Universal, additive effect of temperature on the rheology of amorphous solids
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Universal, additive effect of temperature on the rheology of amorphous solids
Joyjit Chattoraj (1) , Christiane Caroli (2) , and Ana¨el Lemaˆıtre (1) (1)
Universit´e Paris Est – Laboratoire Navier, ENPC-Paris, LCPC,CNRS UMR 8205 2 all´ee Kepler, 77420 Champs-sur-Marne, France and (2)
INSP, Universit´e Pierre et Marie Curie-Paris 6, CNRS,UMR 7588, 140 rue de Lourmel, 75015 Paris, France (Dated: September 5, 2018)Extensive measurements of macroscopic stress in a 2D LJ glass, over a broad range of temperatures( T ) and strain rates ( ˙ γ ), demonstrate a very significant decrease of the flowing stress with T , evenmuch below the glass transition. A detailed analysis of the interplay between loading, thermalactivation, and mechanical noise leads us to propose that over a broad ( ˙ γ, T ) region, the effectof temperature amounts to a mere lowering of the strains at which plastic events occur, whilethe athermal avalanche dynamics remains essentially unperturbed. Temperature is then shown tocorrect the athermal stress by a (negative) additive contribution which presents a universal form,thus bringing support to and extending an expression proposed by Johnson and Samwer [1]. Ourprediction is shown to match strikingly well numerical data up to the vicinity of T g . It is now well accepted that macroscopic plastic defor-mation of amorphous solids is the net result of an ac-cumulation of local rearrangements (“shear transforma-tions”, or simply “flips”) involving small clusters of atomsor particles [2]. However, what controls the occurrence ofthese flips, ie when and where they take place, remains acontroversial question, which needs to be answered priorto the formulation of any theory of plasticity.Recent numerical works bring fragmentary informa-tion about these mechanisms. In athermal simulations–whether quasi-static or at finite strain-rate–it is possibleto identify, at any time, in the steady-state sheared sys-tem a set of soft zones, which retain their identity forsizeable stretches of strain [3]. As a zone is loaded bythe external drive, it gradually softens until it reachesits instability threshold and flips [4]. This event createsin the surrounding medium a long-ranged elastic field,with quadrupolar symmetry [5], which shifts the strainof other zones, hence may induce secondary events. Thismechanism gives rise to avalanches of flips [6] with astrain-rate dependent average size [7].These results have led us to propose a picture in which,at T = 0, the dynamics of soft zones in the vicinity oftheir instability thresholds plays a critical role in control-ling dissipative events, while structural disorder, whichpermits the existence of soft zones in the first place, isotherwise accessory to the unfolding of plastic events.But we must then ask what is the effect of a finite tem-perature on plastic activity [8]. Indeed, recent worksconcerned with stress and elastic fluctuations, as well asenergy barrier distributions, have shown that strainedsystems under steady flow present a sizeable fraction oflow-energy barriers, most of which are not involved inthe athermal response [9]. It is then possible that, evena small temperature may activate jumps over these manyother barriers, hence disrupting the avalanche processeswhich control the athermal response.Here, we perform extensive measurements of the macroscopic stress in a 2D Lennard-Jones glass, over abroad range of temperatures ( T ) and strain rates ( ˙ γ ).We find that finite temperatures induce a very signifi-cant decrease of the flowing stress. We show that thesedata can be interpreted within the framework providedby a detailed analysis of the interplay between loading,thermal activation, and mechanical noise. This analysispredicts that over a broad region of the ( ˙ γ, T ) parameterspace, the avalanche dynamics should remain basicallyunperturbed, the effect of temperature amounting to amere lowering of the strains at which plastic events oc-cur, leading to an expression for stress of the form: σ ( ˙ γ, T ) = σ ( ˙ γ ) + Φ( ˙ γ, T )where (i) σ ( ˙ γ ) exhibits precisely the dependence foundin athermal systems and (ii) the (negative) shift func-tion Φ ∝ − ( − T ln( C ˙ γ/T / )) / is analogous to thatproposed by Johnson and Samwer [1] to account for anumber of experimental results on metallic glasses. Thisexpression matches strikingly well our numerical data upto the vicinity of the glass transition temperature T g .We thus conclude to the robustness of avalanches, whichalso show up clearly in the same temperature range inthe structure of strain maps.We use the same 2D binary LJ mixture as in Refs. [3,7]: it is composed of large (L) and small (S) parti-cles with radii R L = 0 . , R S = 0 . m = 1 (in standard LJ units), in a number ratio of N L /N S = (1 + √ / π ( N L R L + N S R S ) /L = 0 . ∼
20 [3],the shear wave speed c s ≃ .
