glasses plasticity

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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
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Jamming at rest
Picture suggested by Liu and Nagel
Liu, Nagel Nature 1998
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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)
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• most of the experiments in this domain
• our lecture

• athermal systems
• foams
• simulations

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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
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Eyrings Model
At equilibrium
t
Energy
E
Strain
Energy barrier E waiting time for a hop
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Eyrings Model
under stress s
favourable
unfavourable
Energy
Strain
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Eyrings Model
Energy
Strain
jump
v is the activation volume ( 10 nm3 for
polymers)
jump -
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Eyrings Model
Energy
Strain
shear rate
jump
jump -
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Eyrings Model
ltlt

linear regime
non-linear regime
Viscous fluid
Yield stress fluid
elastic modulus
weak dependance on the shear rate ? measurement
of v
spontaneous relaxation time
spontaneous relaxation time
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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
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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
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Creep experiments- in the linear regime - probe
the spontaneous rearrangements experimental
protocol
Aging systems
Energy
Strain
creep
Quench Or strain cessation
Thermal or mechanical rejuvenation (pre-shear !)
Rheological Test (creep /step-strain/)
Waiting time
time
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weak deformation in colloidal and polymer
glasses, below the onset of yielding
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aging
Creep experiments- in the linear regime - probe
the spontaneous rearrangements experimental
protocol
Quench Or strain cessation
Thermal or mechanical rejuvenation (pre-shear !)
Rheological Test (creep /step-strain/)
Waiting time
time
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Spontaneous rearrangements are getting slower and
slower
tw in days
Colloïdal suspensions
Glassy polymer
Borrega, Cloitre, Monti, Leibler C.R. Physique
2000
Struik Book 1976
Linear Creep flow reveals spontaneous
rearrangements
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leading to self-similar compliance evolution
J(t,tw)j(t/twm) where m1
Seen also by step-strain, light scattering
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• It reveals a self-similar evolution of the time
  relaxation spectrum

r
Time elapsed after  quench 
lttgt t wm
Log t
Dynamical measurements are very sensitive to aging
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scaling argument for aging
Simple argument lets D be the inverse of the
relaxation time ta. D 1/ ta
ta tends towards a time gtgt experimental time
scale
Thus D relaxes towards 0, with a time scale equal
to ta
and ta tw
Thus
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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 )
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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
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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
1 s
0.1 s
60 s
Classical aging 100 60s
Classical aging 100 0.1s
Viasnoff, Lequeux PRL,Faraday Discuss 2002
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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
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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
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Mechanical/Thermal effect on polymer glasses
memory of annealing
Test cycle
reference cycle
Tmax 423 K
Tg
time
Tstep , tdef
Tmin 313K
Grefc(w0)
Grefh(w0)
Gmcg1 (w0)
annealing
Montes, Bodiguel, Lequeux, in preparation
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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
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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
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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 ta
• In order to take into account all the memory
  effects, introduce a stretched exponential reponse

This model described quantitatively most of the
effects of complex thermal history
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Use of the memory effect to probe small amplitude
plasticity effect
First cycle
Second cycle
Tmax 423 K
Tg
Tstep , tdef
Tmin 313K
mechanics
Grefc(w0)
Grefh(w0)
Gmcg1 (w0)
annealing at rest
effect of mechanics (-1)
Montes, Bodiquel, Lequeux, in preparation
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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
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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.
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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.
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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)
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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
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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
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Munch et al PRL 2006
Dynamical aspect of the internal stress softening.
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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 ???
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structure after plastic flow
Under plastic deformation, An enhancement of the
density fluctuation is observed (X, Positron
Annihilation Spectroscopy (Hasan, Boyce)
Munch PRL 2006
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structure after plastic flow
Structure factor of labelld chains Affine
motion S(q)?S(q)
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structure after plastic flow
(a)
affine
Towards isotropic
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/dt0.001 s-1.l1.8 (b) scattered intensity
in reduced q-vector. Deviation from the affine
motion clearly appears for large q-vectors.
Casas, Alba-simionesco, Montes, Lequeux, in
preparation
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structure after plastic flow
Below Tg, there is a crossover q-vector that
doesnt depend neither on strain rate, nor on
temperature. Above Tg this crossover length
decreases ( and tends toward zero if shear rate
ltlt trep-1 )
Casas, Alba-simionesco, Montes, Lequeux, in
preparation
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structure after plastic flow
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
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Isotropic distorted
affine
Crossover few nanometers
The structure remains isotropic at small scale (
think about a liquid). But the chains are
distorded
The motions follow the macroscopic deformation
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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
Natural fluctuations of yield stress
Unstretched domains that are softer are know
strained
Plastic Strain self-homogeneize.
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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 (lt
  10 nm)

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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 arent any
satisfactory models, even if most of the simple
models capture qualitatively most of the effects
for small and intermediate deformations.
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GAME OVER