Study of the heating effect contribution to the nonlinear dielectric response of a supercooled liquid
aa r X i v : . [ c ond - m a t . s o f t ] O c t Study of the heating effect contribution to the nonlineardielectric response of a supercooled liquid
C. Brun , C. Crauste-Thibierge , F. Ladieu ⋆ , and D. L’Hˆote ⋆ ∗ SPEC (CNRS URA 2464), DSM/IRAMIS CEA Saclay,Bat.772, F-91191 Gif-sur-Yvette France and LLB (CNRS UMR 12), DSM/IRAMIS CEA Saclay,Bat. 563, F-91191 Gif-sur-Yvette France (Dated: November 24, 2018)
Abstract
We present a detailed study of the heating effects in dielectric measurements carried out ona liquid. Such effects come from the dissipation of the electric power in the liquid and give acontribution to the nonlinear third harmonics susceptibility χ which depends on the frequencyand temperature. This study is used to evaluate a possible ‘spurious’ contribution to the recentlymeasured nonlinear susceptibility of an archetypical glassforming liquid (Glycerol) . Those mea-surements have been shown to give a direct evaluation of the number of dynamically correlatedmolecules temperature dependence close to the glass transition temperature T g ≈ et al. , Phys. Rev. Lett 104,165703(2010)). We show that the heating contribution istotally negligible (i) below 204K at any frequency; (ii) for any temperature at the frequency wherethe third harmonics response χ is maximum. Besides, this heating contribution does not scale asa function of f /f α , with f α ( T ) the relaxation frequency of the liquid. In the high frequency range,when f /f α ≥
1, we find that the heating contribution is damped because the dipoles cannot followinstantaneously the temperature modulation due to the heating phenomenon. An estimate of themagnitude of this damping is given. ∗ Electronic address: [email protected], [email protected] . INTRODUCTION Upon fast enough cooling, most liquids avoid cristallization and enter a supercooled state.The latter is characterized by the “viscous slowing down” phenomenon [1], an extremely fastincrease of the viscosity when the temperature T is decreased. Below the glass transitiontemperature T g the viscosity is so high that the system is, in practice, a solid. The glasstransition is a longstanding issue of condensed matter physics, since no structural signaturehas ever been detected around T g , e.g. the static neutron spectra do not change at T g ,contrary to what happens for the cristallization [1]. Significant progresses in understandingthe physics of the glass transition were made in the last fifteen years, when the heterogeneousnature of the dynamics of supercooled liquids was established, through various experimental[2–5] as well as numerical works [6]. The important concept of ‘dynamical heterogeneities’has emerged [4, 5], according to which the relaxation comes from the collective motion ofgroups of N corr molecules. All these groups evolve in time, some of them being faster orslower than the average dynamics. The viscous slowing down would come from the factthat this number N corr increases as T decreases towards T g . The temperature dependenceof N corr thus became a crucial issue in the field, triggering new theoretical ideas [7–10].One of them came from an analogy [10] with the well known spin glass physics in whichthe nonlinear suceptibility diverges at the critical temperature T c , reflecting the long rangeamorphous order which sets in at T c . It was proposed in Ref. [10], that a similar effect occursin supercooled liquids, with the key difference that the peak of the nonlinear response χ ( ω )should appear at finite frequencies ω of the order of 1 /τ ( T ) where τ ( T ) is the relaxation timeof the supercooled liquid. For the first time, a (nonlinear) susceptibility was directly relatedto the quantity N corr of interest, allowing to scrutinize its temperature dependence. We haveperformed recently the corresponding experiment for the nonlinear dielectric susceptibilityof glycerol [11, 12] and shown that the N corr ( T ) dependence was indeed an increase as T decreases.The nonlinear susceptibility χ measured in Refs. [11, 12] corresponds to the detectionof the third harmonics of the polarisation P , at three times the frequency of the appliedelectric field E cos( ωt ). For E ∼ P /P is typically around 10 − ,with P the polarisation at the frequency ω (mainly dominated by the linear response of thesystem). Due to the small value of P /P , a thorough analysis is required to see whether2ome ‘spurious’ effects can affect the P measurements. Expanding on the arguments givenin Ref. [13], we study in this paper the contribution of the ac heating of the liquid to our P measurements. This heating comes from the fact that the strong applied electric fieldleads to some dissipated electrical power p with a d.c. component as well as a component p ( t ) oscillating at 2 ω . Before being absorbed by the thermal reservoir which sets thebase temperature T of the experiment (hereafter, the metallic electrodes), the heat hasto travel across the sample of thickness e . Thus the dissipated power leads to a smalltemperature increase, containing a 2 ω component δT ∼ p . From the first order estimate δP ( t ) ≃ ( ∂P ( t ) /∂T ) δT ( t ) which involves a product of two terms oscillating at ω and 2 ω ,one sees that a spurious contribution to the measured P comes from δT ( t ), i.e. from a partof the heating phenomenon.We shall see below that, close to T g , this heating contribution to P vanishes in the limitof extremely thin samples ( e → etal. [15–17]. This ‘heterogeneous heating model’ gives a phenomenological description ofthe intrinsic nonlinear effects in supercooled liquids, and was shown to account for nonlinearexperimental data at ω in Ref. [16]. In this model, each dynamical heterogeneity has its ownfictitious temperature ‘on top’ of the temperature of the phonon bath. At the present time,it is the only model which allows to calculate the nonlinear susceptibility near T g and weintend to give a thorough study of its predictions at 3 ω in a future paper. To emphasize thedifference between this heterogeneous heating model and the study of the present paper, wecall the heating contribution to the nonlinear susceptibility investigated here ’homogeneousheating contribution’.The paper is organised as follows: in section II we derive the relations which allow tocalculate the heating contribution to P . In section III we give the corresponding results,and clearly show the temperature and frequency ranges where the heating contribution isnegligible in our experiments. We eventually show that in the high frequency range, i.e.when ωτ ( T ) ≫
1, the heating contribution is damped by the finite relaxation time τ ( T ) ofthe dipoles, and we estimate the magnitude of this damping.3 I. HEATING EFFECTS CALCULATIONS
In section II A we establish the link between the temperature increase and the homoge-neous heating contribution to P . The detailed calculation of the temperature increase ispostponed to section II B. A. From δT to the homogeneous heating contribution to χ . This subsection is divided into three parts: we first establish an upper bound of theheating contribution to P . We then move to an estimate of the damping of this contributionat high frequencies, due to the finite relaxation time of the dipoles. Last, we show what isthe most natural quantity to plot to compare the heating contribution to P to the resultsof Ref. [12].
