viscosity as a function of tg/tperso.ens-lyon.fr/sergio.ciliberto/teaching/cours_phys...viscosity as...
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Viscosity as a function of Tg/T
• Tg is the glass transition temperature
• At Tg the viscosity is about 1012 Pa s
• For T>Tg the Young modulus falls down of several orders of magnitude
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Mechanical measurements
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Dielectric measurements
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Relaxation times
1/Tg
Vogel-Fulcher-Tamman law for T>Tg
where To<Tg
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Type of glasses
• Structural glasses
• Magnetic glasses
•Colloids
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Frustation
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Energy landscape
Bouchaud trap model
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Aging and Memory effect in a polymer
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Aging of PMMA ( Tg= 388K )
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= dielectric constant measured with continuous ramp
Tg
Tg
Reference curveCurve with a cooling stop
= dielectric constant measured with a cooling stop
Memory effect in PMMA
Evolution of ε at f=0.1Hz as a function of T
Reference curve
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Memory effect in PMMA
Evolution of ε at f=0.1Hz as a function of T
= dielectric constant measured without a cooling stop
= dielectric constant measured with a cooling stop
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Memory effect in spin glasses
From: V. Dupuis, E. Vincent, J.P. Bouchaud, J. Hammann, A.Ito, H. Aruga Katori, Aging, rejuvenation and memory effects in Ising and Heisenberg spin glasses, Phys. Rev B 64 (17),174204,(2001). Also in cond-mat/0104399
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Kovacs Effect
Measure of the Volume
(I) direct quench from T0 to T2
(II) Temperature changefrom T1 to T2
(III) The same as II but for a larger T1
Fast quench
Slow ramp
- The sample reminds its thermal history - The response of the system depends on the quench speed
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Memory effects and trap model
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Smart experimental procedures, based either
on multiple cycles of cooling, heating and waiting times
or
on the modulation of the applied external fields
have shown the existence of spectacular effectsof aging in glassy materials, such as
rejuvenation and memory.
These studies have been extremely useful to fix severalimportant constraints for the phenomenological models of aging.
Aging has been often characterized by studing the response functions of the systems
Question: is the analysis of fluctuations useful ?
Aging in glassy materials
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1) Thermal fluctuations and the Fluctuation Dissipation Relations during aging
2) The electrical thermal noiseof two materials: a) a polymer after a quench b) a colloidal glass during the sol-gel transition.
3) Comparisons of the experimental results with those of other experiments and of models of aging.
4) The mechanical noise.
5) Conclusions
Outline
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V and q are two conjugate variables
δV(ω)
δq(ω)R(ω) = is the response function
The thermal fluctuation spectrum S(ω)= <| V(ω)| 2 > is
S(ω)= Im{ R(ω)}ω
4 K T
Typical examples are :
<|U ( )|> 2
ω = 4 K T Ro
Nyquist
U
Density fluctuations
L
H
M
< M> 2
∼ ΚΤ χm''
ω E''<| L ( )|>
2
δ ω ∼
F
(U,q) (L,F) (M,h)
FLUCTUACTION DISSIPATION THEOREMin thermodynamic equilibrium
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In a glass at T < TG the physical properties of the material depend on the aging time tw after the temperature quench. Thus FDR takes the following form:
FDR can be used to define an effective temperature of the system
Fluctuation Dissipation Relation (FDR)in a weakly out of equilibrium system
(Cugliandolo,Kurchan 1992.)
In terms of correlation function FDR takes the form
At equilibrium
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in an aging Lennard-Jones glass, Europhys. Lett. 46, 637 (1999)
Correlation versus τ
Response versusτ
Response versus correlation
slope =-1/T
slope =-1/Teff
KOB , BARRAT, Fluctuation dissipation ratio
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FDR in out of equilibrium system
Theoretical Background
1) This definition of temperature seems to be appropriate for several systems. Cugliandolo, Kurchan, Peliti (1997), Kob, Barrat (1999)Berthier, Barrat (2002), Liu, Nagel (2002), Sciortino(2002)………………..
2) The robustness of this definition of temperature has been questioned . S. Fielding, P. Sollich, (2002), Perez-Madrid, Reguera,Rubi (2002).
Experiments
- 1970 x-ray scattering on PMMA (Weandorf and Fisher)
- 1999 Grigera, Israeloff, super-cooled liquid
- 2001 Bellon, Ciliberto, sol-gel transition
- 2002 Herisson and Ocio, spin-glass
- 2002 et 2005 Buisson, Ciliberto, polymer
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X-ray experiments
slope =-1/T
?
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Spin Glass experiment
Insulating spin glass
Herisson and Ocio
Tg=16K
Quench at 0.8Tg.
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Spin Glass experiment
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Spin Glass experiment
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Spin Glass experiment