Protocol-dependent shear modulus of amorphous solids
PProtocol-dependent shear modulus of amorphous solids
Daijyu Nakayama,
1, 2
Hajime Yoshino,
1, 2 and Francesco Zamponi Cybermedia Center, Osaka University, Toyonaka, Osaka 560-0043, Japan Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan Laboratoire de Physique Th´eorique, ENS & PSL University,UPMC & Sorbonne Universit´es, UMR 8549 CNRS, 75005 Paris, France
We investigate the linear elastic response of amorphous solids to a shear strain at zero temperature.We find that the response is characterized by at least two distinct shear moduli. The first one, µ ZFC ,is associated with the linear response of a single energy minimum. The second, µ FC , is related tosampling, through plastic events, an ensemble of distinct energy minima. We provide examples ofprotocols that allow one to measure both shear moduli. In agreement with a theoretical predictionbased on the exact solution in infinite spatial dimensions, the ratio µ FC /µ ZFC is found to vanishproportionally to the square root of pressure at the jamming transition. Our results establish thatamorphous solids are characterized by a rugged energy landscape, which has a deep impact on theirelastic response, as suggested by the infinite-dimensional solution.
I. INTRODUCTION
Most solid state textbooks are almost entirely devotedto crystals [1]. The reason is obvious: while the the-ory of crystals is fully developed, the theory of amor-phous solids (glasses, foams, granulars, etc.) is still veryincomplete [2, 3]. Crystals can be understood as per-fect periodic lattices, around which particles performsmall vibrations. This allows one to construct a low-temperature harmonic expansion, and obtain all thermo-dynamic properties in terms of harmonic excitations, i.e.phonons. Moreover, crystal flow (or plasticity) and melt-ing is mediated by defects (mostly dislocations) that arealso quite well understood [1].The situation is very different for glasses, which dis-play all kind of anomalies with respect to crystals: theyshow an enhanced low-frequency density of states (theso-called Boson Peak) [4], leading to anomalous behaviorof specific heat and thermal conductivity [5]. Cruciallyfor our study, they show irreversible “plastic” responseto arbitrarily small perturbations [6–10]: during plasticevents, some part of the system relaxes irreversibly to anew low-energy state by crossing some low-energy bar-rier [11–15].These observations suggest the following picture: crys-tals can be thought as isolated minima of the potentialenergy, around which a well-defined harmonic expansioncan be performed, and that are separated from other min-ima by high enough energy barriers [1]. On the contrary,glasses are “fragile” minima of the potential energy func-tion: they are characterized by many soft modes [16],the harmonic expansion thus works only at extremelylow temperatures [17–19], and very low-energy barriersseparate each glassy minimum from many other neigh-boring, and equivalent, glassy minima [11, 20, 21]. Inthis picture, it is natural that even a very small per-turbation destabilizes a glassy minimum and brings thesystem over a barrier to relax, irreversibly, to anotherminimum [11, 14, 21].The exact mathematical solution of the problem in the abstract limit of an infinite-dimensional space can be thesource of inspiration about some physical properties ofthe solid in three dimensions [22, 23]. In particular, itsuggests that the organisation of the energy minima ishierarchical [21]: glassy minima are organised in clus-ters, or “basins”, themselves organised in larger basins,and so on, as it is well-known to happen in mean field spinglasses [24, 25]. In such a situation, the response of theglass to an external perturbation depends on how muchof the energy landscape can be explored [24–27]. Con-sider elastic response. If only a given energy minimum is
E compression E ⇠ µ FC ( ) E ⇠ µ ZFC ( ) = µ ZFC = µ FC ' FIG. 1. Oversimplified sketch of an energy landscape withtwo distinct shear moduli.