1. The heating contribution without damping
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 linear polarisation P lin reads : P lin ( t ) ǫ E = χ ′ cos( ωt ) + χ ′′ sin( ωt ) , (1)where ǫ is the dielectric permitivity of vacuum. In general, χ ′ and χ ′′ strongly depend onfrequency f = ω/ π . We define the frequency f α which characterizes the relaxation at agiven temperature T , as the frequency where χ ′′ is maximum. This relaxation frequency f α is strongly T dependent, and is of the order of 1 /τ where τ ( T ) is the average relaxation timeof the dipoles of the supercooled liquid.As evoked above, the volumic density of dissipated electrical power p ( t ) contains a d.c.term and a term oscillating at 2 ω (see below Eq. (13)). The resulting heat propagatestowards the ‘thermostat’. The resulting average sample temperature increase δT ( t ) can bewritten: δT ( t ) = δT + δT cos(2 ωt − φ ) , (2)4here the mean dc temperature increase δT is larger than or equal to the ac one δT , thusat any time δT ( t ) ≥ φ is a phase shift related to heat transport that will be given insection II B. As our measurements [12] give the nonlinear dielectric response averaged overthe sample volume, δT ( t ) in Eq. (2) is the temperature increase averaged over the samevolume. Using Eqs. (1),(2), we thus obtain for the nonlinear part of the polarisation due toheating effects: P ( t ) − P lin ( t ) ǫ E = (cid:18) ∂χ ′ ∂T δT ( t ) (cid:19) cos( ωt ) + (cid:18) ∂χ ′′ ∂T δT ( t ) (cid:19) sin( ωt ) . (3)This is an upper limit of the heating contribution to the nonlinear response, since we haveassumed that δT ( t ) induces instantaneously a modification of the susceptibility. As alreadyadvocated in Ref. [15], this is questionable, specially in what concerns the contribution of δT ( t ) which should be damped because of the finite relaxation time τ of the dipoles. Thispoint is adressed in the next section II A 2. From Eqs. (2) and (3), one gets : P ( t ) − P lin ( t ) ǫ E = (cid:20)(cid:18) δT + 12 δT cos( φ ) (cid:19) ∂χ ′ ∂T + 12 δT sin( φ ) ∂χ ′′ ∂T (cid:21) cos( ωt )+ (cid:20)(cid:18) δT − δT cos( φ ) (cid:19) ∂χ ′′ ∂T + 12 δT sin( φ ) ∂χ ′ ∂T (cid:21) sin( ωt )+ (cid:20) δT cos( φ ) ∂χ ′ ∂T − δT sin( φ ) ∂χ ′′ ∂T (cid:21) cos(3 ωt )+ (cid:20) δT sin( φ ) ∂χ ′ ∂T + 12 δT cos( φ ) ∂χ ′′ ∂T (cid:21) sin(3 ωt ) . (4)The four terms in the right hand side of Eq. (4) give the nonlinear response of the systemdue to homogeneous heating: it contains two terms oscillating at ω that we shall disregardsince they contribute to the nonlinear part of P . We shall only keep the two terms of Eq. (4)oscillating at 3 ω to obtain the heating contribution P ,h to the third harmonics P . As weare interested in the heating contribution χ ,h to the nonlinear susceptibility we define, asin Eqs. (4)-(5) of Ref. [11] : P ,h ( t ) ǫ E = E χ ′ ,h cos(3 ωt ) + E χ ′′ ,h sin(3 ωt ) . (5)As a result, χ ′ ,h and χ ′′ ,h should not depend on E : their expression is given by identifi-5ation with the two last terms of Eq. (4) and defines, throughout this work, what we callthe overestimated heating nonlinear susceptibility because it is obtained by neglecting thedamping evoked above.
