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Linear Response Theory for Hard and Soft Glassy Materials Eran Bouchbinder1 and J.S. Langer2 1Chemical Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel 2Department of Physics, University of California, Santa Barbara, CA 93106-9530 Despite qualitative differences in their underlying physics, both hard and soft glassy materials exhibitalmostidenticallinearrheologicalbehaviors. Weshowthatthesenearlyuniversalproperties emergenaturally inashear-transformation-zone(STZ)theoryofamorphousplasticity,extendedto include a broad distribution of internal thermal-activation barriers. The principal features of this 1 barrier distribution are predicted by nonequilibrium, effective-temperature thermodynamics. Our 1 theoretical loss modulus G′′(ω) has a peak at the α relaxation rate, and a power law decay of the 0 form ω−ζ for higher frequencies, in quantitativeagreement with experimental data. 2 n a Qualitatively different kinds of amorphous materials activation barrier ∆. Let the number of ± STZ’s with J – e.g. structural, metallic and colloidal glasses – ex- given ∆ be N (∆), and let the total number of (coarse- ± 8 hibit remarkable similarities in their linear rheological grained) molecular sites be N. The master equation for 1 properties [1, 2] despite their enormous range of internal N (∆) is [6] ± dynamics and intrinsic time scales. In particular, their i] frequency dependent loss moduli G′′(ω) all have peaks τ0N˙±(∆)=R(±sC,∆)N∓(∆)−R(∓sC,∆)N±(∆) c s that rise near a viscous relaxation rate and drop slowly l- over many decades of higher frequencies. We show here +ρ(θ) [Neq(∆)/2−N±(∆)]. (1) r thatthisnearuniversalityemergesnaturallyintheshear- mt transformation-zone(STZ) theory of amorphous plastic- Here, τ0 is a fundamental time scale, for example, a vibration period for molecular glasses or a Brownian dif- . ity [3–6], generalizedto include a distribution ofinternal at barriersheightsofthekindfoundexperimentallyin1980 fusion time for colloidal suspensions. R(±sC,∆))/τ0 is m by Argon and Kuo [7]. We also show that the principal the rate per STZ for thermally activated transitions between ± orientations; and s is the partial stress acting - features of this barrier-height distribution are predicted C d ontheconfigurationalsubsystem. IncontrasttoSGR,we by nonequilibrium, effective-temperature thermodynam- n assume thats is a coarse-grainedstress,determinedby ics [8]. Our analysis differs from both soft glassy rheol- C o externalforcing,andthatthemolecular-levelstressesare c ogy (SGR) [9] and from mode-coupling theory [10]. In accountedfor implicitly by the two-state dynamics. The [ particular, we start with a kinematic formulation that is total stress, s=s +η ∗γ˙, includes (when appropriate) fundamentally different from that used in SGR, and we C K 1 a partial stress acting on the kinetic-vibrational subsys- v insistthattheelementarysheartranformationsbedriven tem. For colloidal suspensions, this additional stress is a 9 by ordinary thermal fluctuations. 3 kinetic viscosity; γ˙ denotes the total shear rate and ηK∗ 5 The STZ theory assumes that the degrees of freedom denotes a convolution over time. 3 of a glassy material can be separated into two weakly The two terms in square brackets on the right-hand- 1. coupled subsystems – the slow configurational degrees side of Eq. (1) are the rates at which STZ’s are cre- 0 of freedom, i.e. the inherent structures, and the fast ated and annihilated by spontaneous thermal fluctua- 1 kinetic-vibrational degrees of freedom. While the lat- tions. Mechanically generated noise, as used in [5, 6], 1 ter are generally in equilibrium with the thermal reser- issecondorderintheappliedstressandthereforecanbe : v voir,theformerarecharacterizedbyaneffectivetemper- neglectedinthislineartheory. Wearemakingadetailed- i ature that may depart from the reservoir temperature balance approximation in which N (∆) is the value ap- X eq [8]. A fundamental premise of STZ theory is that irre- proached by 2N (∆) in steady-state. The sum over all r ± a versible shear deformations occur only at rare,localized, possible ∆’s is determined by a Boltzmann-like factor two-state, flow defects in the configurational subsystem. exp(−e /χ) according to N (∆)=Nexp(−e /χ). Z ∆ eq Z In flowing states, these STZ’s appear and disappear as χ is the effective disorderPtemperature in energy units, thedrivensystemmakestransitionsbetweenits inherent and e is a typical STZ formation energy. ρ(θ) is the Z structures. In jammed states, the STZ’s are configura- thermalnoisestrength, andθ=k T is the bathtemper- B tionally frozen, but they still are able to make transi- ature in units of energy. tions between their internal orientations in response to The contribution to the rate of deformation coming ordinary thermal fluctuations or applied stresses. from STZ transitions with barrier-height∆ is assFuomrisnpgatfoiarllsyimhpolmicoitgyentheoautsthseysStTemZs’s, awreemoraieyntsetadrotnblyy τ0Dpl(∆)= vV0 R(sC,∆)N−(∆)−R(−sC,∆)N+(∆) h i in the ± directions relative to the shear stress s. We thenconsiderSTZ’scharacterizedbyaninternal,thermal =ǫ0Λ(∆)C(sC,∆) T(sC,∆)−m(∆) , (2) h i 2 where V is the volume of the system, and v0 is a molec- Eq. (6) describes only ordinary thermally activated pro- ular volume that sets the size of the plastic strain in- cesses. We then write ν(∆) ≡ 2ρ0(θ,∆) exp(−∆/θ), crement induced by an STZ transition. We expect ǫ0 ≡ p˜(ν) = −p(∆)d∆/dν, and, in most cases, work with Nv0/V to be a number of the order of unity. As usual the kinetic quantity ν instead of the energy ∆ as the [6], we have defined independent variable. To compute the linear oscillatoryresponse,we assume N+(∆)+N−(∆) N+(∆)−N−(∆) Λ(∆)= , m(∆)= , that the total shear rate γ˙ is simply the sum of elastic N N+(∆)+N−(∆) and plastic parts: 1 C(s ,∆)≡ R(s ,∆)+R(−s ,∆) , C 2h C C i γ˙ =s˙C/µ+Dpl(sC), sC =s− ηK ∗γ˙, (7) R(s ,∆)−R(−s ,∆) T(s ,∆)≡ C C . (3) where µ is the shear modulus. We then denote Fourier C R(s ,∆)+R(−s ,∆) C C transformsasfunctions offrequencyω byγˆ etc.,letη ∗ K According to Eq. (1), the equation of motion for m(∆) γ˙ →iωηˆ (ω)γˆ,andusetheprecedingequationstosolve K is for G(ω)=sˆ/γˆ. The result is: τI0nm˙d(e∆ri)v=in2gCE(sqC.,∆(4))h,Tw(esCh,a∆ve)−amssu(∆m)eid−tρh(aθt)mΛ˙((∆∆)).=(40). G(ω)=iωτ0µ (cid:20)iωτ0N+(ωΛ¯)J(ω)(cid:21), Λ¯ = ǫ20av00θµe−eZ/χ, (8) This means that we are focusing on time scales in which where truestructuralagingisnegligible,andinwhichthelinear iω ηˆ (ω) response of the system is time-translationally invariant. N(ω)=1+ ηˆ (ω)+ K Λ¯J(ω) , K However, the onset of structural aging does play an im- µ µτ0 portantroleinwhatfollows. TherateatwhichSTZ’sare iωτ0+ρ J(ω)= dνp˜(ν)ν . (9) spontaneouslycreatedandannihilatedbythermalfluctu- Z (cid:18)iωτ0+ρ+ν(cid:19) ations is proportional to (ρ/τ0) exp(−eZ/χ), which has Thestorageandlossmoduliare,respectively,G′=Re[G] the form of a conventional activation rate with χ play- and G′′=Im[G]. ing the role of the temperature and ρ(θ)/τ0 being the Ininterpretingtheseresults,wenotefirstthatG(ω)in attemptfrequency. The kineticprefactorρ(θ)isasuper- Eq. (8) cannot,inany physicallymeaningfulway,be ex- Arrhenius function of the temperature, associated with pressedasanaverageoverMaxwellmodesasinSGR[9]. the fact that the configurational rearrangements needed This feature is a result of our kinematic assumption in toformorannihilateSTZ-likedefectsinvolvemany-body Eq. (7), plus our