Amorphous Order & Non-linear Susceptibilities in Glassy Materials
AAmorphous Order & Non-linear Susceptibilitiesin Glassy Materials
Giulio Biroli , Jean-Philippe Bouchaud , and Francois Ladieu Laboratoire de Physique de l’Ecole Normale Supérieure, Université PSL, CNRS, Sorbonne Université, Université deParis, F-75005 Paris, France CFM, 23 rue de l’Université, F-75007 Paris, France, & Académie des Sciences, Quai de Conti, F-75006 Paris, France SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay Bat 772, F-91191 Gif-sur-Yvette Cedex, France.
December 2020
Abstract
We review 15 years of theoretical and experimental work on the non-linear response of glassysystems. We argue that an anomalous growth of the peak value of non-linear susceptibilities is a sig-nature of growing “amorphous order” in the system, with spin-glasses as a case in point. Experimentalresults on supercooled liquids are fully compatible with the RFOT prediction of compact “glassites”of increasing volume as temperature is decreased, or as the system ages. We clarify why such a be-haviour is hard to explain within purely kinetic theories of glass formation, despite recent claims tothe contrary.
In the bestiary of material science, glasses play the role of hybrid creatures: half-liquid, half-solid,like the centaur or the mermaid. A glass has the structure factor of a liquid, yet it does not flow andresponds elastically to shear deformations – at least on time scales much shorter than the relaxationtime τ of the system, which itself increases extraordinarily fast as temperature is decreased. As anillustration, the relaxation time of “fragile glasses” (such as Ortho-Terphenyl) increases by a factor10 as temperature drops by a mere 10%.From a general point of view, a non-zero static shear modulus is necessarily associated with a lossof ergodicity, and thus a transition into a state where the dynamics is no longer able to probe theentire phase-space. The fundamental question that has riveted theoreticians for decades is whetherthe physics of glasses is indeed driven by an underlying phase transition into an ergodicity-brokenstate characterized by some amorphous long-range order, or whether the dramatic slowdown is purelyof kinetic origin, with no particular thermodynamic signature.The idea of amorphous order sounds like an oxymoron, but in fact spin-glasses [ ] offer a blueprintfor such a scenario. Randomly interacting spins (some pairs favouring alignment while others favouranti-alignment) are known to undergo a phase transition towards a frozen, “spin-glass” state belowsome critical temperature T c . Each spin then points in a random direction, but this direction remainsconstant in time – whereas at higher temperature T > T c , it flips randomly across time. Much as inglasses, instantaneous snapshots of the spin configurations seem featureless both above and below T c . But whereas there is no long range transmission of information above T c , the spin-glass phaseis rigid , in the sense that localized perturbations have a long range effect on the system – much likethe free-energy per particle of rigid bodies depends on the shape of its boundaries. In this sense, thespin-glass is indeed characterized by long-range amorphous order.Said differently, whereas spins respond more or less independently in the high temperature phase,they respond collectively in the spin-glass phase. As the transition temperature is approached fromabove, the collective nature of the response increases, and in fact diverges at T c . But contrarily toferromagnets where all spins point in the same direction, the linear magnetic susceptibility of a spin-glass does not diverge at T c , because the correlated clusters (whose spins are temporarily inter-lockedby interactions) carry a total magnetisation proportional to the square-root of their volume, and notto their volume (see Eq. (2) below for a more precise statement). However, remarkably, all higher a r X i v : . [ c ond - m a t . d i s - nn ] J a n rder static non-linear susceptibilities diverge at T c , and directly elicit the growth of amorphous orderas the spin-glass phase is approached [ ] .How much of this phenomenology is shared by supercooled liquids and other glassy materials?If glasses are simply hyper-slow liquids, as Kinetically Constrained Models (KCM) posit [
