Third and Fifth Harmonic Responses in Viscous Liquids
Samuel Albert, M. Michl, P. Lunkenheimer, A. Loidl, P. Déjardin, F. Ladieu
TThird and fifth harmonic responses in viscousliquids
S. Albert, M. Michl, P. Lunkenheimer, A. Loidl, P. M. D´ejardin, and F. Ladieu
Most of our everyday materials are glasses, from window glasses to plastic bottles,and from colloids to pastes and granular materials. Yet the formation of the glassystate is still a conundrum and the most basic questions about the nature of the glassystate remain unsolved, e.g., it is still hotly debated whether glasses are genuine solidsor merely hyperviscous liquids.Over the past three decades, the notion evolved that higher-order harmonic sus-ceptibilities are especially well suited to unveil the very peculiar correlations gov-erning the glass formation, yielding information that cannot be accessed by moni-
S. AlbertSPEC, CEA, CNRS, Universit´e Paris-Saclay, CEA Saclay Bat 772, 91191 Gif-sur-Yvette Cedex,France. e-mail: [email protected]
M. MichlExperimental Physics V, Center for Electronic Correlations and Magnetism, University of Augs-burg, 86159 Augsburg, Germany. e-mail: [email protected]
P. LunkenheimerExperimental Physics V, Center for Electronic Correlations and Magnetism, Univer-sity of Augsburg, 86159 Augsburg, Germany. e-mail: [email protected]
A. LoidlExperimental Physics V, Center for Electronic Correlations and Magnetism, University of Augs-burg, 86159 Augsburg, Germany. e-mail: [email protected]
P. M. D´ejardinLAMPS Universit´e de Perpignan Via Domitia - 52 avenue Paul Alduy - 66860 Perpignan Cedex,France. e-mail: [email protected]
F. LadieuSPEC, CEA, CNRS, Universit´e Paris-Saclay, CEA Saclay Bat 772, 91191 Gif-sur-Yvette Cedex,France. e-mail: [email protected] a r X i v : . [ c ond - m a t . s o f t ] J u l S. Albert, M. Michl, P. Lunkenheimer, A. Loidl, P. M. D´ejardin, and F. Ladieu toring the linear response. This is illustrated in Fig. 1 displaying the third harmoniccubic susceptibility χ ( ) -defined in Section 2.1- for four very different kinds ofglasses [1, 2, 3, 4, 5, 6]. In the case of spin glasses [1, 7] -see Fig. 1(A)-, it was dis-covered in the eighties that χ ( ) diverges at the spin glass transition temperature T SG ,revealing the long range nature of the spin glass amorphous order emerging around T SG . Here the expression “amorphous order” corresponds to a minimum of the freeenergy realized by a configuration which is not spatially periodic. Similar non-linearsusceptibility experiments have been performed by Hemberger et al. [2] on an orien-tational glass former. In orientational glasses, electric dipolar or quadrupolar degreesof freedom undergo a cooperative freezing process without long-range orientationalorder [8]. As illustrated in Fig. 1(B), the divergence of | χ ( ) | is not accompanied byany divergence of the linear susceptibility | χ | .We shall show in Eqs. (1)-(2) that this is intimately related to the very notion ofamorphous ordering. For structural glasses, e.g., glycerol, it was discovered [3, 4]less than 10 years ago that | χ ( ) ( ω , T ) | has a hump close to the α relaxation fre-quency f α , the height of this hump increasing anomalously upon cooling. A humpof | χ ( ) | has also been recently discovered in a colloidal glass [5, 6], in the vicinityof the β relaxation frequency f β , revealing that any shear strain connects the systemto a non equilibrium steady state -see [5, 6]-. Of course, as detailed balance does nothold in colloids, the comparison of colloidal glasses with spin glasses, orientationalglasses, and structural glasses cannot be quantitative. However, the four very differ-ent kinds of glasses of Fig. 1 have the common qualitative property that nonlinearcubic responses unveil new information about the glassy state formation.Let us now give an oversimplified argument explaining why nonlinear responsesshould unveil the correlations developing in glasses. We shall adopt the dielectriclanguage adapted to this review devoted to supercooled liquids -where detailed bal-ance holds-, and consider a static electric field E st applied onto molecules carrying adipole moment µ dip . At high temperature T the system behaves as an ideal gas andits polarization P is given by: P = µ dip a d L d (cid:18) µ dip E st k B T (cid:19) (cid:39) µ dip a d (cid:18) µ dip E st k B T (cid:19) − µ dip a d (cid:18) µ dip E st k B T (cid:19) + µ dip a d (cid:18) µ dip E st k B T (cid:19) + ... (1)where a d is the molecular d -dimensional volume, L d is the suitable Langevin func-tion expressing the thermal equilibrium of a single dipole in dimension d , and wherethe numerical prefactors of the linear, third, and fifth order responses correspond tothe case d =
3. Assume now that upon cooling some correlations develop over acharacteristic lengthscale (cid:96) , i.e. molecules are correlated within groups containing N corr = ( (cid:96)/ a ) d f molecules, with d f the fractal dimension characterizing the corre-lated regions. Because these domains are independent from each other, one can useEq. (1), provided that we change the elementary volume a d by that of a domain - hird and fifth harmonic responses in viscous liquids 3 Frequency (Hz) R e [ c ( ) ] ( a . u . ) (A) 𝑇 − 𝑇 𝑐 𝑇 𝑐 (C) (D) (B) c [ c ( ) ] / c c c Fig. 1
From Refs. [1, 2, 3, 6]. Third-harmonic susceptibilities of very different types of glassesapproaching their glass transition. (A) : In the Ag:Mn spin glass [1], the static value of χ ( ) di-verges when approaching the critical temperature T c (cid:39) .
94 K [1]. (B) : Similar arguments are usedto rationalize the third-harmonic dielectric susceptibility of an orientational glass [2]. (C) : In glyc-erol [3, 4], the modulus of the -dimensionless- cubic susceptibility X ( ) has a peak as function offrequency, which increases anomalously upon cooling. (D) : Strain-stress experiment in the col-loidal system studied in Refs. [5, 6]. When increasing the volumic density φ the increasing peak of Q = | χ ( ) / χ | reveals that any shear strain connects the system to a non equilibrium steady state-see [5, 6]-. In all these four examples χ ( ) unveils informations about the nature of the glassy statethat cannot be obtained by studying the linear susceptibility χ lin . namely a d ( (cid:96)/ a ) d -, as well as the molecular dipole µ dip by that of a domain -namely µ dip ( (cid:96)/ a ) ( d f / ) -. Here, the exponent d f / P µ dip / a d (cid:39) (cid:18) (cid:96) a (cid:19) d f − d (cid:18) µ dip E st k B T (cid:19) − (cid:18) (cid:96) a (cid:19) d f − d (cid:18) µ dip E st k B T (cid:19) ++ (cid:18) (cid:96) a (cid:19) d f − d (cid:18) µ dip E st k B T (cid:19) + ... (2) S. Albert, M. Michl, P. Lunkenheimer, A. Loidl, P. M. D´ejardin, and F. Ladieu which shows that the larger the order k of the response, the stronger the increase ofthe response when (cid:96) increases. As d f ≤ d , Eq. (2) shows that the linear response never diverges with (cid:96) : it is always, for any (cid:96) , of the order of µ dip / ( a d k B T ) . Thiscan be seen directly in Eq. (2) in the case d f = d ; while for d f < d one must addto Eq. (2) the polarization arising from the uncorrelated molecules not belonging toany correlated region. This insensitivity of the linear response to (cid:96) directly comesfrom the amorphous nature of orientations that we have assumed when rescalingthe net dipole of a domain -by using the power d f / d f to rescale the moment ofa domain, and we would find that the linear response diverges with (cid:96) as soon as d f > d / (cid:96) , as soon as d f > d /
2. This is why cubic responses -as well as higher orderresponses- are ideally suited to test whether or not amorphous order develops insupercooled liquids upon cooling.For spin-glasses, the above purely thermodynamic argument is enough to relatethe divergence of the static value of χ ( ) -see Fig 1-(A)- to the divergence of theamorphous order correlation length (cid:96) . For structural glasses this argument must becomplemented by some dynamical argument, since we have seen on Fig. 1-(C) thatthe anomalous behavior of χ ( ) takes place around the relaxation frequency f α . Thishas been done, on very general grounds, by the predictions of Bouchaud and Biroli(BB), who anticipated [9] the main features reported in Fig. 1-(C). BB’s predic-tions will be explained in Section 3. Before, we shall review in Section 2 the mainexperimental features of third and fifth harmonic susceptibilities. Because of thegenerality of Eq. (2) and of BB’s framework, we anticipate that χ and χ havecommon anomalous features that can be interpreted as reflecting the evolution of (cid:96) -and thus of N corr - upon cooling. The end of the chapter, Section 4, will be devotedto more specific approaches of the cubic response of glass forming liquids. Beyondtheir apparent diversity, we shall show that they can be unified by the fact that in allof them, N corr is a key parameter -even though it is sometimes implicit-. The Ap-pendix contains some additional material for the readers aiming at deepening theirunderstanding of this field of high harmonic responses. When submitted to an electric field E ( t ) depending on time t , the most generalexpression of the polarisation P ( t ) of a dielectric medium is given by a series ex-pansion : hird and fifth harmonic responses in viscous liquids 5 P ( t ) = ∞ ∑ m = P m + ( t ) (3)where because of the E → − E symmetry, the sum contains only odd terms, andthe ( m + ) -order polarisation P m + ( t ) is proportional to E m + . The most generalexpression of P m + ( t ) is given by: P m + ( t ) ε = (cid:90) ∞ − ∞ ... (cid:90) ∞ − ∞ χ m + ( t − t (cid:48) , ..., t − t (cid:48) m + ) E ( t (cid:48) ) ... E ( t (cid:48) m + ) dt (cid:48) ... dt (cid:48) m + (4)Because of causality χ m + ≡ E ( t ) = E cos ( ω t ) of frequency ω and of amplitude E , it is convenient to replace χ m + by its ( m + ) -fold Fourier transform and to integrate first over t (cid:48) , ..., t (cid:48) m + .Defining the one-fold Fourier transform φ ( ω ) of any function φ ( t ) by φ ( ω ) = (cid:82) φ ( t ) e − i ω t dt (with i = −
