning: supercooled dynamics IS heterogeneous (at least in time)

e.g. R.Richert’s group: PRL, 97, 095703 (2006); PRB 75, 064302 (2007); EPJB, 66, 217, (2008); PRL, 104, 085702, (2010)…

A distribution of t’s exists

e(t,w) : should be globally shifted in w

Many improvements since Schiener, Böhmer, Loidl, Chamberlin Science, 274, 752, (1996)

The central idea in Schiener et al ’s seminal paper in 1996:

… Can nonlinear experiments give MORE than originally expected ??....

Strong field (V0) at W No distribution of t

e(t,w) : should be mainly modified close to W

The prediction of Bouchaud-Biroli (B&B): PRB 72, 064204 (2005)

Natural scale of c3

cs= static value of cLin

a3= molecular volume

H

wta « 1 »

Ncorr= number of dynamic. correlated molec.

ta (T) : typical relaxation time

H: scaling function

...33

0

EEP

Lin

tieEtE )(

)()(),(32

03 THTN

Tk

aT corr

B

s

Systematic c3(w,T) measurements to test the prediction and possibly get Ncorr(T)

ctrprptpptrg ),()0,(),0()0,0(),(

sation characteriDH

4

AND Ncorr « large enough »

The issue of interpretations : Box Model versus B&B

c3(w,T) : Ncorr(T) or not ?

→ Each DH « k » has a Debye dynamics. {tk} chosen to recover clin(w) at each given T.

Box model assumptions (designed for NHB):

→ Applying E: each DH « k » is heated by dTk (ttherm) with ttherm tk.

as{tk}ta heat diffusion over one DH takes a macroscopic time close to Tg.

2~ ETk

)~(

~

,

3,,

EP

ETT

PPP

kLink

kLink

Linkk

c3 does NOT contain Ncorr (Box model is

space free)

)E'' ~density power(heat

c

densitypowerheat)(

2

k

kk

k Tt

T

c3

wta « 1 »

For a pure ac field Eac cos(wt): w and T dependences are

qualitatively similar in the Box model and in B&B

Some experiments done since B&B’s prediction (2005)

Using Est will shed a new light on this interpretation issue

→ Very good fits at 1w (better than at 3 )w

Box model : B&B:

e.g. R.Richert’s group: PRL, 97, 095703 (2006); PRB 75, 064302 (2007); EPJB, 66, 217, (2008); PRL, 104, 085702, (2010)…

Our group: PhD’s of C. Thibierge and C. Brun. PRL (2010); (2012) PRB (2011) (2012); JChem Phys (2011) ,

→ Accounts for the transient regime at 1w

→ Several liquids tested (Richert PRL (2007))

)Im(

)Im(''ln

)1(3

Lin

)3(3

)1(3 aswellas

Augsburg group: PhD of Th. Bauer : 2 PRL’s in (2013); etc…

→ Test of B&B’s prediction: OK

→ Evolution of Ncorr(T) or of Ncorr(ta)

→ Several liquids tested (Lunkenheimer & Loidl)


II) Our specially designed experiment III) Results on Glycerol

• Order of magnitude and comparison to the Box model • Relation to Ncorr

• Tg shift



)cos( tEac

')'()'()(

0

dttEtttP

lin

'3

'2

'1

'3

'2

'1

'3

'2

'13 )()()(,, dtdtdttEtEtEtttttt

Linear term

First non-linear term

tjtjac eeE 3)3(

3)1(

33 )( )(3Re

4

1

0

)(

LinPtP

),,(..)( 3)1(

3 TF

),,(..)( 3)3(

3 TF

46 10-10termlinear

termscubic MV/m,1For E Specially designed setup

Dielectric setup and orders of magnitude

P eSupercooled liquid, controlled T

VS (t)

stE

harmonicseven)(Re3 )1(1;2

2 tjacst eEE

)(tE

),0,0(..)( 3)1(1;2 TF

...For “low enough” E

Our setup to measure cubic susceptibilities

Vmeas

r1

Z2 = thickcapacitor(2×thin)

Z1 = thincapacitor

r2

Vac e j w t + Vst

Bridge with two glycerol-filled capacitors of different thicknessesC. Thibierge et al, RSI 79, 103905 (2008))

DV~ Vst2 Vac

Phase (DV) = cte acV

stV

V2

)1(1;2

when r1Z1=r2Z2 : Plin cancels

ILin +INonlLin

t

PSI

:N.B.