4, 5 ] , oneshould not expect any anomalous response to an external field. But if the glass transition is inti-mately related to the growth of amorphous order – now in the sense of molecules inter-locked inparticular relative positions and orientations – then we should expect, as we have indeed observedexperimentally, a strong increase of non-linear susceptibilities as the system slows down.From a theoretical point of view, the deep analogy between glasses and spin-glasses finds itsroots in the landmark series of papers by Ted Kirkpatrick, Dave Thirumalai and Peter Wolynes in themid 80’s [ ] . Based on the solution of a family of mean-field models of spin-glasses, they proposedtheir “Random First Order Transition” (RFOT) theory [ ] , which appears to capture all the knownphenomenology of supercooled liquids, in particular:• The existence of a cross-over temperature T (cid:63) below which the relaxation function exhibits aplateau as a function of time; such plateau is associated with the appearance of local rigidity(also often called “cage formation”) with a non-zero high frequency shear modulus G hf ;• The existence of an ideal glass transition (Kauzmann) temperature T K where the configurationalentropy vanishes;• An Adam-Gibbs-like correlation between the logarithm of the relaxation time and the inverseof the configurational entropy [ ] .More precisely, the RFOT theory envisages the glass state as a mosaic of “glassites” (i.e. locallyfrozen clusters), with a size (cid:96) inversely related to the configurational entropy – and thus divergingas T ↓ T K [
8, 12 ] . Being of finite size, the life-time τ of these glassites is also finite, but growsexponentially with (cid:96) . These glassites are rigid, in the sense that boundary conditions are able tolock all inside molecules around a fixed position [
12, 13 ] . Hence, glassites respond elastically toan external shear for times less than τ , with a shear modulus G hf , but start flowing for times largerthan τ when the local order finally unravels. Using a simple Maxwell model, the viscosity η of thesupercooled liquid is thus η ≈ G hf τ .Note that the relative positions and orientations of the molecules inside RFOT glassites are frozen(apart from small and fast “cage” oscillations), not because of kinetic constraints but because of actualinter-molecular forces that favour specific low energy configurations and drive the system towards athermodynamic glass when (cid:96) → ∞ . These forces will, by the same token, correlate the re-orientationsof the dipoles induced by an oscillating electric field within each glassite, provided the frequency ω ofthe electric field is somewhat larger than τ − . Hence, we expect that glassites respond collectively toan external field, amplifying, as for spin-glasses, all non-linear susceptibilities. At variance with spin-glasses, however, this collective response only holds when ωτ (cid:166)
1. At lower frequencies, glassitesmelt and all collective effects are lost [ ] .This argument suggests that, if the RFOT picture is correct, all non-linear susceptibilities shouldpeak for ωτ ∼
1, with a peak amplitude growing with the moment of the frozen dipole, and thuswith the size of the glassite itself. On the other hand, purely kinetic effects cannot lead by themselvesto growing non-linear susceptibilities. As we shall show later, in this case one should expect a peak ofnon-linear susceptibilities that shift to lower frequencies as τ increases, but with a roughly constantamplitude, independent of temperature.In the following sections, we formulate more precise theoretical statements about the behaviour ofnon-linear susceptibilities in correlated systems, and review recent experimental results that appearto confirm the analogy between glasses and spin-glasses and the presence of amorphous order thatextends to larger and larger length scales as the system slows down and / or ages. We discuss moreformally why Kinetic Constraints cannot account for the experimental results – although this does notmean that the basic tenet of KCMs, i.e. dynamic facilitation, is irrelevant [ ] . Finally, we discussopen issues and directions for future research, both theoretical and experimental. We first present a phenomenological argument that allows one to understand the growth of sus-ceptibilities in correlated systems. This argument can in fact be fully justified in the proximity of a initially