4, and the glass transitiontemperature (identified as the value at which τ α = 10 ,and computed following [10]) is ∼ .
27. Finite temper-ature MD simulations are performed on square L × L systems, using Lees-Edwards boundary conditions, witha standard velocity rescaling protocol [11] and time steps dt ≤ .
01. All the data presented here are obtained af-ter 100% of strain, which ensures that the system is insteady state. Large sets and long strain intervals are usedfor statistical accuracy (e.g. for L = 40, 25 samples arestrained up to 1300%).We present on Fig. 1-(a) steady state stress σ vs strainrate ˙ γ for different system sizes L = 10, 20, 40, 80, 160,two temperatures T = 0 .
025 and T = 0 .
2, and ˙ γ rangingfrom 4 × − to 10 − . For each temperature we find, asin athermal systems, quick convergence with increasing L , of σ ( ˙ γ ) towards a master curve. In the range of ˙ γ studied, saturation is already reached for L = 40, whichallows us to focus in the following on stress data obtainedfor this system size. L=10L=20L=40L=80L=160T=0.2T=0.025
PSfrag replacements ( a )( b ) σ ˙ γTT ց ˙ γ ց T=0.025T=0.2
PSfrag replacements( a ) ( b ) σ ˙ γTT ց ˙ γ ց FIG. 1. (a): The macroscopic stress σ as a function of strainrate ˙ γ for systems of sizes L = 10, 20, 40, 80, and 160, attemperatures T = 0 .
025 and T = 0 .
2. (b): σ versus ˙ γ for L = 40, compared with fits of the form σ = A + A √ ˙ γ (solidlines). To qualify the effect of temperature, we then attempt(see Fig. 1-(b)) to fit these curves with σ = A + A √ ˙ γ ,a form which was shown to match very well the results ofathermal simulations. For our lowest temperature T =0 . T finite but ≪ T g the dynamics is closely similar to thatof an athermal system. At higher temperature, however,not only does this fit become definitely poorer but, moreimportantly, the overall stress level drops significantly:temperature thus has a pronounced effect on dissipation.Does this mean that it completely modifies the avalanche-dominated dynamics at work in the T → T = 0, each flip is triggered right at the thresholdstrain γ c where instability is reached. At finite T , wemust expect that flips occur in anticipation due to ther-mal activation, which becomes gradually more efficientwhen a zone approaches its γ c [12]. More precisely, con-sider a single zone, initially lying at a strain γ < γ c ,which is loaded at finite strain rate ˙ γ in the presenceof thermal noise. At an external strain γ ∈ [ γ , γ c [,the probability P ( γ ) that it has not yet flipped obeys P ( γ + d γ ) = P ( γ ) (cid:16) − d γ ˙ γ R ( γ ) (cid:17) , or: ∂P∂γ = − γ R ( γ ) P ( γ ; γ )with R ( γ ) the rate of activated jumps. The solution is: P ( γ ; γ ) = exp (cid:20) − γ Z γγ d γ ′ R ( γ ′ ) (cid:21) (1)At low temperature, activation is efficient only close tothe saddle node bifurcation occurring at γ c . There, theenergy barrier and the “attempt frequency” present theuniversal scaling forms: ∆ E = B ( γ c − γ ) / and ω = ν ( γ c − γ ) / [4][13]. Provided that ∆ E/T ≫
1, we canuse the standard Kramers expression for the activationrate: R ( γ ) = ω exp( − ∆ E/T ), so that: P ( γ ; γ ) = exp − ν ˙ γ (cid:18) TB (cid:19) / h Q ( δγ ) − Q ( δγ ) i! where δγ = γ c − γ and Q ( δγ ) = Γ (cid:0) ; BT δγ / (cid:1) , withΓ the upper incomplete gamma function. Our usage ofKramers’ expression implies that ǫ = T /Bδγ / ≪ s ; x ) ∼ x s − e − x , whence Q ( δγ ) ∼ ǫ / exp( − /ǫ ). PSfrag replacements˙ γ Kr . ˙ γ lim . δγ δγ δ γ fl i p ˙ γ PT ( a )( b )( c ) -6 -5 -4 -3 -2 PSfrag replacements ˙ γ Kr . ˙ γ lim . δγ δγδγ flip ˙ γP T ( a )( b ) ( c )FIG. 2. The parameters used come from from the fit of ourstress data (see text). For T = 0 .