1) and using (cid:82) e − i ( ω − ω ) t dt = πδ ( ω − ω ) , where δ is the Dirac delta function, one obtains the expression of P m + ( t ) . This expressioncan be simplified by using two properties: (a) the fact that the various frequencies ω λ play the same role, which implies χ m + ( − ω , ω , ..., ω ) = χ m + ( ω , − ω , ..., ω ) ; (b) the fact that χ m + is real in the time domain implying that χ m + ( − ω , ..., − ω ) is the complex conjugate of χ m + ( ω , ..., ω ) . By using these two properties, we ob-tain the expression of all the P m + ( t ) , and in the case of the third order polarisationthis yields: P ( t ) ε = E | χ ( ) ( ω ) | cos ( ω t − δ ( ) ( ω )) + E | χ ( ) ( ω ) | cos ( ω t − δ ( ) ( ω )) (5)where we have set χ ( ω , ω , ω ) = | χ ( ) ( ω ) | e − i δ ( ) ( ω ) , and χ ( ω , ω , − ω ) = | χ ( ) ( ω ) | e − i δ ( ) ( ω ) .Similarly, for the fifth-order polarisation, we obtain: P ( t ) ε = E | χ ( ) ( ω ) | cos ( ω t − δ ( ) ( ω )) + E | χ ( ) ( ω ) | cos ( ω t − δ ( ) ( ω )) ++ E | χ ( ) ( ω ) | cos ( ω t − δ ( ) ( ω )) (6)where, we have set χ ( ω , ω , ω , ω , ω ) = | χ ( ) ( ω ) | e − i δ ( ) ( ω ) , and similarly χ ( ω , ω , ω , ω , − ω ) = | χ ( ) ( ω ) | e − i δ ( ) ( ω ) as well as χ ( ω , ω , ω , − ω , − ω ) = | χ ( ) ( ω ) | e − i δ ( ) ( ω ) .For completeness we recall that the expression of the linear polarisation P ( t ) is P ( t ) / ε = E | χ ( ω ) | cos ( ω t − δ ( ω )) where we have set χ ( ω ) = | χ ( ω ) | e − i δ ( ω ) .In the linear case, we often drop the exponent indicating the harmonic, since thelinear response P ( t ) is by design at the fundamental angular frequency ω . The onlyexception to this simplification is in Fig. 11 (see below) where for convenience thelinear susceptibility is denoted χ ( ) . S. Albert, M. Michl, P. Lunkenheimer, A. Loidl, P. M. D´ejardin, and F. Ladieu
Up to now we have only considered nonlinear responses induced by a pure acfield E , allowing to define the third harmonic cubic susceptibility χ ( ) and/or thefifth-harmonic fifth-order susceptibility χ ( ) to which this chapter is devoted. InSection 2.3 and Figs. 8-9, we shall briefly compare χ ( ) with other cubic suscep-tibilities, namely χ ( ) already defined in Eq. (5) as well as χ ( ) that we introducenow.This supplementary cubic susceptibility is one of the new terms arising when astatic field E st is superimposed on top of E . Because of E st , new cubic responsesarise, both for even and odd harmonics. For brevity, we shall write only the expres-sion of the first harmonic part P ( ) of the cubic polarization, which now containstwo terms: P ( ) ( t ) ε = | χ ( ) ( ω ) | E cos ( ω t − δ ( ) ( ω )) + | χ ( ) , ( ω ) | E st E cos ( ω t − δ ( ) , ( ω )) (7)where we have defined | χ ( ) , ( ω ) | exp ( − i δ ( ) , ( ω )) = χ ( , , ω ) .For any cubic susceptibility – generically noted χ – or for any fifth-order sus-ceptibility – generically noted χ – the corresponding dimensionless susceptibility X or X is defined as : X ≡ k B T ε ∆ χ a χ , X ≡ ( k B T ) ε ∆ χ a χ (8)where ∆ χ is the “dielectric strength”, i.e. ∆ χ = χ lin ( ) − χ lin ( ∞ ) where χ lin ( ) isthe linear susceptibility at zero frequency and χ lin ( ∞ ) is the linear susceptibility at a-high- frequency where the orientational mechanism has ceased to operate. Note that X as well as X have the great advantage to be both dimensionless and independentof the field amplitude. In this section we review the characteristic features of χ ( ) both as a function offrequency and temperature. We separate the effects at equilibrium above T g andthose recorded below T g in the out-of-equilibrium regime. hird and fifth harmonic responses in viscous liquids 7 Fig. 2
Third-order harmonic component of the dielectric susceptibility of propylene carbonate[10]. Spectra of | χ ( ) | E are shown for various temperatures measured at a field of 225 kV/cm.The yellow-shaded plane indicates the plateau arising in the trivial regime. T g In the α regime:Fig. 2 shows the modulus | χ ( ) | for propylene carbonate [10]. It is an archetypicalexample of what has been measured in glass forming liquids close to T g . For a giventemperature one distinguishes two domains:1. For very low frequencies, f / f α ≤ .
05, a plateau is observed as indicated by theshaded area in Fig. 2, i.e. | χ ( ) | does not depend on frequency. This is reminis-cent of the behavior of an ideal gas of dipoles where each dipole experiencesa Brownian motion without any correlation with other dipoles. In such an idealgas, | χ ( ) | has a plateau below the relaxation frequency and monotonously fallsto zero as one increases the frequency. Because the observed plateau in Fig. 2 isreminiscent to the ideal gas case, it has sometimes [3, 4] been called the “trivial”regime. What is meant here is not that the analytical expressions of the various χ are “simple” –see Section 7–, but that the glassy correlations do not changequalitatively the shape of χ ( ) in this range. Physically, an ideal gas of dipolescorresponds to the high- T limit of a fluid. This is why it is a useful benchmarkwhich allows to distinguish the “trivial” features and those involving glassy cor-relations.2. When rising the frequency above 0 . f α one observes for | χ ( ) | a hump for afrequency f peak / f α (cid:39) c where the constant c does not depend on T and weaklydepends on the liquid (e.g., c (cid:39) .
22 for glycerol and c (cid:39) . S. Albert, M. Michl, P. Lunkenheimer, A. Loidl, P. M. D´ejardin, and F. Ladieu bonate). This hump is followed by a power law decrease | χ ( ) | ∼ f − β where β < | χ | above f α .Qualitatively, this hump is important since it exists neither in the cubic suscep-tibility of an ideal gas of dipoles nor in the modulus of the linear response | χ | of the supercooled liquids. This is why this hump has been termed the “glassycontribution” to χ . On a more quantitative basis, the proportionality of f peak andof f α has been observed for f α ranging from 0 .
01 Hz to 10 kHz -above 10 kHzthe measurement of χ ( ) is obscured by heating issues, see [11] and Section 7-. -1 -101 (b) " E ' E | | E glycerol f (Hz) K -2 -1 -505 E ' E " E K | | E (a) Fig. 3 (a) Modulus, real, and imaginary part of the third-order dielectric susceptibility χ ( ) (times E ) of 1-propanol at 120 K as measured with a field of 468 kV/cm [16]. The solid lines werecalculated according to Refs. [17]. (b) Same for glycerol at 204 K and 354 kV/cm [16]. The consistency of the above considerations can be checked by comparing thethird-order susceptibility of canonical glass formers to that of monohydroxy alco-hols. The linear dielectric response of the latter is often dominated by a Debye relax- hird and fifth harmonic responses in viscous liquids 9 ation process, which is commonly ascribed to the fact that part of the molecules areforming chain-like hydrogen-bonded molecule clusters with relatively high dipolarmoments [12]. This process represents an idealised Debye-relaxation case as it lacksthe heterogeneity-related broadening found for other glass formers. Moreover, cor-relations or cooperativity should not play a significant role for this process, becausecluster-cluster interactions can be expected to be rare compared to the intermolec-ular interactions governing the α relaxation in most canonical glass formers [13].Thus, this relaxation process arising from rather isolated dipolar clusters distributedin a liquid matrix can be expected to represent a good approximation of the ”idealdipole gas” case mentioned above. The monohydroxy alcohol 1-propanol is espe-cially well suited to check this notion because here transitions between differentchain topologies, as found in several other alcohols affecting the nonlinear response[14, 15], do not seem to play a role [15]. Figure 3(a) shows the frequency-dependentmodulus, real, and imaginary part of χ ( ) E for 1-propanol at 120 K [13, 16]. In-deed, no hump is observed in | χ ( ) | ( ν ) as predicted for a non-cooperative Debyerelaxation. The solid lines were calculated according to Refs. [17], accounting forthe expected trivial polarization-saturation effect. Indeed, the spectra of all threequantities are reasonably described in this way. In the calculation, for the molecularvolume an additional factor of 2.9 had to be applied to match the experimental data,which is well consistent with the notion that the Debye relaxation in the monohy-droxy alcohols arises from the dynamics of clusters formed by several molecules.In marked contrast to this dipole-gas-like behavior of the Debye relaxation of1-propanol, the χ ( ) spectra related to the conventional α relaxation of canonicalglass formers exhibit strong deviations from the trivial response, just as expected inthe presence of molecular correlations. As an example, Fig. 3(b) shows the modu-lus, real, and imaginary part of χ ( ) E of glycerol at 204 K. Again the lines werecalculated assuming the trivial nonlinear saturation effect only [17]. Obviously, thisapproach is insufficient to provide a reasonable description of the experimental data.Only the detection of plateaus in the spectra arising at low frequencies agrees withthe calculated trivial response. This mirrors the fact that, on long time scales, theliquid flow smoothes out any glassy correlations.When varying the temperature, two very different behaviors of χ ( ) are observed:1. In the plateau region the weak temperature dependence of χ ( ) is easily capturedby converting χ ( ) into its dimensionless form X ( ) by using Eq. (8): one observes[3, 4] that in the plateau region X ( ) does not depend at all on the temperature .Qualitatively this is important since in an ideal gas of dipoles X ( ) does alsonot depend on temperature, once plotted as a function of f / f α . This reinforcesthe “trivial” nature of the plateau region, i.e. the fact that it is not qualitativelyaffected by glassy correlations.2. In the hump region, | X ( ) ( f / f α ) | increases upon cooling, again emphasizingthe “anomalous” –or “non trivial”– behavior of the glassy contribution to χ ( ) .This increase of the hump of | X ( ) | has been related to that of the apparent Fig. 4