assumption that the plastic strain rate fluctuations that become increasingly complex and un- appearing there is a sum overindependent contributions likely as the temperature decreases. Accordingly, ρ(θ) is fromthetwo-stateSTZ’swithdifferent∆’s. Next,weuse approximatelyequaltounityattemperatureswellabove Eq. (8) to compute the Newtonian viscosity associated the glass temperature, but becomes very small at lower with configurationaldeformations: temperatures,andvanishesbelowaglasstransitiontem- perature if such a transition exists [11]. G(ω) µτ0 η = lim = , (10) Wenowintroducep(∆),anormalizeddistributionover N ω→0 iω Λ¯J(0) barrier heights such that Λ(∆) = exp(−e /χ)p(∆)d∆, Z anddeduce immediately, without yetknowingp˜(ν), that and use Eq. (2) to write the total rate of irreversible the low-frequency structure of G(ω) occurs approxi- deformation in the form Dpl(s )= ǫ0e−eχZ d∆p(∆)C(s ,∆) [T(s ,∆)−m(∆)]. mThautesl,yωiαn tishethneeigvhisbcooruhsoroedlaoxfaωtiαon∼raΛ¯teJ.(0H)/oτw0e=veµr,/ηtNhe. C C C τ0 Z structure of the α peak depends sensitively on p˜(ν), and (5) theapproximationmadeabovecannotreplaceafulleval- To complete the derivation, we must specify the tran- uation of G(ω). sition rate R(s ,∆), which determines C(s ,∆) and C C The distribution p(∆) is a near-equilibrium property T(s ,∆) according to Eqs.(3). In a linear theory C of the configurational subsystem; therefore, it must be C(s ,∆)mustbe independentofs ,andT(s ,∆)must C C C determined by the effective temperature χ. Because ∆ belinearinit. Thesimplestassumptionsforourpurposes is measured downward from some reference energy, we are postulate, at least for some range of values of ∆, that C(sC,∆)≃ρ0(θ,∆)e−∆/θ, T(sC,∆)≃ v0sC. (6) p(∆) has the form a0θ p(∆)∝e+b∆/χ, (11) Here,ρ0(θ,∆)isamany-bodyactivationprefactor,which is analogousto ρ(θ), but whichdoes not necessarilyvan- wherebisadimensionlessnumber. Inthelimitofsmall∆ ish at a glass transition. a0 is a dimensionless num- (largeν),whereρ0≃1,Eq. (11)becomesp˜(ν)≃A˜ν−1−ζ, ber of the order of unity. We stress that C(s ,∆) in where ζ=bθ/χ and A˜ is a normalization factor. C 3 Integrability of p(∆) requires that this distribution be cut off at some ∆ > ∆∗, ν < ν∗. To estimate ν∗, we atarhernlgedautrediaoiνtnste∗ah:spa=ptaet2SaρTwr0iZhn(θigct,.hr∆aTt∗nhh)seeietr−iSeof∆Tno∗rZ/re’θa,st=wethse2ecpmρar(nsoθenp)loovetse−esbetZaeh/reeχsl.oiamwppeproeartt(rha1iann2ngt) (cid:144)(cid:144)ΜΜG',G'' 00001.....0125050000 çççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççç Note that we do not need to know ρ0 in order to deter- 0.02 ççççç ççççç mine ν∗ from the last expression on the right-hand side ç ç of Eq. (12). 10-5 0.001 0.1 10 Since the cutoff at ∆∗ cannot be infinitely sharp, Ω@rad(cid:144)secD we assume that the distribution drops off exponentially, p(∆)∝exp[−(∆−∆∗)/∆1] for ∆>∆∗, where ∆1 is an FIG. 1. (Color online) Experimental data for Vitreloy 4 and energy that may be of the order of θ. We combine these theoretical comparisons for the storage modulus G′(ω) (red) limiting behaviors to write and the loss modulus G′′(ω) (blue). The data points were extracted from Fig. 2 in [1], where very similar curves for A˜ oxideand polymeric glasses can befound. p˜(ν)≃ , (13) ν [(ν/ν∗)ζ +(ν∗/ν)ζ1] with ζ1 = θ/∆1 ∼ 1. This is an oversimplified, three- usuallydenotedby χ0, for systemsdrivenpersistentlyat apnadramζ1etdeertreerpmriensienngtatthioenloafrgp˜e(-νν),awndithsmtwaoll-eνxpliomnietnstsreζ- bsheeaarurnaitveesrsmalucqhuasnlotwityer; itthaunsuτa0−ll1y.tTurhnesroautitotχo0b/eeZinmthaye spectively, and a single crossover value of ν∗. It is re- range 0.1−0.2. We therefore estimate θ /e ∼0.15, im- markably similar to the experimentally deduced distri- plying that Λ(θ )∼10−3, and thus thagt Λ¯Z∼10. Then butionsshownin[7]. The parameterζ controlsthe high- g Eq. (12) tells us that ν∗≃10−3ρ(θ ). frequency behavior of G(ω). A simple scaling analysis g afobrovωeτ0th≫e αρ apnedakν.