1: (a) the function P vs δγ for γ = −∞ (solid line) and γ > ∼ γ ⋆ (dashed); (b) δγ flip vs δγ (solid line) with asymptotes (thin line). (c): ˙ γ lim . and˙ γ Kr . vs T (using a/d = 1 /
3, ∆ ǫ = 5%); shaded area: thehigh- T region where data start to depart from fit. Plots of P ( γ ; γ ) are presented on Fig. 2-(a). Let usfirst consider the formal limit γ → −∞ ( Q ( δγ ) → P is an exponential ofan exponential it presents a very sharp transition from P ∼ P ∼ γ ⋆ such that:23 ν ˙ γ (cid:18) TB (cid:19) / Q ( δγ ⋆ ) = 1 , (2)which is next solved at leading order in ǫ , yielding: δγ ⋆ ≃ " TB ln ν ˙ γ (cid:18) TB (cid:19) / ! / . (3)The width of this transition is of order | ∂P/∂γ ( γ ⋆ ; −∞ ) | − = (2 e/ ǫ ⋆ δγ ⋆ , which is thus O ( ǫ ⋆ ) relative to δγ ⋆ itself.As γ increases from −∞ , the curves P ( γ ; γ ) remainnearly identical to P ( γ ; −∞ ) up to the immediate vicin-ity of γ ⋆ . For γ > γ ⋆ , P ( γ ; γ ) no longer presents aplateau at low values of γ , but drops already sharply at γ , with slope | ∂P/∂γ ( γ ; γ ) | = γ R ( γ ), which is an in-creasing function of γ . Therefore, the solution γ flip of P ( γ flip ; γ ) = 1 /e can always be interpreted as the typicalstrain at which a zone starting from γ flips.The curve γ flip ( γ ) (Fig. 2-(b)) exhibits a sharp tran-sition between two limiting behaviors: (i) when injectedat γ < ∼ γ ⋆ , a zone flips at γ flip ≈ γ ⋆ ; (ii) when injectedat γ > ∼ γ ⋆ , it flips almost immediately: γ flip ≈ γ .The width of the transition can be estimated as be-ing O ( δγ ⋆ ǫ ⋆ ) ≪ δγ ⋆ , which compares with the widthof the drop of P ( γ ; −∞ ) around γ ⋆ . We can thereforeconclude that the competition between thermal activa-tion and drive defines (slightly fuzzy) apparent thresh-olds which are shifted by − δγ ⋆ ( ˙ γ, T ) from the mechanicalyield points.In our previous studies of athermal systems, we foundthat the mechanical noise–that is the stress noise gener-ated by the flips themselves–played a key role by inducingcorrelations between flip events, leading to the emergenceof avalanche behavior. At finite T , a zone embedded in asheared system is thus experiencing both thermal noiseand random strain shifts (ie barrier height fluctuations)originating from prior events. In [7], we proposed to sepa-rate mechanical noise into: (i) a low-frequency part, gen-erated by nearby events (within a sphere of radius ℓ ); (ii)a background noise coming from the rest of the system.Requiring that the near-field signals be non-overlappingand stand out of the background noise defines a singlelength ℓ ( ˙ γ ) which we identified as the avalanche size.An exact treatment of the jump dynamics in the pres-ence of activation, loading, and barrier fluctuations isfor the moment out of reach. However we note thatthe values γ ⋆ can still be interpreted as effective thresh-olds, provided that the unfolding