From Ref. [18]: For several glass formers, N corr ( T ) as extracted from the hump of | X ( ) | (leftaxis) closely follows E act ( T ) , deduced from the temperature dependence of the α -relaxation time[10] (right axis). The abbreviations stand for propylene carbonate (PCA), 3-fluoroaniline (FAN),2-ethyl-1-hexanol (2E1H), cyclo-octanol (c-oct), and a mixture of 60% succinonitrile and 40%glutaronitrile (SNGN). activation energy E act ( T ) ≡ ∂ ln τ α / ∂ ( / T ) -see refs. [10, 18]- as well as to T χ T ≡ | ∂ ln τ α / ∂ ln T | [3, 4, 19, 20]. Note that because the experimental tem-perature interval is not so large, the temperature behavior of E act and of T χ T is extremely similar. Both quantities are physically appealing since they are re-lated to the number N corr ( T ) of correlated molecules: the line of thought where E act ∼ N corr ( T ) dates back to the work of Adam and Gibbs [21]; while anotherseries of papers [22, 23] proposed a decade ago that N corr ∝ T χ T . Fig. 4 illus-trates how good is the correlation between the increase of the hump of | X ( ) | -leftaxis- and E act ( T ) . This correlation holds for 5 glass formers, of extremely differ-ent fragilities, including a plastic crystal, where only the orientational degrees offreedom experience the glass transition [24].In the excess wing regime:In the dielectric-loss spectra of various glass formers, at high frequencies the excesswing shows up, corresponding to a second, shallower power law at the right flankof the α peak [25]. Figure 5(a) shows loss spectra of glycerol, measured at low andhigh fields up to 671 kV/cm [26, 27], where the excess wing is indicated by thedashed lines. (It should be noted that the difference of these loss curves for high andlow fields is directly related to the cubic susceptibility χ ( ) , defined in Eq. (5) [16].)As already reported in the seminal paper by Richert and Weinstein [28], in Fig. 5(a)at the right flank of the α -relaxation peak a strong field-induced increase of thedielectric loss is found while no significant field dependence is detected at its low- hird and fifth harmonic responses in viscous liquids 11 frequency flank. In Ref. [28] it was pointed out that these findings are well consistentwith the heterogeneity-based box model (see Section 4.3). However, as revealed byFig. 5(a), remarkably in the region of the excess wing no significant nonlinear effectis detected. Time-resolved measurements, later on reported by Samanta and Richert[29], revealed nonlinearity effects in the excess-wing region when applying the highfield for extended times of up to several 10000 cycles. Anyhow, the nonlinearity inthis region seems to be clearly weaker than for the main relaxation and the nonlinearbehavior of the excess wing differs from that of the α relaxation.To check whether weaker nonlinearity in the excess-wing region is also revealedin higher-harmonic susceptibility measurements, Fig. 5(b) directly compares themodulus of the linear dielectric susceptibility of glycerol at 191 K to the third-order susceptibility | χ ( ) | (multiplied by E ) [30]. (We show | χ | corrected for χ , ∞ = ε ∞ − | χ ( ) ( ν ) | . Thus, we conclude that possiblenonlinearity contributions arising from the excess wing, if present at all, must be sig-nificantly weaker than the known power-law decay of the third-order susceptibilityat high frequencies, ascribed to the nonlinearity of the α relaxation.The excess wing is often regarded as the manifestation of a secondary relaxationprocess, partly superimposed by the dominating α -relaxation [31, 32]. Thus theweaker nonlinearity of the excess wing seems to support long-standing assumptionsof the absence of cooperativity in the molecular motions that lead to secondaryrelaxation processes [33, 34]. Moreover, in a recent work [35] it was pointed outthat the small or even absent nonlinear effects in the excess-wing region can also beconsistently explained within the framework of the coupling model [34], where theexcess wing is identified with the so-called ”nearly constant loss” caused by cagedmolecular motions. T g Below T g , the physical properties are aging , i.e. they depend on the time t a elapsedsince the material has fallen out of equilibrium, i.e. since the glass transition temper-ature T g has been crossed. The mechanism of aging is still a matter of controversy[36, 37, 38, 39, 40], owing to the enormous theoretical and numerical difficulties in-herent to out-of-equilibrium processes. Experimentally, a few clear cut results havebeen obtained in spin glasses [41] where it was shown, by using nonlinear tech-niques, that the increase of the relaxation time τ α with the aging time t a can be ratherconvincingly attributed to the growth of the number N corr of correlated spins with t a . Very recently extremely sophisticated numerical simulations have been carriedout by the so called Janus international collaboration, yielding, among many otherresults, a strong microscopic support [42] to the interpretation given previously inthe experiments of Ref. [41]. - 3 - 1 - 3 - 2 - 1 | c | k V / c m ( b ) g l y c e r o l K | c c
3| E2 f ( H z ) | c | E k V / c m - 2 - 1 o p e n s y m b o l s , l i n e s : l o w f i e l dc l o s e d s y m b o l s : 6 7 1 k V / c m g l y c e r o l K 1 9 5
K 2 0 4 K e " K ( a ) Fig. 5 (a) Dielectric loss of glycerol measured at fields of 14 kV/cm (open symbols) and671 kV/cm (closed symbols) shown for four temperatures [27]. The solid lines were measuredwith 0.2 kV/cm [26]. The dashed lines indicate the excess wing. (b) Open triangles: Absolute val-ues of χ (corrected for χ , ∞ = ε ∞ −
1) at 14 kV/cm for glycerol at 191 K. Closed triangles: χ ( ) E at 565 kV/cm [30]. The solid lines indicate similar power laws above the peak frequency for bothquantities. The dashed line indicates the excess wing in the linear susceptibility at high frequencies,which has no corresponding feature in χ ( ) ( ν ) . In structural glasses, the aging properties of the linear response have been re-ported more than one decade ago [43, 44]. More recently, the aging properties of χ ( ) were reported in glycerol [45] and its main outputs are summarized in Figs. 6and 7. A glycerol sample previously well equilibrated at T g + T w = T g − t a . The dominant effect is the increase of hird and fifth harmonic responses in viscous liquids 13 Fig. 6
From [45]. During the aging of glycerol -at T g − τ α with the aging time t a is measured by rescaling the aging data -symbols- of χ (cid:48)(cid:48) -right axis- onto the equilibrium data-solid black line-. The corresponding scaling fails for X ( ) ( f , t a ) -left axis- revealing the increaseof N corr during aging. See [45] for details about the quantity z ( t a ) / z ( T ) which is involved in theleft axis but varies by less than 2% during aging-. Fig. 7
From [45]. The values of δ = N corr ( t a ) / N corr ( eq ) extracted from Fig. 6 show the increase of N corr during aging. Inset: different theories are tested gathering equilibrium and aging experiments. the relaxation time τ α with t a . In Ref. [45] τ α increases by a factor (cid:39) T w -i.e. t a =
0- and the finally equilibrated situation reached for t a (cid:29) τ α , eq where τ α is equal to its equilibrium value τ α , eq -and no longer evolves with t a -.This variation of τ α with the aging time t a can be very accurately deduced fromthe shift that it produces on the imaginary part of the linear response χ (cid:48)(cid:48) ( f , t a ) . Thisis summarized in Fig. 6 for 5 different frequencies: when plotted as a function of f / f α ( t a ) ≡ π f τ α ( t a ) , the aging values of χ (cid:48)(cid:48) ( f , t a ) -symbols- are nicely rescaledonto the equilibrium values χ (cid:48)(cid:48) ( f , eq ) -continuous line- measured when t a (cid:29) τ α , eq . The most important experimental result is that this scaling fails for | X ( ) ( f , t a ) | asshown by the left axis of Fig. 6: For short aging times, the difference between agingdata (symbols) and equilibrium values (continuous line) is largest. This has beeninterpreted as an increase of N corr with the aging time t a . This increase of N corr ( t a ) towards its equilibrated value N corr ( eq ) is illustrated in Fig. 7 where the variation of δ = N corr ( t a ) / N corr ( eq ) is plotted as a function of t a . It turns out to be independentof the measuring frequency, which is a very important self consistency check.The increase of N corr during aging can be rather well captured by extrapolatingthe N corr ( T ) variation obtained from the growth of the hump of | χ ( ) | measured atequilibrium above T g and by translating the τ α ( t a ) in terms of a fictive temperature T fict ( t a ) which decreases during aging, finally reaching T w when t a (cid:29) τ α , eq . Thisyields the continuous line in Fig. 7, which fairly well captures the data drawn fromthe aging of χ ( ) . Because this extrapolation roughly agrees with the aging data,one can estimate that the quench from T g + T w = T g − N corr , eq . The approximately 10% increase reported in Fig. 7 is thusthe long time tail of this increase, while the first 90% increase cannot be measuredbecause it takes place during the quench.Beyond the qualitative result that N corr increases during aging, these χ ( ) ( t a ) datacan be used to test quantitatively some theories about the emergence of the glassystate. By gathering, in the inset of Fig. 7, the equilibrium data -symbols lying inthe [
1; 1 . ] interval of the horizontal axis- and the aging data translated in terms of T fict ( t a ) -symbols lying in the [
2; 2 . ] interval-, one extends considerably the exper-imental temperature interval, which puts strong constraints onto theories. Summa-rizing two different predictions by ln ( τ α / τ ) = Y N ψ / corr / ( k B T ) with Y ∼ T ; ψ = / Y ∼ ψ = τ α ∝ N zcorr , yieldingan unrealistic large value of z ∼
20 to account for the experiments-.
