Les Houches 2007 : Flow in glassy systems
glasses plasticity
- Weak deformation in colloidal and polymer glasses, below the onset of yielding •aging (Struik)•effect of aging on yield stress
- Intermediate regimes- rejuvenation ? in colloids and in polymer
- Deformation in polymer glasses above the onset of yielding•mechanics and thermodynamics •structure : where is the internal stress ?
•Conclusion
-Background : -the dynamical “phase “ diagram -linear and non linear mechanics in the Eyring model
Thanks for many discussions to H. Montes, V. Viasnoff, D. Long, L. Bocquet, A. Lemaitre, and many others…………
Les Houches 2007 : Flow in glassy systems
Jamming at rest
Picture suggested by Liu and Nagel
Liu, Nagel Nature 1998
Les Houches 2007 : Flow in glassy systems
in practice, plastic flow can be observed only in limited cases
To study the effect of plastic flow, it is necessary- to avoid fracture- to avoid shear banding or flow localisation
Thus it is possible in practice :- polymer glasses ( but above T- colloidal glasses ( with repulsive particles) only below some volume fraction ( - foams in the absence of coarsening, but is there shear localisation ??)- granular material (but not at constant volume !)- simulation ( but at zero T, or during less than 1 ms)
Les Houches 2007 : Flow in glassy systems
•most of the experiments in this domain• our lecture
athermal systems :•foams•simulations
Les Houches 2007 : Flow in glassy systems
here, we will limit ourselves to the following case :
- glassy polymer or colloidal glasses, in the presence of aging
aging activated motions
Eyring model : the simplest model for glass plasticity
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
At equilibrium
Strain
Energy
E
Energy barrier : Ewaiting time for a hop : kTEe /
0
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
under stress
Strain
Energy
favourable
unfavourable
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
Strain
Energy
jump +
jump -
kT
vE
e.
0
kT
vE
e.
0
v is the activation volume (~ 10 nm3 for polymers)
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
Strain
Energy
jump +
jump -
kT
vE
e.
0
kT
vE
e.
0
shear rate :
kT
v
kT
v
kT
E
eee..
0
.111
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
Viscous fluid :
v
kTe
kT
E
..0
spontaneous relaxation time
linear regime : non-linear regime
Yield stress fluid :
).log()ln( 0
kTE
ev
kT
elastic modulus spontaneous
relaxation time
weak dependance on the shear rate measurement of velastic
modulus
kT
vE
e.
0
kT
vE
e.
0
~ kT
vE
e.
0
<< kT
vE
e.
0
Les Houches 2007 : Flow in glassy systems
Memo
•Linear regime is governed by spontaneous rearrangement ( that are slightly modified - biased - by the stress)
•In the non-linear regime, rearrangements – that are not present at rest - are induced by stress
in glass the energy landscape is more complex
Les Houches 2007 : Flow in glassy systems
from Eyring to glasses
spontaneous rearrangements at experimental time scale
Energy
Strain
Yielding
creep
Les Houches 2007 : Flow in glassy systems
Glassy systems are non-ergodic : they do not explore spontaneously enough phase space to flow ( at a given time scale) As a consequence they exhibit a Yield Stress
At opposite, ergodic systems exhibit a Newtonian flow regime - as a consequence of the fluctuation/dissipation theorem
Energy
Strain
Yielding
Les Houches 2007 : Flow in glassy systems
Creep experiments- in the linear regime - probe the spontaneous rearrangements :
experimental protocol
Aging systems
Thermal or mechanical rejuvenation (pre-shear !)
Rheological Test(creep /step-strain/…)
time
QuenchOr strain cessation
Waiting time
Energy
Straincreep
Les Houches 2007 : Flow in glassy systems
weak deformation in colloidal and polymer glasses, below the onset
of yielding
Les Houches 2007 : Flow in glassy systems
aging
Creep experiments- in the linear regime - probe the spontaneous rearrangements :
experimental protocol
Thermal or mechanical rejuvenation (pre-shear !)