2. Damping of heating contribution: an estimate
We now move to the problem evoked above, namely the fact that the finite relaxationtime τ of the dipoles which contribute to the dielectric susceptibility should damp the mod-ification of this susceptibility due to the oscillating δT ( t ), specially in the case ωτ ≥
1. Asa consequence, the heating contribution to χ ,h should be multiplied by a complex factor R ( ωτ ), with a modulus | R ( ωτ ) | which is expected to be lower than 1. For a precise calcula-tion of R ( ωτ ), one should replace Eq. (3) by an equation accounting for the dynamics of thedipoles in the case of a thermal bath where the temperature has an oscillating component,which is of great complexity. For an 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 knownthat χ ( ω ) is “stretched” with respect to a simple Debye law. In fact the Debye dynamicsholds only at much higher temperatures, where the molecular motions are independent ofeach other, which allows to describe the non inertial rotational Brownian motion by theSmoluchowski equation for the probability distribution function of the orientations of thedipoles in configuration space [14, 18]. After an ensemble averaging of this equation, onegets the well known Debye equation for the dynamics of the average polarisation P [14] : τ ∂P∂t + P = ǫ ∆ χ E cos( ωt ) , (6)where ∆ χ = χ ( ω = 0) − χ ( ω → ∞ ) is the part of the static linear susceptibilitycorresponding to the slow relaxation process we consider.(ii) We assume that the main effect of the temperature variation δT ( t ) is to modulate intime the value of τ while leaving unchanged the (Debye) dynamics. This can be justified bythe fact that the temperature oscillation modulates the viscosity η , thus also the relaxationtime τ which is proportional to η [19]. Considering the temperature variations of Eq. (2), τ ( t ) is now given by 6 ( t ) = τ lin + (cid:18) ∂τ lin ∂T (cid:19) δT + (cid:18) ∂τ lin ∂T (cid:19) δT ( t ) , (7)where τ lin is the value of τ at zero field. In the following, δτ will denote the amplitudeof the 2 ω modulation of τ due to δT ( t ) and corresponding to the last term of Eq. (7).Of course, using Eq. (7) for τ ( t ) assumes that δT ( t ) instantaneously fully affects τ . Athorough modelization of this problem could lead to a more involved expression where δτ would be weaker than in the above expression. As we shall find that the third harmonicsis proportionnal to δτ /τ lin , see below Eq. (9), we are led to the conclusion that our newestimate should, again, be slightly overstimated.We now insert τ ( t ) in Eq. (6) and set : P ( t ) = P lin cos( ωt − Ψ lin ) + δP cos( ωt − Ψ ) + P cos(3 ωt − Ψ ) + ... (8)where P lin , Ψ lin , δP , Ψ , P , Ψ are to be determined. As we are only interested in the onsetof nonlinear effects, P lin ∝ E is much larger than δP ∝ E and than P ∝ E . This allowsto neglect higher order harmonics (denoted by the dots in Eq. (8)) and to resolve Eq. (6) byidentification of the terms which have the same frequency and the same power of E . Thisyields: P = ǫ ∆ χ E δτ τ lin ωτ lin p ωτ lin ) p ωτ lin ) Ψ = φ + arctan ( ωτ lin ) + arctan (3 ωτ lin ) + π , (9)where we remind that δτ is the amplitude of the 2 ω modulation of τ due to δT ( t ) in thelast term of Eq. (7): thus δτ ∝ E , see Eq. (14) below, which yields the expected P ∝ E .We now have to compare with the result obtained if we start from Eq. (3) and use aDebye linear susceptibility. A straightforward calculation shows that in that case the thirdharmonics of the polarisation is given by the solution of Eqs. (6-8) divided by the function R ( ωτ ) introduced above. The expression for R ( ωτ ) is written using the complex notation : R ( ωτ ) = p ωτ lin ) p ωτ lin ) × exp (cid:20) i × arctan (cid:18) − ωτ lin ωτ lin ) (cid:19)(cid:21) . (10)7s expected R ( ωτ ≪
1) = 1 (no damping) and the damping arises at high frequenciessince we draw from Eq. (10) | R ( ωτ ≫ | <
1. More precisely | R ( ωτ → ∞ ) | = 1 /
3, whichcomes from the fact that τ ( t ) enters in Eq. (6) as a factor of ∂P/∂t : this gives, in Eq.(6), a weight 3 ωτ to P , contrary to the case where one starts from Eq. (3) where thisweight is simply ωτ . Let us note that when the similar analysis is made for δP , by usingthe second term ( ∝ δT ) of the right hand side of Eq. (7), no reduction is found at anyfrequency: The solution found for δP ( t ) starting with Eq. (3), is exactly the same, inmodulus and phase, as the one found by using Eqs. (6-7). This shows that the reductionof the effect of δT ( t ) on the polarisation comes from the fact that, in Eq. (6), τ and P oscillate together in time. Thus, Eq. (10) can be seen as a first estimate of the fact that thedipoles damp the temperature oscillations, this damping being strong at high frequencies, asphysically expected. Of course, one could build a much more thorough model of this effect,but the reduction given by Eq. (10) will be shown to be quite realistic with respect to ourexperimental data (see below section III D). In practice, to compute the damped heatingcontribution to χ , we first compute the overestimated contribution defined in section II A 1and then multiply by the complex factor R ( ωτ ) defined above in Eq. (10).
3. How to single out the anomalous part of the nonlinear response
Before moving to the calculation of the temperature increase, let us remind the re-lation that Bouchaud and Biroli predict [10], on quite general grounds, between χ and N corr ( T ) -where N corr ( T ) denotes the T -dependent average number of dynamically corre-lated molecules-. In Ref. [10], one finds the following scaling form for χ : χ ( ω, T ) ≈ ǫ (∆ χ ) a k B T N corr ( T ) H ( ωτ ) , (11)where a the volume occupied by one molecule, and H a certain complex scaling functionthat reaches its maximum at ωτ ∼ both for small and large arguments.This ’humped’ shape of |H| is due to the glassy correlations: In the ‘no correlation case’[13, 14], N corr ( T ) H ( ωτ ) in Eq. (11) should be replaced by a function which reaches itsmaximum value at ω = 0. Thus χ ( ω, T ) can always be considered as the product of a gen-eral prefactor ǫ (∆ χ ) a /k B T times a dimensionless term which summarizes the physicsof the system. This is why a natural way to express the various contributions χ ,i ( i indi-8ates the kind of contribution) to χ is to divide them by this prefactor. We thus definea normalized nonlinear susceptibility X ,i = k B T / ( ǫ (∆ χ ) a ) χ ,i . We shall consider thenormalized heating contribution X ,h by dividing the heating contributions χ ,h by the pref-actor. Clearly, when no heating contribution or any other spurious contribution is present,we expect X ( ω, T ) = N corr ( T ) H ( ωτ ). B. Calculation of the temperature increase