We now come back to equilibrium measurements -i.e. above T g - and compare thebehavior of the third-harmonic cubic susceptibility χ ( ) as well as the first-harmoniccubic susceptibilities χ ( ) and χ ( ) introduced in Eq. (7). We remind that χ ( ) corre-sponds to the case where a static field E st is superimposed to the ac field E cos ( ω t ) .Figs. 8 and 9 show the modulus and the phases of the three cubic susceptibilitiesfor glycerol and for propylene carbonate.1. For the modulus: At a fixed temperature, the main features of the frequency de-pendence of | χ ( ) | and of | χ ( ) | are the same as those of | χ ( ) | : when increasingthe frequency, one first observes a low frequency plateau, followed by a hump in hird and fifth harmonic responses in viscous liquids 15 - 2 - 1 - 2 - 1 - 2 0 0- 1 0 001 0 02 0 0 ( a ) | X ( 1 )2 ; 1 || X ( 3 )3 | (cid:1) | X ( 1 )3 | (cid:15) (cid:9) (cid:2) (cid:5) (cid:3) (cid:6) (cid:8) (cid:5) (cid:15) (cid:4)(cid:1) (cid:15) (cid:9) (cid:2) (cid:5) (cid:3) (cid:7) (cid:15) (cid:4)(cid:1) (cid:15) (cid:9) (cid:2) (cid:7) (cid:3) (cid:7) (cid:15) g l y c e r o l ( b ) A r g [ X ( 3 )3 ] A r g [ X ( 1 )2 , 1 ]A r g [ X ( 1 )3 ] (cid:1) f / f a (cid:13)(cid:12)(cid:10) (cid:14) (cid:11) (cid:1) (cid:2) (cid:16) (cid:3) Fig. 8
From [48]: For glycerol and f α (cid:39) Arg [ X ( ) ] + π or + π and support Eqs. (9) and (10). the vicinity of f α and then by a power law decrease ∼ f − β . The most importantdifferences between the three cubic susceptibilities are the precise location of thehump and the absolute value of the height of the hump. As for the temperaturedependence one recovers for | χ ( ) | and for | χ ( ) | what we have already seen for | χ ( ) | : once put into their dimensionless forms X the three cubic susceptibilitiesdo not depend on T in the plateau region, at variance with the region of the humpwhere they increase upon cooling typically as E act ( T ) ≡ ∂ ln τ α / ∂ ( / T ) whichin this T range is very close to T χ T ≡ | ∂ ln τ α / ∂ ln T | [3, 4, 10, 19, 20, 48].2. The phases of the three cubic susceptibilities basically do not depend explicitlyon temperature, but only on u = f / f α , through a master curve that depends onlyon the precise cubic susceptibility under consideration. These master curves havethe same qualitative shape as a function of u in both glycerol and propylene car-bonate. We note that the phases of the three cubic susceptibilities are related to - 2 - 1 ( b )( a ) | X ( 1 )2 ; 1 | | X ( 1 )3 || X ( 3 )3 | A u g s b u r g S a c l a y p r o p y l e n e c a r b o n a t e (cid:15) (cid:9) (cid:2) (cid:5) (cid:3) (cid:6) (cid:8) (cid:5) (cid:15) (cid:4)(cid:1) (cid:15) (cid:9) (cid:2) (cid:5) (cid:3) (cid:7) (cid:15) (cid:4)(cid:1) (cid:15) (cid:9) (cid:2) (cid:7) (cid:3) (cid:7) (cid:15) - 2 - 1 - 2 0 0- 1 0 001 0 02 0 0 A r g [ X ( 3 )3 ] A r g [ X ( 1 )2 , 1 ]A r g [ X ( 1 )3 ] (cid:1) (cid:13)(cid:12)(cid:10) (cid:14) (cid:11) (cid:1) (cid:2) (cid:16) (cid:3) f / f a Fig. 9
From [48]. Same representation as in Fig. 8 but for propylene carbonate. each other. In the plateau region all the phases are equal, which is expected be-cause at low frequency the systems responds adiabatically to the field. At higherfrequencies, we note that for both glycerol and propylene carbonate (expressingthe phases in radians):Arg (cid:104) X ( ) (cid:105) ≈ Arg (cid:104) X ( ) , (cid:105) + π for f / f α ≥ .
5; (9)Arg (cid:104) X ( ) (cid:105) ≈ Arg (cid:104) X ( ) (cid:105) for f / f α ≥ χ ( ) of propylene carbonate (Fig. 9), a jump of π is observedwhich is accompanied by the indication of a spikelike minimum in the modulus-see [48] for more details-. A similar jump may also be present in glycerol (Fig.8). This jump in the phase happens at the crossover between the T -independent“plateau” and the strongly T -dependent hump. More precisely in the “plateau”region one observes a reduction of the real part of the dielectric constant χ (cid:48) ,while around the hump χ (cid:48) is enhanced. At the frequency of the jump, both effects hird and fifth harmonic responses in viscous liquids 17 compensate and this coincides with a very low value of the imaginary part of X ( ) . In this section, we first explain why measuring χ ( ) is interesting for a better under-standing of the glass transition. We then see the characteristic features of χ ( ) as afunction of frequency and temperature. In the previous sections, we have seen that the increase of the hump of | X | uponcooling has been interpreted as reflecting that of the correlation volume N corr a .However in practice, this increase of N corr remains modest -typically it is an increaseby a factor 1 .
5- in the range 0 .
01 Hz ≤ f α ≤
10 kHz where the experiments aretypically performed. Physically this may be interpreted by the fact that an increaseof N corr changes the activation energy, yielding an exponentially large increase ofthe relaxation time τ α . Now if one demands, as in standard critical phenomena, tosee at least a factor of 10 of increase of | X | to be able to conclude on criticality,one is lead to astronomical values of τ α : extrapolating the above result, e.g., | X | ∝ | ∂ ln τ α / ∂ ln T | and assuming a VFT law for τ α , one concludes that the experimentalcharacteristic times corresponding to an increase of | X | by one order of magnitudeis 0 . ≤ τ α ≤ s. This means experiments lasting longer than the age of theuniverse.This issue of astronomical time scales can be circumvented by using a less com-monly exploited but very general property of phase transitions: close to a criticalpoint all the responses diverge together [49], since the common cause of all thesedivergences is the growth of the same correlation length. Showing that all the re-sponses of order k behave as a power law of the first diverging susceptibility is an-other way of establishing criticality. For glasses, we have seen in Eq. (2) that, apartfrom χ which is blind to glassy correlations, all other responses χ k ≥ grow as powerlaws with the amorphous ordering length (cid:96) : χ ∝ ( l / a ) d f − d and χ ∝ ( l / a ) d f − d .Therefore, assuming that the main cause for the singular responses appearing in thesystem is the development of correlations, there should be a scaling relation betweenthe third and fifth order responses, namely one should observe χ ∝ χ µ ( d f ) where µ ( d f ) = ( d f − d ) / ( d f − d ) .Measuring χ is of course extremely difficult, because, for the experimentallyavailable electric fields, one has the hierarchy | χ | E (cid:29) | χ | E (cid:29) | χ | E . Howeverthis was done in Ref. [50] and we shall now briefly review the corresponding results. Fig. 10
From Ref. [50]. Measured values of | χ ( ) | for glycerol - upper panel- and propylene car-bonate - lower panel- (the spheres and cubes in the upper panel indicate results from two differ-ent experimental setups). The hump lies at the same frequency as for | χ ( ) | and has significantlystronger variations in frequency and in temperature, see Figs 11 and 12. The arrows indicate thepeak positions f α in the dielectric loss. The yellow-shaded planes indicate the plateau arising inthe trivial regime. The modulus | χ ( ) | of glycerol and propylene carbonate [50] can be seen in Fig.10 as a function of frequency and temperature. Similarly to what has been seen insection 2.2 on | χ ( ) | , the frequency dependence can be separated in two domains(see also Fig. 11):1. For very low reduced frequencies ( f / f α ≤ . X ( ) de-pends neither on frequency nor on temperature. In this plateau, the behavior ofthe supercooled liquid cannot be qualitatively distinguished from the behavior hird and fifth harmonic responses in viscous liquids 19 -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 Fig. 11
From Ref. [50]. For glycerol, comparison of the fifth, third and linear susceptibilities -thelatter is noted | χ ( ) | -. The hump for | χ ( ) | is much stronger than that of | χ ( ) | . The dashed lines arethe trivial contribution -see [50] for details-. expected from a high temperature liquid of dipoles, depicted by the ”trivial” X ( k ) k curves represented as dotted lines in Fig. 11.2. At higher frequencies, we can observe a hump of | X ( ) | that remarkably occursat the same peak frequency f peak as in | χ ( ) | in both glycerol and propylene car-bonate. Again one finds that, for the five temperatures where the peak is studied, f peak / f α = c , where the constant c does not depend on T and weakly changeswith the liquid. This peak is much sharper for | X ( ) | than for | X ( ) | : this is clearlyevidenced by Fig. 11 where the linear, cubic and fifth-order susceptibilities arecompared, after normalisation to their low-frequency value. This shows that theanomalous features in the frequency dependence are stronger in | X ( ) | than in | X ( ) | : This may be regarded as a sign of criticality since close to a critical point,the larger the order k of the response, the stronger the anomalous features of X k .A second, and more quantitative indication of incipient criticality is obtained bystudying the temperature dependence of | X ( ) | and by comparing it with that of | X ( ) | :1. In the plateau region at f / f α ≤ .