Rheological Test(creep /step-strain/…)
time
QuenchOr strain cessation
Waiting time
Les Houches 2007 : Flow in glassy systems
Spontaneous rearrangements are getting slower and
slower
Colloïdal suspensions
Borrega, Cloitre, Monti, Leibler C.R. Physique 2000
Linear Creep flow reveals spontaneous rearrangements
Struik Book 1976
Glassy polymer
tw in days
Les Houches 2007 : Flow in glassy systems
leading to self-similar compliance evolution J(t,tw)=j(t/tw
) where ~1
Seen also by step-strain, light scattering……
Les Houches 2007 : Flow in glassy systems
It reveals a self-similar evolution of the time relaxation spectrum
Log
Time elapsed after « quench »
<> ~ t w
Dynamical measurements are very sensitive to aging
Les Houches 2007 : Flow in glassy systems
scaling argument for aging
Simple argument :lets D be the inverse of the relaxation time D =
lim D(tw) = tw
Thus D relaxes towards 0, with a time scale equal to
tends towards a time >> experimental time scale
2
)(D
t
D
t
D
ww
Thus : ctet
Dw
1 and ~ tw
Les Houches 2007 : Flow in glassy systems
scaling argument for aging
In practice, this argument is robust for any systems that are getting slower and slower
There are little deviations ( is not egal to 1- but always about 1).This is because there is a spectrum of relaxation time and not a single time
Otherwise, the scaling in t/tw is observed in any system that tends towards an infinitely slow dynamics – and is thus not specific of glasses
( counter example : floculating suspensions )
Les Houches 2007 : Flow in glassy systems
The drift of the relaxation time leads also to slow – logarithmic - drift of other
properties - yield stress, elastic modulus, density….
Time evolution of the transient stress overshoot for polymer (left) and colloidal suspensions (right) under strain
Derec, Ajdari, Lequeux Ducouret. PRE 2000Nanzai JSME intern. A 1999
Les Houches 2007 : Flow in glassy systems
aging and other properties
The same behavior – a logarithmic drift – is observed for yield stress and for other properties ( here calorimetry scanning).
The yield stress is thus a signature of the structure of the glass at rest.
Nanzai JSME intern. A 1999
Les Houches 2007 : Flow in glassy systems
- deformation around yielding
colloids(overaging)
polymer(cyclic plasticity )
There is a temptation to estimate that stress (or strain) has an effect opposite to annealing.(mechanical rejuvenation)
This is qualitatively OK for large strain, but ……
Les Houches 2007 : Flow in glassy systems
small deformations on colloidal glasses
7
8
9
0.1g 2(t
)-1
0.001 0.01 0.1 1 10 100t /s
0.1 s 1 s 60 s
100s, 1 Hz, 5.9%
Classical aging 100+ 0.1s
Classical aging 100+ 60s
Aging for tw0(=100s)+.1sfor tw0+60sWith stress at tw0 +.1sWith stress at tw0 +1sWith stress at for tw0+60s
Viasnoff, Lequeux PRL,Faraday Discuss 2002
Les Houches 2007 : Flow in glassy systems
small deformations on colloidal glasses
The time relaxation spectrum is deeply modified :
Its stretched both in the small and the large time part.
Log
before shear
after shearrejuvenation
overaging
Les Houches 2007 : Flow in glassy systems
Cyclic plasticity of polymer
In this state, the response is apparently linear, but the apparent modulus decreases with the amplitudeAfter sollicitation, the glass recovers slowly its initial properties.
When a polymer glass submitted a periodic strain of small amplitude, its structure evolves and reach a stationary state.
Small, but non-linear deformation brings the glass in a new state.This effect is poorly documented
Rabinowitch S. and Beardmore P. Jour Mat Science 9 (1974) p 81
Les Houches 2007 : Flow in glassy systems
Mechanical/Thermal effect on polymer glasses
Tmax =423 K
Tg
G*refc()
Tmin = 313K
Tstep , tdef
G*refh()G*mc
1 ()
Test cyclereference cycle
2
3
4
567
109
2
G' m
, G
' ref
(P
a)
400380360340320 T (K)
time
annealing
« memory » of annealing
Montes, Bodiguel, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
2
3
4
567
109
2
G' m
, G
' ref
(P
a)
400380360340320 T (K)
120
100
80
60
40
20
0
G' m
-G' re
f ( M
Pa
)
420400380360340320 T (K)
[2nd cycle] – [1st Cycle]
This effect is called the memory effect, and is observed in spin glasses.