In this section, we now calculate the expression of δT ( t ) that one has to introduce inEq. (4) in order to obtain the nonlinear response in Eq. (5).The supercooled liquid is characterized by its thermal conductivity κ th and its specificheat c . As in Ref. [20], we consider that c is frequency dependent due to the fact that theslow degrees of freedom cannot contribute to c for frequencies much larger than f α . Forsimplicity we neglect the small imaginary part of c [20, 22], and we consider also that κ th depends neither on the frequency nor on the temperature T [20, 22]. Let us define ( x, y ) asthe plane of our copper electrodes [12, 13], with z = 0 for the lower electrode and z = e forthe upper one. Due to their very high thermal conductivity and to their large thickness (6mm), the two electrodes can be considered, to a very good approximation, as a thermostat[21]. The temperature increase δθ ( x, y, z, t ) of the supercooled liquid at point ( x, y, z ) andtime t thus vanishes for z = 0 and z = e . As the diameter D = 2 cm of the electrodes istypically one thousand times larger than e ∼ − µ m (see III), 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 ∂ δθ ( z, t ) ∂z + p ( t ) , (12)where the dissipated power is given by : p ( t ) = 12 ǫ χ ′′ ωE (1 + cos(2 ωt − φ )) with φ = − π + 2 arctan (cid:18) χ ′′ χ ′ − χ ′ ( ω → ∞ ) (cid:19) . (13)This expression of p ( t ), where the fast, non relevant, degrees of freedom contributingto χ ′ ( ω → ∞ ) are separated from the slow degrees of freedom corresponding to glassydynamics, deserves some comments. While the prefactor of the right hand side of Eq. (13)is ‘textbook’ knowledge, the expression of φ is far less obvious (see the Appendix of [15]).9his phase comes from the fact that the dissipation arises due to the friction of the dipoleswith the surrounding molecules, and that this friction force F is proportional to ∂P/∂t .The simplest example is the case of Debye dynamics where F is proportionnal to the firstterm of Eq. (6), see [24]. Then, the power p corresponding to F must be given by F v where v ∼ ∂P/∂t is the ‘speed’ of the dipoles. It follows that p ∼ ( ∂P/∂t ) and by usingEq. (1), with the appropriate prefactor for p , the expression given in Eq. (13) comes out.Note that if we consider the limit χ ′′ ≫ χ ′ − χ ′ ( ω → ∞ ), we get φ →
0. This is the caseof a metal where the response is not due to dipoles, but to electrons motion, and for whichit is well known that there is no dephasing between p and E . On the other hand, in thelimit χ ′′ ≪ χ ′ − χ ′ ( ω → ∞ ), which happens in liquids when f ≪ f α , one finds that the 2 ω component of p and of E are in phase opposition. In the case of a Debye dynamics, Eq. (13)can be rigourously derived [15] because the expression of the friction force is explicit. Whenthe dynamics does not follow the Debye ’s law, which is the case of supercooled liquids closeto T g , the dynamical equation obeyed by each dynamical heterogeneity is not known [5].However, Eq. (13) should still remain valid since the assumption F ∼ ∂P/∂t amounts tothe lowest order development of the general idea that there is no friction if the dipoles donot move.Coming back to the heat propagation equation, due to the boundary conditions δθ ( z =0 , t ) = 0 = δθ ( z = e, t ), Eq. (12) is solved by decomposition in a series of spatial modeslabelled by their wave vector K = mπ/e with m an odd integer. The mode m = 1 dominatesthe temperature increase, and we keep only the m = 1 and m = 3 modes since it is enoughto get an accuracy of the order of 1%. By averaging spatially these two modes, we obtainthe δT ( t ) to be used in Eq. (4): δT = δT ⋆ (cid:18) (cid:19) with δT ⋆ = ǫ χ ′′ ωE e κ th δT ( t ) = δT ⋆ cos(2 ωt − φ ,a ) p ωτ th ) + cos(2 ωt − φ ,b )3 p ωτ th / ) ! , (14)where τ th = ce / ( κ th π ) and φ ,a = φ + arctan(2 ωτ th ) are involved in the dominant m = 1mode, while φ ,b = φ + arctan(2 ωτ th / ) appears in the much less important m = 3 spatialmode evoked above [23]. 10ollowing Eqs. (4)-(5), the two terms arising in δT ( t ) in Eq. (14) give a contribution to χ ,h . These contributions are added, yielding the overestimated value of χ ,h , as well as thedamped value of χ ,h (after multiplying by the function R ( ωτ ) given in Eq. (10)). Then, asexplained above, Eq. (11) is used to convert these values in terms of a contribution to X . III. RESULTSA. Behavior at low and high temperature
We shall first get some insight into the heating contribution X ,h to the total nonlinearnormalized susceptibility X by extracting from the previous equations its frequency andthickness dependences. From Eqs. (4), (5), (14), we keep the leading term to obtain: | X ,h | ∼ (cid:12)(cid:12)(cid:12)(cid:12) ∂χ ∂T (cid:12)(cid:12)(cid:12)(cid:12) χ ′′ ωe p ω/ω th ) , (15)where we have defined ω th = 2 πf th = 1 / (2 τ th ). In Eq. (15) two characteristic frequenciesappear: f α , which strongly depends on T , and the thermal frequency which is inverselyproportional to the thickness of the sample. For a given experiment, and thus a given e , wehave to distinguish two regimes: the low temperature regime where f α < f th and the hightemperature regime where f α > f th . Besides, for glycerol, which is the liquid of interesthere, the frequency dependence of χ ′′ is χ ′′ ∼ f /f α below f α and χ ′′ ∼ ( f /f α ) − . above f α .Last, | ∂χ /∂T | ∼ ( f /f α ) . below f α and | ∂χ /∂T | ∼ ( f /f α ) − . above f α . This allows todraw from Eq. (15) the frequency and thickness dependencies of | X ,h | . (i) In the low temperature regime, one gets : | X ,h | ∼ f . e when 0 ≤ f ≤ f α | X ,h | ∼ f − . e when f α ≤ f ≤ f th | X ,h | ∼ f − . e when f th ≤ f. (16)In Eq. (16), the maximum over frequency of | X ,h | arises when the exponent of f changesits sign: this maximum value is thus proportionnal to e , i.e. the heating contributionvanishes in the limit of very thin samples. This shows the non intrinsic (or spurious) nature11 -2 -1 -5 -4 -3 -2 -1 and T=217,8Kand T=204,5K | X | o v e r e s t i m a t ed hea t i ng f/f FIG. 1: (Color on line) Modulus of the overestimated heating contribution X ,h for T = 204 . T = 217 .
8K (circles), for glycerol and e = 19 µ m, as a function of the frequencynormalized to the relaxation frequency f α ( T ). The open symbols are for a ‘one sample experiment’:the very flat maximum of | X ,h | at 204K is typical of the ‘low temperature regime’ (see text), whilethe ‘sharp’ maximum close to f = f α at 217 .
