Nonlinear susceptibilities of interacting polar molecules in the self-consistent field approximation.

The nonlinear stationary response of an assembly of long range interacting electric dipoles is calculated via Berne's nonlinear rotational diffusion equation [J. Chem. Phys. 62 1154 (1975)]. Analytical formulas are derived, showing that the behavior of ω and 3ω components of the nonlinear external field response spectra in such polar dielectrics strongly deviates from the Coffey-Paranjape formulas [Proc. R. Ir. Acad., Sect. A 78, 17 (1978)] as the long range dipole-dipole interactions increase, while the linear response remains qualitatively unaffected. By qualitatively comparing with recent experimental measurements on glycerol, it is further demonstrated that nonlinear dielectric response experiments provide a powerful tool for the characterization of short range intermolecular interactions in polar liquids.

Simultaneously measuring the dichroism and the dielectric response of an azobenzene-doped organic glass former.

We have designed an experimental setup allowing to simultaneously measure both the dielectric response of a supercooled liquid and the dynamics of azobenzene chromophores dispersed in it. Both the azobenzene chromophores and the organic glass former have been synthesized with similar reaction paths: they are chemically similar, apart from the azobenzene group responsible for the strong optical absorption in the [350; 450 nm] range for the chromophores, while the embedding supercooled liquid is optically transparent. This material is deposited on transparent electrodes with an inter-electrode gap as small as 4 µm-obtained thanks to optical lithographic techniques. We show that our setup is sensitive enough to measure the coupling between the dielectric macroscopic response and the isomerization dynamics of 1% of chromophores excited by a 0.5-5 mW/cm2 light beam. We demonstrate that this coupling neither comes from the heating of the sample due to the light absorption nor from changes of the sample shape with light. Finally, we discuss the few physical effects, which may give rise to this coupling, and show that our experiment could test some recent predictions done in the framework of random first order transition theory of the glassy state.

Evidence of growing spatial correlations during the aging of glassy glycerol.

We have measured, as a function of the age t(a), the aging of the nonlinear dielectric susceptibility χ(3) of glycerol below the glass transition. Whereas the linear susceptibility can be accurately accounted for in terms of an age dependent relaxation time τ(α)(t(a)), this scaling breaks down for χ(3), suggesting an increase of the amplitude of χ(3). This is a strong indication that the number N(corr) of molecules involved in relaxation events increases with t(a). For T=0.96×T(g), we find that N(corr) increases by ~10% when t(a) varies from 1 to 100 ks. This sheds new light on the relation between length scales and time scales in glasses.

Third harmonics nonlinear susceptibility in supercooled liquids: a comparison to the box model.

The box model, originally introduced to account for the nonresonant hole burning (NHB) dielectric experiments in supercooled liquids, is compared to the measurements of the third harmonics P(3) of the polarisation, reported recently in glycerol, close to the glass transition temperature T(g) [C. Crauste-Thibierge, C. Brun, F. Ladieu, D. L'Hôte, G. Biroli, and J.-P. Bouchaud, Phys. Rev. Lett. 104, 165703 (2010)]. In this model, each box is a distinct dynamical relaxing entity (hereafter called dynamical heterogeneity (DH)) which follows a Debye dynamics with its own relaxation time τ(dh). When it is submitted to a strong electric field, the model posits that a temperature increase δT(dh), depending on τ(dh), arises due to the dissipation of the electrical power. Each DH has thus its own temperature increase, on top of the temperature increase of the phonon bath δT(ph). Contrary to the "fast" hole burning experiments where δT(ph) is usually neglected, the P(3) measurements are, from a thermal point of view, fully in a stationary regime, which means that δT(ph) can no longer be neglected a priori. This is why the version of the box model that we study here takes δT(ph) into account, which implies that the δT(dh) of the DHs are all coupled together. The value of P(3), including both the "intrinsic" contribution of each DH as well as the "spurious" one coming from δT(ph), is computed within this box model and compared to the P(3) measurements for glycerol, in the same range of frequencies and temperatures T. Qualitatively, we find that this version of the box model shares with experiments some nontrivial features, e.g., the existence of a peak at finite frequency in the modulus of P(3) as well as its order of magnitude. Quantitatively, however, some experimental features are not accounted for by this model. We show that these differences between the model and the experiments do not come from δT(ph) but from the "intrinsic" contribution of the DHs. Finally, we show that the interferences between the 3ω response of the various DHs are the most important issue leading to the discrepancies between the box model prediction and the experiments. We argue that this could explain why the box model is quite successful to account for some kinds of nonlinear experiments (such as NHB) performed close to T(g), even if it does not completely account for all of them (such as the P(3) measurements). This conclusion is supported by an analytical argument which helps understanding how a "space-free" model as the box model is able to account for some of the experimental nonlinear features.