of the activated jumpof a zone near its γ flip is not perturbed by the ambi-ent noise (which includes nearby and far-field signals).For this purpose, we recall that the strain field due toa shear transformation has the form f esh . = a r ∆ ǫ [5],where a is a zone size and ∆ ǫ an elementary plasticstrain. In a system of size L , the flip rate is R =˙ γL / ( a ∆ ǫ ), and the strain fluctuation due to the noiseincoming from the whole system during a time τ ver-ifies ∆ γ ( τ ) = τ R L R Ld f . d r = ˙ γ τ a ∆ ǫ d , with d atypical inter-zone distance [7]. In order for the activa-tion process to be negligibly perturbed, we must ensurethat the strain fluctuation accumulated during the acti-vation time 1 /R ( γ flip ) < /R ( γ ⋆ ), remains much smallerthan the thermal strain shift δγ ⋆ itself. This is guar- anteed as soon as: ˙ γ a ∆ ǫ / ( R ( γ ⋆ ) d ) ≪ ( δγ ⋆ ) . As R ( γ ⋆ ) = BT ˙ γ √ δγ ⋆ , this also writes:˙ γ ≪ ˙ γ lim . = 2 ν (cid:18) TB (cid:19) / e − (cid:0) BT (cid:1) / (cid:0) a ǫ d (cid:1) / (4)When this condition is satisfied, the process of flipactivation disentangles from the response to incomingmechanical noise signals. The basic elements of a phe-nomenology of avalanche dynamics, namely the advec-tion of zones towards (shifted) thresholds and the pres-ence of mechanical noise signals, are preserved. In par-ticular, the separation between nearby, correlated, signaland background noise proceeds exactly along the samelines as in the T = 0 limit [7], thus defining the samevalue for the avalanche size ℓ ( ˙ γ ). As flip events occur atshifted thresholds, the macroscopic stress should thus beof the form: σ ( ˙ γ, T ) = σ ( ˙ γ ) − µ δγ ⋆ ( ˙ γ, T ) (5)where µ is the shear modulus. From athermal simu-lations [7], we know that σ is of the form: σ ( ˙ γ ) = A + A √ ˙ γ . The average δγ ⋆ accounts for the fact thatthe values of B and ν appearing in the above calculationare distributed due to structural disorder. Assuming thatln B and ln ν are well-centered, we can finaly write: σ ( ˙ γ ) = A + A p ˙ γ − A T / h ln (cid:16) A T / / ˙ γ (cid:17)i / (6)with A = µB − / and A = ν/B / .To test this prediction, we now fit the stress data shownon Fig. 1 using this four parameter expression. As seenon Fig. 3, the fit is remarkable over quite a broad rangeof parameters, T varying from nearly 0 up to ∼ . T g = 0 . γ ranging over more thantwo decades. The values of the parameters are stronglyconstrained by the fit: A and A by the low- T data; A by the low- ˙ γ data. It clearly confirms the form of σ ( ˙ γ ) aswell as the functional form of the correction term, except PSfrag replacements σ ˙ γTT ց ˙ γ ց PSfrag replacements σ ˙ γ TT ց ˙ γ ց FIG. 3. Macroscopic stress σ (filled symbols) as a functionof strain rate ˙ γ (left) and T (right) compared with the fitobtained using Eq. (6), with parameters: A = 0 . A =2 . A = 0 .