anticipated by Phil Anderson, who wrote that some -– but not all — transitions to rigid, glass-like states, may entail ahidden, microscopic order parameter which is not a microscopic variable in any usual sense, and describes the rigidity of the system. igure 1: Adapted from Ref. [ ] . Third-harmonic susceptibilities in the Ag:Mn spin glass [ ] : the static value of χ ( ) divergeswhen approaching the critical temperature T c (cid:39) [ ] . At a given T > T c , χ as a plateau below the characteristic frequency1 /τ ( T ) with a power law decrease at high frequencies. phase transition, where the correlation length becomes very large [ ] . Suppose that N corr = ( (cid:96)/ a ) d f molecules are strongly correlated over a distance (cid:96) , with a the molecular size and d f the fractal di-mension of the correlated clusters. The corresponding dipoles are then essentially locked togetherduring a time τ , resulting in a total polarisation of the cluster scaling as N ζ corr , with ζ = ζ = / One expectsthat in the presence of an external field E oscillating at frequency ω ∼ τ − , the dipolar degrees offreedom of the molecules contribute to the polarisation per molecule p as: p = µ ( (cid:96)/ a ) ζ d f ( (cid:96)/ a ) d (cid:70) (cid:18) µ E ( (cid:96)/ a ) ζ d f kT (cid:19) , (1)where µ is an elementary dipole moment, (cid:70) a scaling function such that (cid:70) ( − x ) = −(cid:70) ( x ) , and d = Expanding in powers of E , one readily finds that the field induced polarisation p is given by: p µ = (cid:70) (cid:48) ( ) (cid:129) (cid:96) a (cid:139) ζ d f − d (cid:129) µ EkT (cid:139) + (cid:70) ( ) ( ) (cid:129) (cid:96) a (cid:139) ζ d f − d (cid:129) µ EkT (cid:139) + (cid:70) ( ) ( ) (cid:129) (cid:96) a (cid:139) ζ d f − d (cid:129) µ EkT (cid:139) + . . . (2)• When 2 ζ d f > d , which is the case of usual ferromagnets for which ζ = one finds that thelinear susceptibility χ diverges when (cid:96) increases, revealing an incipient ferromagnetic order.• When ζ = /
2, on the other hand, the linear susceptibility cannot diverge since d f ≤ d . There-fore, amorphous order cannot be revealed by the linear susceptibility. But the second term,contributing to the third-order susceptibility, does grow with (cid:96) provided 2 d f > d (or more gen-erally when 4 ζ d f > d ). Such a divergence is indeed observed as the spin-glass transition isapproached, see Fig. 1 for experimental results and [ ] for state of the art numerical results.This is a clear indication that the weird concept of amorphous order is indeed relevant for thisclass of materials: the spin-glass phase is a quite an exotic phase of matter.Whereas d f < d close to the spin-glass transition, RFOT theory suggests that for supercooledliquids, glassites are compact ( d f = d ) [
8, 32 ] . Assuming further that dipoles are randomly orientedin a typical frozen state leads to ζ = / | χ | ∝ (cid:96) ζ d f − d = (cid:96) d = N corr a d when ωτ ∼ Intermediate values of ζ could be expected near a tri-critical point separating a ferromagnetic, paramagnetic and spin-glassphases. In the spin-glass case, Eq. 1 is in fact equivalent to the scaling arguments of Ref. [ ] . In this case, 2 d f = d + − η , where η is the standard critical exponent for the decay of correlations [ ] . The resultingdivergence of the linear susceptibility χ as (cid:96) − η can also be obtained directly using the Fluctuation-Dissipation Theorem. For 3D Ising spin-glasses, experiments indicate that d f ≈ [ ] whereas large scale numerical simulations give d f ≈ [ ] . he third term of Eq. 2 reveals that the fifth-order susceptibility χ should diverge as (cid:96) ζ d f − d .Hence, the joint measurement of χ and χ provides – in principle – a direct way to estimate ζ d f experimentally, through the following relation: | χ | ∝ | χ | γ ; γ = ζ d f − d ζ d f − d . (4)where γ is equal to 2 when the dynamically correlated regions are compact ( d f = d ) and frozendipoles are randomly oriented ( ζ = / The measurement of non-linear dielectric susceptibilities of supercooled liquids such as glycerol orpropylene carbonate is not easy. Yet it has now been reliably achieved by several groups [ ] .Measuring χ is even more difficult, but after many years of efforts two groups (in Saclay and inAugsburg) have reported similar results that they published in a joint paper [ ] . Here, we summarizethe salient