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

The ac nonlinear dielectric response chi3(omega,T) of glycerol was measured close to its glass transition temperature T(g) to investigate the prediction that supercooled liquids respond in an increasingly nonlinear way as the dynamics slows down (as spin glasses do). We find that chi3(omega,T) indeed displays several nontrivial features. It is peaked as a function of the frequency omega and obeys scaling as a function of omega tau(T), with tau(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 chi3 decays as a power law of omega 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).

Study of the heating effect contribution to the nonlinear dielectric response of a supercooled liquid.

We present a detailed study of the heating effects in dielectric measurements carried out on a liquid. Such effects come from the dissipation of the electric power in the liquid and give contribution to the nonlinear third harmonics susceptibility χ(3), which depends on the frequency and temperature. This study is used to evaluate a possible "spurious" contribution to the recently measured nonlinear susceptibility of an archetypical glassforming liquid (glycerol). Those measurements have been shown to give a direct evaluation of the number of dynamically correlated molecules temperature dependence close to the glass transition temperature T(g) ≈ 190 K [Crauste-Thibierge et al., Phys. Rev. Lett. 104, 165703 (2010)]. We show that the heating contribution is totally negligible (i) below 204 K at any frequency; (ii) for any temperature at the frequency where the third harmonics response χ(3) is maximum. Besides, this heating contribution does not scale as a 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 follow instantaneously the temperature modulation due to the heating phenomenon. An estimate of the magnitude of this damping is given.

Spatial correlations in the dynamics of glassforming liquids: experimental determination of their temperature dependence.

We use recently introduced three-point dynamic susceptibilities to obtain an experimental determination of the temperature evolution of the number of molecules Ncorr that are dynamically correlated during the structural relaxation of supercooled liquids. We first discuss in detail the physical content of three-point functions that relate the sensitivity of the averaged two-time dynamics to external control parameters (such as temperature or density), as well as their connection to the more standard four-point dynamic susceptibility associated with dynamical heterogeneities. We then demonstrate that these functions can be experimentally determined with good precision. We gather available data to obtain the temperature dependence of Ncorr for a large number of supercooled liquids over a wide range of relaxation time scales from the glass transition up to the onset of slow dynamics. We find that Ncorr systematically grows when approaching the glass transition. It does so in a modest manner close to the glass transition, which is consistent with an activation-based picture of the dynamics in glassforming materials. For higher temperatures, there appears to be a regime where Ncorr behaves as a power-law of the relaxation time. Finally, we find that the dynamic response to density, while being smaller than the dynamic response to temperature, behaves similarly, in agreement with theoretical expectations.

Unifying different interpretations of the nonlinear response in glass-forming liquids.

This work aims at reconsidering several interpretations coexisting in the recent literature concerning nonlinear susceptibilities in supercooled liquids. We present experimental results on glycerol and propylene carbonate, showing that the three independent cubic susceptibilities have very similar frequency and temperature dependences, for both their amplitudes and phases. This strongly suggests a unique physical mechanism responsible for the growth of these nonlinear susceptibilities. We show that the framework proposed by two of us [J.-P. Bouchaud and G. Biroli, Phys. Rev. B 72, 064204 (2005)PRBMDO1098-012110.1103/PhysRevB.72.064204], where the growth of nonlinear susceptibilities is intimately related to the growth of glassy domains, accounts for all the salient experimental features. We then review several complementary and/or alternative models and show that the notion of cooperatively rearranging glassy domains is a key (implicit or explicit) ingredient to all of them. This paves the way for future experiments, which should deepen our understanding of glasses.

Fifth-order susceptibility unveils growth of thermodynamic amorphous order in glass-formers.

