P Bouchaud

C Brun

F Ladieu

D L'hôte

M Tarzia

G Biroli

J.-P Bouchaud

Nonlinear dielectric susceptibilities: Accurate determination of the growing correlation volume in a supercooled liquid

Keywords: number(s): 64.70.kj, 77.22.-d, 89.75.Da

Nonlinear dielectric susceptibilities: Accurate determination of the growing correlation volume in a

supercooled liquid

I. INTRODUCTION

The glass transition remains a topic of strong interest because it is characterized by an increase of the viscosity when the temperature decreases that is yet unexplained. 1 A picture which could account for this liquid-solid transition is that of growing dynamical heterogeneities (DH's): 2,3 The extremely fast rise of the relaxation time when the temperature T is lowered would be related to an increase of the DH's sizes. For this reason, the temperature dependence of the average number N corr (T ) of correlated molecules in a DH when T decreases toward the glass transition temperature T g recently has become a major issue of the field. [4][5][6] Two methods for evaluating N corr (T ) from experimental data have been put forward in the past several years. The first one uses the temperature derivative of a two-point correlation function φ(t,T ) and yields N corr (T ) ∝ T χ T , with χ T = ∂φ(t,T )/∂T . 4,6 The second one is based on the alternative current (a.c.) nonlinear susceptibility χ 3 , 7,8 which describes the a.c. response of the fluid at a frequency 3 times that of the excitation (the latter can be of any nature, e.g., dielectric as it is the case in practice 8 ). It has been shown 7 that χ 3 is related to N corr (T ) through

χ 3 (ω,T ) ≈ 0 ( χ 1 ) 2 a 3 k B T N corr (T ) H (ωτ α ) , ( 1
)
where ω is the angular frequency, χ 1 = χ lin (ω = 0)χ lin (ω → ∞) is the part of the static linear susceptibility corresponding to the slow relaxation process we consider, a3 the volume occupied by one molecule, and H a complex scaling function going to zero both for small and large arguments, the humped shape of |H(ωτ α )| being a distinctive feature of the glassy correlations. τ α (T ) is the typical relaxation time at temperature T . As mentioned in Refs. 8 and 9, the method using the nonlinear susceptibility has the advantage over that based on T χ T to be free of some assumptions that are not well under control. In addition, being inspired by the spin glass phase transition physics, [START_REF] Lévy | [END_REF] the method using χ 3 can be considered as an indicator of the possible existence of an underlying phase transition for structural glasses. Recently, we have measured the third-harmonic nonlinear susceptibility in supercooled glycerol close to T g , using a specially dedicated experiment based on a two-capacitors bridge. 11 Our data verified the scaling prediction of Ref. 7 given by Eq. (1), and the dependence of N corr (T ) could be determined experimentally from them. In practice, investigating the scaling predicted by Eq. (1) consists in verifying that the complex quantity χ 3 (ω,T ) depends only on the product ωτ α (T ), except that the modulus of χ 3 (ω,T ) should depend on T for a given value of ωτ α (T ). However, in doing so, it appears that one of the main limitations to this study of the scaling is the fact that two physical mechanisms contribute to the nonlinear susceptibility. The first one stems from the dynamical correlations among the molecules and is given by Eq. (1). In what follows, this contribution to the nonlinear susceptibility will be called singular. It was theoretically shown in Ref. 7 that on quite general grounds this is related to dynamical correlations that emerge when decreasing T . Several physical mechanisms aimed at explaining the growth of dynamical correlations have been put forward in the literature (see, e.g., the book quoted in Ref. 9). The second contribution is due to the standard well-known saturation of the microscopic polarization of the dipoles. 12,13 In what follows, this contribution to the nonlinear susceptibility will be called "trivial," as it should be present whatever the system under study. On general grounds, one expects χ 3,trivial to be negligible with respect to χ 3,singular only in the case where N corr becomes infinite. This is not the case of the glass transition, since the various estimates yield N corr of a few tens close to T g . 2,6,14 In this paper, we investigate the consequences of this double contribution for the study of the scaling predicted in Ref. 7 and given by Eq. (1) and for the extraction of the N corr (T ) dependence from experimental data. We present the results of our measurements of two distinct nonlinear susceptibilities on supercooled glycerol: χ (1) 3 and χ 3 (that will be called χ (3) 3 below) corresponding respectively to the first-and the third-harmonic responses. We show how they can be used to disentangle the trivial and singular contributions to the nonlinear response, thus allowing an accurate verification of the scaling predicted by Eqs. (1) and ( 6) and a precise determination of the N corr (T ) dependence. The χ (1) the first time by us: Our previous results concerned χ (3) 3 only. Measurements to which χ (1) 3 can be related have been presented by the R. Richert group: 13,15 The variation of the imaginary part of the dielectric susceptibility when the field is increased should be proportional to the imaginary part of χ (1) 3 . We checked that this is indeed the case, even if the comparison is not so straightforward since the temperature and field ranges are not the same in our work and in Refs. 13 and 15; this will be addressed in detail in the forthcoming paper in which we shall also compare our χ (1) 3 data with the model put forward by the R. Richert group in a series of recent papers. 13,15,16

II. EXPERIMENT

A. From measurements to nonlinear susceptibilities

We now turn to the detailed definition of the a.c. nonlinear susceptibility and to the two related quantities that can be reached experimentally, χ (3) 3 (ω) and χ (1) 3 (ω), which govern respectively the contribution of the nonlinear susceptibility to the third-and first-harmonic responses. We consider the case where the excitation is an a.c. electric field at angular frequency ω and first write the response [polarization P (t)] to the excitation E(t) as a series expansion in E(t) [note that no even terms are allowed because of the symmetry with respect to field reversal E(t) → -E(t)]:

P (t) 0 = ∞ -∞ χ 1 (t -t )E(t )dt + ∞ -∞ χ 3 (t -t 1 ,t -t 2 ,t -t 3 ) ×E(t 1 )E(t 2 )E(t 3 )dt 1 dt 2 dt 3 + • • • , ( 2
)
where 0 is the dielectric constant of vacuum, χ 1 the linear susceptibility, and χ 3 the cubic nonlinear susceptibility in the time domain. The dots in Eq. (2) correspond to an infinite sum involving higher-order nonlinear susceptibilities χ 5 , and so on. The Fourier transform of Eq. (2) for a purely a.c. field E(t) = E 0 cos(ωt) gives

P (ω ) 0 = E 0 2 χ 1 (ω) + 3E 2 0 4 χ 3 (-ω,ω,ω) + • • • δ(ω -ω) + E 0 2 χ 1 (-ω) + 3E 2 0 4 χ 3 (ω, -ω, -ω) + • • • ×δ(ω + ω) + E 3 0 8 χ 3 (ω,ω,ω)δ(ω -3ω) + E 3 0 8 χ 3 (-ω, -ω, -ω)δ(ω + 3ω) + • • • , ( 3
)
where the polarization P and the susceptibilities χ i are now written in the frequency domain and the dots indicate again infinite sums involving higher-order terms. The response P (t) to E(t) can thus be written

P (t)/ 0 = Re E 0 χ 1 (ω) + 3/4E 3 0 χ (1) 3 (ω) + • • • e -iωt + Re 1/4E 3 0 χ (3) 3 (ω)e -i3ωt + • • • + • • • . (4)
We have used the fact that because χ 1 and χ 3 are real in the time domain, their Fourier transform verifies χ * 1 (ω) = χ 1 (-ω) and

χ * 3 (ω 1 ,ω 2 ,ω 3 ) = χ 3 (-ω 1 , -ω 2 ,
ω 3 ) (the star denotes the complex conjugate) and the invariance of χ 3 by permutation of its arguments. For simplicity, we write χ (3) 3 (ω) = χ 3 (ω) = χ 3 (ω,ω,ω) and χ (1) 3 (ω) = χ 3 (-ω,ω,ω). Equation (4) can be written:

P (t)/ 0 = E 0 |χ 1 | cos(ωt -δ 1 ) + 3/4E 3 0 χ (1) 3 cos ωt -δ (1) 3 + • • • + 1/4E 3 0 χ (3) 3 cos 3ωt -δ (3) 3 + • • • , (5)
where the susceptibilities χ (j ) i are written as |χ

(j ) i |e iδ (j )
i and where -δ (3) 3 and -δ (1) 3 are the corresponding phases (reported below in this work).