8K characterizes the ‘high temperature regime’ (seetext). The filled symbols correspond to the ‘two samples bridge’ described in Refs. [11, 12], wherethe heating contribution of the two samples, e thin = 19 µ m and e thick = 41 µ m, cancel each otherat low enough frequency, i.e. for f ≪ f th ( e thick ) = 65Hz, see text. The net heating contributionis thus reduced in this setup: at 204 .
5K this reduction is so strong that the maximum is shiftedup slightly above 65Hz (the up arrow corresponds to 65Hz at 204 . .
8K the weakerreduction is present only below 65Hz (see the down arrow). of the heating contribution which is studied in this paper. We note that for e ≃ µ m,which corresponds to the experiment of Ref. [12], one typically gets f th = 300Hz. Since thestandard definition of the glass transition temperature T g corresponds to f α ( T g ) = 0 . e ≃ µ m, the heating contribution disappears close enough to T g . This would not be truefor samples where the thickness e is millimetric. (ii) In the high temperature regime, one gets :12 -2 -1 -1000100200300400500 and T=217,8Kand T=204,5K P ha s e o f X , o v e r e s t. hea t i ng [ D eg ] f/f FIG. 2: (Color on line) Phase of the overestimated heating contribution X ,h for T = 204 . T = 217 .
8K (circles), for glycerol and e = 19 µ m, as a function of the frequencynormalized to the relaxation frequency f α ( T ). The open symbols are for a ‘one sample experiment’.The filled symbols correspond to the ‘two samples bridge’ technique of Refs. [11, 12], where theheating contribution of the two samples, e thin = 19 µ m and e thick = 41 µ m, cancel each other atlow enough frequency, i.e. for f ≪ f th ( e thick ) = 65Hz, see text. The net heating contribution inthis setup thus depends on f /f th ( e thick ). At 204 . f ≃ f th ( e thick ) corresponds to f /f α ≃ . f /f α ≃ . f ≃ f th ( e thick ) which corresponds to f /f α ≃ .
13, see the down arrow. In the limit f ≪ f th ( e thick ), the shift of the phase in the two samples setup with respect to the one samplesetup is π/ | X ,h | ∼ f . e when 0 ≤ f ≤ f th | X ,h | ∼ f . e when f th ≤ f ≤ f α | X ,h | ∼ f − . e when f α ≤ f. (17)Eq. (17) shows that the maximum over frequency of | X ,h | arises for f = f α and isproportional to e , i.e. independent of the thickness of the sample, just as the intrinsic nonlinear response. With f th ≃ T ≥ | X ,h | could13lay a role for the highest temperatures reported in [12]. The open circles of Fig. 1 showthe ‘overestimated value’ of | X ,h | for T = 217 . K . The exponents predicted in Eq. (17),for f ≤ f th and f ≥ f α , are well observed. Besides, by taking into account the prefactornot explicitly written in Eq. (17), one can check that the maximum over frequency of | X ,h | should arise for f ≃ f α and give a value of order 0 .
15, as observed on Fig. 1. The comparisonto the experimental data will be presented in section (III C).
B. Heating contribution cancellation at low frequency with the bridge technique
Before moving to the detailed study of X ,h ( ω, T ), we investigate here the consequencesof the fact that our experiment reported in Ref. [12] was performed with a bridge technique.Two samples of different thicknesses, e thin ≃ µ m and e thick ≃ µ m were used in a bridgeto suppress the 3 ω voltage due to the voltage source imperfection and to the (small) nonlinearity of the voltage detector [11, 12]. Once the bridge is equilibrated, the voltages V applied, ω applied to each of the two samples are stricly equal. The field applied onto thethin sample is thus larger than that applied onto the thick sample by a factor e thick /e thin .The subtraction operated by the bridge does not cancel the sought nonlinear response, sincethe latter goes as E . We show now that this ‘two samples technique’ strongly reduces thevalues of | X ,h | as long as f ≪ f th . Indeed, from Eq. (14), one finds, in the limit f ≪ f th : | δT | = ǫ χ ′′ ωE e κ th ∼ ( V applied, ω ) , (18)i.e. the value of δT is the same for the thin and the thick sample. By using Eqs. (4),(5),this implies that the two heating contributions in the bridge setup perfectly cancel eachother in the limit f ≪ f th .This is illustrated in Fig. 1 where the filled symbols are the result corresponding to thebridge technique, while the open symbols are for a ‘one sample’ experiment. At T = 204K,one sees that | X ,h | is strongly reduced by the bridge technique as long as f ≤ f α whichamounts to f ≤ f th ( e thick ) ≃ | X ,h | comes from correctionswhich depend on f /f th to Eq. (18), and Fig. 2 reveals that the leading correction, in thelimit f ≪ f th , mainly produces a π/ X ,h . On the contrary, for f ≫ f th , one sees on Figs. 1-2 that the heating contributions are similar, both in phase andmagnitude, for a bridge setup and a ‘one sample’ experiment.14he same happens at T = 218K where f ≤ f th ( e thick ) = 65Hz corresponds again to thefrequency range where the bridge technique reduces (and phase shifts) the heating contribu-tion. Since the distinction between the ‘high’ and ‘low’ temperature regimes introduced insection (III A) involves a comparison between f th and f α , one concludes that the reductionof the heating contribution will be very important in the low temperature regime and muchless important in the high temperature regime. More precisely, in a one sample experiment,for the low temperature range, the heating contribution has a very flat maximum around afew times f α . The example of the 204K curve of Fig. 1 shows that the two samples setupreduces the heating contribution so strongly that its maximum over frequency is shiftedslightly above f th ( e thick ) = 65Hz, i.e. at f ≫ f α . For the same reasons, the bridge techniqueextends slightly upwards the low temperature regime: the high temperature regime, charac-terised by a well defined maximum of X ,h located around f = f α , only arises at T ≥ C. Main features of the heating contribution to χ We shall now discuss the main features of the overestimated heating contribution(Figs. 3,4), as well as those of the damped heating contribution (Figs. 5-6). They willbe compared to the experimental X ( ≃ N corr H ) values at T = 210 .