features of these results, without doing justice to the intricate details of the experiments.In order to elicit the temperature dependence of collective effects, one introduces the followingdimensionless quantities: X : = kT ε ∆ χ a χ , X : = ( kT ) ε ∆ χ a χ (5)where ε is the permittivity of free space, ∆ χ = χ ( ω = ) − χ ( ω → ∞ ) is the “dielectric strength”(removing high frequency contributions irrelevant for the glass physics), a is the molecular volumeand k is the Boltzmann constant. The main advantage of these dimensionless non-linear susceptibil-ities is that in the trivial case of an ideal gas of dipoles, the values of X and X are independent oftemperature once plotted as a function of ωτ ( T ) . Hence, their experimental variation can be ascribedto the non-trivial dynamical correlations in the supercooled liquid [
14, 31 ] .The experimental results on X and X for several supercooled liquids can be summarized asfollows:• For a fixed temperature T in the supercooled regime, X has a humped shape as a function offrequency, as shown in Fig. 2, with a maximum obtained at a frequency ≈ f α where therelaxation frequency is f α = / ( πτ ( T )) .• The height of the peak of X increases as the temperature decreases, whereas X ( ω → ) isroughly constant, see Fig. 2.• The rescaled fifth-order susceptibility X is also hump-shaped but with a much stronger increasein peak value (compared to X ) as temperature decreases, as shown in Fig. 3.• The relation between X peak3 and X peak5 is compatible with Eq. (4) with γ ≈ ± f α , X ( ω ) and X ( ω ) decrease as clean power-laws, symptomatic ofa broad spectrum of relaxation times [ ] .Aging experiments were also performed, with the conclusion that the amplification of χ alsoincreases with the age of the system, at fixed temperature [ ] . This is in agreement with the ideathat amorphous order, remarkably, propagates over larger and larger distances as the system ages.All these features are compatible with the RFOT picture of temporarily frozen glassites. The factthat X ( ω ) is humped shape corresponds to the unraveling of the amorphous order at long times (i.e. ω → X ( ω ∼ τ − ) grows as temperature is decreased is compatible with Eq. (3) with (cid:96) (or N corr ) increasing as the system is cooled down. Furthermore, the fact that γ is close to 2 meansthat (assuming ζ = /
2) glassites are compact objects with d f = d =
3, as predicted by theory, atleast sufficiently deep below T (cid:63) [ ] .Finally, the power-law dependence of X and X for ωτ (cid:29) For an early attempt, see L. Wu, Phys. Rev. B 43, 9906 (1991). igure 2: Adapted from Ref. [ ] . Dimensionless third-harmonics susceptibility X ( ) in glycerol (with a glass transition T g (cid:39) f / f α . X ( ) has a plateau at very low frequencies which does not appreciably dependson T . By contrast the maximum value of X ( ) systematically increases upon cooling. The solid line X ( ) D is the dimensionlesscubic susceptibility in a ideal gas of dipoles, and does not depend at all on T when plotted as a function of f / f α : this “trivial”case thus serves as a benchmark to elicit the glassy correlations (see [ ] for more details). I nset: same data excepted that X ( ) is normalized by its maximum value over frequency. -2 -1 | χ ( k ) k ( f ) | / | χ ( k ) k ( ) | f / f α k = 5 k = 3 k = 1 k = 5, trivial k = 3, trivialglycerol 204K Figure 3:
Adapted from Ref. [ ] . Comparison of the susceptibilities of various orders χ ( k ) k (scaled by their value at zerofrequency) in glycerol at T = (cid:39) T g + ω (resp. 5 ω ). For convenience the linear susceptibility has been denoted here χ ( ) . Similar resultswere obtained in propylene carbonate [ ] . Two points are noteworthy: ( i) the humped shape in frequency in only present fornonlinear susceptibilities (in fact, their modulus); (ii) the humped shape is much more pronounced for χ ( ) than for χ ( ) . Forcomparison the dashed lines show the corresponding curves for the “trivial” case of an ideal gas of dipoles (see [ ] for moredetails). Following up on these non-linear dielectric susceptibilities experiments, non-linear mechanicalresponses have been studied in glassy colloids [ ] . In that case, one can probe experimentally amilder regime of super-cooling in which the growth of nonlinear responses can be well explained byMode Coupling Theory and its extensions [
36, 37 ] .
