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:
The most emblematic claim of this work :
Tg(Est)-Tg(0) , [mK]
Glycerol (Tg=187K at Est=0)
6
 Est is a new control
parameter in Glycerol.
4
 Previously, the unique
way to change Tg was the
Pressure P
2
0
-4
-2
0
2
4
Est , [MV/m]
 Small effect: discovered through a nonlinear technique
 The most interesting is not dTg but what we learn when varying P, Est.
I)
Outline:
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.
What happens around Tg ???
tα
Relaxation
time ta
Glass
Supercooled
liquid
Liquid
(Crystal)
ta=100s
tα~ 10-12s
T
Tm
Tg
Angell, Science 267 (1995)
 Correlations 
when T 
10
8
6
S(q) [a.u.]
Ea  when T 
12
Log (viscosity)  ta
 ~t a ~ e
Ea
T
Polybutadiene, Tg 180K
4
2
0
-2
-4
0
0.2
0.4T / 0.6
g T
Tg / T
0.8
1.0
No (static) cristalline order
How to combine the existence of correlations with the absence of order ?
Dynamical Heterogeneities in supercooled liquids
• Ncorr = average number of dynamically correlated molecules : N corr   3
… directly observed in granular matter
or in numerical simulations.
…Experimentally, the heterogeneous nature
of the dynamics has been established
through various breakthroughs:
Example : numerical simulations on soft
spheres :
• NMR experiments
• Local measurements
Hurley,
Harowell,
PRE, 52,
1694, (1995)
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
 Hole burning experiments
When T: Ncorr would , which would explain why ta increases so much
Dynamical Heterogeneities and NHB.
 Many improvements since Schiener, Böhmer, Loidl, Chamberlin Science, 274, 752, (1996)
e.g. R.Richert’s group: PRL, 97, 095703 (2006); PRB 75, 064302 (2007); EPJB, 66,
217, (2008); PRL, 104, 085702, (2010)…
 The central idea in Schiener et al ’s seminal paper in 1996:
No distribution of t
Strong field (V0) at W
A distribution of t’s exists
e(t,w) : should be globally
shifted in w
e(t,w) : should be mainly
modified close to W
Non Res Hole Burning: supercooled dynamics IS heterogeneous (at least in time)
… Can nonlinear experiments give MORE than originally expected ??....
The prediction of Bouchaud-Biroli (B&B): PRB 72, 064204 (2005)
DH characterisation 

       
g 4 (r , t )  p (0,0) p (0, t ) p (r ,0) p (r , t )
AND
c
E (t )  E eiw t
c 3 (w , T ) 
e0c a
2
s
k BT
3
Ncorr « large enough »
P
e0
 c Lin E  c 3 E 3  ...
N corr (T ) H wt a (T ) 
Natural scale of c3 Ncorr= number of dynamic. correlated molec.
H
cs= static value of cLin ta (T) : typical relaxation time
a3= molecular volume H: scaling function
«1»
Systematic c3(w,T) measurements to test the prediction and
possibly get Ncorr(T)
wta
The issue of interpretations : Box Model versus B&B
Box model assumptions (designed for NHB):
→ Each DH « k » has a Debye dynamics.
{tk} chosen to recover clin(w) at each given T.
→ 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.
tk
 (dTk )
heat power density
 dTk 
t
c
(heat power density ~ c k ' ' w E 2 )
c3
dTk ~ E 2
( Pk , Lin
For a pure ac field Eac cos(wt):
w and T dependences are
qualitatively similar in the Box
model and in B&B
«1»
Pk  Pk , Lin 
Pk , Lin
dTk ~ E 3
T
~ ck E)
c3 does NOT contain
Ncorr (Box model is
space free)
wta
c3(w,T) : Ncorr(T) or not ?
Some experiments done since B&B’s prediction (2005)
 Im(c 3(1) )
Box model : d ln e ' ' 
Im(c Lin )
e.g. R.Richert’s group: PRL, 97, 095703
(2006); PRB 75, 064302 (2007); EPJB, 66,
217, (2008); PRL, 104, 085702, (2010)…
(1)
( 3)
B&B: c 3 as well as c 3
Our group: PhD’s of C. Thibierge and C. Brun.
PRL (2010); (2012) PRB (2011) (2012); JChem Phys (2011) ,
Augsburg group: PhD of Th. Bauer : 2 PRL’s in
(2013); etc…
→ Very good fits at 1w (better than at 3w
→ Accounts for the transient regime at 1w
→ Several liquids tested (Richert PRL (2007))
→ Test of B&B’s prediction: OK
→ Evolution of Ncorr(T) or of Ncorr(ta)
→ Several liquids tested (Lunkenheimer & Loidl)
Using Est will shed a new light on this interpretation issue
I)
Outline:
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.
Dielectric setup and orders of magnitude
VS (t)
Supercooled liquid,
controlled T
P(t )
e0

