- Email: [email protected]
Polarization fluctuations near the glass transition
Journal of Non-Crystalline Solids 352 (2006) 4920–4927 www.elsevier.com/locate/jnoncrysol
Polarization fluctuations near the glass transition M. Lucchesi a
a,c,* ,
A. Dominjon a, S. Capaccioli
a,b
, D. Prevosto
a,c
, P.A. Rolla
a,c
Dipartimento di Fisica and INFM, Universita` di Pisa, Largo Bruno Pontecorvo, 3, I-56127 Pisa, Italy b CNR-INFM, CRS SOFT, Piazzale Aldo Moro 2, I-00185 Roma, Italy c CNR-INFM, Polylab, Largo Bruno Pontecorvo, 3, I-56127 Pisa, Italy Available online 7 September 2006
Abstract Measurements of polarization fluctuations were performed for an epoxy glass former. The voltage noise produced by polarization fluctuations of the sample filling a capacitor was acquired via a home-made very high input impedance current–voltage converter, in series with an ultra low noise pre-amplifier, achieving high sensitivity and accuracy in the range 0.1–1000 Hz. The temperature and frequency dependence of polarization noise was investigated above and below the glass transition temperature Tg. The sample was driven to the glassy state with different cooling rate and then isothermally aged, while the noise spectral density was measured at different times and compared with the Johnson–Nyquist noise determined by the sample impedance measured by conventional dielectric spectroscopy. At thermodynamic equilibrium the polarization noise agreed with the predictions of the fluctuation–dissipation theorem linking noise spectral density to susceptibility. On the contrary, a strong violation of the theorem was observed after a fast cooling below Tg: an intense polarization noise was detected, with a power spectral density following an inverse power law frequency behavior, whose intensity was decreasing with aging time. At the same time, the amplitude of polarization fluctuations showed a non-Gaussian distribution, whose width reduced during the aging process, up to recover the Gaussian statistics on approaching the equilibrium. Ó 2006 Elsevier B.V. All rights reserved. PACS: 64.70.Pf; 77.22.Gm; 05.40.a Keywords: Dielectric properties; Relaxation, electric modulus; Glass transition; Fluctuation-dissipation theorem
1. Introduction Systems characterized by slow dynamics can be driven out of thermodynamic equilibrium by means of a rapid change of appropriate external variables, for instance a temperature jump taking a liquid down to a glassy state. After such treatment, the system spontaneously tends to recover the equilibrium state, and its properties (dynamic, thermodynamic, mechanical) depend on the waiting time tw, elapsed after the achievement of the out-of-equilibrium state [1]. The evolution towards equilibrium with time is commonly called ‘aging’. In an out-of-equilibrium system * Corresponding author. Address: Dipartimento di Fisica and INFM, Universita` di Pisa, Largo Bruno Pontecorvo, 3, I-56127 Pisa, Italy. Tel.: +39 0502214514; fax: +39 0502214333. E-mail address: [email protected] (M. Lucchesi).
