Random quench predicts universal properties of amorphous solids
RRandom quench predicts universal properties of amorphous solids
Masanari Shimada
Graduate School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902, Japan
Eric De Giuli
Department of Physics, Ryerson University, M5B 2K3, Toronto, Canada
Amorphous solids display numerous universal features in their mechanics, structure, and response.Typically, these are rationalized with distinct models, leading to a profusion of control parameters.Here we propose a universal field-theoretic model of an overdamped quench, and compute structural,mechanical, and vibrational observables in arbitrary dimension d . We show that previous results aresubsumed by our analysis and unify spatial fluctuations of elastic moduli, long-range correlations ofinherent state stress, universal vibrational anomalies, and localized modes into one picture. At Kelvin-scale temperatures, glasses universallypresent mechanical and vibrational anomalies with re-spect to crystalline solids: below 1 K , the heat capacitybehaves as C ( T ) ∝ T , to be compared with C ( T ) ∝ T for crystals [1]. It is accepted that the anomalous behav-ior of C ( T ) in glasses is caused by quantum mechanicaltunneling between nearby energy minima in phase space,the two-level systems (TLS) initially proposed as a phe-nomenological model [2]. Recently, a microscopic pictureof the TLS has begun to emerge, thanks to numerical sim-ulations in which quasi-localized vibrational modes havebeen identified and counted [3–8].Moreover, near 10 K , glasses display an excess of vibra-tional modes over phonons, the so-called ’boson peak.’ Itis not agreed what is the cause of the modes near the bo-son peak: the glass-specific behavior has variously beenattributed to soft localized modes [9, 10], generic stiff-ness disorder [11, 12], proximity to jamming [13–15], andproximity to elastic instability [15].The jamming approach predicts a regime in which thedensity of vibrational states g ( ω ) scales as g ( ω ) ∝ ω inany dimension d , sufficiently close to jamming and elasticinstability [15]. The corresponding modes are extendedbut not plane waves, instead showing vortex-like motionat the particle scale. This law has been confirmed in nu-merical simulations [15–18], but is found to break downbelow some frequency ω , below which there are onlyphonons and quasi-localized modes [6, 16, 18]. The lat-ter, now confirmed to be present in many glass models,has a density following g ( ω ) ∝ ω α where 3 ≤ α ≤ α = 4 in the ther-modynamic limit [20], while others argue that smaller ex-ponents are possible due to interactions between localizedinstabilities [7]. Microscopic theory is needed to clarifythese results.The frequency ω setting the lower-limit of the jam-ming regime is controlled by the distance to elastic in-stability [15]. It was proposed that glasses dominated byrepulsive interactions lay close to elastic instability dueto the quench dynamics [14, 15]. This suggests that amodel faithful to the physics of the quench might shedlight on the density of quasi-localized modes and the fre-quency ω below which they become important. Ideally, any such model should also reproduce the universal vi-brational features captured in prevous models [12, 15],such as the dip in sound speed and crossover in acousticattenuation observed in many experiments [21–24].In this Letter we present such a model. Following acrude but principled model of an overdamped quenchinto an inherent state (IS), we derive universal vibra-tional properties characterized by complex elastic mod-uli and the density of vibrational states. As a bonus,our model predicts other universal mechanical features,namely short-range correlations in elastic moduli andlong-range correlations in the IS stress, as observed re-cently in simulations and experiments [25–30]. Random quench model:
Consider the elasticityequation ρ d u i dt − ∂ j [ S ijkl ∂ k u l ] = F i , (1)where ρ is density, u i is a displacement field, and S ijkl = C ijkl + δ ik σ jl in terms of the elastic modulus tensor C ijkl and the IS stress σ jl . The elastic Green’s function G ij is the solution to (1) for a Dirac-delta function forcing, F j ( (cid:126)r ) = f j δ ( (cid:126)r ), that is, u i ( (cid:126)r ) = G ij ( (cid:126)r ) f j .At the mesoscale, the elastic moduli and the stress ten-sor σ ij can be considered to be spatially fluctuating fields.Vibrational properties can be derived from the disorder-averaged Green’s function G ij , for different models of ran-dom disorder. The model of Schirmacher corresponds torandom Gaussian fluctuations of elastic moduli [11, 12].From the jamming approach, Ref. [15] employed amicroscopic lattice model, which does not directly cor-respond to (1). However, elastic instability is caused bythe destabilizing effect of stress in particulate matter withshort-range repulsive interactions [14, 15]. Its proposedimportance highlights stress as an important control pa-rameter for vibrational properties. Moreover, it has beenshown that the stress also plays a crucial role in the emer-gence of quasi-localized modes [5]. These works suggestthat one should consider a model in which the IS stress σ ij is randomly fluctuating. Such an effort must immedi-ately confront a no-go theorem of Di Donna and Lubensky[31]. In a comprehensive treatment of non-affine correla-tions in random media, the latter authors showed that a a r X i v : . [ c ond - m a t . d i s - nn ] A ug (a) (b) (c) ˜ σ ij σ ij FIG. 1. Illustration of random quench model. To an initialhomogeneous continuum (a), we add a quench stress ˜ σ ij , (b).This then relaxes to an inherent state with stress σ ij , (c).Illustrated here are ˜ σ xy and σ xy for µ/λ = 0 .