∗≪2 predicts that G′′(ω) ∼ (ωτ0)−ζ usiTnhgeζ1th=eo1r,eρti(cθagl)/cuτ0rv=es1.i2n5F×i1g0.−12sheacv−e1baenednΛ¯co=m2p5u.teIdn We now compareour theoreticalresults to experimen- effect, we have set ǫ0/a0∼2.5 in Eq. (8), which is well withinourtheoreticaluncertainty. Itisimportanttonote tal data on the rheology of both hard and soft glasses. that these parameters imply that ν∗ ∼10−17, which is We first focus on the oscillatory response of structural extremely small comparedto its upper limit at ν=2. So and metallic glasses, for which τ0 is of the order of pi- far as we can tell from numerical exploration, this small coseconds,ωτ0≪1, andηK is negligible. The interesting value of ν∗ is sharply determined by the experimental behavior occurs at temperatures near or slightly above data. The major discrepancy between the theoretical the glass temperature. curve and the experimental data is that the measured Ourprincipalsourcesofinformationaboutthe oscilla- G′(ω) is approximately linear in ω at low frequencies, tory responses of structural and metallic glasses are the instead of being proportional to ω2 as predicted by our papers by Gauthier et al., in particular [1]. These au- theory and by Maxwell models. thors show that the functions G(ω), for a wide variety of noncrystalline materials at their glass temperatures, We turn now to soft glasses, in particular, to ther- have very similar behaviors. Specifically, the loss modu- mosensitive colloidal suspensions, whose rheology differs lus G′′(ω) has a broad peak at ω and drops off at high from that of structural and metallic glasses in at least α frequencies like ω−ζ as predicted above. For metallic two important respects. First, in colloidal systems, the glasses, Gauthier et al. find ζ≃0.4. approachtojammingnearaglasstransitioniscontrolled In Fig. 1, we show G′(ω)/µ and G′′(ω)/µ as predicted moresensitivelybythevolumefractionthanbythetem- by Eq. (8), along with data from Fig. 2 of [1] for the perature. Second, the microscopically short molecular metallic glass Vitreloy 4 at its glass temperature T . vibration period in structural glasses is replaced in col- g In estimating the theoretical parameters, we have used loids by the very much longer time scale for Brownian Tg≃600K, τ0≃2×10−12sec and µ≃50 GPa. To make motion of the particles. As a result, the high-frequency approximations for the other parameters in Eq. (8), we cutoff at ν=2 is probed in rheological experiments, and note that, if the volume v0 is of the order of a few cu- the kinetic viscosity ηˆK(ω) is relevant at accessibly high bic nanometers, then the ratio v0µ/θg (the prefactor of values of ω. Λ¯ in Eq. (8)) is approximately 104. Then, to estimate Inorderto evaluateG(ω) for colloidalsuspensions,we Λ(χ≃θ )∼exp(−e /θ ), we assume that θ is the same need an expression for the frequency-dependent kinetic g Z g g as the steady-state value of the effective temperature, viscosity ηˆ (ω). Here, we follow [12] and write ηˆ (ω)= K K 4 decreasesslightlyasthesystembecomesstiffer. Λ¯ isvery large for the liquidlike example in the top panel, imply- 100 ing that the STZ density is large in this system. On the HLG',G''Pa 101 ççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççç ppaontarherdceaeρrbdlhfeionartgnotddht,ihestechgueleasssstvisiaomylnuasey–tesodstfeefmΛos¯pr,iittnahenetdthhbeteuhblfekoatcrmtteoeltamthtaailpoltinactnhbgeelelatauswrsneediencenortmlνhy∗e-- çç ç 0.1 çç ç ing time scales for these systems differ by nine orders of çç ç 0.01 çç ç magnitude. 