3K presented in Ref.[12]. Two properties reported in [12] are of interest here. First, when the temperature T isvaried between 225 .
3K and 194 . max f ( | X | ) over frequency of themeasured | X | increases by a factor ≃ .
5. Second, in this temperature interval, the fre-quency dependence of | X | at each temperature, once rescaled vertically by its T -dependent max f ( | X | ) value, fall onto a master curve depending only of f /f α . Besides, the phase of X also depends only of f /f α . This ‘Time Temperature Superposition’ (TTS) property of X is analogous to the similar properties of other well studied observables, e.g. χ , for manysupercooled liquids. Considering the fact that the above mentioned 1 . T variations of | X | are hardly visible in these two plots.15 -2 -1 -5 -4 -3 -2 -1 Glycerol 210.3K 196,8K 198,5K 201,3K 204,5K 207,5K 210,3K 213,3K 215,6K 217,8K 219,2K 222,3K 225,3K | X | o v e r e s t i m a t ed hea t i ng f/f FIG. 3: (Color on line) Modulus of the overestimated heating contribution | X ,h | for the twoglycerol samples setup e thin = 19 µ m, e thick = 41 µ m of Ref. [12], as a function of the frequencynormalized to the relaxation frequency f α ( T ). For comparison the measured values of | X | [12] aregiven for T = 210 .
3K (filled red circles). Several features of this overstimated heating contributionare at odds with those measured in [12], e.g. the maximum over frequency increases with T , itoccurs at a temperature dependent value of f /f α and TTS is not obeyed (see text). From Figs. 3-6, one gets the five following features, common both to the overstimatedand damped heating contributions: (i)
For a given f /f α , X ,h mainly increases with T , at odds with the experimental behaviorreported in Ref. [12]. Besides the magnitude of the T -dependence of X ,h is much largerthan that of X reported in Ref. [12]. (ii) The results for the modulus and the phase show that the heating contributions donot obey TTS. For the reasons explained in section III B, the maximum over frequency of X ,h arises for f ≫ f α at low temperature, and for f ≃ f α in the 218K − | X | ≃ | N corr H| arises at a very different frequency f ≃ . f α -see Ref. [12]-. Summarising points (i)-(ii) ,the frequency and T dependence of the heating contributions do not, at all, look like thoseof X . (iii) For f ≃ . f α , the heating contribution is always negligible with respect to thevalues of X reported in Ref. [12]. More precisely, the heating correction to max f ( | X | ) is16 -2 -1 -1000100200300400500 Glycerol 210.3K 196,8K 198,5K 201,3K 204,5K 207,5K 210,3K 213,3K 215,6K 217,8K 219,2K 222,3K 225,3K P ha s e o f X , o v e r e s t. hea t i ng [ D eg ] f/f FIG. 4: (Color on line) Phase of the overestimated heating contribution X ,h for the two glycerolsamples setup e thin = 19 µ m, e thick = 41 µ m of reference [12], as a function of the frequencynormalized to the relaxation frequency f α ( T ). For comparison the measured values of X at T = 210 .
3K [12] are given (filled red circles): they do not significantly vary with T since the datareported in [12] obey TTS (see text), contrarily to the heating contribution phases. smaller than 1 .
5% for T ≤ . K , and reaches 5% for the highest temperature of 225 . decrease max f ( | X | ) by an amount equal to the error bar given in the Figure 3 ofRef.[12]. (iv) For f ≫ f α , the heating contributions decrease significantly faster with frequency(as f − . ) than | X | . Thus the influence of the heating contributions disappears in the limit f ≫ f α . (v) For T ≤ − | X | by less than the 5% errorbar reported in Ref. [12]. This means that in the range [194K , -2 -1 -5 -4 -3 -2 -1 Glycerol 210.3K 196,8K 198,5K 201,3K 204,5K 207,5K 210,3K 213,3K 215,6K 217,8K 219,2K 222,3K 225,3K | X | da m ped hea t i ng f/f FIG. 5: (Color on line) Modulus of the damped heating contribution X ,h for the two glycerolsamples setup e thin = 19 µ m, e thick = 41 µ m of Ref. [12], as a function of the frequency normalizedto the relaxation frequency f α ( T ). For comparison the measured values of X at T = 210 .
3K [12]are given (filled red circles). The comments made in the caption of Fig. 3 still apply in this case,with the noticeable difference that the values of | X ,h | are now smaller than those of Fig. 3 due tothe fact that one takes into account the damping effect arising from the finite relaxation time ofthe dipoles (see text section II A 2 and Eq. (10)) . D. Evidence of a damping of the heating contribution to χ In the range [194K , N corr H ,obeying TTS, is well measured. In the [204K , N corr H ) glyc values of glyc-erol can be investigated by subtracting the heating contribution to the measured X ( f, T )reported at various temperatures in [12]. As we have seen, the heating contribution is weakor negligible in many cases, but it is interesting to study to what extent the expected TTSis better verified after such a subtraction. We define :( N corr H ) glyc = X − X ,h . (19)We performed this (complex) subtraction for the three temperatures above 204K in Ref.[12], i.e. 210 . . . -2 -1 -1000100200300400500 Glycerol 210.3K 196,8K 198,5K 201,3K 204,5K 207,5K 210,3K 213,3K 215,6K 217,8K 219,2K 222,3K 225,3K P ha s e o f X , da m ped hea t i ng [ D eg ] f/f FIG. 6: (Color on line) Phase of the overestimated heating contribution X ,h for the two glycerolsamples setup e thin = 19 µ m, e thick = 41 µ m of reference [12], as a function of the frequencynormalized to the relaxation frequency f α ( T ). The comment of the caption of Fig. 4 applies alsohere. three fits appear as continuous lines in Figs. 7-10. Then, the subtraction of X ,h to themeasured X is performed, X ,h being either the overestimated contribution(see Figs. 7, 8)or the damped one (see Figs. 9, 10). The results of this subtraction appear as open symbolsin Figs. 7-10.When the overestimated value of X ,h is used, ( N corr H ) glyc may differ strongly from themeasured X . Fig. 7 shows that the shape of the curve giving the modulus of ( N corr H ) glyc vs frequency is modified at f /f α ≃
20 for T = 210 . . f /f α ≃
2, and at 225 .
