glasses plasticity Background the dynamical phase diagram linear

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 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 : 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 + jump - v is the activation volume (~ 10 nm 3 for polymers) Les Houches 2007 : Flow in glassy systems

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

Eyring’s Model << ~ linear regime : Viscous fluid : spontaneous relaxation time non-linear regime Yield stress fluid : elastic modulus 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 Les Houches 2007 : Flow in glassy systems

from Eyring to glasses Yielding Energy creep Strain 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 Les Houches 2007 : Flow in glassy systems

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 Les Houches 2007 : Flow in glassy systems

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 Thus : 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. Les Houches 2007 : Flow in glassy systems

- 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

small deformations on colloidal glasses 100 s, 1 Hz, 5. 9% 0. 1 s Aging for tw 0(=100 s)+. 1 s for tw 0+60 s With stress at tw 0 +. 1 s With stress at tw 0 +1 s With stress at for tw 0+60 s 1 s 60 s Classical aging 100+ 60 s Classical aging 100+ 0. 1 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 Tmax =423 K Test cycle Tstep , tdef Tg time Tmin = 313 K G*refc(w 0) G*refh(w 0) G*mcg 1 (w 0) annealing Montes, Bodiguel, Lequeux, in preparation 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 Les Houches 2007 : Flow in glassy systems

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: • 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

Use of the memory effect to probe small amplitude plasticity effect First cycle Tmax =423 K Second cycle Tstep , tdef Tg Tmin = 313 K mechanics G*refc(w 0) G*refh(w 0) G*mcg 1 (w 0) G*mhg 1 (w 0) n annealing at rest effect of mechanics * (-1) Montes, Bodiquel, Lequeux, in preparation 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 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 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 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*) Les Houches 2007 : Flow in glassy systems

structure after plastic flow 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 qvectors. Casas, Alba-simionesco, Montes, Lequeux, in preparation Les Houches 2007 : Flow in glassy systems

structure after plastic flow Below Tg, there is a crossover qvector 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

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

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