Thus, experimentally, the third-order nonlinear susceptibility is given by two quantities, χ (3) 3 (ω) and χ (1) 3 (ω), which are the Fourier transforms of χ 3 (t 1 ,t 2 ,t 3 ) at points (ω,ω,ω) and (-ω,ω,ω), respectively. They govern the third-and the first-harmonic contributions to the nonlinear response. Both correspond to contributions to P which are proportional to the third power of the excitating electric field magnitude E 0 . As a consequence, in a dielectric spectroscopy experiment using a capacitor with the supercooled liquid as dielectric layer, they can be obtained by performing the difference between the responses obtained at two different values of the excitating voltage. 13,15 With the bridge technique that we use, 8,11 each of them can be obtained in a single experiment, just by selecting the measurement harmonics. We verified that the magnitude of the measured signal was indeed proportional to the third power of the excitation voltage. An equation relating N corr to χ (1) 3 , similar to Eq. ( 1), holds:

χ (1) 3 (ω,T ) ≈ 0 ( χ 1 ) 2 a 3 k B T N corr (T ) G (ωτ α ) . ( 6
)
Where G is a complex scaling function going to zero both for small and large arguments, with the humped shape of |G(ωτ α )| being a distinctive feature of the glassy correlations.

B. Experimental setup

The liquid used in this experiment is ultrapure (99.5%) glycerol (T g 190 K) purchased from VWR (BDH Prolabo). Special care was taken to avoid water absorption during sample preparation. Two samples were prepared between metallic electrodes 2 cm in diameter, with polymer spacers ensuring interelectrode distances of 12 and 26 μm or of 8 and 27 μm. The samples were placed in a cryogenerator, and their temperature T was regulated with a precision much better than 100 mK in an interval T g + 4 K to T g + 35 K. A low-harmonic-distortion voltage source yielded a field E 0 3 × 10 6 V/m (rms) in the thinnest capacitor. χ (3) 3 (ω,T ) and χ (1) 3 (ω,T ) were respectively obtained from the third-and first-harmonic currents. For the third-harmonic measurements, our high-sensitivity method based on a bridge containing the two glycerol capacitors 11 allows us to get rid of the harmonic distortion of our electronic apparatus. For the first-harmonic measurements, a very careful equilibration of the bridge at low voltage source allowed us to make the effect of χ 1 negligible with respect to that of χ (1) 3 [see Eq. ( 5)] for E 0 in the range 1.5-3 × 10 6 V/m.

C. Getting rid of possible artifacts: Heating and mechanical effects

We briefly explain how we managed to avoid any distortion of the data reported here by heating or mechanical effects. Heating effects come from the dissipated electrical power ρ that raises the temperature T of the liquid by an amount δT . For a field E(t) = E 0 cos(ωt), one finds that ρ ∼ E 2 0 contains a constant term and a term oscillating at frequency 2ω; see Ref. 17. As a result, δT (t) = δT dc + δT 2 (t), where δT dc ∼ E 2 0 is a constant and δT 2 (t) ∼ E 2 0 oscillates at 2ω. The heating affects the polarization by an amount given, as a first-order estimate, by δP (t) [∂P lin (t)/∂T ]δT (t) ∝ E 3 0 , where P lin (t) is the linear part of the polarization. Thus, one sees that δT 2 (t) enters in products of two terms oscillating at ω and 2ω, yielding a heating contribution to the third harmonics. This contribution was carefully studied in Ref. 17 where it was shown that it only plays a minor role at high frequencies and high temperatures. In the range of parameters reported here, this heating contribution to the third harmonics can be safely neglected. The smallness of the heating effect on third harmonics comes mainly from the "skin effect," i.e., from the fact that the oscillating part of heating is exponentially damped in the thick metallic electrodes of the samples. This contrasts with δT dc that is not damped in the electrodes and yields the main contribution of heating to the nonlinear part of the first harmonics. Fortunately, the thermal time constant τ dc ∼ 300 s associated with the settlement of δT dc is really long since it corresponds to heat transfer across the whole experimental cell. Thus, the effect of δT dc can be avoided by carrying out the measurements of the nonlinear part of the first harmonics over a time scale τ 1 much smaller than τ dc . As in Refs. 13,15, and 16 we checked that different values of τ 1 did not change the measured χ (1) 3 noticeably, provided that the condition τ 1 τ dc was obeyed. We note here that the heating effects we consider must not be confused with those studied in the box model framework 13,15,16,18 because the latter are a possible description of the physics of the connection between the nonlinear susceptibilities and the DH's.

Mechanical effects would come from a time dependence of the thickness e of the capacitance due to electrical forces between the electrodes that would yield a nonlinear component in the response. We considered this possibility by calculating its contribution to χ (1) 3 and χ (3) 3 . The oscillations of the capacitance can be rather easily obtained by taking into account the electric attractive force (∼E 2 0 ) between the electrodes and the elasticity of the spacers and glycerol layer. As for heating effects of Ref. 17 evoked above, one finds that the variation δe(t) of the thickness contains a constant term and a term oscillating at angular frequency 2ω, i.e., δe(t) = δe dc + δe 2 (t), where both terms are proportional to E 2 0 . The result of our calculation is that such an effect can safely be neglected since it gives a contribution that is less than 1.5% of the values of |X (i) 3 | reported here. In addition, we note that, in our experiments, we checked [for X (3) 3 ] that the measured values are the same whatever the surface or shape of the spacers. As their size and shape contribute to the possible spurious nonlinear response, we consider this experimental result as an important clue that the contribution of the attractive force between the electrodes to the nonlinear response can be neglected. Note that a similar calculation has been performed by the authors of Ref. 13. Their result confirms that electrode attraction can be disregarded as a significant source of nonlinearity.

III. RESULTS AND DISCUSSIONS

In Fig. 1, the dimensionless value X (3) 3 (ω,T ) = χ (3) 3 (ω,T ) × k B T /[( χ 1 ) 2 a 3 0 ] is plotted as a function of the frequency f = ω/(2π ) for six temperatures. [START_REF]All the values of X (3) 3 reported here are 10% smaller than those reported in our Ref. 8, the difference[END_REF] The definition of X (3) 3 is chosen in order that according to Eq. (1), |X (3) 3 | is N corr (T ) |H(ωτ α )|. Indeed, we find that for each temperature, the frequency dependence exhibits the expected humped shape, while the overall magnitude of the curves increases when T decreases as expected from Eq. (1) since this magnitude 7 should be proportional to N corr (T ). Figure 2 shows, for two temperatures, the dimensionless value X (1) 3 (ω,T ) = χ (1) 3 (ω,T ) × k B T /[( χ 1 ) 2 a 3 0 ] as a function of the frequency. As for X (3) 3 , a humped shape is found and the magnitude of |X (1) 3 (ω,T )| increases as T decreases, as expected from Eq. ( 6). We now investigate the possible scaling behavior of X (3) 3 (ω,T ) and X (1) 3 (ω,T ).

A. Scaling of X (3) 3

The main part of Fig. 3 shows the same results as Fig. 1 but plotted as a function of the ratio f/f α , where f α (T ) = 1/[2πτ α (T )] is the relaxation frequency at temperature T , obtained as the frequency for which χ 1 is maximum. As found in Ref. 8, for each temperature, the maximum value of |X (3) 3 (ω,T )| is obtained for f/f α 0.21. In the inset of Fig. 3, for each temperature, the values of |X (3) 3 (ω,T )| have been divided by their maximum value over frequency to test the scaling predicted by Eq. (1). It can be seen that around the peak of |X (3) 3 | the predicted scaling is well obeyed. The phase

FIG. 1. (Color online) |X

(3) 3 (ω,T )|, the modulus of the dimensionless third-harmonic nonlinear susceptibility χ (3) 3 (ω,T ) measured in supercooled glycerol, is plotted as a function of the frequency for six temperatures labeling the curves. The arrows indicate the relaxation frequency f α for each temperature, f α being the frequency for which χ 1 (ω,T ) is maximum.

FIG. 2. (Color online) |X (1)

3 (ω,T )|, the modulus of the dimensionless first-harmonic nonlinear susceptibility χ (1) 3 (ω,T ) measured in supercooled glycerol, is plotted as a function of the frequency for two temperatures labeling the curves. The arrows indicate the relaxation frequency f α for each temperature. of X (3) 3 (see Fig. 4) confirms this scaling, since, as found in Ref. 8, the phases measured at different temperatures depend only on f/f α .

