Transcript glasses plasticity

glasses plasticity
-Background :
-the dynamical “phase “ diagram
-linear and non linear mechanics in the Eyring model
- 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
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 Tb)
- colloidal glasses ( with repulsive particles) only below some volume fraction ( Fb ?)
- 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
t
Energy
E
Strain
Energy barrier : E
waiting time for a hop : t  t 0e E / kT
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
under stress s
favourable
unfavourable
Energy
Strain
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
Energy
Strain
jump + t   t 0e
jump - t   t 0e
E s .v
kT
E s .v
kT
v is the activation volume
(~ 10 nm3 for polymers)
Les Houches 2007 : Flow in glassy systems
Eyring’s Model
Energy
Strain
jump + t   t 0e
jump - t   t 0e
shear rate :
E s .v
kT
E s .v
kT
 
1

1
t t

1
t0
.e

E
kT
s .v

 skT.v

 e  e kT 




Les Houches 2007 : Flow in glassy systems
Eyring’s Model
t   t 0e
E s .v
kT
~
t   t 0e
E s .v
kT
linear regime :
Viscous fluid :
<<
t   t 0e
E s .v
kT
non-linear regime
Yield stress fluid :
E
kT
kT
  t 0 .e .
v
spontaneous
relaxation time
t   t 0e
E s .v
kT
elastic
modulus
kT
s
v
elastic
modulus
ln( )  log(t .e E kT )
0


weak dependance
on the shear rate 
measurement of v
spontaneous
relaxation time
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
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from Eyring to glasses
Yielding
Energy
Strain
creep
spontaneous rearrangements at experimental time scale
Les Houches 2007 : Flow in glassy systems
Yielding
Energy
Strain
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
Les Houches 2007 : Flow in glassy systems
Aging systems
Energy
Strain
creep
Creep experiments- in the linear regime - probe the
spontaneous rearrangements :
experimental protocol
Thermal or mechanical
rejuvenation (pre-shear !)
Quench
Or strain
cessation
Waiting time
Rheological Test
(creep /stepstrain/…)
time
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 !)
Quench
Or strain
cessation
Waiting time
Rheological Test
(creep /stepstrain/…)
time
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tw in days
Colloïdal suspensions
Glassy polymer
Borrega, Cloitre, Monti, Leibler C.R. Physique 2000
Struik Book 1976
Linear Creep flow reveals spontaneous rearrangements
Spontaneous rearrangements
are getting slower and slower
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leading to self-similar compliance evolution
J(t,tw)=j(t/twm) where m~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
r
Time elapsed after « quench »
<t> ~ t wm
Log t
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 ta.
D =1/ ta
lim D(tw) = e ~0
tw
ta tends towards a time >> experimental time scale
Thus D relaxes towards 0, with a time scale equal to ta
D
D

 D2
t w
t a (t w )
1
Thus : D 
t w  cte
and ta ~ 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 (m 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 2000
Nanzai JSME intern. A 1999
Les Houches 2007 : Flow in glassy systems
aging and other properties
Nanzai JSME intern. A 1999
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.
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- deformation around yielding
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 ……
colloids
(overaging)
polymer
(cyclic plasticity )
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small deformations on colloidal
glasses
100s, 1 Hz, 5.9%
0.1 s
1s
60 s
g2(t)-1
Classical aging
100+ 60s
0.1
9
Aging for tw0(=100s)+.1s
for tw0+60s
With stress at tw0 +.1s
With stress at tw0 +1s
With stress at for tw0+60s
8
Classical aging 100+ 0.1s
7
0.001
0.01
0.1
1
10
100
t /s
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.
r
before shear
rejuvenation
after shear
overaging
Log t
Les Houches 2007 : Flow in glassy systems
Cyclic plasticity of polymer
Rabinowitch S. and Beardmore P. Jour Mat Science 9 (1974) p 81
When a polymer glass submitted
a periodic strain of small
amplitude, its structure evolves
and reach a stationary state.
In this state, the response is apparently
linear, but the apparent modulus
decreases with the amplitude
After sollicitation, the glass recovers
slowly its initial properties.
Small, but non-linear deformation brings the glass in a new state.
This effect is poorly documented
Les Houches 2007 : Flow in glassy systems
Mechanical/Thermal effect on
polymer glasses
« memory » of annealing
reference
cycle
Test cycle
Tmin = 313K
G*mc1 (w0)
G*refc(w0)
G*refh(w0)
time
9
10
G'm , G'ref
Tstep , tdef
Tg
(Pa)
2
Tmax =423 K
7
6
5
4
3
2
320
340
360 380
T (K)
400
annealing
Montes, Bodiguel, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
[2nd cycle] – [1st Cycle]
9
10
G'm , G'ref
(Pa)
2
7
6
5
100
80
60
40
20
0
4
3
320
340
360
380
400
420
T (K)
2
320
G'm-G'ref ( MPa)
120
340
360 380
T (K)
400
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
G'mh-G'refh (MPa)
120
Tstep = 343K
Tstep = 353K
Tstep = 363K
100
80
60
40
20
0
-20
320
340
360
380
400
420
T (K)
Montes, Bodiquel, Lequeux, in preparation
Nanzai JSME intern. A 1999
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
Les Houches 2007 : Flow in glassy systems
Phenomenological TNM model
•
•
•
•
A fictive temperature Tf described
the state of the system.
The relaxation time is:
t a t 0.e
A
T .(T f T0 )
T f
Tf tends towards T with a typical
t
time ta