0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.02.153
the response to a small external perturbation, especially at long time, is no longer related, via the temperature of the thermal bath, to the spontaneous fluctuations of the observable under test, as predicted by the fluctuation–dissipation theorem (FDT) [2], depending themselves on the thermal history of the system. In such a situation the concept of temperature of the system is not well defined any more, and the response of the system to an external perturbation is unable to give new insight into the molecular dynamics. Therefore, from an experimental point of view, it is of importance not only to study the response of the system to external fields but also its thermal fluctuations. Another significant reason of interest in the simultaneous study of response and fluctuations of a system out-of-equilibrium involves the definition of temperature. Indeed a description of weakly out-of-equilibrium dynamics has been recently proposed, extending the concept of temperature by
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
a generalization of the fluctuation–dissipation theorem (FDT) [3]. According to FDT the response R(t) to an external perturbation h is related to the normalized time autocorrelation function C(t) of the observable A to which the perturbation field h is conjugated: for systems at equilibrium, the time derivative of the correlation function is proportional to the response: dCðtÞ ¼ k B TRðtÞ dt
ð1Þ
where kB is the Boltzmann constant and T is the temperature of the thermal bath contacting the system. Eq. (1) provides an alternative definition for the concept of ‘temperature of the system’ [2]. As FDT is derived by assuming time-translation invariance [2], its violation is expected for out-of-equilibrium systems: in this case, a generalization of FDT has been proposed [3,5] by introducing an effective temperature, Teff, that should replace T in Eq. (1) as a proportionality constant between the time derivative of the correlation function and the response function dCðt; tw Þ T eff ðt; tw Þ ¼ ð2Þ k B Rðt; tw Þ dt At equilibrium, of course Teff = T. It is noteworthy that Teff is expected to depend on the time-scale t (or on the angular frequency x) of observation, on the waiting time tw during aging and to be higher than the temperature of the thermal bath contacting the system [5]. Indeed in a ‘slow dynamics’ system driven out-of-equilibrium it is possible to find fast processes, that equilibrate with thermal bath and attain Teff = T, and slower processes, that slowly relax towards the equilibrium and keep memory of higher temperature configurations for a longer time, and that are characterized by Teff higher than the temperature of thermal bath. A study of evolution of Teff(x, tw) and its dependence on the observation time-scale and aging time could verify the conflicting predictions on FDT violation of the different models of aging and out-of-equilibrium dynamics [3,4,6–8]. A number of recent simulation works deal with the study of FDT violation in out-of-equilibrium state for aging systems [9–11] and also for stationary systems undergoing strong external perturbation (e.g. shear deformations) [12,13]: numerical simulations, in agreement with theoretical models, predict that, in an out-of-equilibrium system, fluctuations of dynamic variables on very long time scales (comparable to those of structural relaxation) feel a fictive temperature Teff (defined by the generalized FDT) higher than T (thermal bath temperature), whereas on short time scales (comparable to those of vibrational or cage rattling motions) Teff = T [3,9–18]. The value of Teff depends on tw (for aging systems) or, in the case of stationary out-of-equilibrium states, on the frequency of the external driving [3,10–18]. On the contrary, few experimental works verifying the violation or the generalization of FDT in out-of-equilibrium systems have been performed (for an extensive review see Ref.[4]). Some examples concern aging spin glasses [19]
4921
and colloidal solutions during the liquid-gel transition [20,21], but studies on structural glasses are almost lacking. Concerning this issue, the measurements of polarization at low frequencies (or long times), by means of dielectric spectroscopy to characterize the response function and by means of electric noise measurements to acquire the fluctuations, proved to be well suitable, so that up till now the only experimental checks of FDT violation or generalization in structural glasses deal with polarization as observable under test. Usually the voltage spontaneous fluctuations of a capacitor filled by the material under test (with complex permittivity e = e 0 je00 and empty capacitance C0) are acquired and compared with the Johnson– Nyquist noise determined by electric impedance Z(x) of the capacitor assessed by dielectric spectroscopy [22–27]. In this case FDT has the form of Nyquist’s theorem that for the voltage fluctuation spectral density SV(x) = hjV(x)j2i predicts S V ðx; tw Þ ¼ 4k B T eff ðx; tw ÞRe½Zðx; tw Þ ¼ 4k B T eff ðx; tw Þ
e00 ðx; tw Þ 1 e0 ðx; tw Þ e0 ðx; tw ÞC 0 x
1 00 2 e ðx;tw Þ 1 þ e0 ðx;tw Þ
ð3Þ
Investigations performed above the temperature of glass transition Tg on molecular and polymeric glass-formers verified Eq. (3) with Teff = T [22,23]. On the contrary, for glycerol, studies done few degrees below Tg during the instants following a quenching revealed a weak FDT violation, characterized by Teff slowly decreasing with time until the value of the temperature T was reached [25]. In a broadband study on polycarbonate, a stronger violation of FDT was observed after fast cooling from liquid to glassy state [26,27]. The amplitude and the persistence time of this violation resulted to be decreasing functions of frequency and to depend on the cooling rate. The origin of such a violation has been ascribed to the presence of a highly intermittent dynamics characterized by large fluctuations with amplitude distributed according to a strongly non-Gaussian statistics [21,26,27]. The aim of our paper is to present a new experimental setup able to acquire voltage fluctuations and impedance and to assess the ratio Teff over a wide frequency range. Measurements have been performed on an epoxy glass-former in both equilibrium liquid state and out-of-equilibrium aging glass. 2. Experimental procedures Triphenylolmethane triglycidyl ether (TPMTGE) was commercially available1 and used as received. Its molecular weight is 460.5 g mol1 and Tg is 287 K. Further details on 1
Sigma–Aldrich.