In Fig. 3, the most important departure from scaling arises for f/f α < 3 × 10 -2 where the |X (3) 3 (ω,T )| curves fall on the same plateau value ∼0.16 at low frequency. We explain this deviation to complete scaling by the fact that at low frequency χ (3) 3 (ω) is dominated by the trivial contribution χ (3) 3,trivial (due to the saturation of the polarization of the dipoles). To confirm this explanation, we evaluate χ (3) 3,trivial by using the analytic calculation of Ref. 12. In this calculation that is free of any glassy correlations, the nonlinear response χ (3) 3,D (ω,T ) is calculated for a system made of independent 3 (ω,T )|, the moduli of the dimensionless measured nonlinear susceptibilities χ (3) 3 (ω,T ) and χ (1) 3 (ω,T ), are plotted as a function of f/f α for two temperatures. The lines without symbols X (3) 3,D (ω) (black solid) and X (1) 3,D (ω) (purple dashed) are estimates of the "trivial" contributions to the nonlinear susceptibilities. (Lower graph) Phases (in degrees) corresponding to the moduli of the upper graph (same symbols).

rigid polar molecules with a Brownian rotational motion in a viscous medium. To the best of our knowledge, Ref. 12 is the only framework yielding the frequency dependence of the nonlinear effects for polar molecules under rotational Brownian motion. This is, of course, a very simplified view of the trivial contribution of glycerol where each molecule contains several dipoles. Following Ref. 12, we calculate χ (3) 3,D (ω,T ). χ (3) 3,D yields, as for χ (3) 3 , the dimensionless nonlinear susceptibility X (3) 3,D (ω,T ) = |χ (3) 3,D (ω,T )| × k B T /[( χ 1 ) 2 a 3 0 ]. We find that X (3) 3,D (ω,T ) depends only on the ratio f/f α and that the magnitude of X (3) 3,D (ω,T ) is independent of the temperature. The continuous line in Fig. 3 shows the frequency dependence of |X (3) 3,D |: |X (3) 3,D | has no peak at finite frequency. This confirms that the humped shape of the measured |X (3) 3 | is a distinctive feature of glassy correlations, as predicted by Bouchaud and Biroli in Ref. 7. Besides, X (3) 3,D reaches its maximum plateau value for f/f α < 10 -1 and decreases to zero for f/f α > 10 -1 , becoming negligible with respect to the measured |X (3) 3 (ω,T )| of glycerol at f/f α > 1. Let us emphasize that the values of |X (3) 3,D (ω,T )| reported in Fig. 3 are those calculated in Ref. 12 times a factor 0.8. This factor was chosen to enforce the agreement between our model of trivial effects and the experimental measurements in the plateau regime occuring for f/f α < 3 × 10 -2 . Due to the fact that the analytic calculation of Ref. 12 is only a first crude model of the trivial nonlinear effects of glycerol, the use of an adjustable factor close to unity is reasonable. Thus, the fact that χ (3) 3 results from both the contributions of χ (3) 3,singular (due to glassy correlations 7 ) that should be dominant for f/f α > 0.1 and of χ (3) 3,trivial that is dominant for f/f α < 5 × 10 -2 can well explain that the scaling predicted by Eq. ( 1) is verified only for f/f α > 0.05, as found in our previous paper. 8

B. Scaling of X (1) 3

In what concerns X (1) 3 , similar results are obtained: in Fig. 4 (closed symbols), we can see that for two different temperatures, the curves |X (1) 3 (f/f α )| have the same shape but not the same magnitude. In addition, the phases depend only on f/f α as can be seen in Fig. 4. Thus the scaling predicted by Eq. ( 6) holds for X (1) 3 . Comparing the measured |X (1) 3 | and |X (3) 3 | in Fig. 4 reveals that (i) the former is peaked at a value of f/f α larger than the latter and (ii) both quantities decrease at large frequencies with similar power laws whose exponent is -0.61±0.04. These two features can be accounted for by using (and extending) the considerations of Ref. 20 where χ (3) 3 and χ (1) 3 were computed analytically within the mode coupling theory (MCT) framework. As MCT is valid at temperatures much higher than those of our experiments, 21 the fact that some of the features predicted within MCT are still observed close to T g might come from the dynamical amorphous order that emerges in supercooled liquids as the temperature is decreased. Comparisons with theoretical models, in particular the one developed by the R. Richert group, 13,15,16 was thoroughly done very recently in Ref. 18 for X (3) 3 and will be done in a forthcoming publication for X (1) 3 . As for χ (3) 3 , we have evaluated the contribution to χ (1) 3 due to the polarization saturation χ (1) 3,trivial by calculating χ (1) 3,D in the model of Ref. 12. The resulting dimensionless nonlinear susceptibility X (1) 3,D , multiplied by the same factor 0.8 as in Fig. 3, is shown in Fig. 4 (purple dashed line). As for |X (3) 3,D |, its modulus has no peak and reaches a maximum plateau value at low frequencies, which does not depend on T , and decreases to negligible values at high frequencies. Even if, in Fig. 4, the measured |X (1) 3 | is close to the trivial estimate |X (1) 3,D | for frequencies as high as f/f α = 0.3, the phase of the measured X (1) 3 is still well above the value of ±180 • which is expected, on general grounds, for the trivial contribution. Thus, the trivial contribution does not fully dominate the measured X (1) 3,D in the range f/f α = 0.1-0.3 and we hypothesize that this happens at lower frequencies, say f/f α 0.05, just as for χ (3) 3 . Finally, from Fig. 4 we conclude that concerning the scaling and the extraction of N corr (T ) from the magnitude of the dimensionless nonlinear susceptibility, χ (1) 3 is a better candidate than χ (3) 3 because, at the peak frequency [f/f α 0.21 for χ (3) 3 and f/f α 2.5 for χ (1) 3 ], the value of the estimated trivial relative contribution is much smaller for χ (1) 3 than for χ (3) 3 .

C. Extraction of N corr (T )

Considering the results and discussions presented above, we can now look for the best way to estimate the N corr (T ) dependence from the χ (1) 3 and χ (3) 3 data, taking into account the fact that the glassy correlations contribution to the nonlinear susceptibility is dominant only at "large" frequencies. 3 (ω,T )|, according to Eq. ( 6) (open lozenges). τ α is defined as τ α (T ) = 1/[2πf α (T )]. The filled triangles correspond to |X (3) 3 (f/f α = 2.5,T )|, which should be free of any trivial contribution (see text). As in Fig. 5, all the data are normalized to 1 at T = 204.7 K.

The best estimate of the T dependence of N corr (T )

We see in Fig. 5 that |X (1) 3 (f/f α = 2.5)| has the same temperature dependence as |X (3) 3 (f/f α = 2.5)|. This confirms that, in the range where the trivial contribution is fully negligible, both |χ (1) 3 | and |χ (3) 3 | have the same temperature evolution, as expected from Eqs. (1) and ( 6). This is why the most precise way to extract the T dependence of N corr (T ) is to follow |X (1) 3 (f/f α = 2.5)| or |X (3) 3 (f/f α = 2.5)|. This yields a T dependence faster than that of T χ T : we note here that T χ T is defined as T χ T = max ω [T ∂(χ lin (ω)/ χ 1 )/∂T ], see Ref. 6, where χ lin (ω) is the real part of the linear dielectric response. Let us emphasize that T χ T is an approximate estimator of N corr (T ). It has the great advantage of being easily obtained from experiments, but its ability to reproduce the T dependence of "true" N corr (T ) guaranteed. For example, in of strong glasses where one expects N corr to be independent of temperature, one T χ T ∝ 1/T . For fragile liquids, the T dependencies of T χ T and of N corr (T ) are only the same provided a prefactor sensitive to microscopic details can be assumed to be temperature independent. 22 Since this assumption is not needed to relate the nonlinear response to the "true" N corr (T ), the T dependence extracted from |X (3) 3 (f/f α = 2.5)| and |X (1) 3 (f/f α = 2.5)| is, in our opinion, the most reliable estimate to date of the T dependence of N corr (T ).

This increase of |X (1) 3 (f/f α = 2.5)| and of |X (3) 3 (f/f α = 2.5)| when approaching the glass transition is plotted as a function of ln τ α in Fig. 6, exhibiting a linear behavior in the experimentally accessible region (but note that the relative change of ln(τ α /τ 0 ) is only 20%, assuming τ 0 = 10 -13 s. This means that a nonlinear dependence, if any, is hard to detect). Plotting the corresponding quantities for other supercooled liquids on such a plot would be interesting as it would allow us to compare directly the rate of increase of N corr (T ) for glasses with different fragilities and possibly put some experimental constraints onto the microscopic theories aiming at describing the glass transition. Finally, we note that such a logarithmic increase of N corr (T ) with τ α has already been considered, e.g., in Refs. 6 and 23: The relaxation time would be related exponentially to an activation energy or a configurational entropy that would be proportional to a power of N corr . This is why it is conceivable that an increase of N corr (T ) by 35% corresponds to an increase of τ α by 2 decades, as in Fig. 6.

IV. CONCLUSION

In order to determine experimentally the important physical information provided by the dependence of the number of dynamically correlated molecules N corr on temperature, we have measured the two a.c. nonlinear dielectric susceptibilities χ (3) 3 and χ (1) 3 in supercooled glycerol. According to Ref. 7, those quantities contain a contribution χ (i) 3,singular due to glassy correlations that should have a scaling behavior in its frequency and temperature dependencies and be proportional to N corr . By performing a detailed study of the trivial contributions, χ (3) 3,trivial and χ (1) 3,trivial , due to standard polarization saturation, we have shown that they indeed affect the scaling, primarily at the lowest frequencies. As a consequence, an accurate determination of the temperature dependence of N corr from the nonlinear susceptibilities needs an appropriate subtraction of the trivial contributions in order to extract the singular part of the nonlinear susceptibility. It seems difficult to have a precise theoretical prediction of the trivial contribution and on its interplay with the glassy contribution to the nonlinear susceptibility. This is why, at present, the most reliable way to get rid of the trivial contribution is to compare the T dependence of χ (3) 3 and of χ (1) 3 . This allowed us to conclude that in glycerol the T dependence of N corr is faster than previously determined using the dynamic susceptibility, T χ T , a method used in many previous publications. 4,6,23

FIG. 3 .

