glasses plasticity

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…………


Jamming at rest

Picture suggested by Liu and Nagel

Liu, Nagel Nature 1998


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)


•most of the experiments in this domain• our lecture

athermal systems :•foams•simulations


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


Eyring’s Model

At equilibrium

Strain

Energy

E

Energy barrier : Ewaiting time for a hop : kTEe /

0


Eyring’s Model

under stress

Strain

Energy

favourable

unfavourable


Eyring’s Model

Strain

Energy

jump +

jump -

kT

vE

e.

0

kT

vE

e.

0

v is the activation volume (~ 10 nm3 for polymers)


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


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


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


from Eyring to glasses

spontaneous rearrangements at experimental time scale

Energy

Strain

Yielding

creep


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


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


weak deformation in colloidal and polymer glasses, below the onset

of yielding


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


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


leading to self-similar compliance evolution J(t,tw)=j(t/tw

) where ~1

Seen also by step-strain, light scattering……


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


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


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 )


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


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


- 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 ……


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


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


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


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


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


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


Nanzai JSME intern. A 1999


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


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



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


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.


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%.


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


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


Temperature of plastic deformation

Exothermic heat inducedby plastic deformation

0.A. Hassan and M.C. Boyce Polymer 1993 34, p 5085


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


Dynamical aspect of the internal stress softening.

Munch et al PRL 2006


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 ???


structure after plastic flow

Under plastic deformation,An enhancement of the density fluctuation is observed (X, Positron Annihilation Spectroscopy (Hasan, Boyce)

Munch PRL 2006



Structure factor of labelld chains

Affine motion S(q)S(q*)

qq�

.*



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



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




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.



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


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



• 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)


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.


GAME OVER

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glasses plasticity

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