3K the dip is present at f /f α ≃ .
8. These features come from thefrequency range where the values of | X ,h | are close to or larger than those of | X | , as shownin Fig. 3. Depending on the difference between the phases of X and of X ,h , the effect on | N corr H| glyc is more or less pronounced: at T = 217 . K the dip at f /f α ≃ vanishing value of | N corr H| glyc , as shown by the π jump on the phase (see Fig. 8).At 210 .
3K and 225 . | N corr H| glyc is related to a less pronouncedeffect on the phase (see Fig. 8). To summarize, using the overstimated values of X ,h yieldsnon TTS features in the resulting ( N corr H ) glyc . We note that this is true already for the19 -2 -1 -2 -1 | X | f/f FIG. 7: (Color on line) Solid lines: interpolated values of the experimental | X | values reported in[12] for glycerol, obtained with the ‘two samples bridge setup’. The intrinsic | N corr H| glyc of glycerolis plot as symbols and results from the complex subtraction X − X ,h where the overestimatedheating contribution of X ,h (displayed in Figs. 3- 4) is used. The resulting | N corr H| glyc curves,which display a dip or a strong variation of the f dependence for the three temperatures consideredhere, are non TTS, contrary to what is expected in the case of no heating effect. .
3K curve corresponding to a temperature close to 204 .
7K for which the curve of Ref.[12] verifies TTS. The fact that our experimental X ( f, T ) curves verify TTS is a strongindication of the overestimated character (already anticipated) of the heating contributionused to calculate the results presented in Figs. 7- 8. We are thus led to the conclusion thatthe heating contribution has to be damped.We now perform the same analysis as above, using the damped X ,h ( f, T ) values displayedin Figs. 5- 6. At first glance, Figs. 9-10 reveal that for the damped heating estimate, thecomplex subtraction no longer produces the strong non TTS features depicted above. Theresulting | N corr H| glyc does not strongly differ from the original | X | : the value of the exponentgiving the decay at f /f α > . | N corr H| glyc is only 5% larger than the corresponding onefor | X | at 217 . K and 225 . .
3K and f /f α ≃
20 hasalmost disappeared. Last, the phase of ( N corr H ) glyc is not significantly ‘less TTS’ than theoriginal phase of X . We thus conclude that the damped X ,h values, introduced in section(II A 2) and displayed in Figs 5- 6, meet the requirement resulting from the last point of20 -2 -1 -100-50050100150200 f/f P ha s e o f X [ D eg ] FIG. 8: (Color on line) Solid lines: interpolated values of the phase of the experimental X reportedin [12] for glycerol, obtained with the ‘two samples bridge setup’. Symbols: phases correspondingto the complex subtraction evoked in the caption of Fig. 7 (with the same symbols used). The π jump occuring for 217 .
8K around f /f α = 2 reveals that | N corr H| glyc vanishes at this point. section (III C), namely that ( N corr H ) glyc should be TTS. This is an experimental indicationthat the damping factor introduced in Eq. (10) is reasonnable, despite the two simplifyingassumptions made along its derivation.We leave for future work the very difficult experimental task of isolating the heatingcontribution in itself. Two main ideas could be conceivable. First, one could try to pushthe 225 K experiment in the range f /f α ≫ N corr H ) glyc if one extrapolates the calculations of Fig. 10. In practice it isextremely difficult to measure accurately X in this range of parameters, because of the d.c.heating contribution coming from the whole thermal circuit between each sample and theexperimental cell [25]. This d.c. heating contribution does not cancel in our two samples’bridge, and thus the balance condition changes at each value of the voltage source. Thesecond idea would be to use the fact that the canceling effect of the heating contribution inthe bridge arises only for f ≤ f th ( e thick ). As a result, the heating contribution to X shouldbe different for two setups, say A and B , where e thick is different. In the frequency range f th ( e thick,A ) ≤ f ≤ f th ( e thick,B ), one may hope that the differences between the values of X ,h -2 -1 -2 -1 | X | f/f FIG. 9: (Color on line) (Color on line) Solid lines: interpolated values of the experimental | X | values reported in [12] for glycerol, obtained with the ‘two samples bridge setup’. The intrinsic | N corr H| glyc of glycerol is plot as symbols and results from the complex subtraction X − X ,h where the damped heating contribution of X ,h (displayed in Figs. 5- 6) is used. The dips visiblein Fig. 7 are no longer visible, and the resulting | N corr H| glyc are approximately TTS. This suggeststhat the damped heating estimate is reasonable. could be detectable. Putting numbers, in the case e thick,A = 41 µ m and e thick,B = 25 µ m (see[13]), shows that this difference does not reach 10% of X (even for the optimal frequency),and is therefore very difficult to single out unambiguously. IV. CONCLUSION