c

P

lin

e
For “low enough” E


(t  t ' ) E (t ' )dt '    c 3 t  t1' , t  t 2' , t  t3' E (t1' ) E (t 2' ) E (t3' ) dt1' dt 2' dt3'
 ...

Linear term
First non-linear term
E (t )  Eac cos(wt )  Est
P(t )  PLin
e0





1 3
2
(1)
 j wt
 even harmonics
Eac Re 3c (31) (w )e  jwt  c (33) (w )e  j 3wt  3Est Eac Re c 2;1 (w )e
4
c 3(1) (w )  F .T .{c 3 (w , w , w )
c 3(3) (w )  F .T .{c 3 (w , w , w )
For E  1 MV/m,
c 2(1;1) (w )  F .T .{c 3 (0,0, w )
cubic terms
10 6 -10  4
linear term
 Specially designed setup
Our setup to measure cubic susceptibilities
C. Thibierge et al, RSI
79, 103905 (2008))
Bridge with two glycerol-filled capacitors of different thicknesses

3
 P  PLin 

  Eac3 Re c (31) (w )e  jwt
4
 e 0 1w
Vac e j w t + Vst
r1
N.B. : I  S
r2
 3Est2 Eac Re c (21;)1 (w )e  jwt
DV = Vmeas(Vac,Vst)
- Vmeas(Vac,0)
2 ILin +
8 INonlLin
P
t
 when r1Z1=r2Z2 : Plin cancels
gives PNonLin
-3
10
300
Modulus, [V]
Vmeas
ILin +
INonlLin

Z1 = thin
capacitor
-4
200
10
100
-5
10
0
Phase, [deg]
Z2 = thick
capacitor
(2×thin)

-100
-6
10
1
DV~ Vst2 Vac
Phase (DV) = cte
10
Vst , [V]
c 2(1;1) 
DV
V2V
st ac

I)
Outline:
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.
 NB: wta  f/fa
fa peak of clin’’(w)
|clin(w)| has no peak
C or G/w [Farads]
1,E-08
~ c Lin '
1,E-09
~ c Lin ' '
1,E-10
1,E+00
1,E+02
1,E+04
frequency [Hz]
1,E+06
Main features of c 2(;11) (w , T )
50
-15
10
At constant T:
-16
(1)
10
-100
-150
-200
-17
10
(1)
2
197K (fa = 0.3 Hz)
202K (fa = 2.1 Hz)
211K (fa = 60 Hz)
218K (fa = 520 Hz)
Arg(c2;1), [Deg]
-50
2
|c2;1|, [m / V ]
0
 humped shape
for |c2;1(1)|
 maximum
happens in the
range of fa
Scaling of the
hump in T
-250
0.1
1
f/fa
10
100
Same qualitative trends as for c3(1) and c3(3)
(1)
c
Comparing c (w , T ) and 3 (w , T )
(1)
2;1