2 ILin +8 INonlLin

10

LinPP tjac eE )(Re

4

3 )1(3

3

tjacst eEE )(Re3 )1(

1;22

DV = Vmeas(Vac,Vst)- Vmeas(Vac,0)

1 1010-6

10-5

10-4

10-3

-100

0

100

200

300

Mod

ulus

, [V

]

Vst , [V]

Pha

se, [

deg]

gives PNonLin

Outline:I) Motivations for nonlinear experimentsII) Our specially designed experiment

III) Results on Glycerol• Order of magnitude and comparison to the Box model • Relation to Ncorr

• Tg shiftIV) Some naives questions/ideas.


NB: wta f/fa

1,E-10

1,E-09

1,E-08

1,E+00 1,E+02 1,E+04 1,E+06frequency [Hz]

C o

r G

/

[Far

ads]

'~ Lin

''~ Linfa peak of clin’’(w)

|clin(w)| has no peak

0.1 1 10 10010-17

10-16

10-15

-250

-200

-150

-100

-50

0

50

197K (f = 0.3 Hz) 202K (f = 2.1 Hz) 211K (f = 60 Hz) 218K (f = 520 Hz)

|

(1)

2;1|,

[m2 / V

2 ]

f/f

Arg

((1

)2;

1),

[Deg

]

At constant T:

humped shape for |c2;1

(1)|

maximum happens in the range of fa

Scaling of the hump in T

Main features of ),()1(1;2 T

Same qualitative trends as for c3(1) and c3

(3)

Comparing and ),()1(1;2 T ),()1(

3 T

For the first time, Box Model is unable to account for a cubic response:

Decisive point for the interpretation issue …

10

LinPP tjac eE )(Re

4

3 )1(3

3 tjacst eEE )(Re3 )1(

1;22 Compare |c3

(1)|/4 and |c2;1(1)|

→ Same order of magitude

10-1 100 101 1022x10-172x10-17

2x10-16

|X2,1

(1)|, T=202 K

|X3

(1)|/4

|(1)

2;1| o

r |(1

)

3|/4

f/f

, [m

²/V²]

10-1 100 101 1022x10-172x10-17

2x10-16

|X2,1

(1)|, T=202 K

|X3

(1)|/4

|(1)

2;1| o

r |(1

)

3|/4

f/f

→ Measurements ( ) are in the stationnary regime (tst>> ta)

Est

tst

tVarying Est ZERO dissipated power

powerdissipated~

:ModelBox In the

kTBox model’s prediction :|c2;1(1)|<< |c3

(1)|

Box model’s prediction is too small by a factor 300 for |c2;1(1)|

(ions)


II) Our specially designed experiment

III) Results on Glycerol• Order of magnitude and comparison to the Box model • Relation to Ncorr

• Tg shift



Comparing the w dependences of and of ),()1(1;2 T

0

stET

Lin

Direct link withdT

dχTn Linestim

corr ~ Berthier et al., Science (2005); JCP, (2007); PRE (2007).

For f/fa > 0.2:

,,0

),()1(1;2

TTT Lin

80.0, ²

²16102.1V

Kmwith

expected from

for both Re and Im parts T

For f/fa < 0.2: “Trivial” dominatesReshuffling Ideal gas at t >>ta

T

T Lin ),()1(1;2

0.1 1 10 10010-17

10-16

10-15

-250

-200

-150

-100

-50

0

50

|(1

)

2;1| o

r |

d L

in/d

T|, [

m2 /

V2 ]

f/for f/f

Arg

((1)

2;1)

or

Arg

(- d L

in/d

T),

[D

eg

]

218K

197K

T-dependences of the dimensionless nonlinear suscept. )(knX

195 200 205 210 215 2200.6

0.8

1.0

1.2

1.4

1.6

10-1 100 1010.05

0.1

|X(1

)

2;1|

|Z (T

)|/|Z

(202

K)|

T, [K]

|X2;1

(1)|

|TT|

|X3

(1 or 3)(T<204.7K)|

f/f

Tk

a

TTX

B

s

knk

n 320

)()( ,

),(

is T-independent in the trivial limit (ideal gas)

))(()( THTN kncorr if B&B’s prediction holds

“Trivial”)1(

1;2X looks OK

Similar T dependences for TTX Lin

kn forand)(

w and T dependences consistent with X2;1(1) ~ Ncorr (OK within MCT)

Each Dyn.Het. µ = µm

Can we fit nonlinear resp. ? The ‘‘toy model’’ as an attempt :