 3 FIG. 3. (Color online) |X (3) 3 (ω,T )| is plotted as a function of f/f α for six temperatures, where f α is the relaxation frequency at the considered temperature. The data (and the temperatures) are the same as those shown in Fig. 1. The continuous line is X (3) 3,D (ω), an estimate of the "trivial" contribution to the nonlinear susceptibility. (Inset) The same data, normalized to test the scaling predicted by Eq. (1): |X (3) 3 (ω,T )|/max ω |X (3) 3 (ω,T )| is plotted as a function of f/f α (trivial estimate not shown).

FIG. 4

 4 FIG. 4. (Color online) (Upper graph)(3) 3 (ω,T )| and |X(1) 3 (ω,T )|, the moduli of the dimensionless measured nonlinear susceptibilities χ(3) 3 (ω,T ) and χ (1) 3 (ω,T ), are plotted as a function of f/f α for two temperatures. The lines without symbols X(3) 3,D (ω) (black solid) and X(1) 3,D (ω) (purple dashed) are estimates of the "trivial" contributions to the nonlinear susceptibilities. (Lower graph) Phases (in degrees) corresponding to the moduli of the upper graph (same symbols).

Figure 5

 5 Figure5shows the maximum values reached by |X(3) 3 (ω)| and |X(1) 3 (ω)| when ω varies as a function of the temperature. According to Eqs. (1) and (6), those quantities are proportional

results we give in this paper are presented for

(ω,T )|, according to Eq. (6) (open lozenges). The triangles pointing down correspond to the maximum of X 3 (ω,T ) that is obtained from X(3) 3 (ω,T ) by subtracting an estimated trivial contribution. The filled triangles correspond to |X(3) 3 (f/f α = 2.5,T )|, which should be free of any trivial contribution (see text). The continuous line is a simplified estimator of the number of correlated molecules estimated from T × χ T (see Ref.6 and text). All the data are normalized to 1 at T = 204.7 K to allow an accurate comparison of their relative evolution with temperature.

ACKNOWLEDGMENTS

We thank R. Tourbot for realizing the experimental cell and P. Pari for the cryogenic setup. We acknowledge interesting discussions with C. Alba-Simionesco, A. Lefèvre, and R. Richert and support by ANR grant DynHet.

to N corr , the average number of correlated molecules in a DH. The comparison of the rates of increase of max ω [|X (3) 3 (ω,T )|] and max ω [|X (1) 3 (ω,T )|] when T decreases below 205 K confirms our previous discussion on the two contributions. Indeed, as at the peak frequency the relative contribution of the trivial component is weaker for X (1) 3 than for X (3) 3 , the maximum of |X (1) 3 | increases faster than the maximum of |X (3) 3 |. Figure 5 shows the quantity max ω [|X 3 (ω,T )|], where X 3 is the dimensionless nonlinear susceptibility from which the estimate of the trivial contribution has been subtracted:

We can see in Fig. 5 that when T decreases, the increase of max ω [|X 3 (ω,T )|] is steeper than that of max ω [|X (3) 3 (ω,T )|], which again confirms the importance of getting rid of the trivial contribution to χ (3) 3 for estimating the N corr (T ) dependence from the nonlinear susceptibility. The difference between the rates of increase of max ω [|X 3 (ω,T )|] and of max ω [|X (1) 3 (ω,T )|] is yet unexplained. This suggests that we should clarify our two assumptions. First, the estimate of the trivial contribution could be refined, since the detailed shapes of the X (3) 3 (ω) and X (1) 3 (ω) functions compared to that of the X (i) 3,trivial X (i) 3,D play a major role in the comparison of Fig. 5. Second, our assumption that the trivial contribution simply adds to the singular one is questionable, especially when the two contributions are of the same order of magnitude. A refined theoretical analysis would be needed on that point.


C Crauste-Thibierge

C Brun

F Ladieu

D L'

Hôte 1⋆

G Biroli

J-P Bouchaud

Evidence of growing spatial correlations at the glass transition from nonlinear response experiments

The ac nonlinear dielectric response χ 3 (ω, T ) of glycerol was measured close to its glass transition temperature T g to investigate the prediction that supercooled liquids respond in an increasingly non-linear way as the dynamics slows down (as spin-glasses do). We find that χ 3 (ω, T ) indeed displays several non trivial features. It is peaked as a function of the frequency ω and obeys scaling as a function of ωτ (T ), with τ (T ) the relaxation time of the liquid. The height of the peak, proportional to the number of dynamically correlated molecules N corr (T ), increases as the system becomes glassy, and χ 3 decays as a power-law of ω over several decades beyond the peak. These findings confirm the collective nature of the glassy dynamics and provide the first direct estimate of the T dependence of N corr .

perature T decreases to their glass transition temperature T g . This transition is similar for a vast variety of systems such as polymeric, colloidal, molecular liquids. This ubiquity echoes the universality of critical phenomena, where the emergence of long range order makes irrelevant most of the microscopic details, yielding the same critical behavior in various systems. This is why it has been argued for a long time that some collective effects should be associated to the glass transition, ultimately related to the proximity of a phase transition [1].

An immediate problem with this idea is that standard equilibrium correlation and response functions remain those of a normal liquid when T → T g [2]. In the two past decades, however, the heterogeneous nature of the dynamics close to T g was established [3,4], and a length scale associated to these dynamic heterogeneities was estimated [5]. The increase of the number of correlated molecules N corr when T decreases towards T g is expected to explain the main aspect of the glass transition, i.e. the huge increase of the relaxation time;

N corr (T ) has therefore become a central concern in the field. The first estimate of this dependence [6][7][8] relied on the temperature derivative of a two-point correlation function. Its relation with the number of dynamically correlated molecules N corr (T ) is however ambiguous because it involves a temperature dependent prefactor difficult to estimate and control [7,9]. Furthermore, this estimator leads to a low-temperature divergence of N corr (T ) for a purely Arrhenius system, for which no collective behaviour is expected, at least at first sight [7]. An indisputable experimental estimate of the T -dependence of the dynamical correlation volume is thus still lacking. In this work, we fill this gap by measuring the anomalous increase of a physical susceptibility. It leads to a direct estimate of N corr (T ), up to a numerical prefactor that is now temperature independent. Our findings show that supercooled liquids respond in an increasingly non-linear way approaching the glass transition. For many systems, e.g. spin glasses, such an increase is related to criticality. In the present case, it suggests that an underlying phase transition could possibly be present as well, although it does not necessarily imply it.

The spin glass transition taught us that two-point functions can be blind to the amorphous long range order and that this critical behavior is revealed in particular by the third order nonlinear magnetic susceptibility [10,11]. It was recently argued [12,13] that the frequency and temperature dependent nonlinear dielectric susceptibility χ 3 (ω, T ) in supercooled liquids plays a role similar to the nonlinear magnetic susceptibility in spin glasses and that its increase when T decreases would be a signature of the incipient long range amorphous order. Accordingly, χ 3 (ω, T ) should display a peak for ωτ (T ) ∼ 1 whose height grows as one approaches T g . Here τ (T ) is the relaxation time of the supercooled liquid which increases rapidly as T decreases toward T g . Contrary to spin glasses, however, χ 3 (ω = 0, T ) should remain trivial since at long times glasses still behave as disordered liquids [13]. One expects that when τ (T ) is large, χ 3 takes the following scaling form:

χ 3 (ω, T ) ≈ ǫ 0 (∆χ 1 ) 2 a 3 k B T N corr (T ) H (ωτ ) , (1)
where N corr (T ) is the T -dependent average number of dynamically correlated molecules,

∆χ 1 = χ 1 (ω = 0) -χ 1 (ω → ∞)
is the part of the static linear susceptibility corresponding to the slow relaxation process we consider, a 3 the volume occupied by one molecule, and H a certain complex scaling function that goes to zero both for small and large arguments. The humped shape of |H (ωτ ) | is a distinctive feature of the glassy correlations. Indeed, in the 'no correlation case ' [14, 15], χ 3 (ω, T ) is given by the prefactor of Eq. ( 1) times a function that, for all T , reaches its maximum value at ω = 0. Furthermore N corr (T ) does not show any temperature dependence in that case.

We have devised an experiment to measure, for the first time, χ 3 (ω, T ) of a classic glassformer (glycerol), via third harmonics measurements. Our findings are in good agreement with the predictions of Eq. ( 1): the scaling, the growth of the number of dynamically correlated molecules with decreasing T , the humped shape of |H| and its power-law dependence for ωτ ≫ 1 are all confirmed experimentally. The anomalous increase of χ 3 is compatible with the existence of incipient critical fluctuations when T → T g , even if our measurements are too far from the possible critical point to be a clue of criticality.

