glasses plasticity

1. 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

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

3. 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

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

5. 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

6. Eyring’s Model At equilibrium t Energy E Strain Energy barrier : E waiting time for a hop : Les Houches 2007 : Flow in glassy systems

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

8. Eyring’s Model Energy Strain jump + v is the activation volume (~ 10 nm3 for polymers) jump - Les Houches 2007 : Flow in glassy systems

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

10. elastic modulus Eyring’s Model << ~ 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 Les Houches 2007 : Flow in glassy systems

11. 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

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

13. 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

14. Aging systems Creep experiments- in the linear regime - probe the spontaneous rearrangements : experimental protocol Energy Strain creep Quench Or strain cessation Thermal or mechanical rejuvenation (pre-shear !) Rheological Test (creep /step-strain/…) Waiting time time Les Houches 2007 : Flow in glassy systems

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

16. 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 Les Houches 2007 : Flow in glassy systems

17. tw in days Spontaneous rearrangements are getting slower and slower Colloïdal suspensions Glassy polymer Borrega, Cloitre, Monti, Leibler C.R. Physique 2000 Struik Book 1976 Linear Creep flow reveals spontaneous rearrangements Les Houches 2007 : Flow in glassy systems

18. 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

19. r Time elapsed after « quench » It reveals a self-similar evolution of the time relaxation spectrum <t> ~ t wm Log t Dynamical measurements are very sensitive to aging Les Houches 2007 : Flow in glassy systems

20. lim D(tw) = e ~0 tw scaling argument for aging Simple argument : lets D be the inverse of the relaxation time ta. D =1/ ta ta tends towards a time >> experimental time scale Thus D relaxes towards 0, with a time scale equal to ta and ta ~ tw Thus : Les Houches 2007 : Flow in glassy systems

21. 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

22. 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

23. 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. Les Houches 2007 : Flow in glassy systems

24. - 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 ) Les Houches 2007 : Flow in glassy systems

25. Aging for tw0(=100s)+.1s for tw0+60s With stress at tw0 +.1s With stress at tw0 +1s With stress at for tw0+60s 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 Les Houches 2007 : Flow in glassy systems

26. 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

27. 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

28. Mechanical/Thermal effect on polymer glasses « memory » of annealing reference cycle Test cycle Tmax =423 K time Tg Tstep , tdef Tmin = 313K G*refc(w0) G*refh(w0) G*mcg1 (w0) annealing Montes, Bodiguel, Lequeux, in preparation Les Houches 2007 : Flow in glassy systems

29. [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

30. 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

31. 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 Les Houches 2007 : Flow in glassy systems

32. G*mhg1 (w0) n Use of the memory effect to probe small amplitude plasticity effect First cycle Second cycle Tmax =423 K Tg Tstep , tdef mechanics Tmin = 313K G*refc(w0) G*refh(w0) G*mcg1 (w0) annealing at rest effect of mechanics * (-1) Montes, Bodiquel, Lequeux, in preparation Les Houches 2007 : Flow in glassy systems

33. 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

34. 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

35. 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

36. 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

37. 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 Les Houches 2007 : Flow in glassy systems

38. 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

39. 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

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

41. 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

42. 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

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

44. 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/dt=0.001 s-1.l=1.8 (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 Les Houches 2007 : Flow in glassy systems

45. structure after plastic flow Below 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 << trep-1 ) Casas, Alba-simionesco, Montes, Lequeux, in preparation Les Houches 2007 : Flow in glassy systems

46. 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 Les Houches 2007 : Flow in glassy systems

47. 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 Les Houches 2007 : Flow in glassy systems

48. 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. Les Houches 2007 : Flow in glassy systems

49. 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) structure after plastic flow Les Houches 2007 : Flow in glassy systems

50. 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