Glasses are ubiquitous in daily life and technology. However, the microscopic mechanisms generating this state of matter remain subject to debate: Glasses are considered either as merely hyperviscous liquids or as resulting from a genuine thermodynamic phase transition toward a rigid state. We show that third- and fifth-order susceptibilities provide a definite answer to this long-standing controversy. Performing the corresponding high-precision nonlinear dielectric experiments for supercooled glycerol and propylene carbonate, we find strong support for theories based on thermodynamic amorphous order. Moreover, when lowering temperature, we find that the growing transient domains are compact--that is, their fractal dimension d(f) = 3. The glass transition may thus represent a class of critical phenomena different from canonical second-order phase transitions for which d(f) < 3.

A simple device for dielectric spectroscopy of polymers with temperature regulation close to 300 K based on a Peltier junction.

We present a simple thermostat device for performing dielectric spectroscopy measurements on polymers close to their glass transition temperature. By using a vacuum chamber containing a Peltier junction with its regulator, we show that a very simple setup yields a temperature accuracy which is good enough for accurate studies of polymer dielectric properties. This technique is also more cost effective than standard setups using cryogenic fluids.

A method for measuring the nonlinear response in dielectric spectroscopy through third harmonics detection.

We present a high sensitivity method allowing the measurement of the nonlinear dielectric susceptibility of an insulating material at finite frequency. It has been developed for the study of dynamic heterogeneities in supercooled liquids using dielectric spectroscopy at frequencies 0.05 Hz < or = f < or = 3x10(4) Hz. It relies on the measurement of the third harmonics component of the current flowing out of a capacitor. We first show that standard laboratory electronics (amplifiers and voltage sources) nonlinearities lead to limits on the third harmonics measurements that preclude reaching the level needed by our physical goal, a ratio of the third harmonics to the fundamental signal about 10(-7). We show that reaching such a sensitivity needs a method able to get rid of the nonlinear contributions both of the measuring device (lock-in amplifier) and of the excitation voltage source. A bridge using two sources fulfills only the first of these two requirements, but allows to measure the nonlinearities of the sources. Our final method is based on a bridge with two plane capacitors characterized by different dielectric layer thicknesses. It gets rid of the source and amplifier nonlinearities because in spite of a strong frequency dependence of the capacitor impedance, it is equilibrated at any frequency. We present the first measurements of the physical nonlinear response using our method. Two extensions of the method are suggested.

Ground state of the kagomé-like S=1/2 antiferromagnet volborthite Cu3V2O7(OH)2 x 2H2O.

The volborthite compound is one of the very few realizations of S=1/2 quantum spins on a highly frustrated kagomé-like lattice. Low-T SQUID measurements reveal a broad magnetic transition below 2 K which is further confirmed by a peak in the 51V nuclear spin relaxation rate (1/T1) at 1.4 K +/- 0.2 K. Through 51V NMR, the ground state (GS) appears to be a mixture of different spin configurations, among which 20% corresponds to a well defined short-range order, possibly of the sqrt(3) x sqrt(3) type. While the freezing involves all the Cu2+ spins, only 40% of the copper moment is actually frozen which suggests that quantum fluctuations strongly renormalize the GS.

Direct experimental evidence of a growing length scale accompanying the glass transition.

Understanding glass formation is a challenge, because the existence of a true glass state, distinct from liquid and solid, remains elusive: Glasses are liquids that have become too viscous to flow. An old idea, as yet unproven experimentally, is that the dynamics becomes sluggish as the glass transition approaches, because increasingly larger regions of the material have to move simultaneously to allow flow. We introduce new multipoint dynamical susceptibilities to estimate quantitatively the size of these regions and provide direct experimental evidence that the glass formation of molecular liquids and colloidal suspensions is accompanied by growing dynamic correlation length scales.

Dielectric constant of glasses: evidence for dipole-dipole interactions.

The 1 kHz real part chi(') of the dielectric constant of a structural glass (a-SiO(2+x)C(1+y)H(z)) was measured at low temperature T. Reducing the sample thickness h below 100 nm weakens the slope /delta(chi('))/delta (T) for T less than or approximately equal 0.1 K, for all measuring fields E. This contrasts with the predictions of the two-level system (TLS) model but is in agreement with the recently proposed delocalization of excitations derived from a field-induced TLS-TLS interaction mechanism. For small h this interaction is screened, which explains the h effects on chi('). Hence, interactions must play a key role in standard thick samples as soon as T less than or approximately equal 0.1 K.