95 200 205 210 215 2201,02,0
Y(T) = Max (X (5)5 Sing (T)) Augsburg Y(T) = Max (X (5)5 Sing (T)) Saclay Y(T) = [Max (X (3)3 Sing (T))] SaclayHatched area : = 2.2 ± 0.5 glycerol Y ( T ) / Y ( K ) T(K)
160 164 1680,61,01,5 Y ( T ) / Y ( K ) T(K)propylene carbonate Y(T) = Max (X (5)5 Sing (T)) Y(T) = [Max (X (3)3 Sing (T))] Hatched area : = 1.7 ± 0.4 Figure 4:
Adapted from Ref. [ ] . Temperature evolution of the singular parts of fifth and third order responses. All quantitiesare normalized at a given temperature, namely 207 K for glycerol, upper panel, and 164 K for propylene carbonate, bottom panel.This allows one to determine the exponent γ relating | X | and | X | γ and to conclude that the amorphously ordering domains arecompact (see text). The hatched areas represent the uncertainty on γ . Kinetically Constrained Models encode the view that the main physical ingredient explaining thespectacular slowing down of glasses is the rarefaction of point defects that act as facilitators forstructural rearrangements [ ] . Relaxation is postulated to take place only when one defect passesclose-by. In this scenario thermodynamics plays only a minor role: The system is simply a liquid thatcannot flow, only because of kinetic constraints. There is no driving force towards any kind of locallypreferred structure or amorphous order. In between two relaxation events, the system is in a statetypical of the liquid phase, i.e. without any special type of correlations between molecules.The density c ( T ) of facilitation defects is assumed to be small (for dynamics to be sluggish) andto go down quickly with temperature: c ( T ) = c e − J / kT , (6)where J is the energy needed to create a defect. This translates, for sufficiently complex facilitationrules [
4, 5 ] , into a relation between relaxation time τ and density of the form τ ( T ) ∼ [ c ( T )] − z ( T ) , (7)with a temperature dependent exponent z ( T ) that increases as temperature decreases, accountingfor the super-Arrhenius dependence of τ ( T ) on temperature. Ref. [ ] argues that z ( T ) = ( T o − ) / T , where T o is an “onset” temperature where local rigidity appears (in fact quite analogous tothe incipient rigidity temperature T (cid:63) mentioned above, which already suggests that rather non-trivialcorrelations must underpin the KCM picture, see below). Furthermore, each independent defectunlocks the dynamics for a cluster of size ∼ c − / d . This happens sequentially (when the defect passesby) for the molecules belonging to the same cluster. In consequence, KCMs predict the presence ofdynamical heterogeneities, with molecules in clusters of size ∼ c − / d unlocking in a correlated way. Note however that while molecules within each cluster do unlock in a correlated fashion (due tothe presence of the same defect wandering about), these molecules are still basically free to reorganizeeach in their own way. So in the presence of an oscillating electric field, each dipole carried by thesemolecules reorients in a direction that is roughly independent of the direction in which other dipolesin the cluster decide to point. Contrarily to the case of spin-glasses, or of frozen RFOT glassites, thereis no spatial propagation of information when slowing down is purely of kinetic origin. Another wayto see that KCMs do not lead to super-dipoles responding to the oscillating field is to go back to Eq.(1) and ask what is the energy scale that couples to the field when thermodynamics is trivial anddynamics is unlocked by wandering defects. Since there is no spatial correlation and relaxation takesplace sequentially, during an oscillation the external field sequentially couples to the local dipolesencountered by the defect. Hence, for KCM-like dynamics, the scaling function of Eq. (1) can onlya function of µ E / kT : there is no amplification of the effect of the field in this case, at variance withwhat happens in the presence of amorphous order extending over some length scale (cid:96) .In conclusion, while one still expects all susceptibilities to peak around ω ∼ τ − in KCMs, thereis no possibility for an increase of the peak value of X or X in the absence of any thermodynamicrigidity, in spite of claims to the contrary. For example, T. Speck [ ] has recently argued that theexperimental result X ∝ X is fully compatible with the KCM picture. Our explanation above showthat this cannot be the case.Let us repeat here Speck’s argument for the sake of clarity: when ω ∼ τ − , there is a number ∝ c ( T ) of active clusters of size (cid:96) ( T ) that can respond to an external electric field. Each such clustercarries a dipole ∼ (cid:96) d / and thus, according to an argument very similar to the one leading to Eq. (2),the k -th order peak susceptibility should scale as: χ KCM k ∼ µ k + T k c ( T ) [ (cid:96) ( T )] d ( k − ) / , ( k =
