1. 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…………

2. Les Houches 2007 : Flow in glassy systems Jamming at rest Picture suggested by Liu and Nagel Liu, Nagel Nature 1998

3. 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)

4. Les Houches 2007 : Flow in glassy systems most of the experiments in this domain our lecture athermal systems : foams simulations

5. 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

6. Les Houches 2007 : Flow in glassy systems Eyring’s Model At equilibrium  Strain Energy E Energy barrier : E waiting time for a hop :

7. Les Houches 2007 : Flow in glassy systems Eyring’s Model under stress  Strain Energy favourable unfavourable

8. Les Houches 2007 : Flow in glassy systems Eyring’s Model Strain Energy jump + jump - v is the activation volume (~ 10 nm 3 for polymers)

9. Les Houches 2007 : Flow in glassy systems Eyring’s Model Strain Energy jump + jump - shear rate :

10. Les Houches 2007 : Flow in glassy systems Eyring’s Model Viscous fluid : spontaneous relaxation time linear regime :non-linear regime Yield stress fluid : elastic modulus spontaneous relaxation time weak dependance on the shear rate  measurement of v elastic modulus ~ <<

11. 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

12. Les Houches 2007 : Flow in glassy systems from Eyring to glasses spontaneous rearrangements at experimental time scale Energy Strain Yielding creep

13. 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

14. 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 Quench Or strain cessation Waiting time Energy Strain creep

15. Les Houches 2007 : Flow in glassy systems weak deformation in colloidal and polymer glasses, below the onset of yielding

16. 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 Quench Or strain cessation Waiting time

17. 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 t w in days

18. Les Houches 2007 : Flow in glassy systems leading to self-similar compliance evolution J(t,t w )=j(t/t w  ) where  ~1 Seen also by step-strain, light scattering……

19. 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

20. 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(t w ) =  twtw Thus D relaxes towards 0, with a time scale equal to     tends towards a time >> experimental time scale Thus : and   ~ t w

21. 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/t w is observed in any system that tends towards an infinitely slow dynamics – and is thus not specific of glasses ( counter example : floculating suspensions )

22. 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

23. 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

24. 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 ……

25. Les Houches 2007 : Flow in glassy systems small deformations on colloidal glasses 0.1 s 1 s 60 s 100s, 1 Hz, 5.9% Classical aging 100+ 0.1s Classical aging 100+ 60s Aging for t w0 (=100s)+.1s for t w0 +60s With stress at t w0 +.1s With stress at t w0 +1s With stress at for t w0 +60s Viasnoff, Lequeux PRL,Faraday Discuss 2002

26. 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 shear rejuvenation overaging

27. 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 amplitude After 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

28. Les Houches 2007 : Flow in glassy systems Mechanical/Thermal effect on polymer glasses T max =423 K TgTg G* refc (   ) T min = 313K T step, t def G* refh (   ) G* mc  1 (   ) Test cyclereference cycle time annealing « memory » of annealing Montes, Bodiguel, Lequeux, in preparation

29. Les Houches 2007 : Flow in glassy systems [2 nd cycle] – [1 st 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

30. Les Houches 2007 : Flow in glassy systems 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

31. Les Houches 2007 : Flow in glassy systems Phenomenological TNM model A fictive temperature T f described the state of the system. The relaxation time is: T f tends towards T with a typical time   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

32. Les Houches 2007 : Flow in glassy systems n G* mh  1 (   ) T max =423 K TgTg G* refc (   ) T min = 313K T step, t def G* refh (   ) G* mc  1 (   ) Second cycle First cycle mechanics annealing at rest effect of mechanics * (-1) Use of the memory effect to probe small amplitude plasticity effect Montes, Bodiquel, Lequeux, in preparation

33. Les Houches 2007 : Flow in glassy systems 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

34. 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.

35. 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 T g) or cold-drawing, of about 200%.

36. Les Houches 2007 : Flow in glassy systems Oleynik Dissipated heat Irreversibly stored energy Reversibly 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

37. 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

38. Les Houches 2007 : Flow in glassy systems Temperature of plastic deformation Exothermic heat induced by plastic deformation 0.A. Hassan and M.C. Boyce Polymer 1993 34, p 5085

39. Les Houches 2007 : Flow in glassy systems Retraction of polymer at zero stress after cold-drawing, while increasing temperature, exhibiting Spontaneous rearrangements Mechanical dissipation observed in the same condition Munch et al PRL 2006

40. Les Houches 2007 : Flow in glassy systems Dynamical aspect of the internal stress softening. Munch et al PRL 2006

41. 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 ???

42. 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

43. Les Houches 2007 : Flow in glassy systems structure after plastic flow Structure factor of labelld chains Affine motion S(q)  S(q * )

44. Les Houches 2007 : Flow in glassy systems structure after plastic flow (a ) 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. affine Towards isotropic Casas, Alba-simionesco, Montes, Lequeux, in preparation

45. Les Houches 2007 : Flow in glassy systems structure after plastic flow Below T g, there is a crossover q- vector that doesn’t depend neither on strain rate, nor on temperature. Above T g this crossover length decreases ( and tends toward zero if shear rate <<  rep   Casas, Alba-simionesco, Montes, Lequeux, in preparation

46. Les Houches 2007 : Flow in glassy systems 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

47. Les Houches 2007 : Flow in glassy systems affine Isotropic distorted 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

48. 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

49. 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)

50. 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.

51. Les Houches 2007 : Flow in glassy systems GAME OVER