1. The long-range correlations in σ xy are apparent. random IS stress alone does not yield non-affine motion,and therefore cannot give rise to anomalous vibrationalproperties: a material that behaves affinely is a homoge-neous continuum, the continuum limit of a crystal. Howcan we reconcile the importance of destabilizing stresswith its apparently mild effect on non-affine motion?We propose that the solution is to consider the quenchitself. Indeed, as shown in [31], if random forces are addedto an initially featureless continuum, then the relaxationto an IS will produce both a random IS stress and fluc-tuations in elastic moduli. The latter cause anomalousvibrational properties. Our model is the following: weconsider a homogeneous elastic continuum with elasticconstants ˜ λ and ˜ µ , which can be considered as the IS ofthe melt. To model the finite-temperature melt we add tothis continuum a random quench stress ˜ σ ij ( (cid:126)r ). Under theaction of this stress, the material will relax to a new stateof mechanical equilibrium, as depicted in Figure 1. Wewill then compute the disorder-averaged Green’s functionof the IS.Since we aim to describe universal properties and wework in the continuum, the relevant distribution of ˜ σ ij ( (cid:126)r )can be inferred using field-theoretical arguments [32, 33].In fact, the arguments from [32, 33] can be applied verba-tim, so that for a generic glass, it is sufficient to consider aGaussian distribution of the symmetric tensor field ˜ σ ij ( (cid:126)r ): P [˜ σ ] ∝ exp (cid:18) − (cid:90) r (cid:104) s ˜ (cid:54) σ ij ˜ (cid:54) σ ij + s ˜ σ ii ˜ σ jj (cid:105)(cid:19) , (2)where ˜ (cid:54) σ ij = ˜ σ ij − d δ ij ˜ σ kk is the deviatoric stress, and s and s are parameters related to the magnitude of quenchstress by ˜ (cid:54) σ ij ( (cid:126)r )˜ (cid:54) σ ij (0) = d − s δ ( (cid:126)r ) , (3)˜ σ ii ( (cid:126)r )˜ σ jj (0) = 1 s δ ( (cid:126)r ) (4)in d dimensions. The constant component of the quenchstress will be treated separately.The distribution of quench stress (2) has no spatialcorrelations, thus naturally modelling Brownian forces.Renormalization arguments [32, 33] predict that correc-tions to (2) will introduce a length scale a which is on the order of the relevant microscopic length, namely the par-ticle radius. In our continuum treatment, we will employa cutoff Λ ≈ π/a in momentum space, so that correc-tions to (2) need not be explicitly incorporated.Consider the displacement field u i ( (cid:126)r ) along the quench.At any moment, there is a stress field σ ij [ (cid:126)u ]( (cid:126)r ), itself afunction of the quench stress field. In an overdampedquench, a new IS will be found as soon as σ ij is in me-chanical equilibrium. Di Donna and Lubensky foundthe new IS stress σ ij ( (cid:126)r ) and the elastic modulus tensor C (cid:48) ijkl ( (cid:126)r ) = C ijkl ( (cid:126)r ) + δC ijkl ( (cid:126)r ) around the IS, to lead-ing order in u i , for arbitrary quench stress fields ˜ σ ij ( (cid:126)r )with zero spatial average. The result is a pair of linearfunctionals δC ijkl ( (cid:126)q ) = S ijklmn ( (cid:126)q )˜ σ mn ( (cid:126)q ) (5) σ ij ( (cid:126)q ) = P ijkl ( (cid:126)q )˜ σ kl ( (cid:126)q ) (6)in Fourier space, depending on the elastic moduli and themomentum (cid:126)q .Ref. [31] did not include any constant component ofthe quench stress, but this is easily added. To leadingorder in u , we find that if ˜ σ ij ( (cid:126)r ) = pδ ij , then this isequivalent to replacing the Lam´e moduli by λ = ˜ λ − dp and µ = ˜ µ + p . Stress correlations & elastic moduli fluctua-tions:
Combining Eqs.(5),(6) with (2) immediatelyyields predictions for the distribution of local elastic mod-uli and the distribution of IS stress. The latter is conve-niently represented in terms of a gauge field [32–34]. In d = 2 we can write σ ij = (cid:15) ik (cid:15) jl ∂ k ∂ l ψ and we predict P [ σ [ ψ ]] ∝ exp (cid:18) − ˜ η (cid:90) r tr σ (cid:19) , d = 2 , (7)with ˜ η = (1 + 2 µ ) s s µ ) s + 2 µ s , d = 2 . (8)Similarly, in d = 3 we can write σ ij = (cid:15) ikl (cid:15) jmn ∂ k ∂ m Ψ ln and we predict P [ σ [Ψ]] ∝ exp (cid:18) − (cid:90) r (cid:2) η tr ( σ ) + g tr σ (cid:3)(cid:19) , d = 3 , (9)where η = − s µ s + (9 − µ ) s µ s + 3(3 + 2 µ ) s d = 3 (10) g = s . (11)Eq.(7) and Eq.(9) are in precise agreement with [32, 33]when boundary effects are neglected. We emphasize that σ is a functional of ψ in d = 2 and Ψ in d = 3 andthus these distributions are nontrivial. They predictanisotropic long-range correlations in the stress field, asdiscussed at length in [32, 33].We can also determine the distribution of local elas-tic moduli. We focus here on the bulk modulus fluc-tuation δK = δC iikk /d and shear modulus fluctuation δµ = [ dδC ijij − δC iijj ] / ( d + d − d ). These are pre-dicted to be Gaussian, with fluctuations (cid:104) δK ( (cid:126)r ) δK (0) (cid:105) = 2 s d (cid:2) ¯ d + (5 c ¯ d + 4) (12)+ s − ds d s ( d + 1 + 5 c ¯ d ) (cid:21) δ ( (cid:126)r ) (cid:104) δµ ( (cid:126)r ) δµ (0) (cid:105) = 2 s d ( d + 2) ¯ d (cid:2) ( d + 1) (13)+¯ d (2 d + 4 + c ( d − d − +¯ d ( d − d + 5 + c ( d − d − s − ds d s (cid:21) δ ( (cid:126)r )with ¯ d = d − c = 1 / (1 + 2 µ ). The strictly localnature of these correlations is a consequence of the lo-cal quench stress correlations. We expect corrections tothese correlations only at the particle scale, as observedin generic models [25, 27].Eqs. (8), (10), (11), (12), and (13) can be used to re-late the strength of stress correlations and elastic modulifluctuations to the quench stress, and to each other. Effective medium theory:
We now proceed to de-termine the disorder-averaged Green’s function G ij . Inour model, this cannot be done exactly. We employ theeffective medium theory (EMT), a sophisticated mean-field approximation that determines the optimal complexelastic moduli µ E ( ω ) and λ E ( ω ) to represent the effectof scattering by disorder. The disorder-averaged elasticGreen’s function is thus taken to be of the form G ij ( (cid:126)r ) = (cid:88) α = T,L (cid:90) q G α ( q, ω ) P αij e i(cid:126)q · (cid:126)r , (14)where G T ( q, ω ) = 1 / ( − ρω + µ E ( q, ω ) q ) , G L ( q, ω ) =1 / ( − ρω + λ E ( q, ω ) q ), P Tij = δ ij − ˆ q i ˆ q j , P Lij = ˆ q i ˆ q j ,and (cid:82) q = (cid:82) d d q/ (2 π ) d over the region q < Λ. Defining∆ S ijkl = S ijkl − S Eijkl and ∆ A il ( (cid:126)k, (cid:126)r ) = k k k j ∆ S ijkl ( (cid:126)r ) − ik k ( ∂ j ∆ S ijkl ( (cid:126)r )), in EMT µ E and λ E are fixed by [35]0 = (cid:20) v (cid:90) k ∆ ˆ A ( (cid:126)k, (cid:126)r ) · G α ˆ P α ( (cid:126)k ) (cid:21) − · ∆ ˆ A ( (cid:126)q, (cid:126)r ) · ˆ P α ( (cid:126)q ) , for α = T, L , where v is the correlation volume overwhich the EMT G ij is attained; we take v = 1 / (cid:82) q δW [ J ] /δJ α ( q ) | J =0 in terms of the generatingfunctional W [ J ] = log det (cid:20) v (cid:90) k ∆ ˆ A ( (cid:126)k, (cid:126)r ) · ( G α + J α ) ˆ P α ( (cid:126)k ) (cid:21) Our main result is that W [ J ] can be exactly evaluated interms of a d × d random matrix in the Gaussian orthogonal ensemble (GOE) [36]. We find W [ J ] = (cid:90) dλ σ ( λ ) log (cid:0) ˜ C + 2 ˜ C √ Bλ (cid:1) , (15)where σ ( λ ) = (cid:82) dλ (cid:48) √ − π ˜ A/ ˜ B ρ ( λ (cid:48) ) e ( λ − λ (cid:48) ) ˜ B/ ˜ A is the convo-lution of a Gaussian with the eigenvalue density ρ ( λ (cid:48) ) inGOE, which is completely characterized in all dimensions[36]. In these equations, ˜ A, ˜ B , and ˜ C are complicatedfunctionals of G T and G L . Here we focus on the limit µ (cid:28)