Top : elastic energy E versus shearstrain γ . Bottom : same illustration using stress σ = dE/dγ asa function of strain. Within a single energy minimum (left col-umn), the energy increase behaves elastically ∆ E ∝ (∆ γ ) fora small enough increment of the shear strain ∆ γ . When theenergy landscape becomes bifurcated (right column), energyminima (red line) are organised in basins (blue line). Eachof the dotted line represents a region where a minimum is lo-cally stable. If the system can sample the basin, a lower shearmodulus µ FC is observed in the γ → a r X i v : . [ c ond - m a t . s o f t ] A ug explored, the system responds linearly with certain elas-tic coefficients. If a larger cluster of minima can be ex-plored, the response is still linear, but elastic coefficientsare different (see Fig. 1 for an illustration). Precise com-putations can be performed in the infinite-dimensionallimit [27, 28].In this paper, inspired by this idea, we explore theelastic response of the simplest amorphous solid, a zero-temperature jammed assembly of soft spheres at pressure P , to the simplest perturbation, shear strain. We focuson the vicinity of the jamming transition, which hap-pens at the density where P = 0 for the first time upondecompression [29]. By analogy with spin glasses [24],we use two distinct measurement protocols to determinethe linear shear modulus: in the “zero-field compression”(ZFC) protocol, one first reaches the target pressure andthen applies the shear strain; in the “field compression”(FC) one first applies the shear strain, and then com-presses to target pressure P . The terminology comesfrom spin glasses where the strain is replaced by a mag-netic field [24].We obtain three main results. (i) We show that in theZFC protocol, the response is elastic with a shear mod-ulus µ ZFC that characterizes a single glassy minimum.In the FC protocol the response turns out to be stillelastic but with a distinct shear modulus µ FC < µ ZFC due to plastic events or inter-valley transitions, similarlyto what happens with magnetic susceptibility in spinglasses [24, 25, 30, 31]. This result suggests a non-trivialorganisation of glassy minima (but does not prove thatit is hierarchical as in the infinite-dimensional solution). (ii)
Infinite-dimensional calculations predict that in thelimit in which the solid unjams, and P →
0, the hierar-chical organisation of basins becomes fractal [21]; in thislimit, it is predicted that µ ZFC ∝ P / while µ FC ∝ P ,thus µ FC (cid:28) µ ZFC resulting in a sharp separation of thetwo shear moduli [27]. Our numerical data agree with thetheoretical prediction. (iii)
We find that µ FC decreaseswith increasing system size, suggesting that µ FC = 0 inthe thermodynamic limit. This finding is not consistentwith the most naive expectation based on the infinite-dimensional solution, and could be due to several aspectsof our numerical simulation protocol, as we discuss below. II. METHODSA. Details of the system
We study a 3-dimensional system of N = 1000 − U = (cid:88) i 4. This is a standard choice in studiesof jamming [29, 32].In thermal equilibrium, the control parameters arereduced temperature ˆ T = k B T /(cid:15) and volume fraction ϕ = ( π/ D + D ) ρ , where ρ = N/V is the numberdensity and V is the volume of the system. Note thatinflating the particles by increasing the diameters D , is completely equivalent to reducing the volume V : bothoperations amount to an increase of ϕ at constant ˆ T , i.e.a compression. The main observables we consider arepressure, which is the response of the system to a changein its volume: P = − V (cid:88) i Each of our N s = O (10 ) independent “samples” isobtained as follows. We start by a random configurationat ϕ init = 0 . 