4922
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
the thermodynamic and dielectric properties of this epoxy compound can be found in Ref. [28]. This sample is particularly suitable for at least three reasons: (a) Tg is almost at room temperature, making easier the environment for the experimental setup; (b) it is very fragile according to Angell classification [29], i.e. very sensitive to temperature, allowing to get a strong out-of-equilibrium state not too much below Tg even for moderate quenching; (c) it has appreciable dielectric losses e00 /e 0 in the investigated frequency–temperature range, so enabling noise power spectral density (see Eq. (3)) to be measured. In the present experiment, polarization fluctuations were observed via voltage thermal noise produced by a capacitor cell filled with the material. The cell consists of two metallic electrodes of 3 cm diameter separated by quartz spacers (100 lm thick). The empty capacitance, C0 of this cell is 62 pF. In order to achieve an enough good signal-to-noise ratio, voltage fluctuations have to be amplified, taking care of reducing the additional noise coming from the amplifier. As a first step, the commercial low noise amplifier Stanford Research 550 was directly used at the end of the capacitor to amplify the voltage fluctuations. In Fig. 1 current (ei) and voltage (ev) spectral densities coming from the electric noise of the SR550 amplifier are compared to the Johnson– Nyquist noise spectral density expected according to Eq. (3) from the impedance of our cell filled with TPMTGE at T = 293 K: for frequencies below 1 kHz, the signal is dominated by the current generator noise of the SR550 amplifier. Then, in order to extend the available spectral band to lower frequencies, we set up a home-made low-noise preamplifier based on an Analog Devices 549 L ultra-low current noise junction field-effect transistor op-amp. The scheme of the electronic setup is shown in Fig. 1, as well as the current (ei) and voltage (ev) spectral densities of the home-made amplifier: the available spectral band for measurements now is extended down to less 1 Hz, as the current
noise level of the AD549 has been reduced. On the other hand, the voltage noise level is increased, limiting the high frequency band at 1 kHz. However, by combining the two setups, it is possible to measure the sample polarization fluctuations on a frequency range from less than 1 Hz up to more than 10 kHz. As in the present experiment our interest was restricted to low frequency, the scheme shown in Fig. 1 is related to the home-made low-noise setup (based on AD549), whose signal was further amplified by the commercial low-noise amplifier. The feedback impedance (Z2) on the AD549 amplifier consists of a 100 GX resistance in parallel of a capacitance of 1 pF. The input capacitance Z1 is the capacitor filled by the sample under test. Before each measure of the output fluctuation, the transfer function of the system was recorded by applying a weak sinusoidal signal to the input capacitance Z1 (switch of the circuit in the position 1 (Fig. 1)). The transfer function usually displayed an almost constant gain over the all bandwidth (1 Hz – 10 kHz). Then, switching the circuit to position 2, the amplified output total voltage fluctuations were measured with a signal analyzer.2 The test cell was connected to the amplification circuit by minimizing the length of the connections and the stray capacitance. All the system, capacitor and electronic circuit, was placed in a shielded cryogenic chamber with double walls (vacuum thermally insulated). The electronic circuit was kept at 298 K in a separate box, whereas the capacitor was placed in a nitrogen flow controlled by a PID controller.3 In order to avoid excess noise introduced by vibrations induced by the gas, the flow was reduced: a flow generated by a gradient of 10–30 mbar had a negligible effect on the noise spectral density measurement but it was enough to control the temperature in the range 270–320 K with accuracy better than 0.1 K, high stability (several hours) and reproducibility, and able to cool with rate between 0.05 and 2 K/min (enough to induce strong out-of-equilibrium states in our system). In order to avoid extra electric noise coming from thermocouples or platinum thermometers, the temperature of the capacitor was measured using a fluoroptic thermometer, consisting of an optical fiber (inserted inside a capacitor plate) measuring a T-dependent fluorescence of a phosphor sensor. This technology achieves an accuracy better than ±0.1 K and a measurement range of 170–570 K. 3. Results
Fig. 1. Square root of the power spectral density for current (dotted lines) and voltage (continuous lines) noise of the used amplifiers. The spectral density of the voltage fluctuation of the capacitor filled with TPMTGE at 293 K is shown as open triangles. In the inset the scheme of the homemade low-noise amplifier is shown.