Our measurements were performed at temperatures between T g + 4 K and T g + 35 K, with

T g ≈ 190 K. The linear susceptibility χ 1 (ω) = χ ′ 1 (ω) + iχ ′′ 1 (ω)
quantifies the first harmonic response of polar molecules to a periodic excitation field Ee -iωt . The polarization, P (ω)e -iωt , of the system is given to first order by P (ω) = ǫ 0 χ 1 (ω)E. We define the relaxation time τ (T ) as the inverse of the frequency f α where χ ′′ 1 is maximum. The nonlinear response is dominated by χ 3 (ω) which gives to first order the magnitude of the third harmonics response to the field at pulsation ω [14, 16], P (3ω)e -3iωt = ǫ 0 χ 3 (ω)[Ee -iωt ] 3 (the second harmonics is zero because of the E → -E symmetry). While χ 1 (ω, T ) has been widely studied in supercooled liquids [17][18][19], the T and ω dependence of χ 3 has not been measured so far, for any liquid, glassy or not [START_REF]An attempt to measure χ 3 of a glass former was reported in L. Wu[END_REF].

In a closed cell, two high purity glycerol samples were prepared between gold plated copper electrodes 2 cm in diameter, with mylar R spacers ensuring inter-electrode distances of 19 µm and 41 µm [14,16]. The cell was placed in a cryogenerator, and the temperature T of the samples was regulated with a precision of 50 mK. A low harmonic distortion voltage source yielded a field E ∼ 7 × 10 5 V/m (rms) in the thinnest capacitor, and χ 3 (ω, T ) was obtained from the measured third harmonics current I(3ω). This nonlinear signal was 10 -7 to 10 -5 of the linear one, well below the harmonic distortion of electronic devices. To get rid of them, we used a high sensitivity method based on a bridge containing the two glycerol capacitors [16] . 1), normalized to its maximum value at each T , plotted as a function of the ratio f /f α (T ). The straight line is a power-law with an exponent -0.65. Inset: corresponding phase of χ 3 (ω, T ).

Figure 1 shows, for five temperatures, the ω dependence of X(ω, T ) = |χ 3 (ω, T )| × k B T /[(∆χ 1 ) 2 a 3 ǫ 0 ]. According to Eq. ( 1), this quantity should be N corr (T ) |H(ωτ )|. Two striking results, confirming the predictions of Eq. (1), can be seen in Fig. 1. First, X(ω, T ) is actually peaked at a frequency f * of the order of 1/τ (T ); more precisely f * ≃ 0.21/τ (T ).

Second, the height of the peak, which should be proportional to N corr (T ), increases significantly as T decreases. The small value of X(ω, T ) can be understood from calculations of χ 3 (ω, T ) for molecules undergoing independent rotational Brownian motion [14,15], which should be accurate for simple supercooled liquids at high T . They indicate that for the 'no correlation' case, X(ω, T

) ≤ 0.2.
To what extent is the scaling predicted by Eq. (1) verified by χ 3 (ω, T )? Fig. 2 shows the same data as in Fig. 1 but plotted as a function of log(f /f α ) with f α = τ (T ) -1 , and normalized by their maximum at each T . We also show the phase of χ 3 . Both the modulii ] plotted vs. temperature. The continuous line is the number of correlated molecules estimated from T χ T (see [8]) with an arbitrary normalization chosen so that it coincides with the experimental points at 202 K.

and the phases collapse fairly well on a single master curve, as predicted by Eq. (1). A weak departure from scaling occurs at low ω for the highest T . This is not surprising: scaling should not hold far above T g , where the dynamical correlations become short-ranged [12,13].

Furthermore, the nonlinear response results both from trivial dielectric saturation effects [15,21,[START_REF] Böttcher | Theory of electric polarization[END_REF] (always present for ωτ < 1), and from the non trivial dynamical correlations contribution. As the latter vanishes when ω → 0 ( [12,13]), the contribution of the trivial saturation effect should dominate at high T and low ω. The observed departure from scaling can thus be explained by the different T and ω dependencies of the two contributions.

Figure 3 gives the T dependence of the maximum value of X(ω, T ), reached for ω = ω * = 2πf * . Since scaling is well obeyed, Eq. ( 1) tells us that this maximum value is proportional to N corr (T ). A clear increase of this quantity as T decreases toward T g is visible in Fig.

3

. This is one of the main results of our study, and a strong experimental evidence of a growth of the dynamical correlation length close to T g . We now investigate how our results are related to another estimate of the size of the dynamical heterogeneities, using the temperature derivative of χ 1 (ω).

In [7,8], the T -derivative of the two-body correlation C(t) (or of χ 1 (ω)) was introduced as a new response function χ T (t) = ∂C(t)/∂T . It turns out that T χ T (t) is also proportional to a dynamical correlation volume, but with an unknown, and possibly T dependent prefactor [7,9]. The line in Fig. 3 is the number of correlated molecules estimated from max ω (T ∂(χ ′ 1 (ω)/∆χ 1 )/∂T ) where, due to the unknown value of H, the normalization was chosen such that it coincides with X(ω * , T = 202K). We see that the T dependencies of these two quantities are close. However, whereas the increase of X(ω * , T ) is a proof of a growing correlation length, one needs some extra assumptions to infer this from χ T [7]. That χ 3 and χ T lead to a similar T dependence for N corr (T ) suggests that these assumptions are indeed warranted (at least for glycerol) and validates the simpler procedure advocated in [6], and used extensively in [8], to extract N corr (T ).

A precise identity relating χ T and χ 3 has been obtained in [13]. It holds at low enough ω when the linear response depends only on external parameters (T , density, electric field, . . . ) via the dependence of τ (T ) on these parameters, a property called time temperature superposition (TTS), and reads:

χ 3 (ω) ≈ κ χT (2ω) ωτ ≪ 1 , (2)
where κ is independent of ω and χT is the Fourier transform of χ T . Corrections to this relation in the regime ωτ ≪ 1 are due to weak violations of TTS and to the additional term that corresponds to ω independent non-cooperative degrees of freedom. The continuous line on Fig. 4 presents X(ω, T ) calculated from the linear susceptibility, according to Eq. ( 2)

with κ = 7.6 × 10 -17 K(m/V) 2 . This value of κ was chosen to be consistent with Fig. 3, data exhibit a clear power-law behavior with a fitted exponent -0.65 ± 0.04, see also Fig. 2, which is close to that describing the decay of χT (2ω). This behavior is predicted by Mode-Coupling Theory [13], and our experimental exponent is compatible with the MCT prediction -b ≃ -0.6 for glycerol [START_REF] Lunkenheimer | [END_REF]. Note however that MCT should in fact be relevant only at temperatures much higher than the ones we focused on. Hence, the existence of a power-law regime for large ω with identical exponents for both χ 1 and χ 3 appears to be a generic property, valid outside the MCT regime. We think that this might be related to the possible incipient criticality of the system.

We now address the question of a possible heating contribution to our results. Such effects would come from the fact that applying the field E(t) = E cos(ωt) across the glycerol sample leads to a dissipated power per unit volume p(t) ∝ ωχ ′′ 1 (ω)E 2 (1 + cos(2ωt -φ))

(see [14]). The resulting temperature increase is the sum of a constant term and a term δT 2ω (t), oscillating at 2ω. This δT 2ω (t) oscillation leads to a modulation of the linear susceptibility δχ 1 (ω)(t) = (∂χ 1 (ω)/∂T )δT 2ω (t), thus to a 3ω modulation of the polarization P (t) ∝ χ 1 E. We calculated precisely this 3ω heating contribution using the heat propagation equation [14]. This calculation gives an upper limit to the heating effects contribution, in particular for f ≥ f α , because the finite relaxation time τ of the molecules prevents them from following instantaneously the temperature oscillation δT 2ω (t) (as already advocated by Richert et al. [24]). A representative example of these calculations is given on Fig. 4 : the heating effects are negligible in most of the frequency range of interest, in particular around the χ 3 (ω) peak. Thus, our estimate of N corr (T ) and the power-law dependence of χ 3 (ω) are not affected by heating.