1, 3, 5, · · · ) (8)from which, using X KCM : = T χ KCM ( χ KCM ) ; X KCM : = T χ KCM ( χ KCM ) , (9)one obtains X KCM ∼ (cid:96) d / c and X KCM ∼ (cid:96) d / c , and hence X KCM ≈ ( X KCM ) γ with γ ≡
2, as found inexperiments.There are various problems with this argument. First, as argued above, it is not because domainsare dynamically correlated that response to an external field is collective. In theories and modelsbased on Kinetic Constraints, rearrangements are synchronized in time, but still local in space: thereis no “super-dipole” of size (cid:96) d / which responds rigidly. But more mundanely, Eq. (8), which is crucialto obtain X KCM = ( X KCM ) , also predicts that the linear susceptibility χ is proportional to c ( T ) / T andshould thus strongly decay as temperature is decreased, at variance with empirical results where χ increases roughly as ∼ / T . If one (rightly) argues that χ is in fact dominated by immobile regionsand not by active regions, then the definitions of X and X in Eq. (9) are inadequate and shouldrather read X = T χ KCM ; X = T χ KCM , (10)where we have used χ ∼ µ/ T . Now, assuming that only active regions contribute to χ and χ as inEq. (8) one finds that (cid:96) factors cancel out and X X = c ( T ) , (11)which should be (i) quite large since the concentration of defects c ( T ) is assumed to be small in theglassy region; and more importantly (ii) strongly increase as temperature is reduced since c ( T ) isassumed to decrease considerably when temperature is lowered. Both facts are in complete contra-diction with our experimental data: X / X is instead found to be roughly independent of temperatureand at most equal to ≈ In some more refined theories, the unlocked clusters are fractal and c − / d is replaced by c − / d f [
5, 33 ] . Note in that respect that χ is indeed age independent , whereas χ does depend on age [ ] . ence, we cannot subscribe to the claim that kinetic pictures “explain” the experimental results onnon-linear susceptibilities. The same remark applies to other non-thermodynamic theories of glasses,such as the “shoving model” [ ] . In fact, as we have argued elsewhere [ ] , RFOT describes the verymechanism that leads to the appearance of local rigidity, and hence to the possibility of extremelylong-lived metastable states in these systems for T < T (cid:63) or T o . Absent such a local caging mechanismthat hinders individual moves, the whole idea of kinetically constrained models falls apart.Therefore, theories based on Kinetic Constraints should first come up with a consistent scenariofor local rigidity – like for example the one proposed in [ ] – before invoking facilitation as themechanism for sluggishness. Whereas facilitation processes are no doubt present in glasses, we be-lieve that the fundamental mechanism leading to metastability for T < T o is of thermodynamic origin,and is necessarily accompanied by some incipient amorphous order.On the other hand, dynamical heterogeneities – characterizing how many molecules rearrange ina correlated way, but not necessarily cooperatively – do extend over length scales that are somewhatlarger than the length (cid:96) over which amorphous order sets in. This is trivially true for KCM, sincedynamical correlations grow in the absence of any local order. More generally, it was shown in [ ] that the dynamical correlation length can only diverge faster than the amorphous order correlationlength (cid:96) that governs non-linear susceptibilities.Finally, we do not agree either with the argument given by Speck [ ] on the role of randompinning on glassiness induced by Kinetic Constraints. Since the size of wandering defects is muchsmaller than (cid:96) , pinning a fraction c pin ∼ (cid:96) − d (cid:28) [ ] . Hence,in absence of a precise analysis or detailed model studies, we do not see how theories only basedon Kinetic Constraints can explain the results of, e.g. Ref. [ ] , where randomly pinning a smallfraction of particles dramatically changes the glassy dynamics. In our view, random pinning is insteada