(T f  T )
ta
 t dt" 
.T (t ' )  T f (t ' )dt'
T f (t )   exp  


 t' ta 
In order to take into account all the
memory effects, introduce a
t
 t dt" b 

stretched exponential reponse T (t )  exp 
f
  t ' t a .T (t ' )  T f (t ' )dt'


t
This model described quantitatively most of the effects of complex thermal history
Les Houches 2007 : Flow in glassy systems
Use of the memory effect to probe small amplitude plasticity effect
First cycle
Second
cycle
Tmax =423 K
Tstep , tdef
Tg
Tmin = 313K
mechanics
G*mh1 (w0)
G*mc1 (w0)
G*refc(w0)

G'mh 1/ G'refh
1.06
8x10

1.08
1= 0 (simple memory effect)
1=0.5%
1=1%
1=1.5%
[ G'mh-G'mh 1]/ G'refh
1.10
n
G*refh(w0)
1.04
1.02
1.00
-2
1=1.5%
6
4
2
0
0.98
simple memory effect
[G'mh - G'refh] / G'refh
-2
0.96
320
320
340
360
T (K)
380
400
340
360
T(°C)
380
400
annealing at rest
effect of mechanics * (-1)
Montes, Bodiquel, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems

[ G'mh-G'mh 1]/ G'refh
8x10
-2
1=1.5%
6
4
2
0
simple memory effect
[G'mh - G'refh] / G'refh
-2
320
340
360
T(°C)
380
400
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 yielding
in 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 heat
Irreversibly stored energy
Reversibly stored energy
0.A. Hassan and M.C. Boyce Polymer 1993 34, p 5085
Oleynik E. Progress in Colloid and Polymer Science 80 p 140 (1989)
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 heat
Irreversibly stored energy
Reversibly stored energy
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Temperature of plastic deformation
Exothermic heat induced
by plastic deformation
0.A. Hassan and M.C. Boyce Polymer 1993 34, p 5085
Les Houches 2007 : Flow in glassy systems
Munch et al PRL 2006
Mechanical dissipation observed
in the same condition
Retraction of polymer at zero
stress after cold-drawing, while
increasing temperature, exhibiting
Spontaneous rearrangements
Les Houches 2007 : Flow in glassy systems
Munch et al PRL 2006
Dynamical aspect of the internal stress softening.
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 the
temperature 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*)
*  
q  e .q
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
affine
(b)
(b)
100
-1
d/d (cm )
-1
d/d (cm )
100
10
1
Tstret=Tg-26
unstretched
q
q//
0.1
10
Towards isotropic
1
Tg-26
1/2
q*// = q/l
q* = ql
unstretched
0.1
0.01
0.01
4 5 6
4 5 6
2
0.01
3 4 5 6
-1
Q (Å )
2
3 4
0.1
2
0.01
3 4 5 6
-1
Q* (Å )
2
3 4
0.1
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. de/dt=0.001 s-1.l=1.8
(b) : scattered intensity in reduced q-vector. Deviation from the affine motion clearly appears for large qvectors.
Casas, Alba-simionesco, Montes, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
0.060
Below Tg, there is a crossover qvector that doesn’t depend neither
on strain rate, nor on temperature.
Tg
0.055
-1
q*c (Å )
0.050
Above Tg this crossover length
decreases ( and tends toward zero
if shear rate << trep1 )
0.045
0.040
0.035
0.030
0.025
370
380
390
400
410
420
T (K)
Casas, Alba-simionesco, Montes, Lequeux, in preparation
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
4.0
4.0
Stretched Tg+30, l=2
q ;
q// ;
unstretched
unstretched
3.5
-1
-1
d/d (cm )
3.5
d/d (cm )
Stretched Tg-26 l=2
q ;
q//
3.0
3.0
2.5
2.5
(b)
(a)
2.0
2.0
5
6
7 8 9
2
1
3
-1
q (Å )
4
5
6
7
5
8 9
10
6
7
8 9
2
1
-1
q (Å )
3
4
5
6
7 8 9
10
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
Isotropic distorted
affine
Crossover ~ few
nanometers
The motions follow the macroscopic
deformation
The structure remains isotropic at small
scale ( 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
Natural fluctuations
of yield stress
Streched domains have
a larger yield stress
Unstretched domains
that are softer are know
strained
Plastic Strain self-homogeneize.
Les Houches 2007 : Flow in glassy systems
structure after plastic flow
• Plastic flow is quite homogeneous in
polymer ( because of local strainhardening)
• 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 conclusion
Yield 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
Les Houches 2007 : Flow in glassy systems