64 and we run molecular dynamics (MD)simulation at ˆ T = 10 − for 30 τ c ol , where τ c ol is the typ-ical collision time. This is done in order to stabilyzethe system against small thermal fluctuations within theinitial energy basin selected by the random configura-tion, similarly to [18]. However, note that ˆ T = 10 − issuch a low temperature that no barrier crossing to otherglassy basins can occur, so our system effectively remainstrapped into a random energy basin, selected by the ini-tial random configuration [18, 32]. This will be a crucialobservation for the following discussion.Next, we bring the system to ˆ T = 0 by en-ergy minimization using the conjugated gradient (CG)method [33], i.e. we reach an energy minimum close tothe thermally stabilized configurations. We obtain inthis way our initial configurations at ˆ T = 0 and ϕ init ,and from now on we always work at zero temperature.Note that ϕ init = 0 . 64 is lower than the jamming density ϕ j ≈ . C. Measurement protocols To each sample we then apply two different protocolsto measure the shear modulus, inspired by the ones usedin spin glasses [25, 30, 31]. They consist in compressingthe samples in presence or in absence of a shear strain.Before describing the protocols, we specify that(de)compression is done in small steps, during which thesystem is subjected to (i) affine deformation (multiply-ing by a common factor all particles’ diameters in such away that ϕ changes by an amount dϕ = 5 . × − ) fol-lowed by (ii) energy minimization via CG. Shear strain γ is also applied in two steps by (i) affine deformation,where x i → x i + γz i for all particles (boundary condi-tion into the z direction are also shifted by the Lees-Edwards scheme [34]), followed by (ii) energy minimiza-tion via CG.In the Field Compression (FC) protocol, the system isfirst subjected to a shear γ at ϕ init . Then it is adiabati-cally compressed (AC) in small steps (affine deformation+ CG) up to ϕ f = 0 . 66 corresponding to a pressure P f (cid:39) . σ ( P, γ ) is measured atfixed values of the pressure P ∈ [0 , P f ], and from it we de-duce the FC shear modulus µ FC ( P, γ ) = σ ( P, γ ) /γ . Next,the system is adiabatically decompressed (AD) back to ϕ init and the same measurements are performed along theway. In the Zero-Field Compression (ZFC), the system isAC up to the same P f and then AD in small steps in ab-sence of any shear. The stress and pressure are measuredafter each step of the compression and decompression. Tomeasure the stress, in the ZFC case we take the currentconfiguration and apply to it a small strain γ , and mea-sure µ ZFC ( P, γ ) = σ ( P, γ ) /γ ; the sheared configurationis then discarded.In both cases, the averages over different samples aredone at constant pressure and not at constant ϕ : infact, due to finite-size effects, the jamming point ϕ j where pressure vanishes depends on the sample [29]. Ifwe want to study the scaling for P → P is done by collecting data in the range[ P, P + dP ] choosing some dP . We examined dP in therange O (10 − ) − O (10 − ) and found that its precise choiceis irrelevant: here we choose it such that we have a goodnumber of samples in each pressure bin.In the ZFC process we also measure the shear modulusdirectly at γ = 0 via the “fluctuation formula” [8]: µ ZFC ( P, γ = 0) = b − V N (cid:88) i =1 Ξ i · ( H − Ξ ) i . (4)Here b is the Born term (affine part of µ ) defined as b = 1 V (cid:88) i We first discuss results obtained with the ZFC proto-col. We note that ZFC is the standard protocol that hasbeen used in a number of previous studies [29, 35], so wecan directly compare our data with previous work.In Fig. 2 we report results for µ ZFC obtained at con-stant pressure P and for several values of shear strain γ .We observe that at large γ there is a strong non-linearcontribution and µ ZFC ∼ P , but upon lowering γ the -4 -3 -2 -1 -4 -3 -2 -1 γ =1.5e-05 γ =6.1e-05 γ =2.4e-04 γ =9.8e-04 γ =3.9e-03 γ =1.6e-02 γ =6.2e-02power 0.5power 1 µ Z F C a / P a FIG. 3. ZFC shear modulus, scaled according to Eq. (6) with a = 4 / 3. Here dP = 2 . × − and O (10 ) samples are ineach bin. µ F C P γ =1.6e-02(AD) γ =1.6e-02(AD-short) γ =1.6e-02(AC) γ =9.8e-04(AD) γ =9.8e-04(AD-short) γ =9.8e-04 FIG. 4. FC shear modulus measured under compression anddecompression. Here data sets of N = 1000 with γ = 2 − (cid:39) . × − and γ = 2 − (cid:39) . × − . For the decompression,data obtained returning from ϕ f = 0 . 66 ( P f (cid:39) . ϕ f = 0 . 