05, the value of | X ( ) | does not depend onthe temperature. This shows that the factor involved in the calculation of thedimensionless X ( ) from χ ( ) -see Eq. (8)- is extremely efficient to remove alltrivial temperature dependences. As the trivial behavior depends on frequency-see the dashed lines of Fig. 11-, the “singular” parts of X and of X are obtainedas follows: X ( ) , sing . ≡ X ( ) − X ( ) , trivial , X ( ) , sing . ≡ X ( ) − X ( ) , trivial (11) Y ( T ) / Y ( K ) Fig. 12
From Ref. [50] Temperature evolution of the singular parts of fifth and third order re-sponses. All quantities are normalized at a given temperature -namely 207 K for glycerol, upperpanel; and 164 K for propylene carbonate, bottom panel-. This allows to determine the exponent µ relating | X | and | X | µ and to conclude that the amorphously ordering domains are compact -seetext-. The hatched areas represent the uncertainty on µ . which correspond in Fig. 11 to a complex subtraction between the measured data-symbols- and the trivial behavior -dashed lines.2. Around the hump, the temperature behavior of | X ( ) , sing . ( f peak ) | is compared to thatof | X ( ) , sing . ( f peak ) | µ where µ is an exponent that is determined experimentally bylooking for the best overlap of the two series of data in Fig. 12 -see [50] fordetails-. This leads us to values of µ = . ± . µ = . ± . | X ( ) | and | X ( ) | would seem to advocate a value of µ ≈
2. With µ = ( d f − d ) / ( d f − d ) as seen in Eq. (2) -see also Eq. (13) below-, this corresponds to afractal dimensions of d f ≈ hird and fifth harmonic responses in viscous liquids 21 Having shown the experimental data for the nonlinear responses, we now move tothe interpretation part and start with Bouchaud-Biroli’s approach (BB), which isthe most general one. The more specific and/or phenomenological approaches ofnonlinear responses will be detailed in Section 4. χ k + To illustrate the general relations existing between the susceptibility χ k + andthe correlation function of order 2 k + k ≥
0- in a system at thermal equi-librium, let us consider a sample, submitted to a constant and uniform magneticfield h , containing N spins with an Hamiltonian H that depends on the spinconfiguration “c”. The elementary relations of statistical physics yield the mag-netisation M ≡ ∑ i < S i > / ( Na ) where a is the elementary volume and wherethe thermal average < S i > is obtained with the help of the partition function Z = ∑ c exp ( − β H + β h ∑ k S k ) by writing < S i > = ∑ c S i exp ( − β H + β h ∑ k S k ) / Z with β = / ( k B T ) . The linear response χ ≡ ( ∂ M / ∂ h ) h = is readily obtained: Na χ = β Z (cid:18) ∂ Z ∂ h (cid:19) h = − β (cid:18) ∂ ZZ ∂ h (cid:19) h = = β (cid:32) ∑ i i < S i S i > − ( ∑ i < S i > ) (cid:33) (12)which shows that the linear response is related to the connected two-point cor-relation function. Repeating the argument for higher-order responses -e.g. χ ∝ ( ∂ M / ∂ h ) h = -, one obtains that χ k + is connected to the ( k + ) points cor-relation function -e.g., χ is connected to a sum combining < S i S i S i S i > , < S i S i S i >< S i > , < S i S i >< S i S i > etc...-. Spin glasses are characterized by the fact that there is frozen disorder, i.e. the setof the interaction constants { J i ; j } between two given spins S i and S j is fixed onceand for all, and has a random sign -half of the pairs of spins are coupled ferromag-netically, the other half antiferromagnetically-. Despite the fact that the system isneither a ferromagnet, nor an antiferromagnet, upon cooling it freezes, below a crit-ical temperature T SG , into a solid -long range ordered- state called a spin glass state.This amorphous ordering is not detected by χ which does not diverge at T SG : this is because the various terms of ∑ i i < S i S i > cancel since half of them are posi-tive and the other half are negative. By contrast the cubic susceptibility χ containsa term ∑ i i < S i S i > which does diverge since all its components are strictlypositive: this comes from the fact that the influence < S i S i > of the polarizationof spin S i on spin S i may be either positive or negative, but it has the same sign asthe reverse influence < S i S i > of spin S i on spin S i . This is why the amorphousordering is directly elicited by the divergence of the static value of χ when decreas-ing T towards T SG , as already illustrated in Fig. 1-(A). By adding a standard scalingassumption close to T SG one can account for the behavior of χ at finite frequencies,i.e. one easily explains that χ is frequency independent for ωτ α ≤
1, and smoothlytends to zero at higher frequencies. Finally, similar scaling arguments about corre-lation functions easily explain the fact that the stronger k ≥ χ k + in spin glasses, as observed experimentally by Levy et al [51]. The case of glass forming liquids is of course different from that of spin glasses forsome obvious reasons (e.g. molecules have both translational and rotational degreesof freedom). As it has been well established that rotational and translational degreesof freedom are well coupled in most of liquids, it is tempting to attempt a mappingbetween spin glasses and glass forming liquids by replacing the spins S i by the localfluctuations of density δ ρ i or by the dielectric polarisation p i . As far as nonlinearresponses are concerned, this mapping requires a grain of salt because (a) there isno frozen-in disorder in glass forming liquids, and (b) there is a nonzero value ofthe molecular configurational entropy S c around T g .The main physical idea of BB’s work [9] is that these difficulties have an effectwhich is important at low frequencies and negligible at high enough frequencies:1. Provided f ≥ f α , i.e. for processes faster than the relaxation time, one cannotdistinguish between a truly frozen glass and a still flowing liquid. If some amor-phous order is present in the glass forming system, then non-trivial spatial cor-relations should be present and lead to anomalously high values of non-linearsusceptibilities: this holds for very general reasons -e.g., the Langevin equationfor continuous spins which is used in Ref. [9] needs not to specify the detailedHamiltonian of the system- and comes from an analysis of the most divergingterm in the four terms contributing to χ ( ω ) . If the amorphous correlations ex-tend far enough to be in the scaling regime, one can neglect the subleading termsand one predicts that the nonlinear susceptibilities are dominated by the glassycorrelations and given by [9, 50]: X glass k + ( f , T ) = [ N corr ( T )] α k × H k (cid:18) ff α (cid:19) with α k = ( k + ) − d / d f (13)where the scaling functions H k do not explicitly depend on temperature, butdepend on the kind of susceptibility that is considered, i.e. X ( ) , X ( ) or X ( ) , in hird and fifth harmonic responses in viscous liquids 23 the third order case k =
1. We emphasize that in Ref. [9] the amorphously ordereddomains were assumed to be compact, i.e. d f = d , yielding α = X ∝ N corr .The possibility of having a fractal dimension d f lower than the spatial dimension d was considered in Ref. [50] where the fifth order response was studied. Asalready shown in Section 2.4.2, the experimental results were consistent with d f = d , i.e. X ∝ N corr .2. In the low frequency regime f (cid:28) f α , relaxation has happened everywhere in thesystem, destroying amorphous order [52] and the associated anomalous responseto the external field and H k ( ) =
0. In other words, in this very low frequencyregime, every molecule behaves independently of others and X k + is dominatedby the “trivial” response of effectively independent molecules.Due to the definition adopted in Eq. (8), the trivial contribution to X k + shouldnot depend on temperature (or very weakly) . Hence, provided N corr increases uponcooling, there will be a regime where the glassy contribution X glass k + should exceedthe trivial contribution, leading to hump-shaped non-linear susceptibilities, peakingat f peak ∼ f α , where the scaling function H k reaches its maximum. We now briefly recall why all the experimental features reported in section 2 arewell accounted for by BB’s prediction:1. The modulus of both the third order susceptibilities | χ ( ) | , | χ ( ) | , | χ ( ) | and of | χ ( ) | have a humped shape in frequency, contrary to | χ | .2. Due to the fact that H k does not depend explicitly on T , the value of f peak / f α should not depend on temperature, consistent with the experimental behavior.3. Because of the dominant role played by the glassy response for f ≥ f peak , the T -dependence of | X k + | will be much stronger above f peak than in the triviallow-frequency region.4. Finally, because non-linear susceptibilities are expressed in terms of scaling func-tions, it is natural that the behavior of their modulii and phases are quantitativelyrelated especially at high frequency where the ”trivial” contribution can be ne-glected, consistent with Eqs. (9)-(10) –see below for a more quantitative argu-ment in the context of the so-called “Toy model”– [53].Having shown that BB’s prediction is consistent with experiments, the temper-ature variation of N corr can be drawn from the increase of the hump of X uponcooling. It has been found [3, 4, 10, 19, 20] that the temperature dependence of N corr inferred from the height of the humps of the three X ’s are compatible withone another, and closely related to the temperature dependence of T χ T , which wasproposed in Refs. [22, 23] as a simplified estimator of N corr in supercooled liq-uids. The convergence of these different estimates, that rely on general, model-freetheoretical arguments, is a strong hint that the underlying physical phenomenon is indeed the growth of collective effects in glassy systems – a conclusion that will bereinforced by analyzing other approaches in Section 4.Let us again emphasize that the BB prediction relies on a scaling argument, wherethe correlation length (cid:96) of amorphously ordered domains is (much) larger than themolecular size a . This naturally explains the similarities of the cubic responses inmicroscopically very different liquids such as glycerol and propylene carbonate, aswell as many other liquids [10, 20]. Indeed the microscopic differences are likely tobe wiped out for large (cid:96) ∝ N / d f corr , much like in usual phase transitions. χ and χ as tests of thetheories of the glass transition. We now shortly discuss whether N corr , as extracted from the hump of | X | , mustbe regarded as a purely dynamical correlation volume, or as a static correlationvolume. This ambiguity arises because theorems relating (in a strict sense) nonlinearresponses to high-order correlation functions only exist in the static case, and thatsupplementary arguments are needed to interpret the humped shape of X (and of X ) observed experimentally. In the original BB’s work [9] it was clearly stated that N corr was a dynamical correlation volume since it was related to a four point timedependent correlation function. This question was revisited in Ref. [50] where it wasargued that the experimental results could be accounted for only when assumingthat N corr is driven by static correlations. This statement comes from an inspectionof the various theories of the glass transition [50]: as we now briefly explain, onlythe theories where the underlying static correlation volume is driving the dynamicalcorrelation volume are consistent with the observed features of nonlinear responses.As a first example, the case of the family of