3 3
Eac Re c (31) (w )e  jwt  3Est2 Eac Re c (21;)1 (w)e jwt  Compare |c3(1)|/4 and |c2;1(1)|
4
→ Same order of magitude
→ Measurements (  ) are in the
stationnary regime (tst>> ta
2x10
2;1
(1)
3
|c2;1| or|c(1)
|c|3 or|/4|c,(1)[m²/V²]
|/4
 P  PLin 

 
e
0

1w
2x10
Est
-16
-16
tst
(1)
|X2,1 |, T=202 K
(1)
(1)
|X3 |/4
2x10
-17
(1)
10
2x10
-1
10
|X2,1 |, T=202
0
1 K
f/fa
(1)
|X3 |/4
10
10
-17
2
t
Varying Est  ZERO dissipated power
(1)|<<2 |c (1)|
-1 prediction
0
1
Box model’s
:|c
3
10
10
10 2;1
10
f/fa
In the Box Model :
dTk ~ dissipated power
(ions)
Box model’s prediction is too small by a factor 300 for |c2;1(1)|
For the first time, Box Model is unable to account for a cubic response:
Decisive point for the interpretation issue …
I)
Outline:
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.
0
197K
-50
10
-100
-16
218K
-150
(1)
-200
10
-17
-250
0.1
1
10
f/faor f/fa
estim
~T
Direct link with ncorr
100
(1)
10
2
2
|c2;1| or | dcLin/dT|, [m / V ]
50
-15
Arg(c2;1) or Arg(- dcLin/dT), [Deg]
Comparing the w dependences of c 2(;11) (w , T ) and of
 c Lin 


 T  E  0
st
 For f/fa > 0.2:
 c Lin 
c (T , w )   

 T 0, T , w

Km
²
 16
,   0.80
with   1.2 10
(1)
2;1
V²
T
for both Re and Im parts
dχ Lin
expected from Berthier et al., Science (2005);
dT
JCP, (2007); PRE (2007).
 For f/fa < 0.2: “Trivial” dominates
Reshuffling  Ideal gas at t >>ta
 c Lin 
(1)
c
(
w
,
T
)






2;1
 T 
T-dependences of the dimensionless nonlinear suscept. X n(k )
c n( k ) w , T 
(k )
X n (w , T ) 
 e 0 c s2 a 3 


 k BT 
is T-independent in the trivial limit (ideal gas)
 N corr (T ) H nk (wt a (T )) if B&B’s prediction holds
(1)
|X2;1|
|Z (T)|/|Z (202K)|
1.6
1.4
(1)
 “Trivial” X 2;1 looks OK
0.1
1.2
0.05
-1
0
10
1.0
10
(1)
f/fa
|X2;1 |
0.8
1
10
|TcT|
 Similar T dependences
(k )
for X n and for T c Lin T
(1 or 3)
|X3
0.6
195
(T<204.7K)|
200
205
210
215
220
T, [K]
w and T dependences consistent with X2;1(1) ~ Ncorr (OK within MCT)
Can we fit nonlinear resp. ? The ‘‘toy model’’ as an attempt :
Each Dyn.Het. 
µ = µm 𝑵𝒄𝒐𝒓𝒓
in a double well
(to get long t),
of assymetry D
Simplest example: D=0=q1
t P
ch e t
 P  M th e
µm N corr E (t )
where e 
~ N corr
k BT
µm N corr
1
M
~
3
N corr
N corr a
𝐸 //z
PLin ~ Μ e ~
P3 ~ Μ e
3