68 ( P f (cid:39) . dP =10 − − × − and the number of samples for each bin is O (10 ) for AC, O (10 ) for AD. linear response regime emerges, because the curves con-verge towards the result obtained using the fluctuationformula. This result confirms that µ ZFC is indeed a prop-erty of a single energy minimum, as it can be measuredwith the fluctuation formula while the system sits in theminimum without applying any perturbation.Also, µ ZFC is found, as in previous work [29], to scaleproportionally to P / when P → 0. In order to havea clean demonstration of this behavior, we collapse allcurves at finite γ using the form proposed in [36] in theframework of a proposed scaling theory of the jamming transition : µ ZFC ( P, γ ) = γ a/ F ( P/γ a ) . (6)Here, F ( x → ∞ ) ∼ x / , while F ( x → ∼ x . Thisimplies µ ZFC ( P, γ → ∼ P / , while µ ZFC ( P → , γ > ∼ P γ − a/ . In Fig. 3 we report a very good datacollapse using the value of a = 4 / µ ZFC ( P, γ = 0) ∼ P / [29] and extends it to the non-linear regime [36–38]. Although finite size effects mayappear at lower values of the pressure approaching theunjamming point, we checked that there is no signature offinite size effects within the range of pressure we studieddown to P = O (10 − ) and system sizes N = 500 − en passant that, although this is not the main focusof this paper, the result in Eq. (6), proposed in [36], tothe best of our knowledge has not been tested numericallybefore, and therefore it is an original result of this work. B. Field compression We now turn to the discussion of the FC shear modu-lus, which is reported in Fig. 4. First, we note that whilefor µ ZFC adiabatic compression (AC) and decompression(AD) give identical results, this is not the case for µ FC .During AC, µ FC grows linearly at small P and then satu-rates at larger P ; it remains larger during AC than duringAD. By comparing results for two different final valuesof pressure P f (see Fig. 4), we note that the AD curvesconnects perfectly linearly the final value of stress justbefore starting the decompression and 0, which seems tobe a general feature of the AD curves. We also foundthat the AD curve is reversible, i. e. the stress followsexactly the AD curve under re-compression.Moreover, while µ ZFC deviates from the linear regimefor quite small γ ( γ ∼ − , see Fig. 2), here we observethat µ FC is almost independent of γ in a regime of shearstrain that is larger by two orders of magnitude, γ (cid:46) − . We conclude that µ FC ( P ) is measured in the linearresponse regime, and is proportional to P at low pressureboth in AC and AD protocols (although with differentcoefficients).Having established the existence of a linear regime forboth FC and ZFC shear moduli, in Fig. 5 we comparethe two. We find that at all pressures, µ FC < µ ZFC , with µ FC ∼ P (cid:28) µ ZFC ∼ P / in the jamming limit. C. Comparison between glassy and crystallinesolids In Fig. 5, we compare the results for the amorphoussolids with the results for the FCC crystal, whose closepacking density is ϕ c (cid:39) . 72. The ZFC shear modulusof the FCC system is obtained using the fluctuation for-mula. We found that the non-affine correction term isabsent in the FCC case. On the other hand the FC shearmodulus of the FCC system is obtained performing pre-cisely the same analysis as we did for the glassy system.As expected, we find no difference between the FC andZFC shear moduli in the crystalline system. D. Finite size effects In Fig. 6 and Fig. 7 we study the system size depen-dence of µ FC . We observe that µ FC decreases upon in-creasing N and that N α ( γ ) µ FC is approximately constantwith a shear strain-dependent exponent α ( γ ), which sug-gests that µ FC = 0 in the thermodynamic limit for oursamples. However, we only obtained data on three sizesand the proposed N − α behavior works with an exponent α that is not so big. There might be still the opportunitythat the infinite size limit of µ FC is compatible also witha finite value, or that we are in a pre-asymptotic limitfor the finite size scaling. Finally, recall that we do notobserve significant finite size effects for µ ZFC . IV. COMPARISON WITH MEAN FIELDTHEORY As discussed in the introduction, the design of our nu-merical simulation has been inspired by the analyticalsolution of the infinite dimensional problem, which pro-vides a mean field theory for the problem and establishesan analogy with mean field spin glasses. Although ourwork is not meant to be a precise test of the theory, it isstill useful to compare our findings to what is expectedon the basis of the mean field picture.First of all, we have found that µ FC < µ ZFC , as sug-gested by the theory. Moreover, upon approaching jam-ming, µ FC ∼ P (cid:28) µ ZFC ∼ P / , which is also consistentwith the theoretical expectation. The fact that µ FC isstrongly dependent on the protocol (e.g. it is differentfor AC and AD protocols, as shown in Fig. 4) is also ex-pected from the theory: because µ FC results from an av-eraging over minima within a basin, its value depends onthe way this minima are sampled out of equilibrium [27].All these results are thus qualitatively consistent with thetheoretical expectation.There is one result, however, that deserves a more de-tailed discussion: the system size dependence of µ FC .Mean field theory predicts that glassy states preparedby slow annealing from the liquid should fall withinhierarchically-organised basins [39, 40], having a FCshear modulus that remains finite in the thermodynamiclimit. Another theoretical approach is to put a uniformweight ( `a la Edwards ) over all possible glassy states: thisleads to similar results [21]. Our preparation protocolinstead starts from totally random initial configurations,which are instantaneously cooled to very low tempera- ture, and then to zero temperature (see Section II B):this is very different from preparing a glass by slow an-nealing or by giving the same weight to all possible glassystates [22, 41–43]. This is a possible reason that explainswhy µ FC is observed to vanish for N → ∞ : our sampleshave been prepared very far from equilibrium and there-fore do not belong to fully stabilized glassy metabasins.How to compute properties of these states within meanfield theory remains an open question: it could be possi-ble using dynamical methods such as the ones introducedin [44]. It would thus be interesting to repeat this studyusing fully stabilised glasses for which theoretical predic-tions have been succesfully tested [40]: but this is muchmore computationally demanding and we leave it for thefuture. Another possibility that we cannot exclude is thatspecific finite-dimensional effects lead to the vanishing of µ FC in all glassy states: to test this idea, one should re-peat the present study in different spatial dimensions tosee if a systematic trend with dimension emerges [45]. V. DISCUSSION In this paper we have shown that the shear modulusof a simple amorphous solid at zero temperature is pro-tocol dependent: there are at least two distinct shearmoduli in the linear response regime. The FC protocol,in which strain is applied before compression, leads tosofter glasses than the ZFC protocol, in which strain isapplied after compression. The infinite-dimensional so-lution of the problem provides a natural interpretationof this result [27, 28]. In the ZFC protocol the system isfirst prepared in a minimum of the energy, then strain isapplied. In this way one probes the response of a singleenergy minimum. We confirm this by showing that µ ZFC can be equivalently obtained by the fluctuation formula,i.e. without applying strain but using linear responsein the vicinity of a single minimum. In the FC instead,the strain is applied before compression, and during com-pression the system is allowed to explore, through plas-tic events, some part of a larger “basin” composed byseveral energy minima. In this way more stress can berelaxed, leading to a softer response, µ FC < µ ZFC . Notethat while plastic events themselves are non-linear pro-cesses from the microscopic point of view, they give riseto a “renormalized”, softer linear response at the macro-scopic level. This result suggests the presence of at leasttwo “structures” in the energy landscape: minima, andbasins of minima (Fig. 1).We also find that upon approaching the jamming pointwhere pressure vanishes, the ratio µ FC /µ ZFC ∝ P / van-ishes. This result is consistent with the theoretical pre-diction obtained in infinite spatial dimensions where thestructure of minima inside clusters is hierarchical andfractal [21, 27]. It thus hints at a very complex land-scape characterized by many nested “structures”.Finally, we find that for the numerically investigatedsamples, µ FC → γ =6.1e-5FC γ =9.7e-4F-formula 'µ γ =9.8e-04FC γ =1.6e-02F-formula ' FIG. 5. Comparison of the ZFC (blue) and FC (red) shear modulus of the FCC crystal (left) and the glassy system (right).For the crystal and glassy systems N = 864 and N = 1000 are used respectively, and we use the AD protocol for the FC shearmodulus of the glass. In both cases, the fluctuation formula is used to obtain the ZFC shear-modulus. For the glassy system,binning is done by dP = 5 . × − for FC and dP = 10 − for ZFC by which the number of samples for each bin becomes O (10 ). µ FC < µ ZFC for all pressures in the glassy system. µ F C P FIG. 6. Finite size effects of the FC shear modulus. γ =2 − (cid:39) . × − . dP = 10 − and the number of samples foreach bin is O (10 ). to what the theory predicts for fully stabilized glassybasins. We tentatively attribute the discrepancy to thefact that theory focuses on fully stabilized glasses, whilein the numerical simulation we used glasses preparedfrom totally random configurations, see the discussionin Section IV.Our results are related to other works and can be ex-tended in several directions. Explorations of plasticityin amorphous solids have been reported in many stud-ies [6–10], where the instability of energy minima un-der strain have been characterised in terms of soft en-ergy modes [12–15]. In particular it has been suggestedthat plastic events happen for values of strain that vanishwhen N → ∞ as power laws, δγ ∼ N − β [12, 14, 15, 17],which suggests a non-trivial linear response even in the P µ F C N ↵ FIG. 7. Finite size scaling of the FC shear modulus. Here thedata shown in Fig. 6 are used. The exponent is α = 0 . vicinity of a single minimum. It would be interesting tocheck whether this is consistent with our results and withtheoretical predictions. It is interesting to note that thecartoon in Fig. 1 immediately suggests that if one defines • as the average over states, then dσ/dγ (cid:54) = dσ/dγ , consis-tently with the results of [46]. Furthermore, our resultsimply that there is dissipation even at zero frequency,hence the dissipative part of the frequency-dependentshear modulus does not go to zero at low frequency, as insolid friction. This is one of the signatures of soft glassyrheology [47], and is typical of energy landscapes withcusps like the one studied in [48]. Another interesting is-sue is that of non-linear responses, which are suggested tobe strongly anomalous both by theory [49] and numericalsimulations [12, 37, 46], in close relation with the com-plexity of the landscape suggested by our results. Finally,a crucial question is whether, upon adding temperature,the difference µ FC < µ ZFC persists until the glass melts,or there is a well defined temperature (a Gardner tem-perature) above which the glass becomes a normal solidwith µ FC = µ ZFC [21, 40]. ACKNOWLEDGMENTS We warmly thank Jean-Philippe Bouchaud for bring-ing this problem to our attention and for many inspiring discussions, and we thank two anonymous referees forsuggesting important improvements to a previous ver-sion of this manuscript. We also thank Giulio Biroli,Hisao Hayakawa, Andrea Liu, Michio Otsuki, GiorgioParisi, Corrado Rainone, Pierfrancesco Urbani and Yu-liang Jin for many useful discussions. This work wassupported by KAKENHI (No. 25103005 “Fluctuation& Structure” and No. 50335337) from MEXT, Japan,by JPS Core-to-Core Program “Non-equilibrium dynam-ics of soft matter and informations”. The Computationswere performed using Research Center for ComputationalScience, Okazaki, Japan. [1] N. W. Ashcroft and N. D. 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