promising avenue for gaining more insights about the glass transition, the ideal glass phase and thevalidity of the RFOT theory. We have shown that a growing static length associated to an emerging amorphous order naturallyleads to growing non-linear susceptibilities. This generally holds true when there is an underlyinggrowing static length, as the one envisioned in RFOT theory but also in the case of locally preferredstructures [ ] or frustration-limited domains [ ] . In contrast, we have argued that a purely dy-namical mechanism for the slowing-down of the dynamics, such as the one put forward in KineticConstraints based theories [ ] , cannot be responsible for the growth of the non-linear dynamical sus-ceptibilities, mainly because there is no spatial propagation of information in such a framework. Itwould actually be quite instructive to study numerically the non-linear susceptibilities of model sys-tems and confirm that KCMs do behave trivially in that respect, whereas more realistic models indeedexhibit an increase of non-linear response as the glass transition is approached.The experimental results reviewed in this paper clearly demonstrate that non-linear susceptibilitiesdo grow as the system slows down. The growth is stronger for higher non-linearities, as expected fora critical phenomenon. This is solid evidence that the glass transition is accompanied by a growingstatic length of the type predicted by RFOT theory. Such a conclusion however does not contradictthe idea that dynamical facilitation plays a role in the dynamics of supercooled liquids. Even withinRFOT, it is clear that the relaxation of a given glassite will trigger relaxation in neighbouring glassites,and hence that dynamical correlations must extend beyond the size of the glassites (cid:96) (see [ ] , and [ ] for a recent discussion of this point). Whereas non-linear susceptibilities probe the increase of (cid:96) , dynamical correlations should extend over a possibly much larger length-scale.Hence the question is not whether dynamical facilitation, rather than amorphous order, can ex-plain the increase of the non-linear susceptibilities. As the theoretical and experimental results pre-sented here show, such a growth must be due to an increase of a static length. The central issue isinstead assessing the relative role of dynamical constraints versus amorphous order in the momentousincrease of the relaxation time in glassy system. [ Note that such an increase in time-scale amounts toa multiplication of the activation energy, for fragile glasses, by a factor 2 to 5 between the incipientrigidity temperature T (cid:63) and the glass transition temperature T g ] .This question has been debated recently in relation with the spectacular acceleration of “swap”Monte Carlo simulations [ ] ; we refer to [
16, 17 ] for two opposite conclusions on this matter. Inorder to settle this issue, the most pressing theoretical is to elicit the precise nature and geometry f the activated events that unlock the amorphous order inside glassites [
12, 42 ] . In particular, aquantitative understanding of the activation energy, and its dependence on the size of glassites (cid:96) [ ] , is still very much needed to patch together all the pieces of the puzzle – in particular the power-law behaviour of χ and χ for ωτ (cid:29)
1. Note that the aging experiments of [ ] offer some importantinsights, again compatible with the prediction of RFOT.Finally, let us end where we started, i.e. with spin-glasses. Although many experiments have beenperformed to probe the divergence of non-linear susceptibilities above the phase transition, and theaging properties of the linear susceptibility below the transition, we are not aware of any experimentalinvestigation of the non-linear susceptibility in the aging regime. This would provide very interestinginformation on the nature of the spin-glass phase, and, possibly, an alternative approach to the vexingquestion of the existence of a de Almeida-Thouless line in three dimensional spin-glasses (on thispoint, see the discussion in [ ] ). Acknowledgments
We thank Dave Thirumalai for inviting us to put together our thoughts andwrite this piece. We also want to acknowledge the tremendous influence of his (and his collaborators’)ideas on our own research trail. We also thank all our collaborators on these topics for their numerousand invaluable inputs.
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