We have presented a thorough study of the ‘homogeneous’ heating contribution to thethird harmonics experiments carried out on glycerol in Ref. [12] between T g + 4K and T g + 35K. We have emphasized the ‘spurious’ nature of this heating contribution by showingthat it vanishes for thin enough samples and low enough temperatures. We have shownthat the ‘two samples’ bridge technique, presented in Ref. [12], widens the temperatureinterval over which the heating contribution is totally negligible with respect to the mea-sured N corr H values reported in [12]. Globally, the heating contribution exhibits behaviorsvery different from those of the nonlinear normalized susceptibility X reported in [12]: It22 -2 -1 -100-50050100150200 f/f P ha s e o f X [ D eg ] FIG. 10: (Color on line) Solid lines: interpolated values of the phase of the experimental X reported in [12] for glycerol, obtained with the ‘two samples bridge setup’. Symbols: phases corre-sponding to the complex subtraction evoked in the caption of Fig. 9 (with the same symbols used).The phase of the resulting ( N corr H ) glyc values are not significantly ’less TTS’ than the originalones. This suggests that the damped heating estimate, displayed in Figs. 5- 6, is reasonnable. (mainly) increases with T for a given f /f α , its peak arises at much higher frequencies thanin [12], its frequency dependence is faster, it does not obey TTS (except for the highest tem-peratures). At a quantitative level, we have shown that one can safely neglect the heatingcontribution to the maximum over frequency of | X | for all the temperature interval studiedin [12]. Thanks to the quite large temperature interval where the absence of any heatingcontribution is guaranteed, we have shown that the intrinsic N corr H of glycerol obeys TTSin this temperature range. Extrapolating this TTS feature up to 225 K allows to put anexperimental constraint on the homogeneous heating contribution. We obtain the followingimportant result: The fact that the homogeneous heating contribution must be dampedbecause of the finite relaxation time of the dipoles is confirmed by an investigation of theshape and TTS property of the X ( f, T ) curves. In addition, we may conclude that the23amping factor given by Eq. (10) is reasonnable. [1] P.G. Debenedetti, F.H. Stilinger, Nature , 259-267 (2001).[2] B. Schiener, R. B¨ohmer, A. Loidl, and R. V. Chamberlin, Science, , 752, (1996).[3] U. Tracht et al. , Phys. Rev. Lett. , 2127 (1998).[4] M.D. Ediger, Annu. Rev. Phys. Chem. , 99 (2000).[5] R. Richert, J. Phys.: Condens. Matter R703 (2002).[6] M.M. Hurley, P. Harowell, Phys. Rev. E, , 1694, (1995).[7] L. Berthier et al. , Science , 1797 (2005).[8] L. Berthier, et al. , J. Chem. Phys. , 184503 (2007).[9] C. Dalle-Ferrier et al. , Phys. Rev. E , 041510 (2007).[10] J.-P. Bouchaud, G. Biroli, Phys. Rev. B , 064204 (2005).[11] C. Thibierge, D. L’Hˆote, F. Ladieu, R. Tourbot, Rev. Scient. Instrum. , 103905 (2008).[12] C. Crauste-Thibierge, C. Brun, F. Ladieu, D. L’Hˆote, G. Biroli, J-P. Bouchaud, Phys. Rev.Lett. , 165703 (2010).[13] C. Crauste-Thibierge, C. Brun, F. Ladieu, D. L’Hˆote, G.Biroli, J-P. Bouchaud, EPAPS of Ref. [12], available athttp://prl.aps.org/epaps/PRL/v104/i16/e165703/Crauste-EPAPS-modified.pdf.[14] J.L. D´ejardin, Yu.P. Kalmykov, Phys. Rev. E , 1211 (2000).[15] W. Huang, R. Richert, Eur. Phys. J. B , 217 (2008).[16] R. Richert, S. Weinstein, Phys. Rev. Lett. , 095703 (2006).[17] S. Weinstein, R. Richert, Phys. Rev. B , 064302 (2007).[18] The Debye dynamics can be also obtained by starting with the non inertial Langevin equa-tion for the rotational Brownian motion of a particule, by appropriate transformation of thevariables and direct averaging of the stochastic equation so obtained.[19] Since the relative thermal variation of τ is much larger than that of ∆ χ , the latter quantityis taken as a constant when solving Eq. 6.[20] N. O. Birge, Phys. Rev.B , 2674 (1986); N. O. Birge, S. R. Nagel Phys. Rev. Lett. , 1631(1985).[21] Standard (but tedious) thermal calculations taking into account the thermal circuit connect- ng the electrodes to the metallic experimental cell do show that the approximation madeby considering electrodes as a perfect thermostat produce an error smaller, in any case, than2% for δT . Besides, we also neglect the possible thermal resistance associated to the inter-face glycerol/electrode: this is supported by the thorough study of Minakov et al. who hasshowed, for the Glycerol/Macor interface, that such an effect was responsible of only 5% ofthe temperature increase [22].[22] A. A. Minakov, S. A. Adamovsky and C. Schick, ThermochimicaActa , , 89 (2003).[23] In [12, 13], the m = 3 mode was taken into account into the calculations yielding the dottedcurve of Fig.4 of [12], even if, for simplicity, the m = 3 part of the temperature increasewas not explicitly written in Eq. (15) of [13]. Besides, for simplicity we suppressed the factor96 /π ≃ .
985 in the present definition of δT ⋆ .[24] H. Fr¨olich, Theory of Dielectrics , Oxford at the Clarendon Press, 1958.[25] The approximation that the electrodes are a perfect thermostat are in general valid for com-puting δT , see [21], but it is in general not valid for computing the d.c. temperature increase.The difference lies in the fact that, because of the ‘skin effect’, the temperature oscillationsare efficiently damped in the thick electrodes, while the d.c. component is not damped at all.The d.c. heat flux is thus fully transmitted by the electrodes, and the unavoidable insulatingpiece between the electrode and the experimental cell is responsible for the major part of thed.c. temperature increase. The only way to get rid of the d.c. heating is to perform extremelyfast experiments, in the spirit of what is made in the seminal reference [16], so as to measurethe sample properties in a time scale much lower than the thermal diffusion time between thesample and the experimental cell., see [21], but it is in general not valid for computing the d.c. temperature increase.The difference lies in the fact that, because of the ‘skin effect’, the temperature oscillationsare efficiently damped in the thick electrodes, while the d.c. component is not damped at all.The d.c. heat flux is thus fully transmitted by the electrodes, and the unavoidable insulatingpiece between the electrode and the experimental cell is responsible for the major part of thed.c. temperature increase. The only way to get rid of the d.c. heating is to perform extremelyfast experiments, in the spirit of what is made in the seminal reference [16], so as to measurethe sample properties in a time scale much lower than the thermal diffusion time between thesample and the experimental cell.