kinetically constrained models(KCMs) [56] is especially interesting since dynamical correlations, revealed by,e.g., four-point correlation functions, exist even in the absence of a static corre-lation length. However in the KCM family, one does not expect any humped shapefor nonlinear responses [50]. This is not the case for theories (such as RFOT [46]or Frustration theories [57]) where a non-trivial thermodynamic critical point drivesthe glass transition: in this case the incipient amorphous order allows to account[50] for the observed features of X and X . This is why it was argued in [48, 50]that, in order for X and X to grow, some incipient amorphous order is needed,and that dynamical correlations in strongly supercooled liquids are driven by static(“point-to-set”) correlations [55] –this statement will be reinforced in section 4.2. We now review the various other approaches that have been elaborated for the non-linear responses of glass forming liquids. We shall see that most of them -if not hird and fifth harmonic responses in viscous liquids 25 all- are consistent with BB’s approach since they involve N corr as a key -implicit orexplicit- parameter. The “Toy model” has been proposed in Refs. [19, 58] as a simple incarnation ofthe BB mechanism, while the “Pragmatical model” is more recent [59, 60]. Bothmodels start with the same assumptions: (i) each amorphously ordered domain iscompact and contains N corr molecules, which yields a dipole moment ∝ √ N corr andleads to an anomalous contribution to the cubic response X glass3 ∝ N corr ; (ii) thereis a crossover at low frequencies towards a trivial cubic susceptibility contribution X triv which does not depend on N corr . More precisely, in the “Toy model” eachamorphously ordered domain is supposed to live in a simplified energy landscape,namely an asymmetric double-well potential with a dimensionless asymmetry δ ,favoring one well over the other. The most important difference between the Toy andthe Pragmatical model comes from the description of the low-frequency crossover,see Refs. [58] and [60] for more details.On top of N corr and δ , the Toy model uses a third adjustable parameter, namelythe frequency f ∗ below which the trivial contribution becomes dominant. In Ref.[58], both the modulus and the phase of X ( ) ( ω , T ) and of X ( ) ( ω , T ) in glycerolwere well fitted by using f ∗ (cid:39) f α / δ = . T =
204 K, N corr = X ( ) and N corr =
15 for X ( ) . Fig. 13 gives an example of the Toy model predictionfor X ( ) in glycerol. Besides, in Ref. [19], the behavior of X ( ) , ( ω , T ) in glycerolwas fitted with the same values of δ and of f ∗ but with N corr =
10 (at a slightlydifferent temperature T =
202 K). Of course, the fact that a different value of N corr must be used for the three cubic susceptibilities reveals that the Toy model is over-simplified, as expected. However, keeping in mind that the precise value of N corr does not change the behavior of the phases, we note that the fit of the three exper-imental phases is achieved [19, 58] by using the very same values of f ∗ / f α and of δ . This means that Eqs. (9) and (10) are well accounted for by the Toy model bychoosing two free parameters. This is a quantitative illustration of how the BB gen-eral framework does indeed lead to strong relations between the various non-linearsusceptibilities, such as those contained in Eqs. (9) and (10).Let us mention briefly the Asymmetric Double Well Potential (ADWP) model[61], which is also about species living in a double well of asymmetry energy ∆ , ex-cepted that two key assumptions of the Toy and Pragmatical models are not made:the value of N corr is not introduced, and the crossover to trivial cubic response is notenforced at low frequencies. As a result, the hump for | X ( ) | is predicted [61, 62]only when the reduced asymmetry δ = tanh ( ∆ / ( k B T )) is close to a very specificvalue, namely δ c = (cid:112) /
3, where X vanishes at zero frequency due to the compen-sation of its several terms. However, at the fifth order [62] this compensation hap-pens for two values of δ very different from δ c : as a result the model cannot predict Fig. 13
From Ref. [58]. Fit of the values of X ( ) measured in glycerol -symbols- at 204 K by usingthe Toy model with N corr = δ = . f ∗ (cid:39) f α /
7. The prediction of the Toy model is givenby the two thick solid lines (main panel for the modulus of X ( ) and inset for its phase). a hump happening both for the third and for the fifth order in the same parametricregime, contrarily to the experimental results of Ref. [50]. This very recent calcu-lation of fifth order susceptibility [62] reinforces the point of view of the Toy andPragmatical models, which do predict a hump occurring at the same frequency andtemperature due to their two key assumptions ( N corr and crossover to trivial nonlin-ear responses at low frequencies). This can be understood qualitatively: because theToy model predicts [58] an anomalous contribution X glass k + ∼ [ N corr ] k , provided that N corr is large enough, the magnitude of this contribution is much larger than that ofthe small trivial contribution X triv . k + ∼
1, and the left side of the peak of | X k + | arisesjust because the Toy model enforces a crossover from the large anomalous responseto the small trivial response at low frequencies f (cid:28) f α . As for the right side of thepeak, it comes from the fact that | X k + | → f (cid:29) f α for the simple reasonthat the supercooled liquid does not respond to the field at very large frequencies. A contribution to nonlinear responses was recently calculated by Johari in Refs [63,64] in the case where a static field E st drives the supercooled liquid in the nonlinearregime. Johari’s idea was positively tested in the corresponding χ ( ) experiments inRefs [65, 66, 67, 68] -see however Ref. [69] for a case where the agreement is notas good-. It was then extended to pure ac experiments -and thus to χ ( ) - in Refs.[70, 71]. The relation between Johari’s idea and N corr was made in Ref. [48]. hird and fifth harmonic responses in viscous liquids 27 E st is applied Let us start with the case of χ ( ) experiments, i.e. with the case where a static field E st is superimposed onto an ac field E cos ( ω t ) . In this case, there is a well definedvariation of entropy [ δ S ] E st induced by E st , which, for small E st and a fixed T , isgiven by: [ δ S ] E st ≈ ε ∂ ∆ χ ∂ T E st a , (14)where a is the molecular volume. Eq. (14) holds generically for any material. How-ever, in the specific case of supercooled liquids close enough to their glass transitiontemperature T g , a special relation exists between the molecular relaxation time τ α and the configurational contribution to the entropy S c . This relation, first anticipatedby Adam and Gibbs [21], can be written as :ln τ α ( T ) τ = ∆ T S c ( T ) (15)where τ is a microscopic time, and ∆ is an effective energy barrier for a molecule.The temperature dependence of T S c ( T ) quite well captures the temperature varia-tion of ln ( τ α ) , at least for a large class of supercooled liquids [72].Following Johari [63, 64] let us now assume that [ δ S ] E st is dominated by thedependence of S c on field, –see the Appendix of Ref. [48] for a further discussion ofthis important physical assumption-. Combining Eqs. (14) and (15), one finds that astatic field E st produces a shift of ln ( τ α / τ ) given by: [ δ ln τ α ] E st = − ∆ T S c [ δ S ] E st (16)As shown in Ref. [48] this entropic effect gives a contribution to X ( ) , , which we call J ( ) , after Johari. Introducing x = ωτ α , the most general and model-free expressionof J ( ) reads: J ( ) , = − k B ∆ S c (cid:20) ∂ ln ( ∆ χ ) ∂ T (cid:21) (cid:34) ∂ χ lin ∆ χ ∂ ln x (cid:35) ∝ S c (17)where χ lin is the complex linear susceptibility.Eq. (17) deserves three comments:1. | J ( ) , | has a humped shaped in frequency with a maximum in the region of ωτ α (cid:39)
1, because of the frequency dependence of the factor ∝ ∂ χ lin / ∂ ln x in Eq. (17).2. The temperature variation of J ( ) , is overwhelmingly dominated by that of S − c because S c ∝ ( T − T K ) -with T K the Kauzmann temperature-.3. The smaller S c , the larger must be the size of the amorphously ordered domains-in the hypothetical limit where S c would vanish, the whole sample would be trapped in a single amorphously ordered sate and N corr would diverge-. In otherwords, there is a relation between S − c and N corr , which yields [48]: J ( ) , ∝ N qcorr , (18)where it was in shown in Ref. [48] that:a. the exponent q lies in the [ /
3; 2 ] interval when one combines the Adam-Gibbsoriginal argument with general constraints about boundary conditions [48].b. the exponent q lies in the [ /
3; 3 / ] interval [48] when one uses the RFOTand plays with its two critical exponents Ψ and θ . Notably, taking the “recom-mended RFOT values” - Ψ = θ = / d =
3- gives q =
1, which preciselycorresponds to BB’s prediction. In this case, entropic effects are a physicallymotivated picture of BB’s mechanism -see [48] for a refined discussion-. E cos ( ω t ) is applied Motivated by several works [65, 66, 67, 68] showing that Johari’s reduction of en-tropy fairly well captures the measured χ ( ) in various liquids, an extension of thisidea was proposed in Refs. [70, 71] for pure ac experiments, i.e. for χ ( ) and χ ( ) .This has given rise to the phenomenological model elaborated in Refs. [70, 71]where the entropy reduction depends on time, which is nevertheless acceptable inthe region ωτ α ≤ | χ ( ) | at three temperatures for glycerol. The calculation fairly well reproduces thehump of the modulus observed experimentally -the phase has not been calculated-.As very clearly explained in Ref. [71], the hump displayed in Fig. 14 comes directlyfrom the entropic contribution and not from the two other contributions included inthe model (namely the “trivial” -or “saturation”- contribution, and the Box modelcontribution -see Section 4.3 below-).Summarizing this section about entropy effects, we remind the two main as-sumptions made by Johari: (i) the field-induced entropy variation mainly goes intothe configurational part of the entropy; (ii) its effects can be calculated by using theAdam-Gibbs relation. Once combined, these two assumptions give a contributionto χ ( ) reasonably well in agreement with the measured values in several liquids[65, 66, 67, 68]. An extension to χ ( ) is even possible, at least in the region ωτ α ≤ | χ ( ) | in glycerol [70, 71] -a fig-ure similar to Fig. 11 for | χ ( ) ( ω ) / χ ( ) ( ) | is even obtained in Ref. [71]-. As shownin Eq. (18), this entropy contribution to cubic responses is related to N corr , whichis consistent with the general prediction of BB. Additionally, because S c is a staticquantity, Eq. (18) supports the interpretation that the various cubic susceptibilities χ are related to static amorphous correlations, as discussed in Section 3.3. hird and fifth harmonic responses in viscous liquids 29 Fig. 14