This effect is often invoked to justify a spatial arangement of the dynamics(Bouchaud et al)
Montes, Bodiquel, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
120
100
80
60
40
20
0
-20
G' m
h-G
' refh
(M
Pa
)
420400380360340320 T (K)
Tstep = 343K Tstep = 353K Tstep = 363K
Indeed, this effect is described by the simple phenomenological model T.N.M.
It does not reveal anything else expect the fact that there is a large distribution of relaxation time
Montes, Bodiquel, Lequeux, in preparation
Nanzai JSME intern. A 1999
Les Houches 2007 : Flow in glassy systems
Phenomenological TNM model
• A fictive temperature Tf described the state of the system.
• The relaxation time is:
• Tf tends towards T with a typical time
• In order to take into account all the memory effects, introduce a stretched exponential reponse
).(0
0. TTTA
fe
')'()'(."
exp)(
)(
'
dttTtTdt
tT
TT
t
T
f
t t
t
f
ff
')'()'(."
exp)('
dttTtTdt
tT f
t t
t
f
This model described quantitatively most of the effects of complex thermal history
Les Houches 2007 : Flow in glassy systems
nG*mh1 ()
Tmax =423 K
Tg
G*refc()
Tmin = 313K
Tstep , tdef
G*refh()G*mc
1 ()
Second cycle
First cycle
1.10
1.08
1.06
1.04
1.02
1.00
0.98
0.96
G' m
h 1/ G
' refh
400380360340320 T (K)
1= 0 (simple memory effect)
1=0.5%
1=1%
1=1.5%
8x10-2
6
4
2
0
-2
[ G
' mh-
G' m
h 1]/
G' re
fh
400380360340320 T(°C)
1=1.5%
simple memory effect[G'mh - G'refh] / G'refh
mechanics
annealing at rest
effect of mechanics * (-1)
Use of the memory effect to probe small amplitude plasticity effect
Montes, Bodiquel, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
8x10-2
6
4
2
0
-2
[ G
' mh-
G' m
h 1]/
G' re
fh
400380360340320 T(°C)
1=1.5%
simple memory effect[G'mh - G'refh] / G'refh
annealing at rest
effect of mechanics * (-1)
Mechanics has not en effect opposite to simple thermal annealing. Under small amplitude mechanical sollicitation, the system undergoes a widening of its relaxation spectrum
Les Houches 2007 : Flow in glassy systems
deformation around yielding
The experimental situation is complex :
Strain is not equivalent to rejuvenation, but has the tendency to stretched the spectrum of relaxation time.
However, these experiments may be very good tests for future models.
Les Houches 2007 : Flow in glassy systems
deformation far above yieldingin polymers
Glassy polymer can be strained up to a few hundred %, without fracture, and homogeneously. In fact it is the reason why they are so often used in our everyday life !
It is well-known that a large strain erases the history.
Here we focuss on deformation ( below Tg) or cold-drawing, of about 200%.
Les Houches 2007 : Flow in glassy systems
Oleynik
Dissipated heatIrreversibly stored energyReversibly stored energy
Oleynik E. Progress in Colloid and Polymer Science 80 p 140 (1989)
0.A. Hassan and M.C. Boyce Polymer 1993 34, p 5085
Les Houches 2007 : Flow in glassy systems
A large amount of energy is irreversibly stored during cold-drawing.This energy is likely stored in internal stresses modes.
Its is transformed into heat while heating the sample, or during aging.