Our results should allow significant progress of models which predict or assume dynamical correlations in supercooled liquids. Such models should not only reproduce the increase of the correlated volume we found as T → T g (see Fig. 3), but they should also account for the magnitude and shape of χ 3 (ω, T ), given by the H function in Eq. ( 1), which carries an important information on the physics of the dynamical correlations [12,13]. Recently, Richert et al. put forward a phenomenological model [21,24,25] which accounts for their nonlinear susceptibility data at 1ω. In this model, the supercooled liquid is thought of as a collection of "Debye-like" dynamical heterogeneities (dh), each of them having its own relaxation time τ dh . The model posits that the electrical power absorbed by each dh raises its temperature above that of the phonon bath. A non-trivial and crucial assumption is that heat exchange between the slow degrees of freedom and the phonon bath is set by the local relaxation time τ dh , and not by a microscopic vibration time. As a result, each dh has its own fictive temperature with a dc and an ac component. Again, the ac component δT dh,2ω (t) leads to a 3ω "heating" contribution to the polarization P (t). Such a contribution to χ 3 should however not be considered to be in competition with the critical dynamical correlations related to the glassy dynamics [12,13]. As this model assumes a priori the existence of dynamical heterogeneities, it should be seen as a phenomenological description of the influence of dynamical correlations on the nonlinear susceptibility. We calculated the prediction of this model for the 3ω nonlinear response and found that χ 3 (ω) is indeed peaked at a frequency of the order of 1/τ (T ). However, some assumptions are needed to generalize the model to the 3ω response. These assumptions influence the magnitude and position of the peak and deserve further scrutiny. A detailed comparison of this model to our data will be presented in a subsequent paper. In any case, we believe that for a proper prediction of χ 3 (ω, T ) close to T g , a theory of supercooled liquids able to account for dynamical correlations is required.

To conclude, we have provided the first direct experimental evidence that a supercooled liquid responds in an increasingly non-linear way approaching the glass transition. By measuring the frequency dependent third harmonics response χ 3 (ω, T ) to a periodic electric field, which is tightly related to the dynamical correlation length, we showed that the number of correlated molecules increases as T decreases towards T g , confirmed the validity of previous estimates, and found that χ 3 scales as a function of ωτ . This opens a new path for probing the spatial correlations in both fragile and strong supercooled liquids and in the aging regime of glasses and spin-glasses, by systematic studies of non-linear responses. Future investigations along these lines might help to unveil the possible critical nature of the glass transition.

Electronic Physics Auxiliary Publication Service: Supplementary Information for :

Evidence of growing spatial correlations at the glass transition from nonlinear response experiments Caroline Crauste-Thibierge, Coralie Brun, Francois Ladieu, Denis L'Hôte, Giulio Biroli, Jean-Philippe Bouchaud

For the sake of completness, we detail in the four sections hereafter: (i) the connexion between the third harmonics of the fundamental response and the nonlinear part of the response; (ii) what can be expected for the third harmonics response of a dielectric liquid where no (glassy) correlations are present; (iii) some additionnal information about our experimental setup; (iv) how the upper limit of the heating contribution was computed (Fig. 4 of the letter) and the reduction which can be expected from a more refined estimate.

I) NONLINEAR RESPONSE MEASUREMENTS THROUGH THIRD HARMON-ICS DETECTION

As in the main paper, we consider the nonlinear response of a dielectric system to a time dependent electric field E(t). The most general relationship between the polarisation P (t) and the excitation E(t) is a series expansion in E [1]. For a purely ac field, the even terms are forbidden because of the E(t) → -E(t) symmetry, which yields :

P (t) ǫ 0 = ∞ -∞ χ 1 (t-t ′ )E(t ′ )dt ′ + ∞ -∞ χ 3 (t-t ′ 1 , t-t ′ 2 , t-t ′ 3 )×E(t ′ 1 )E(t ′ 2 )E(t ′ 3 )dt ′ 1 dt ′ 2 dt ′ 3 +.... (3)
In this equation ǫ 0 is the dielectric constant of vacuum, χ 1 the linear susceptibility and χ 3 the cubic nonlinear susceptibility in the time domain. The dots in Eq. 3 indicate an infinite sum involving higher order nonlinear susceptibilities χ 5 , etc. Note that causality implies χ i (t < 0) = 0. The Fourier transform of Eq. 3 for a purely ac field E(t) = E cos(ωt) gives:

P (ω ′ ) ǫ 0 = E 2 χ 1 (ω) + 3E 2 4 χ 3 (-ω, ω, ω) + ... δ(ω ′ -ω) + E 2 χ 1 (-ω) + 3E 2 4 χ 3 (ω, -ω, -ω) + ... δ(ω ′ + ω) + E 3 8 χ 3 (ω, ω, ω)δ(ω ′ -3ω) + E 3 8 χ 3 (-ω, -ω, -ω)δ(ω ′ + 3ω) + ..., (4)
where the polarization P and the susceptibilities χ i are now taken in the frequency domain and the dots indicate again infinite sums involving higher order terms. The response P (t)

to E(t) = E cos(ωt) can thus be written

P (t)/ǫ 0 = Re (Eχ 1 (ω) + 3/4E 3 χ3(ω) + ...)e -iωt + Re 1/4E 3 χ 3 (ω)e -i3ωt + ... + .... (5)
To obtain Eq. ( 5), we have used the fact that because χ 1 and χ 3 are real in the time domain, their Fourier transform verify χ * 1 (ω) = χ 1 (-ω) and χ * 3 (ω 1 , ω 2 , ω 3 ) = χ 3 (-ω 1 , -ω 2 , -ω 3 ) (the star denotes the complex conjugate), and the invariance of χ 3 by permutation of its arguments. For simplicity, we write χ 3 (ω) = χ 3 (ω, ω, ω) and χ3(ω) = χ 3 (-ω, ω, ω). Eq. ( 5) can be written

P (t)/ǫ 0 = E(χ ′ 1 cos ωt + χ ′′ 1 sin ωt) + 3/4E 3 (χ ′ 3 cos ωt + χ ′′ 3 sin ωt) + ... + 1/4E 3 (χ ′ 3 cos 3ωt + χ ′′ 3 sin 3ωt) + ..., (6)
where the susceptibilities χ i are given as a function of their real and imaginary parts χ ′ i and χ ′′ i . For practical applications, the modulii and arguments |χ i | and δ i are rather used:

P (t)/ǫ 0 = E |χ 1 | cos(ωt -δ 1 ) + 3/4E 3 |χ3| cos(ωt -δ3)+ + ... + 1/4E 3 |χ 3 | cos(3ωt -δ 3 ) + ... (7)
We see in the rhs in Eqs 5-7 that the nonlinear susceptibility of interest, namely χ 3 appearing in Eq. 1 of the letter, is directly given by the measurement of the third harmonics of the polarisation.

II) THIRD HARMONICS FOR A LIQUID WITHOUT GLASSY CORRELATIONS.

We briefly summarize here what can be expected for the third harmonics in the case of a liquid without any glassy correlations. We use the work of Déjardin and Kalmykov [2] which studies the nonlinear dielectric relaxation of an assembly of rigid polar symmetric particles. The particles are assumed to be diluted enough in a non polar solvent, so that any interaction between them can be safely neglected. A strong electric field is applied to the system, which biases the (noninertial) rotational Brownian motion of the polar particles and yields a net dielectric response. This response is expressed by expanding the relaxation functions as a Fourier series in the time domain. Then the infinite hierarchy of recurrence equations for the Fourier components is obtained in terms of a matrix continued fraction.

We focus on the case where the field is purely ac, namely E(t) = E cos ωt. In the following equations, the subscript "D" refers to the calculation of Déjardin and Kalmykov. By using their Eq. ( 21), one gets :

χ 3,D (ω) χ 3,D (ω = 0) = -3 3 -17ω 2 τ 2 D + iωτ D (14 -6ω 2 τ 2 D ) (1 + ω 2 τ 2 D )(9 + 4ω 2 τ 2 D )(1 + 9ω 2 τ 2 D ) , ( 8
)
where τ D is the usual Debye time since one finds χ ′′ 1,D ∝ ωτ D 1+ω 2 τ 2 D , i.e. χ ′′ 1 is maximum for the frequency f D = 1/(2πτ D ). By using Eqs. ( 7) and ( 19) of Déjardin and Kalmykov, one can calculate all the prefactors and obtain χ 3,D (ω = 0):

χ 3,D (ω = 0) = ǫ 0 (∆χ 1 ) 2 k B T N 0 1 5 , ( 9
)
where N 0 is the volumic concentration of polar particles, and where, as in the letter, ∆χ

1 = χ ′ 1,D (0) -χ ′ 1,D (∞).
To get a benchmark for the behavior of χ 3,trivial for the 'no glassy correlations' case, we set f D = f α as well as N 0 = 1/a 3 with a 3 the molecular volume in glycerol. We thus obtain from the above equations: The last term in the rhs of the latter equation depends only on f /f D , i.e. f /f α . Thus, the latter equation and Eq. ( 1) of the letter are formally similar: The quantity X trivial = χ 3,trivial (ω)k B T ǫ 0 (∆χ 1 ) 2 a 3 plays, for the 'no glassy correlations' case, the same role as X = χ 3 (ω)k B T ǫ 0 (∆χ 1 ) 2 a 3 in the main letter. The behavior of X trivial is shown on Figs S1-S2. Three main features are noticeable:

χ 3,trivial (ω) = ǫ 0 (∆χ 1 ) 2 a 3 k B T 1 5 χ 3,D (ω) χ 3,D (ω = 0) . ( 10
)
(i) On Fig. S1, one sees that |X trivial | has no peak in frequency, at variance with what is observed on Figs ( 1),( 2),(4) of the main letter. This strongly support that, as claimed in the letter, the peak in frequency of |X| is directly linked to the effect of glassy correlations.