From Ref. [71]. The model elaborated in Refs. [70, 71] includes three contributions -entropy reduction, Box model, and trivial-. It predicts for | χ ( ) | the solid lines which account verywell for the measured values in glycerol in frequency and in temperature. The peak of | χ ( ) | arisesbecause of the entropy reduction effect (noticed “sing. T fic . ”) which completely dominates the twoother contributions in the peak region, as shown by the inset. The “Box model” is historically the first model of nonlinear response in supercooledliquids, designed to account for the Nonresonant Hole Burning (NHB) experiments[73]. When these pionneering experiments were carried out, a central question waswhether the dynamics in supercooled liquids is homogeneous or heterogeneous. Inthe seminal ref. [73] it was reported that when applying a strong ac field E of angularfrequency ω , the changes in the dielectric spectrum are localised close to ω and thatthey last a time of the order of 1 / ω . These two findings yield a strong qualitativesupport to the heterogenous character of the dynamics, and the Box model wasdesigned to provide a quantitative description of these results. Accordingly, the Boxmodel assumes that the dielectric response comes from “domains” -that will be latercalled Dynamical Heterogeneities (DH)-, each domain being characterized by itsdielectric relaxation time τ and obeying the Debye dynamics. The distribution ofthe various τ ’s is chosen to recover the measured non Debye spectrum by adding thevarious linear Debye susceptibilities χ , dh = ∆ χ / ( − i ωτ ) of the various domains.For the nonlinear response, the Box model assumes that it is given by the Debyelinear equation in which τ ( T ) is replaced by τ ( T f ) where the fictive temperature T f = T + δ T f is governed by the constitutive equation -see e.g. [28, 77]-: c dh ∂ ( δ T f ) ∂ t + κδ T f = ε χ (cid:48)(cid:48) , dh ω E (19) with c dh the volumic specific heat of the DH under consideration, κ the thermalconductance (divided by the DH volume v ) between the DH and the phonon bath, τ therm = c / κ the corresponding thermal relaxation time. In Eq. (19), only the con-stant part of the dissipated power has been written, omitting its component at 2 ω which is important only for χ ( ) -see e.g. [77]-. From Eq. (19) one easily finds thestationary value δ T (cid:63) f of δ T f which reads: δ T (cid:63) f = τ therm τ ε ∆ χ E c dh ω τ + ω τ (20)As very clearly stated in the seminal Ref. [73] because the DH size is smaller than5nm, the typical value of τ therm is at most in the nanoseconds range: this yields,close to T g , a vanishingly small value of τ therm / τ , which, because of Eq. (20), givesfully negligible values for δ T (cid:63) f . The choice of the Box model is to increase τ therm by orders of magnitude by setting τ therm = τ , expanding onto the intuition that thisis a way to model the “energy storage” in the domains. The main justification ofthis choice is its efficiency : it allows to account reasonably well for the NHB ex-periments [73] and thus to bring a strong support to the heterogeneous character ofthe dynamics in supercooled liquids. Since the seminal ref. [73], some other workshave shown [28, 74, 75, 76] that the Box model efficiently accounts for the mea-sured χ ( ) ( f > f α ) in many glass forming liquids. It was shown also [77] that theBox model is not able to fit quantitatively the measured X ( ) (even though somequalitative features are accounted for), and that the Box model only provides a van-ishing contribution to X ( ) , – see [19].The key choice τ therm = τ made by the Box model has two important conse-quences for cubic susceptibilities: it implies a) that χ ( ) mainly comes from theenergy absorption (since the source term in Eq. (19) is the dissipated power) and b) that χ ( ) does not explicitly depend on the volume v = N corr a of the DH’s (see[28, 77]). However, alternative models of nonlinear responses are now available[58, 60] where, instead of choosing τ therm , one directly resolves the microscopicpopulation equations, which is a molecular physics approach, and not a macroscopiclaw transferred to microscopics. The population equations approach is equivalent tosolving the relevant multidimensional Fokker-Planck equation describing the col-lective tumbling dynamics of the system at times longer than the time between twomolecular collisions (called τ c in Appendix 3). By using this molecular physics ap-proach one obtains that χ ( ) is governed by N corr and not by energy absorption. For χ ( ) , writing loosely P ( ) ≈ ∂ P / ( ∂ ln τ ) δ ln τ , one sees that the pivotal quantity isthe field induced shift of the relaxation time δ ln τ . Comparing the Box model (BM)and, e.g., the Toy model (TM), one gets respectively: δ ln τ BM (cid:39) − χ T ε ∆ χ E c dh ; δ ln τ TM (cid:39) − N corr T ε ∆ χ E k B / a (21) hird and fifth harmonic responses in viscous liquids 31 where we remind our definition χ T = | ∂ ln τ α / ∂ T | and where the limit ωτ (cid:29) χ ( ) ( f > f α ) was taken in the Box model, while the simplest case(symmetric double well with a net dipole parallel to the field) was considered forthe Toy model. Eq. (21) deserves two comments:1. one sees that the two values of δ ln τ are similar provided N corr and T χ T areproportional – which is a reasonable assumption as explained above and in Refs.[4, 22, 23]. Taking reasonable values of this proportionality factor, it was shownin Ref. [48] that χ ( ) ( f > f α ) is the same in the two models. This sheds a newlight on the efficiency of the Box model and on consequence b) .2. Let us shortly discuss consequence a) . In the Toy model, δ ln τ directly expressesthe field induced modification of the energy of each of the two wells modeling agiven DH. It comes from the work produced by E onto the DH and this is why itinvolves N corr : the larger this number, the larger the work produced by the fieldbecause the net dipole of a DH is ∝ √ N corr and thus increases with N corr . It iseasy to show that the dissipation -i.e. the “energy absorption”- is not involvedin δ ln τ because dissipation depends only on χ (cid:48)(cid:48) , which in the Toy model doesnot depend on N corr . In the Toy model, as in the Pragmatical model [60] and theDiezemann model [61], the heating is neglected because at the scale of a givenDH it is vanishingly small as shown above when discussing τ therm . Of course,at the scale of the whole sample, some global heating arises for thick samplesand/or high frequencies because the dissipated power has to travel to the elec-trodes which are the actual heat sinks in dielectric experiments [11]. This purelyexogeneous effect can be precisely calculated by solving the heat propagationequation, see e.g. ref.[11] and Appendix 2, and must not be confused with whatwas discussed in this section. As explained above, in Refs. [70, 71], the three experimental cubic susceptibilitieshave been argued to result from a superposition of an entropic contribution and ofan energy absorption contribution coming from the Box model (plus a trivial con-tribution playing a minor role around the peaks of the cubic susceptibilities). Moreprecisely, the hump of | X ( ) , | and of | X ( ) | would be mainly due to the entropy effect,contrarily to the hump of | X ( ) | which would be due to the Box model contribution.As noted in Ref. [48], this means that very different physical mechanisms wouldconspire to give contributions of the same order of magnitude, with phases that haveno reason to match as they do empirically, see Eqs. (9) and (10): why should X ( ) and X ( ) have the same phase at high frequencies if their physical origin is different?This is why it was emphasized in Ref. [48] that there is no reason for such a sim-ilarity if the growth of X ( ) and X ( ) are due to independent mechanisms. Becauseentropic effects have been related to the increase of N corr -see Eqs. (17) and (18)-,everything becomes instead very natural if the Box model is recasted in a frame- work where X ( ) is related to the glassy correlation volume. As evoked above, a firststep in this direction was done in Ref. [48] where it was shown that the Box modelprediction for X ( ) at high frequencies is identical to the above Toy model predic-tion, provided N corr and T χ T are proportional. In all, it is argued in Ref. [48] thatthe only reasonable way to account for the similarity of all three cubic susceptibili-ties, demonstrated experimentally in Figs. 8 and 9, is to invoke a common physicalmechanism. As all the other existing approaches, previously reviewed, relate cubicresponses to the growth of the glassy correlation volume, reformulating the Boxmodel along the same line seems to be a necessity. We have reviewed in this chapter the salient features reported for the third and fifthharmonic susceptibilities close to the glass transition. This is a three decades longstory, which has started in the mid-eighties as a decisive tool to evidence the solid,long range ordered, nature of the spin glass phase. The question of whether thisnotion of “amorphous order” was just a curiosity restricted to the -somehow exotic-case of spin glasses remained mostly theoretical until the seminal work of Bouchaudand Biroli in 2005. This work took a lot from the spin glass physics, and by takinginto account the necessary modifications relevant for glass forming liquids, it hasanticipated all the salient features discovered in the last decade for the three cubicsusceptibilities X . This is why, in most of the works, the increase of the hump of X upon cooling has been interpreted as reflecting that of the glassy correlation volume.Challenging alternative and more specific interpretations have been proposed, butwe have seen that most -if not all- of them can be recasted into the framework ofBB. The avenue opened by BB’s prediction was also used to circumvent the issueof exponentially long time scales -which are the reason why the nature of the glasstransition is still debated-: this is how the idea of comparing the anomalous featuresof X and of X has arisen. The experimental findings are finally consistent with theexistence of an underlying thermodynamic critical point, which drives the formationof amorphously ordered compact domains, the size of which increases upon cooling.Last we note that this field of nonlinear responses in supercooled liquids has beeninspiring both theoretically [79, 5] and experimentally, e.g. for colloidal glasses: thevery recent experiments [6] have shed a new light on the colloidal glass transitionand shown interesting differences with glass forming liquids.All these progresses open several routes of research. On the purely theoreticalside, any prediction of nonlinear responses in one of the models belonging to theKinetically Constrained Model family will be extremely welcome to go beyondthe general arguments given in Refs. [50]. Moreover, it would be very interestingto access χ (and χ ) in molecular liquids at higher temperatures, closer to theMode Coupling Transition temperature T MCT , and/or for frequencies close to thefast β process where more complex, fractal structures with d f < d may be antici-pated [80, 81]. This will require a joined effort of experimentalists -to avoid heating hird and fifth harmonic responses in viscous liquids 33 issues- and of theorists -to elicit the nature of nonlinear responses close to T MCT -.Additionally, one could revisit the vast field of polymers by monitoring their non-linear responses, which should shed new light onto the temperature evolution of thecorrelations in these systems. Therefore there is likely much room to deepen our un-derstanding of the glass transition by carrying out new experiments about nonlinearsusceptibilities.
ACKNOWLEDGEMENTS
We thank C. Alba-Simionesco, Th. Bauer, U. Buchenau,A. Coniglio, G. Johari, K. Ngai, R. Richert, G. Tarjus, and M. Tarzia for interest-ing discussions. The work in Saclay has been supported by the Labex RTRA grantAricover and by the Institut des Syst`emes Complexes ISC-PIF. The work in Augs-burg was supported by the Deutsche Forschungsgemeinschaft via Research UnitFOR1394.