0.01 0.1 1 10 100 1000 104 Thelinearoscillatorymeasurementsdiscussedhereare Ω@rad(cid:144)secD sensitive probes of the internal structure of glassy materials, revealing the broad range of activation mecha- nisms that occur within them and the relation between HLG',G''Pa 101001 çççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççç tdtbirnhheyegleatna,swtxaeiewmnmeteoinwcewhshciaanhiocvrtafdheengrnilttnsahhoamseetlsrsysaSmladdTonodwiZsddroeycrdtsnodhsynaeeenmfidrae.giffmhucOeericrnecaetqseti,iuvooahiefnnlo-itadtbwhelremetidvurdepuemgereer,e.repasiestWtesrustutheorceefqthufutrarheerevsealeaetrlditmobaoiongomens--- 0.1 çç gun to address the latter issues and will report on them 0.01 in a forthcoming publication [14]. 10-5 0.001 0.1 10 1000 Ω@rad(cid:144)secD We thank M. Siebenburger for sending us the data shown in Fig. 2. JSL was supported in part by the FIG. 2. Experimental data and theoretical comparisons for Division of Materials Science and Engineering, Office of thestorage modulusG′(ω)(red)andtheloss modulusG′′(ω) BasicEnergySciences,DepartmentofEnergy,DE-AC05- (blue) for two different suspensions of thermosensitive parti- 00OR-22725,throughasubcontractfromOakRidgeNa- cles,asreportedin[13]. Thevaluesoftheparametersforthe top and bottom panels, respectively, are: ρ=0.04, 3×10−4; tional Laboratory. EB was supported in part by the ν∗=0.001, 10−3ρ;ζ=1.0, 0.5;Λ¯=200, 40;µ=12, 35Pa;and Harold Perlman Family Foundation and in part by a τK=0.004, 0.002 sec. In all cases, ζ1=1 and c=0.1. grant from the Robert Rees Applied Research Fund. µτ (c+iωτ )−1/2, where τ is a viscoustime scale, and K K K cisadimensionlessconstantthatwehaveaddedinorder to regularize this formula at small ω. [1] C.Gauthier,J.-M.Pelletier,Q.WandandJ.J.Blandin, J. Non-Crystalline Solids, 345&346, 469 (2004). In Fig. 2 we show two examples of how the STZ the- [2] D.T.N.Chenetal.,Annu.Rev.Condens.MatterPhys. ory developed here is capable of reproducing the exper- 1, 301 (2010). imental results of Siebenburger et al. [13]. These au- [3] M. L. Falk and J. S. Langer, Phys. Rev. E, 57, 7192 thors explored a range of effective volume fractions φeff (1998). and a wide range of frequencies ω by using suspensions [4] E.Bouchbinder,J.S.LangerandI.Procaccia,Phys.Rev. of thermosensitive particles (polystyrene cores with at- E 75, 036108 (2007). tached networks of thermosensitive isopropylacrylamide [5] J. S.Langer, Phys. Rev.E 77, 021502 (2008). [6] M. L. Falk and J. S. Langer, arXiv:1004.4684 (2010). molecules). The Brownian time scale in these experi- [7] A. S. Argon and H. Y. Kuo, J. Non-Crystalline Solids ments is τ0≃0.003 sec. The volume fraction for the top 37, 241 (1980). panel is φeff = 0.518, while that for the bottom panel [8] E. Bouchbinder and J. S. Langer, Phys. Rev. E 80, is 0.600. The theoretical parameters, deduced by fitting 031132 (2009). the data, are listed in the figure caption. [9] P. Sollich, F. Lequeux, P. Hebraud and M. Cates, Phys. The trends are interesting. The example in the top Rev. Lett.78, 2020 (1997). panelofFig. 2isasystemwhoserelativelysmallvolume [10] J.M.Brader,T.Voigtmann,M.Fuchs,R.G.Larsonand M. Cates, Proc. Natl. Acad. Sci106, 15186 (2009). fraction puts it well away from the glass transition. It is [11] J. S.Langer, Phys. Rev.Lett. 97, 115704 (2006). effectivelyaliquid; ρ=0.04meansthatthereisrelatively [12] R. A. Lionberger and W. B. Russel, J. Rheol. 38, 1885 littlesuper-Arrheniussuppressionofthestructuralrelax- (1994). ation rate. In contrast, the system in the bottom panel [13] M. Siebenburger, M. Fuchs, H. Winter and M. Ballauff, is much more glassy, and ρ decreases significantly. The J. Rheol. 53, 707 (2009). shear modulus µ increases and the viscous time scale τ [14] J. S.Langer and E. Bouchbinder, arXiv:1101.2015v1. K