//z

corrB

corr NTk

tENmµe ~

)(where

corrcorr

corr

NaN

Nmµ 1~

3M

corr

Lin

N~

1~

3

corr

corr

corr

corrLin

N

ENeP

EN

NeP

3~~

~~

33 Μ

Μ

ePt

P

eth

chM

Simplest example: D=0=q1

can it fit the data ? …

Two key points

Amorphous Order («as» in S.G.)corrNµ ~

Crossover to trivial is enforced at f<<fa

in a double well (to get long t),of assymetry D

Fits at Tg+17K:

Ncorr has the right order of magnitude good fits for ALL the Xn

(k)

… but with different values of Ncorr (toy model)

Ncorr=10d=0.60

Ncorr=15d=0.60

Ladieu, Brun, L’Hôte, PRB 85, 184207, (2012)

L’Hôte, Tourbot, Ladieu, Gadige to appear in PRB (2014)



III) Results on Glycerol• Order of magnitude and comparison to the Box model • Relation to Ncorr• Tg shift



Hensel-Bielowka et al. , PRE (2004)Translating c2,1(1) as a dTg shift

Pressure experiments:

dTg(P) is drawn from :

Slight trivial distortion of

c2,1(1) l≠1

)(;0;;; gTTPTP

Same method for Est:

)(;0;;; stst ETTPTEP g

1with,,0

),()1(1;2

TT

T Lin

dTg=3kEst2 Est is a new control parameter in glycerol

2st

E3)st

(Eg

T

A picture: D.H. overcrowded subway

Density … S and ta

Increasing Est …

Est … S and ta

Increasing Pressure …

Ncorr



III) Results on Glycerol• Order of magnitude and comparison to the Box model • Relation to Ncorr• Tg shift



Does TK follow Tg ? We find dTg (Est) > 0

|Est |

TTK Tg

« true » glass

« Supercooled liquid »|Hst |

Tc

Spin Glass Paramagnet

S.G: dTg (Hst) < 0 or “very negative”

Is it allowed that dTK(Est)>0 ? e.g. peculiarity of 1RSB ?

T

Est slows down the dynamics

S.G.: Hst speeds up the relaxation

Not a contradiction since T>Tg differs from T<Tc

Y.G. Joh et al, PRL 82, 438 (1999)

and Saclay data

Can we test theories with cubic responses?L. Levy et T. Oglieksi PRL57, 3288 (1986), L. Lévy, PRB, 38, 4963 (1988)

c 3 (a

.u.)

0.01 0.1 1 10

0.8

1.0

1.2

1.4

(s)

Nco

rr [a

.u.]

Above Tg Too narrow to test

theories relating ta and Ncorr

Quench at Tg-7K

In Spin Glasses: critical

behavior nicely evidenced

Nco

rr /N

corr(e

q)

ta, [s]

Brun et al. PRL109, 175702 (2012)

Brun et al. PRB 84, 104204 (2011)

ta, [s]

Quasi equilibrium

𝑇 −𝑇𝑐𝑇 𝑐

Can we test theories with cubic responses?

AND

We can test theories

0.01 0.1 1 10

0.8

1.0

1.2

1.4

(s)

Nco

rr [a

.u.]

Above Tg

Brun et al. PRB 84, 104204 (2011)

ta, [s]10000 20000 30000

0.90

0.95

1.00

12mHz 36mHz 106mHz 1.1Hz 11Hz

Nco

rr [a

.u.]

ta, [s]

Brun et al. PRL109, 175702 (2012)

Consistent with RFOT

NB: yields z>20 : not reasonnable

Tk

N

B

corr3/

0Ln

2/3

TkB

)20(w

)(w)(

msW

0Ln)w(

10

0Ln)v(

T)20(v

)(v)(

msV

Consistent with

Cammarota et al, JCP (2009)

zNcorr )(~0

)20(N

)(N

mscorr

corr

Below Tg

Recent works on cubic responses

kjikji

kji SSStJH

,,

,,

How to choose a good model ? corrNµ ~

A model accounting for c3 will ALSO capture dyn. spatial aspects ☺

→ In colloids: experiments and MCT done for oscillatory shear : See Brader et al, PRE, 82, 061401, (2010) MCT for oscillatory stress is coming: see Matthias Fuchs papers (Konstanz univ.)

t / ta 1

tJJ kjikji ,,,, 0

→ In supercooled liquids: Bouchaud Biroli PRB 72, 064204 (2005) and Tarzia et al, JCP (2010) in MCT· G. Diezemann : PRE, (2012); JCP(2013); arXiv: 1407.4333v1

Þ no general link between c3 and four times correlation functions (trap models…)Þ This link depends on the chosen model and chosen observable.

AND crossover to trivial

withWhy not simulate ? (d=3)