(ii) From Eqs. ( 8)- (10), one sees that the magnitude of |X trivial | does not depend on the temperature T . The T dependence of X trivial is only through τ D (T ), since we have set dominates over the singular part in the limit of low frequencies f ≪ f α . Of course the role of the trivial part of χ 3 is larger when the non trivial part is lower, i.e. when the temperature is high. This naturally accounts for the departure to the scaling of X as a function of f /f α when both the temperature is high and the frequency f is much lower than f α .

f D = f α .

III) MORE INFORMATION ABOUT OUR EXPERIMENTAL SETUP

The principle of the experiment was that of a bridge including two capacitors in which the dielectric layer was the supercooled liquid under study (here, glycerol) as described in section V of ref. [1]. These two glycerol samples with different inter-electrode distances were placed in a closed cell filled with high purity glycerol (purity > 99.6 %) in a dessicated atmosphere to prevent water absorption. The temperature of the cell was regulated by a LakeShore R temperature controller with a 50 Ohms heater fixed on the cell. The cell and its heater were placed in a cryostat in which the cold stage remained at ∼ 20 K and was connected to the cell through a thermal impedance. This device allowed a temperature regulation with a precision of ∼ 50 mK, and a temperature quench from 300 to 200 K in ∼ 1.5 hour.

The electrodes of the two capacitors are 6 mm thick rods of high purity copper on which 200 nm of gold was evaporated to prevent the formation of any oxydized layer. A highly specialised surface treatment reduced any surface defects: The final roughness was about 10 nm, while the planeity, i.e. the difference between the highest point of the electrodes and the mean plane, was better than 1 µm.

Most of the results presented in the letter were obtained with Mylar R spacers depicted in the letter, giving inter-electrode distances of 19 µm and 41 µm for the two samples. For completeness, we have added a few results obtained very recently by using spacers made of resin patterned by photolithography (with a thickness L = 12 µm for the thin sample and 25 µm for the thick sample).

The bridge was fed by a low distortion voltage source (Stanford Research Systems R DS360) which gave a rms excitation voltage ≤ 14V at a frequency f = ω/2π ranging from 10 mHz to 200 kHz. The nonlinear susceptibility χ 3 was determined after having carefully checked the cubic dependence of the measured signal at 3ω with respect to the excitation at 1ω. Last, for a significant fraction of the frequencies and temperatures investigated here, we have checked the quantitative consistency between the 3ω signal measured in our "two sample bridge" and the one found with the "twin T notch filter" depicted in Ref. [1], which allows to measure χ 3 by a different method.

IV) HEATING EFFECTS CALCULATIONS

Let us consider a sample made of a supercooled liquid excited by an oscillating field E cos(ωt). For small enough E values, the resulting polarisation P lin is linear and reads :

P lin (t) ǫ 0 E = χ ′ 1 cos(ωt) + χ ′′ 1 sin(ωt). ( 11
)
When E is increased, a part of the nonlinear response comes from the dissipated electrical power, the volumic density of which is p(t) = 1 2 ǫ 0 χ ′′ 1 ωE 2 (1 + cos(2ωt -φ)), with φ = -π + 2 arctan(χ ′′ 1 /χ ′ 1 ) (see e.g. Appendix of Ref. [3]). The resulting heat propagates towards a "thermostat" which in our case can be considered to be the copper electrodes (thickness 6 mm, diameter 20 mm). The resulting average sample temperature increase δT (t) is

δT (t) = δT 0 + δT 2 cos(2ωt -φ 2 ), (12)
where the mean dc temperature increase δT 0 is larger or equal than the ac one δT 2 , thus at any time δT (t) ≥ 0. φ 2 is a phase shift related to heat transport. As our measurements give the nonlinear dielectric response averaged over the sample volume, δT (t) in equation ( 12)

is the temperature increase averaged over the same volume. Using equations ( 11)-( 12), we thus obtain for the nonlinear part of the susceptibility due to heating effects:

P (t) -P lin (t) ǫ 0 E = ∂χ ′ 1 ∂T δT (t) cos(ωt) + ∂χ ′′ 1 ∂T δT (t) sin(ωt). ( 13
)
This is an upper limit on the contribution of heating to the nonlinear response, since we have assumed that δT (t) induces instantaneously a modification of the susceptibility. As already advocated in Ref. [3], this is questionable, specially in what concerns the contribution of δT 2 (t) which should be averaged out because of the finite relaxation time τ of the system (see after eq. 17). By using equations (12)(13), one gets :

P (t) -P lin (t) ǫ 0 E = δT 0 + 1 2 δT 2 cos(φ 2 ) ∂χ ′ 1 ∂T + 1 2 δT 2 sin(φ 2 ) ∂χ ′′ 1 ∂T cos(ωt) + δT 0 - 1 2 δT 2 cos(φ 2 ) ∂χ ′′ 1 ∂T + 1 2 δT 2 sin(φ 2 ) ∂χ ′ 1 ∂T sin(ωt) + 1 2 δT 2 cos(φ 2 ) ∂χ ′ 1 ∂T - 1 2 δT 2 sin(φ 2 ) ∂χ ′′ 1 ∂T cos(3ωt) + 1 2 δT 2 sin(φ 2 ) ∂χ ′ 1 ∂T + 1 2 δT 2 cos(φ 2 ) ∂χ ′′ 1 ∂T sin(3ωt). ( 14
)
The four terms in the right hand side of equation ( 14) give the nonlinear response of the system due to heating that we define in general by

P (t) -P lin (t) ǫ 0 E = (δχ ′ 1 ) h cos(ωt) + (δχ ′′ 1 ) h sin(ωt) + E 2 4 χ ′ 3,h cos(3ωt) + E 2 4 χ ′′ 3,h sin(3ωt). ( 15
)
In the right hand side of equation ( 15) each of the four terms are proportional to E 2 . χ ′ 3,h and χ ′′ 3,h do not depend on E, while (δχ ′ 1 ) h and (δχ ′′ 1 ) h are proportional to E 2 . Our notations have been chosen in order to remain as close as possible to the notations of refs [3][4][5].

We now calculate the expression of δT (t) that one has to introduce in equation ( 14) in order to obtain the nonlinear response in equation (15). We just consider the simple heating effect, when the dynamical heterogeneities are not considered: The supercooled liquid is thus just characterized by its thermal conductivity κ th and its total specific heat c. As in ref. [6], we consider that c is frequency dependent due to the fact that the slow degrees of freedom cannot contribute to c for frequencies much larger than f α = 1/τ . Let us define (x, y) as the plane of our copper electrodes, with z = 0 for the lower electrode and z = L for the upper one. Due to their very high thermal conductivity and to their large thickness (6 mm), the two electrodes can be considered, to a very good approximation, as a thermostat.

The temperature increase δθ(x, y, z, t) of the supercooled liquid at point (x, y, z) and time t thus vanishes for z = 0 and z = L. As the diameter D = 2 cm of the electrodes is typically one thousand times larger than L, we may consider that δθ does not depend on (x, y). We obtain δθ(z, t) by solving the heat propagation equation:

c ∂δθ(z, t) ∂t = κ th ∂ 2 δθ(z, t) ∂z 2 + p(t). (16)
By averaging spatially the solution of equation ( 16), we obtain the δT (t) to be used in equation ( 14):

δT 0 = ǫ 0 χ ′′ 1 ωE 2 L 2 24κ th δT 2 (t) = δT 0 cos(2ωt -φ 2 ) 1 + (2ωτ th ) 2 , (17)
where τ th = cL 2 /(κ th π 2 ) and φ 2 = φ + arctan(2ωτ th ). By using these expressions in equation ( 14) and identifying it with equation ( 15), we obtain the upper limit to the heating contribution to χ 3 (dotted line in the Fig. 4 of the Letter).

We now move to the problem evoked above, namely the fact that the finite relaxation time τ of the dipoles which contribute to the dielectric susceptibility should average out the modification of this susceptibility due to the oscillation δT 2 (t), specially in the case ωτ ≥ 1. As a consequence, the heating contribution to χ 3 should be reduced by a factor R(ωτ ) ≤ 1. For a precise calculation of R(ωτ ), one should replace Eq. ( 13) by a microscopic equation accounting for the dynamics of the dipoles in the case of a thermal bath where the temperature has an oscillating component, which is of great complexity. For a first estimate, we make two very simplifying assumptions:

(i) We assume that the dipoles have a Debye dynamics with a given characteristic time τ (T ). This is a simplifying assumption in the sense that when T is close to T g , it is well known that χ 1 (ω) is "stretched" with respect to a simple Debye law. In fact the Debye dynamics holds only at much higher temperatures, where the molecular motions are independent of each other, which allows to describe the non inertial rotational Brownian motion by the Smoluchowski equation for the probability distribution function of the orientations of the dipoles in configuration space [2,7]. After an ensemble averaging of this equation, one gets the well known Debye equation for the dynamics of the average polarisation [2] :

τ ∂P ∂t + P = ǫ 0 ∆χ 1 E cos(ωt) (18)
(ii) we assume that the main effect of δT 2 (t) is to modulate in time the value of τ while leaving unchanged the (Debye) dynamics. This can be justified by the fact that the temperature oscillation modulates the viscosity η, thus the relaxation time τ which is proportional to η. Considering the heatings of eq. ( 17), τ (t) is now given by τ (t) = τ lin + ∂τ lin ∂T δT 0 + ∂τ lin ∂T δT 2 (t) ,

where τ lin is the value of τ at zero field. In the following, δτ 2 will denote the amplitude of the 2ω modulation of τ due to δT 2 (t) and corresponding to the last term of Eq. (19).