Dissipated heatIrreversibly stored energyReversibly stored energy
Les Houches 2007 : Flow in glassy systems
Temperature of plastic deformation
Exothermic heat inducedby plastic deformation
0.A. Hassan and M.C. Boyce Polymer 1993 34, p 5085
Les Houches 2007 : Flow in glassy systems
Retraction of polymer at zero stress after cold-drawing, while increasing temperature, exhibitingSpontaneous rearrangements
Mechanical dissipation observedin the same condition
Munch et al PRL 2006
Les Houches 2007 : Flow in glassy systems
Dynamical aspect of the internal stress softening.
Munch et al PRL 2006
Les Houches 2007 : Flow in glassy systems
deformation far above yielding
Conclusion
Plastic flow generates internal stress that stored a lot of energy.
This internal stress is released under any increase of temperature from thetemperature of cold drawing.
How is stored the energy ???
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
Under plastic deformation,An enhancement of the density fluctuation is observed (X, Positron Annihilation Spectroscopy (Hasan, Boyce)
Munch PRL 2006
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
Structure factor of labelld chains
Affine motion S(q)S(q*)
qq�
.*
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
Figure 2 : (a) Intensity scattered of a cold-drawn sample compared to the unstretched sample. Measurements were performed on a sample composed by 90% of crosslinked hydrogenated chains mixed to 10% of deuterated chains. d/dt=0.001 s-1.=1.8(b) : scattered intensity in reduced q-vector. Deviation from the affine motion clearly appears for large q-vectors.
0.01
0.1
1
10
100
d/d
(cm
-1)
4 5 60.01
2 3 4 5 60.1
2 3 4
Q (Å-1
)
Tstret=Tg-26
unstretched q
q//
(b)
0.01
0.1
1
10
100
d/d
(cm
-1)
4 5 60.01
2 3 4 5 60.1
2 3 4
Q* (Å-1
)
Tg-26 q*// = q/1/2
q* = q unstretched
(b)
affine
Towards isotropic
Casas, Alba-simionesco, Montes, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
0.060
0.055
0.050
0.045
0.040
0.035
0.030
0.025
q*c
(Å-1
)
420410400390380370
T (K)
TgBelow Tg, there is a crossover q-vector that doesn’t depend neither on strain rate, nor on temperature.
Above Tg this crossover length decreases ( and tends toward zero if shear rate << rep
Casas, Alba-simionesco, Montes, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
4.0
3.5
3.0
2.5
2.0
d/d
(cm
-1)
5 6 7 8 91
2 3 4 5 6 7 8 910q (Å
-1)
Stretched Tg+30, =2 q ; q// ;
unstretched
(a)
4.0
3.5
3.0
2.5
2.0d
/d
(cm
-1)
5 6 7 8 91
2 3 4 5 6 7 8 910q (Å
-1)
unstretched
Stretched Tg-26 =2 q ; q//
(b)
On the opposite, at the monomer scale, the structure is nearly isotropic !There is a slight « distortion » of the chains.
Casas, Alba-simionesco, Montes, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
affineIsotropic distorted
Crossover ~ few nanometers
The motions follow the macroscopic deformation
The structure remains isotropic at smallscale ( think about a liquid).But the chains are distorded
Les Houches 2007 : Flow in glassy systems
Probably, strain-hardening due to polymer topological contraints is responsible for the flow homogeneity at intermerdiate scale
Macroscopic strain-hardening
Streched domains have a larger yield stress
Unstretched domains that are softer are know strained
Plastic Strain self-homogeneize.
Natural fluctuations of yield stress
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
• Plastic flow is quite homogeneous in polymer ( because of local strain-hardening)
• At small scale the chains are nearly iscotropic but distorded
• The internal stress is stored at small scale (< 10 nm)
Les Houches 2007 : Flow in glassy systems
General conclusionYield stress and creep are signature of the structure of a glass ( and of its history)
Cyclic strain of small amplitude generates a new structure. It has the tendancy to widen the relaxation spectrum
Large deformations generate a lot of internal stress that is stored at small length-scale.Strain-hardening, which is specific to polymer glasses, tends to make large deformation homogeneous.
There aren’t any satisfactory models, even if most of the simple models capture qualitatively most of the effects for small and intermediate deformations.
Les Houches 2007 : Flow in glassy systems
GAME OVER