1, for which˜ A = − vI T d ( d + 2) s µ ˜ C (1 + O ( µ )) (16)˜ B = + vI T d + 2) s µ ˜ C (1 + O ( µ )) (17)˜ C = 1 + v ¯ dd (cid:18) − µ E µ (cid:19) µI T + vd (1 − λ E ) I L + O ( µ ) , (18)where I α = (cid:82) q q G α ( q ). From this it follows that λ E =1 + O ( µ ): to leading order, the longitudinal Lam´e mod-ulus is not modified by the quench. Introducing the keycontrol parameter e = (cid:0) v ( d + 2) s µ / (cid:1) − , (19)and a fluctuating shear modulus µ r ( λ ) = µ + e / µλ ¯ d/d we find that µ E satisfies0 = (cid:90) dλ σ ( λ )( µ E − µ r ( λ ))1 − (1 − µ r ( λ ) µ E ) ¯ dd (cid:16) ρω v (cid:82) q G T ( q, ω ) (cid:17) (20)We emphasize that the GOE matrix whose spectrum ap-pears in (15) and (20) is not put into the model, butemerges from its solution.Remarkably, Eq.(20) exactly matches the form of anequation derived in [15], under the identifications µ E → k (cid:107) , an effective longitudinal stiffness; µ r → k α , a fluctuat-ing stiffness; and z = 2 d / ¯ d , a lattice parameter. In [15],and in companion works [37, 38], the stiffnesses k α weremicroscopic spring constants, whose distribution P ( k α )was assumed to take simple tractable forms. In contrast,here we derive the relevant distribution σ ( λ ) from ourmodel of quench dynamics. Wigner semicircle:
The simplest limit is d → ∞ ,for which we expect EMT to be exact [39]. In d = ∞ the GOE spectrum is given by the Wigner semicircle law ρ W ( λ ) = √ d − λ / ( πd ) and σ ( λ ) = ρ ( λ ) up to irrelevantcorrections of relative order 1 /d . Using the fact that ρ W is supported on a finite interval ( −√ d, √ d ), we candetermine the relevant scaling e ∼ /d and x ∼ d . Defining ˜ ω = ω (cid:112) A d /µ with A d = v ρ (cid:82) q q − ,we take ˜ ω ∼
1. Then we can derive a cubic equation for x = µ E /µ : 0 = x − x + d e (cid:104) x + A d µ ω (cid:105) , (21)The same equation has recently been derived for a latticeEMT, and analyzed in detail [40]. Translating these re-sults, we find: (i) the solid is stable for e < e c = 1 / (2 d );(ii) near e c and at small ω , x satisfies a quadratic equationequivalent to that derived in [15], giving µ E ( ω ) = µ − i (cid:113) µA d ( ω − ω ) , (22)where the onset frequency is ω = (cid:113) µdA d ( e c − e ) (23)In this limit the vibrational properties are thus equiva-lent to those discussed in [15]. In particular, the den-sity of vibrational states is g ( ω ) = (2 ω/π )Im G ii (0) ≈− (2¯ dω/π )Im[ µ E ] (cid:82) q q − / | µ E | , from which it follows thatfor ω > ω Eq.(22) gives the non-Debye law g ( ω ) ∝ ω discussed in the introduction.Scattering experiments measure the sound attentua-tion Γ( ω ) ≈ − ω Im[ µ E ] / Re[ µ E ] [15]. We find Γ( ω ) ≈ ( πdµe/ vρω g ( ω ) / Re[ µ E ]. In deriving Eq.(21), we ig-nored the hydrodynamic pole in the Green’s function,which leads to the Debye law g ( ω ) ∼ ω d − for ω < ω .This thus leads to Γ( ω ) ∼ ω d +1 , which is Rayleigh scat-tering. Its amplitude is proportional to the variance ofelastic moduli fluctuations, as observed [28]. Finite-dimensional corrections:
GOE matri-ces have a spectrum whose bulk resembles the Wignersemicircle in all dimensions, with oscillatory corrections.Eq.(21) and its consequences can then be used in any di-mension d to determine the leading physics. However,for d < ∞ , there is a new phenomenon completely ab-sent in the Wigner semicircle: the spectrum develops anexponentially-decaying tail. This tail, which is Gaus-sian in d = 2 and d = 3, adds new excitations, whichwe expect to be localized. Formally, the tail extends to ±∞ , implying that there are now unstable modes. In-deed, from Eq.(20), one can determine a stability con-dition µ r > µ c = − Re[ µ E ] / ¯ d [40]; the smallest mod-ulus can be negative, but small, scaling as 1 /d . How-ever, our results are derived for systems in the thermo-dynamic limit. Since λ corresponds to a spatially fluc-tuating modulus, in a system of N particles there areapproximately N values of λ sampled from σ ( λ ). Define λ ∗ from 1 /N = (cid:82) λ ∗ −∞ dλ σ ( λ ). When µ r ( λ ∗ ) > µ c , corre-sponding to small systems, the tail is irrelevant and thesystem is stable. In this case, using [41], we find that µ E has a contribution δµ ∼ − iω in the regime 0 < ω < ω ,which will lead to g ( ω ) ∼ ω , on top of the Debye con-tribution. Instead, when µ r ( λ ∗ ) < µ c , our quench hasended in an unstable state, and must further relax to atrue inherent state. If σ ( λ ) is modified ad-hoc to vanishat µ r ( λ c ) = µ c as σ ( λ ) ∼ ( λ − λ c ) β , then instead we findthat g ( ω ) ∼ ω β +1 [40].These results are in qualitative agreement with previ-ous findings, which indeed found a density of localizedmodes g ( ω ) ∼ ω α with α < e . However, we do not enforce local stability of the final state. Since α = 4 is typicallyobserved, these results imply that our assumption of anoverdamped quench cannot be realistic in large systems.Future work should explicitly incorporate a condition oflocal mechanical stability in the IS, to properly predictthe form of σ ( λ ) in realistic systems. Conclusion:
We propose a model for universal prop-erties of amorphous solids based on the quench into aninherent state. Under a single universal distribution ofquench stress, our model predicts (i) short-range correla-tions of elastic moduli, as observed [25, 27, 28]; (ii) long-range correlations of the IS stress, as observed [26, 29, 30];(iii) exact reduction to previous models, shown to repro-duce universal vibrational anomalies [15, 40]; and (iv)a tail of potentially unstable modes, beyond mean-fieldpredictions, which leads to g ( ω ) ∼ ω in small systemsand can rationalize larger exponents g ( ω ) ∼ ω α in large,stable systems.Our model is based on an overdamped quench, whichenforces mechanical equilibrium in the IS, but does notenforce stability. In particular, the Hessian field H ij ( (cid:126)r )that controls local stability could have regions where itis not positive-definite. We predict Gaussian fluctuationsof local elastic moduli, as observed in [25, 27], while ithas very recently been argued that the moduli have apower-law tail due to localized modes [28]. We expectthat these modes are created by local relaxation in re-gions of an unstable Hessian. To rigorously predict thedensity of small-frequency localized modes in large sys-tems, and their potential modifications to elastic modulifluctuations, future work should thus add local stabilityas one remaining feature to the model. Acknowledgments:
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