Of course, using Eq. ( 19) for τ (t) assumes that δT 2 (t) instantaneously fully affects τ . A thorough modelization of this problem could lead to a more involved expression where δτ 2 should be weaker than in the above expression. As we shall find that the third harmonics is proportionnal to δτ 2 /τ lin , see below Eq. ( 21), we are lead to the conclusion that our new estimate should, again, be slightly overstimated.

We now insert τ (t) in Eq. ( 18) and set : P (t) = P lin cos(ωt -Ψ lin ) + δP 1 cos(ωt -Ψ 1 ) + P 3 cos(3ωt -Ψ 3 ) + ...

where P lin , Ψ lin , δP 1 , Ψ 1 , P 3 , Ψ 3 are to be determined. As we are only interested in the onset of nonlinear effects, P lin ∝ E is much larger than δP 1 ∝ E 3 and than P 3 ∝ E 3 . This allows 20 to neglect higher order harmonics (denoted by the dots in Eq. ( 20)) and to resolve Eq. ( 18) by identification of the terms which have the same frequency and the same power of E. This yields:

P 3 = ǫ 0 ∆χ 1 E 2 δτ 2 τ lin ωτ lin 1 + (ωτ lin ) 2 1 + (3ωτ lin ) 2 (21)
where δτ 2 denotes the amplitude of the 2ω modulation of τ arising from δT 2 (t) in the last term of Eq. ( 19): thus δτ 2 ∝ E 2 , see Eq. ( 17), which yields the expected P 3 ∝ E 3 .

We now have to compare this result to that obtained if we start from Eq. ( 13) and use a Debye linear susceptibility. A straightforward calculation shows that in that case the modulus of the third harmonics of the polarisation is given by the expression of Eq. ( 21)

divided by a function R(ωτ ) given by : R(ωτ ) = 1 + (ωτ lin ) 2

1 + (3ωτ lin ) 2

As expected R(ω = 0) = 1 and R(ωτ ≫ 1) is smaller than 1. We note that R(ωτ → ∞) = 1/3, which comes from the fact that τ (t) enters in Eq. ( 18) as a factor of ∂P/∂t: this gives a weight 3ωτ contrarily to the case where one starts from Eq. ( 13) where this weight is simply ωτ . This reduction of the effect of δT 2 (t) on the polarisation can be seen as a first estimate of the fact that the dipoles average out the temperature oscillations. Of course, one could build a much more thorough model of this effect, but the reduction given by Eq. ( 22) is enough to ensure that the power law regime reported in the main letter is not affected by the heating contribution calculated here. Finaly, the lengths of the arrows of the Fig. 4 of the main letter are those predicted by Eq. ( 22).

3 FIG. 1 :

 31 FIG. 1: (Color online) For each of the five temperatures labelling the curves, X(ω, T ) = |χ 3 (ω, T )|× k B T /[ǫ 0 (∆χ 1 ) 2 a 3 ] is given as a function of frequency f = ω/2π. The arrows indicate the five relaxation frequencies f α (T ), for which χ ′′ 1 (ω) is maximum. The dashed lines are guides to the eyes.

FIG. 2 :

 2 FIG.2:(Color online) The quantity |χ 3 (ω, T )| at five temperatures (same data and symbols as in Fig.1), normalized to its maximum value at each T , plotted as a function of the ratio f /f α (T ). The straight line is a power-law with an exponent -0.65. Inset: corresponding phase of χ 3 (ω, T ).

5 FIG. 3 :

 53 FIG. 3: (Color online)The measured quantity max ω [X(ω, T )] = max ω (|χ 3 (ω, T )|) × k B T /[ǫ 0 (∆χ 1 ) 2 a 3 ] plotted vs. temperature. The continuous line is the number of correlated molecules estimated from T χ T (see[8]) with an arbitrary normalization chosen so that it coincides with the experimental points at 202 K.

FIG. 4 :

 4 FIG. 4: (Color online)The measured quantity X(ω, T ) at 210.2 K (filled squares with dashed line) is compared to the same quantity calculated by using Eq. (2) with κ ≈ 7.6×10 -17 K(m/V) 2 ). Dotted line: Calculated heating contribution (upper limit, the arrows indicate the possible reduction [14]). Inset: The phase of χ 3 (ω, T ) at 210.2 K (same symbols as for the main figure, dotted line not shown).

Fig. S 1 :

 1 Fig. S 1: The quantity |X trivial | = | χ 3,trivial (ω)k B T ǫ 0 (∆χ 1 ) 2 a 3 | is plotted as a function of the ratio f /f D . Inset: same graph in log-log plot showing that X trivial decreases as f -3 when f ≫ f D .

Fig. S 2 :

 2 Fig. S 2: The phase of X trivial is plotted as a function of the ratio f /f D .

Acknowledgements

We thank R. Tourbot for realizing the experimental cell, P. Pari for cryogenics, S. Nakamae for her careful reading of the manuscript. We acknowledge interesting discussions with C. Alba-Simionesco, A. Lefèvre, R. Richert, M. Tarzia, and support by ANR grant DynHet. * Electronic address: francois.ladieu@cea.fr, denis.lhote@cea.fr


Valentin Resseguier

Etienne Mémin

UNCECOMP 2015

∂T ∂t + ∇ • Tū = ∇ • -u T

CLASSICAL LARGE EDDY APPROACH

Numericians approach :

∂T ∂t + ∇ • Tū = ∇ • -u T = σ 2 2 T

CLASSICAL LARGE EDDY APPROACH

Numericians approach :

∂T ∂t + ∇ • Tū = ∇ • -u T = σ 2 2 T -> Eddy diffusivity
CLASSICAL LARGE EDDY APPROACH Numericians approach :

∂T ∂t + ∇ • Tū = ∇ • -u T = σ 2 2 T -> Eddy diffusivity -6 -4 -2 0 2 -5 0 5 y x d X t = w d t -6 -4 -2 0 2 -5 0 5 y x d X t = w d t + σ d
u(x, t) ≈ n i=0 b i (t)φ i (x) Chronos : b i ∂u ∂t = I + L(u) + C(u, u) (a PDE) becomes : ∀i, db i dt = Ω φ i • I + n p=0 Ω φ i • L(φ p ) b p + n p,q=0 Ω φ i • C(φ p , φ q ) b p b q (ODEs)
ADDITIONAL REDUCTION : TIME SUB-SAMPLING Large time scale approach we keep only necessary information for the resolved modes :

db i dt = i i + l •i b + b T c ••i b
Nyquist-Shanon criterion for sub-sampling :

f s 2 = 2 max i f max (b i ) meets Shanon. 0 0.1 0.2 0.3 0.4 0 2 4 6 8
x 10

4 n i = 1 | b i | 2 λ i Normalized frequency n = 10 m o d e s

OUR APPROACH

Our approach :

u = w + σdB dt w = n i=0 b i φ i (resolved modes) v res = u -w = N i=n+1 b i φ i a
∂w i ∂t + (w * • ∇)w i = ∇ • a 2 ∇w i - 1 ρ ∂p ∂x i + ν w - σdB t dt • ∇ w i multiplicative noise + ν σ i dB t dt additive noise QUESTIONS Thank you for your attention ESTIMATION OF a DEPENDING OF TIME ∀x ∈ R d , d Xx t = σ(x, t)dB t z i (x) = T 0 b i (t) λ i T a(x, t)dt = T 0 b i (t) λ i T d < Xx , Xx t > t (since a = σσ t ) = P -lim ∆t→0 T t k =0 b i (t k ) λ i T Xx t k+1 -Xx t k Xx t k+1 -Xx t k t v res (x, t)∆t a realization of d Xx t = σ(x, t)dB t
Estimator with good statistical properties (Genon-Catalot 1992) (when the projection subset is well chosen)

  FIGURE : Re=3900 (3D flow), a constant and in the span of Chronos b i .

FIGURE

  FIGURE : n=10, Re=3900 (3D flow), a constant in time.

DATA

Wake behind a cylinder at Reynolds 3900 : 3D flow y

ESTIMATION OF a DEPENDING OF TIME

We look for a decomposition of a

Estimation based on Stochastic Calculus (Genon-Catalot 1992) :