We briefly explain how the nonlinear effects reported here have been shown to be-mainly- free of exogeneous effects:1. The global homogeneous heating of the samples by the dielectric energy dissi-pated by the application of the strong ac field E was shown to be fully negligiblefor X ( ) as long as the inverse of the relaxation time f α is ≤ X ( ) : tominimize them, one can either keep f α below 10 Hz [4], and/or severely limit thenumber n of periods during which the electric field is applied -see, e.g., [28, 78]).2. The contribution of electrostriction was demonstrated to be safely negligible inRefs. [74, 4], both by using theoretical estimates and by showing that changingthe geometry of spacers does not affect X ( ) .3. As for the small ionic impurities present in most of liquids, we briefly explainthat they have a negligible role, except at zero frequency where the ion contri-bution might explain why the three X ’s are not strictly equal, contrarily to whatis expected on general grounds -see, e.g., Figs. 8 and 9-. On the one hand it wasshown that the ion heating contribution is fully negligible in X ( ) , (see Ref. [19]),on the other hand it is well known that ions affect the linear response χ at verylow frequencies (say f / f α ≤ . χ (cid:48)(cid:48) , which diverges as 1 / ω instead of vanishing as ω in an ide-ally pure liquid containing only molecular dipoles. This may be the reason whymost of the χ measurements are reported above 0 . f α : at lower frequencies thenonlinear responses is likely to be dominated by the ionic contribution. As explained in the main text, in the long time limit -i.e. for f / f α (cid:28) τ D - due to the underlying thermalreservoir at temperature T . The linear susceptibility of such an ideal gas of dipolesis given by the Debye susceptibility ∆ χ / ( − i ωτ D ) , hence the subscript “Debye”in the Eq. (22) below. By using Refs. [17], and following the definitions given in themain text, as well as Eqs. (5)-(8) above, one gets for the dimensionless nonlinearresponses of such an ideal gas, setting for brevity x = ωτ D : X ( ) , Debye = (cid:18) − (cid:19) − x + ix ( − x )( + x )( + x )( + x ) X ( ) , Debye = ( − x − x + x ) ( + x )( + x )( + x )( + x )( + x )( + x )+ i x ( − x − x + x ) ( + x )( + x )( + x )( + x )( + x )( + x ) (22)In Ref. [50] the trivial response combined the above X ( k ) k , Debye with a distribution G ( τ ) of relaxation times τ chosen to account for the linear susceptibility of thesupercooled liquid of interest. In Refs [58, 19] a slightly different modelization wasused since G ( τ ) was replaced by the Dirac delta function δ ( τ − τ α ) , i.e. τ D wassimply replaced by τ α for the cubic trivial susceptibilities. In this section we shall rederive the phenomenological Toy model of Ladieu et al.[58] starting from the Langevin-Fokker-Planck equation, which is the starting pointof Bouchaud and Biroli when they illustrate their general theoretical ideas in thelast part of Ref. [9]. We shall idealize the supercooled state of a liquid as follows. Athigh temperatures, the liquid is made of molecules the interactions between whichare completely negligible. On cooling, the molecules arrange themselves in groups,called “dynamical heterogeneities” (DH), between which there are no interactions.Inside a typical group, specific intermolecular interactions manifest themselves dy-namically, by which we mean that in a time larger than a characteristic time τ α ,such interactions lose their coherence and the typical behavior of the liquid is thatof an ideal gas. Before and around τ α , these interactions manifest themselves in afrequency range ω ≈ / τ α . Thus, stricto sensu, our modelling of this specific pro- hird and fifth harmonic responses in viscous liquids 35 cess pertains to the behavior of the various dielectric responses of a DH, linear andnonlinear, near this frequency range. This indeed implies that information regard-ing the “ideal gas” phase must be added to fit experimental data. It may be shownon fairly general grounds that either for linear and non-linear responses, such extrainformation simply superposes onto the specific behavior that has been alluded toabove [82]. Now, we consider that a) a given DH has a given size at temperature T , b) that a DH is made of certain mobile elements that do interact between them-selves, c) that there are no interactions between DHs, d) that the dipole moment of aDH is µ d = µ √ N corr , and e) that all constituents of a DH are subjected to Brownianmotion.In order to translate the above assumptions in mathematical language, we assign toeach constituent of a DH a generalized coordinate q i ( t ) , so that each DH is describedby a set of generalized coordinates q at temperature T , viz. q ( t ) = { q ( t ) , . . . , q n ( t ) } Inside each DH, each elementary constituent is assumed to interact via a multidi-mensional interaction potential V int ( q ) that possesses a double-well structure withminima at q A and q B , and are sensitive both to external stresses and thermal agita-tion. The equations of motion may be described by overdamped Langevin equationswith additive noise, viz. ˙ q i = − ζ ∂ V T ∂ q i ( q , t ) + Ξ i ( t ) (23)where ζ is a generalized friction coefficient, V T = V int + V ext , V ext is the potentialenergy of externally applied forces and the generalized forces Ξ i ( t ) have Gaussianwhite noise properties, namely Ξ i ( t ) = , Ξ i ( t ) Ξ j ( t (cid:48) ) = kT ζ δ i j δ (cid:0) t − t (cid:48) (cid:1) (24)Thus, the dynamics of a DH is represented by the stochastic differential equations(23) and (24), which are in effect the starting point of the Bouchaud-Biroli theory,as stated above. A totally equivalent representation of these stochastic dynamicsis obtained by writing down the Fokker-Planck equation [83] for the probabilitydensity W ( q , t ) to find the system in state q at time t which corresponds to Eqs. (23)and (24), namely ∂ W ∂ t ( q , t ) = τ c ∇ · [ ∇ W ( q , t ) + β W ( q , t ) ∇ V T ( q , t )]= L FP ( q , t ) W ( q , t ) (25)where 2 τ c = ζ / ( kT ) is the characteristic time of fluctuations, ∇ is the del operatorin q space, and L FP ( q , t ) is the Fokker-Planck operator. We notice that Eq. (25) mayalso be written ∂ W ∂ t ( q , t ) = τ c ∇ · (cid:110) e − β V T ( q , t ) ∇ (cid:104) W ( q , t ) e β V T ( q , t ) (cid:105)(cid:111) (26)Now we use the transformation [84] φ ( q , t ) = W ( q , t ) e β V T ( q , t ) (27)so that Eq. (26) becomes ∂ φ∂ t ( q , t ) − β ∂ V T ∂ t ( q , t ) φ ( q , t ) = τ c e β V T ( q , t ) ∇ · (cid:110) e − β V T ( q , t ) ∇ φ ( q , t ) (cid:111) = L † FP ( q , t ) φ ( q , t ) (28)where L † FP ( q , t ) is the adjoint Fokker-Planck operator [83].Next, we make the first approximation in our derivation, namely, we assume thatthe time variation of V T is small with respect to that of W . If the time dependenceof V T is contained in, say, the application of a time-varying uniform AC field only,this implies immediately that neglecting the second term in the left hand side of Eq.(28) means that W is near its equilibrium value, so restricting further calculations tolow frequencies, ωτ c << ∂ φ∂ t ( q , t ) ≈ L † FP ( q , t ) φ ( q , t ) (29)Now, the interpretation of the Fokker-Planck equation (25) (or equally well theLangevin equations (23)) with time-dependent potential in terms of usual popu-lation equations with time-dependent rate coefficients has a meaning, since now Eq.(27) means detailed balancing. The polarization of an assembly of noninteractingDH in the direction of the applied field may then be defined as P ( t ) = ρ µ d (cid:90) cos ϑ ( q ) W ( q , t ) d q (30)where ρ is the number of DH per unit volume, and ϑ ( q ) is the angle a DH dipolemakes with the externally applied electric field. Because of the double-well structureof the interaction potential, we may equally well write Eq. (30) P ( t ) = ρ µ d (cid:90) well A cos ϑ ( q ) W ( q , t ) d q + (cid:90) well B cos ϑ ( q ) W ( q , t ) d q (31)Now, it is known from the Kramers theory of chemical reaction rates [84] that atsufficiently large energy barriers, most of the contributions of the integrands comefrom the minima of the wells, therefore we have P ( t ) ≈ ρ µ d cos ϑ ( q A ) (cid:90) well A W ( q , t ) d q + cos ϑ ( q B ) (cid:90) well B W ( q , t ) d q (32) hird and fifth harmonic responses in viscous liquids 37 Now, the integrals represent the relative populations x i ( t ) = n i ( t ) / N , i = A , B ineach well (we assume that W ( q , t ) is normalized to unity), where n i ( t ) is the numberof DH states in well i , and N the total number of DH. At any time t , we have theconservation law x A ( t ) + x B ( t ) = P ( t ) ≈ ρ µ d [ cos ϑ ( q A ) x A ( t ) + cos ϑ ( q B ) x B ( t )] (34)We assume now for simplicity that ϑ ( q B ) = π − ϑ ( q A ) , so that P ( t ) ≈ ρ µ d cos ϑ ( q A ) [ x A ( t ) − x B ( t )] (35)Finally, since ρ = N / V where V is the volume of the polar substance made of DHonly, we obtain P ( t ) ≈ µ d cos ϑ ( q A ) N υ DH [ n A ( t ) − n B ( t )] (36)where υ DH is the volume of a DH. This is the definition of the polarization in theToy model.In order to determine the polarization (36), we need to calculate the dynamics of n i ( t ) . From the conservation law -Eq. (33)-, we have˙ x A ( t ) = − ˙ x B ( t ) = (cid:90) well A ∂ W ∂ t ( q , t ) d q (37)By using the Fokker-Planck equation (26) and limiting well A to a closed general-ized bounding surface constituting the saddle region ∂ A , we have by Gauss’s theo-rem ˙ x A ( t ) = − ˙ x B ( t ) = τ c (cid:73) ∂ A e − β V T ( q , t ) ∇ φ ( q , t ) · ν q dS q (38)where ν q is the outward normal to the bounding surface and dS q is a generalizedsurface element of the bounding surface, and where we have used Eq. (27). Now,we follow closely Coffey et al. [85] and introduce the crossover function ∆ ( q , t ) viathe equation φ ( q , t ) = φ A ( t ) + [ φ B ( t ) − φ A ( t )] ∆ ( q , t ) (39)where ∆ ( q , t ) = q ∈ well A while ∆ ( q , t ) = q ∈ well B and exhibits stronggradients in the saddle region ∂ A allowing the crossing from A to B (and vice-versa) by thermally activated escape. By combining Eqs. (38) and (39), we haveimmediately ˙ x A ( t ) = − ˙ x B ( t ) = φ B ( t ) − φ A ( t ) τ c (cid:73) ∂ A e − β V T ( q , t ) ∇ [ ∆ ( q , t )] · ν q dS q (40)Now, x i ( t ) = φ i ( t ) x si ( t ) , x si ( t ) = (cid:90) well i W s ( q , t ) d q (41)where W s ( q , t ) is a normalized solution of the Fokker-Planck equation L FP ( q , t ) W s ( q , t ) = τ c ∂ W s ∂ t ( q , t ) ≈ τ c and because the time-dependent part of the po-tential V T is much smaller than other terms in it at any time. We have x sA ( t ) + x sB ( t ) = φ B ( t ) − φ A ( t ) = (cid:18) x sA ( t ) + x sB ( t ) (cid:19) [ x B ( t ) x sA ( t ) − x A ( t ) x sB ( t )] (44)By combining Eqs. (38) and (44), we readily obtain˙ x A ( t ) = − ˙ x B ( t ) = Γ ( t ) ( x B ( t ) x sA ( t ) − x A ( t ) x sB ( t )) (45)where the overall time-dependent escape rate Γ ( t ) is given by [85] Γ ( t ) = τ c (cid:18) x sA ( t ) + x sB ( t ) (cid:19) (cid:73) ∂ A e − β V T ( q , t ) ∇ φ ( q , t ) · ν q dS q (46)Finally, by setting Π AB ( t ) = Γ ( t ) x sB ( t ) , Π BA ( t ) = Γ ( t ) x sA ( t ) (47)we arrive at the population equations˙ n A ( t ) = − ˙ n B ( t ) = − Π AB ( t ) n A ( t ) + Π BA ( t ) n B ( t ) (48)The obtaining of a more explicit formula for the various rates involved in Eq. (47)is not possible, due to the impossibility to calculate the surface integral in Eq. (46)explicitly, in turn due to the fact that V T is not known explicitly. Then, the rates inEqs. (47) and (48) are estimated using Arrhenius’s formula. All subsequent deriva-tions regarding the Toy model of Ladieu et al. 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