27, and A = 0 . for the 5 / T / dependence, which is very visible at low ˙ γ , is a clearsignature that the saddle-node bifurcation controls thebehavior of energy barriers heights over the whole rangeof relevants strains. FIG. 4. The non-affine strain field accumulated over strainintervals ∆ γ = 1% , ,
10% (left to right), when sheared at˙ γ = 10 − , and for T = 0 . , . , . × From the fits we extract typical values ν ∼
50 and B ∼ /ǫ ⋆ = ln((2 ν/ γ ) (cid:0) T /B ) / (cid:1) ≫ γ Kr . and ˙ γ lim . (resp.), which are increasing functions of T (see Fig. 2-(c)). By iteration, we have used for our fit only thepoints which satisfy these conditions. This excludes afew data points at the lowest temperatures and higheststrain rates.The conditions ˙ γ < ˙ γ Kr . ( T ), ˙ γ < ˙ γ lim . ( T ) and T < ∼ . T g , the gradual departureof the measured stress away from this expression shouldbe assigned to the increasing contribution of thermallyactivated events of a different nature than the zone flipsthat we have considered here. We believe, these might berelated to the finite, low-lying barriers shown by Rodneyand Schuh [9] to be present in sheared systems.The present work provides a firm support to the notionof effective threshold and generalizes significantly the ex-pression shown by Johnson and Samwer to fit a large body of experimental data on metallic glasses [1]. In-deed, our derivation clarifies why an additive correctionto stress, with the universal form set by the saddle-nodebifurcation, holds despite the presence of structural dis-order. Moreover, our data show that this correction addsup, as predicted by our model, to a stress σ ( ˙ γ ) which isprecisely that expected in an athermal system. We thusconclude to the robustness of avalanche dynamics up tothe vicinity of T g .In order to gain further insight into this question, wenow show how the non-affine strain field accumulates,at different temperatures, as the system is macroscopi-cally sheared. Strain maps are displayed on Fig. 4 for in-creasing values of the external strain ∆ γ = 1% , , T = 0 . , . , . T , we clearlysee comparable correlated structures. The strain pat-terns at T = 0 .
025 and 0.2 are similar and exhibit, asthose seen previously in athermal simulations, a clear di-rectionality which can only be due to Eshelby-like elasticinteractions [7]. It is only at T = 0 .
3, i.e. in the super-cooled regime, that a qualitative change can be observed,the structures becoming slightly blurred and shorter-ranged. We thus see a continuity of behavior acrossthe glass transition which connects the low-temperatureavalanche behavior with the cooperative dynamics ob-served by Tanaka in the supercooled regime [14]. Morequantitative information on this question might comefrom the detailed analysis of the ( ˙ γ, T )-dependence ofthe diffusion coefficient, a study which is presently underway.This work was supported by the French competitive-ness cluster Advancity and Region ˆIle de France. [1] W. L. Johnson and K. Samwer, Phys. Rev. Lett. ,195501 (Nov. 2005).[2] A. S. Argon, Acta Met , 47 (1979).[3] A. Lemaitre and C. Caroli, Phys. Rev. E , 036104(2007).[4] C. Maloney and A. Lemaˆıtre, Phys. Rev. Lett. , 195501(2004).[5] J. D. Eshelby, Proc. Roy. Soc. London A , 376 (1957).[6] C. Maloney and A. Lemaˆıtre, Phys. Rev. Lett. , 16001(2004); N. P. Bailey, J. Schiotz, A. Lemaitre, and K. W.Jacobsen, ibid . , 095501 (2007).[7] A. Lemaitre and C. Caroli, Phys. Rev. Lett. , 065501(2009).[8] H. G. E. Hentschel, S. Karmakar, E. Lerner, and I. Pro-caccia, Phys. Rev. Lett. (2010).[9] D. Rodney and C. Schuh, Phys. Rev. Lett. , 235503(2009).[10] D. N. Perera and P. Harrowell, J. Chem. Phys. , 5441(1999).[11] M. Allen and D. Tildesley, Computer Simulation of Liq-uids (Oxford Science Publications, 1996).[12] C. Caroli and P. Nozi`eres, in
Physics of sliding friction , Series E: Applied Sciences, Vol. 311, edited by B. Perssonand E. Tosatti, NATO ASI Series (Kluwer Acad. Publ.,Dordrecht, 1996). [13] The vanishing eigenvalue controlling the saddlenode bi-furcation scales as λ ∝ ( γ c − γ ) / ∝ ω .[14] A. Furukawa, K. Kim, S. Saito, and H. Tanaka, Phys.Rev. Lett.102