~
N corr
N corr
N corr E
E
3
N corr
Two key µ ~ N corr  Amorphous Order («as» in S.G.)
points
Crossover to trivial is enforced at f<<fa
c Lin ~ 1
c 3 ~ N corr
can it fit the
data ? …
Fits at
Tg+17K:
L’Hôte, Tourbot, Ladieu,
Gadige to appear in PRB
(2014)
Ncorr=10
d=0.60
Ncorr=15
d=0.60
 Ncorr has the right
order of magnitude
 good fits for ALL
the Xn(k)
 … but with
different values of
Ncorr (toy model)
Ladieu, Brun, L’Hôte, PRB
85, 184207, (2012)
I)
Outline:
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.
Translating c2,1(1) as a dTg shift
Hensel-Bielowka et al. , PRE (2004)
Pressure experiments:
dTg(P) is drawn from :
Pw ; P; T   P w ;0; T  dTg (P ) 
Same method for Est:
Pw ; Est ; T   P w ;0; T  dTg ( Est ) 
dT (E )  3E 2
g st
st
 c 
c 2(;11) (T , w )    Lin 
with   1
w
 T 0, T ,

Slight trivial distortion
of
c2,1(1) ≠1
dTg=3Est2  Est is a new control parameter in glycerol
A picture: D.H.
 overcrowded
subway
Increasing Pressure …
Density … S  and ta
Ncorr
Increasing Est …
Est … S  and ta
I)
Outline:
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.
Does TK follow Tg ?
|Est |
 S.G: dTg (Hst) < 0 or “very negative”
 We find dTg (Est) > 0
« Supercooled liquid »
« true »
glass
|Hst |
Spin
Glass
TK
Tg
T
Paramagnet
T
Tc
Is it allowed that dTK(Est)>0 ?  e.g. peculiarity of 1RSB ?
 Est slows
down the
dynamics
 S.G.: Hst speeds up the relaxation
Y.G. Joh et
al, PRL 82,
438
(1999)
and
Saclay
data
Not a contradiction since T>Tg differs from T<Tc
Can we test theories with cubic responses?
Ncorr [a.u.]
1.4
Too narrow
to test
theories
relating ta
and Ncorr
Above Tg
1.2
1.0
0.8
Ncorr /Ncorr(eq)
0.01
0.1
ta, [s]
ta (s)
1
10
Quench at
Tg-7K
Brun et al. PRL109,
175702 (2012)
ta, [s] Quasi equilibrium
L. Levy et T. Oglieksi PRL57, 3288
(1986), L. Lévy, PRB, 38, 4963 (1988)
𝑇−𝑇𝑐
𝑇𝑐
c3 (a.u.)
Brun et al. PRB 84, 104204 (2011)
In Spin
Glasses:
critical
behavior nicely
evidenced
Can we test theories with cubic responses?
Brun et al. PRB 84, 104204 (2011)
Above Tg
1.00
1.2
AND
1.0
0.8
Ncorr [a.u.]
Ncorr [a.u.]
1.4
Brun et al. PRL109, 175702 (2012)
Below Tg
0.95
12mHz
36mHz
106mHz
1.1Hz
11Hz
0.90
0.01
   k BT
0.1
ta, [s]
ta (s)
  3/ 2
t
w(t )  Ln a
a
t
 0
 
0
 1
t
v(t )  T Ln a
a
t
 0
1
10
t
Ln a
t
 0

 N

k BT

 /3
corr
10000
ta, [s]
20000
w (t ) Consistent
a
W (t ) 
a w (20ms) with RFOT








Consistent with
v(t )
a
V (t ) 
a v(20ms) Cammarota et al,
JCP (2009)
t
NB: a ~ ( N corr ) z yields z>20 : not reasonnable
t
0
We can test theories
N corr (t )
a
N corr (20ms)
30000
Recent works on cubic responses
→ 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.)
→ 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
 no general link between c3 and four times correlation functions (trap models…)
 This link depends on the chosen model and chosen observable.
How to choose a good model ?
µ~
N corr
AND crossover to trivial
A model accounting for c3 will ALSO capture dyn. spatial aspects ☺
Why not simulate ?
(d=3)
H

<i , j , k >
 a S S S
J i , j ,k t t
i
j
J i , j ,k 0J i , j ,k t t a 
k
with
1
t / ta