Measurements of noise power spectral density were performed in the range 0.1 Hz–1 kHz at different temperatures above and below the glass transition temperature of TPMTGE. Fig. 2 shows some selected spectra acquired above Tg. The total spectral density (displayed as symbols) SV = hjVoutj2i measured at different temperatures is shown together with the expected spectra (continuous lines)
2 3
Stanford Research SR785. Lake Shore Model 330.
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
4923
Fig. 3. Noise power spectral density of the signal versus frequency acquired for temperatures: 283 K (black squares), 280 K (open circles), 278 K (black triangles). Continuous lines represent the total expected spectral density. Dashed lines represent the noise spectral density predicted by Johnson–Nyquist’s formula. Arrows indicate the direction of decreasing temperature. Inset shows the evolution of spectral density with aging after a quenching to 278 K for different waiting times tw.
Fig. 2. (a) Noise power spectral density of the signal acquired for different temperatures: square symbols (313, 311, 309, 307, 305, 303 K). Arrow indicates the direction of decreasing temperature. Continuous lines are the total expected signal composed of the instrumental background noise and of the noise spectral density predicted by Johnson–Nyquist’s formula. (b) Noise power spectral density at 303 K together with the total expected signal (continuous line), the Johnson–Nyquist contribution (dashed line) and the instrumental background noise (dotted line). (c) Noise power spectral density of the signal acquired for different temperatures: square symbols (301, 299, 297, 293, 289 K). Arrow indicates the direction of decreasing temperature. Continuous lines are the total expected signal composed of the instrumental background noise and of the noise spectral density predicted by Johnson–Nyquist’s formula. (d) Noise power spectral density at 293 K together with the total expected signal (continuous line), the Johnson–Nyquist contribution (dashed line) and the instrumental background noise (dotted line).
consisting of the sum of the noise spectral density Sa of the amplifier (dotted lines in Fig. 2(b)–(d)) and of the Johnson– Nyquist voltage noise (dashed lines) of the impedance SZ(x) = 4kBTRe[Z(x)], where T is the temperature of the sample and Z(x) is the complex impedance of the capacitor filled by the sample, measured by common dielectric spectroscopy [28]. The instrumental noise background Sa was measured by assessing the voltage and current contribution to the amplifier noise spectral density, using the procedure of replacing the sample capacitor with an empty capacitor with the same value of C at that temperature (see Ref. [24] for details). Spectral density data of Fig. 2 demonstrate
that FDT predictions are quite well satisfied, as expected when the material is at the thermodynamic equilibrium. A good agreement is obtained over most of the frequency–temperature range. It is noteworthy that the Johnson–Nyquist voltage noise of the sample impedance is well detectable over the instrumental noise background for a range of at least two decades, that shifts to lower frequency on decreasing the temperature (see Fig. 2(b)–(d)). An interesting behavior is the increase of the total spectral density with decreasing temperature (up to 303 K), followed by a decrease related to the passing, on cooling, of the maximum of dielectric loss peak through the experimental window. At the lower temperature (289 K), quite close to Tg, the low frequency data show an extra-noise contribution, exceeding the expected signal well above the background noise. Such disagreement with the FDT predictions is even more evident for measurements performed below Tg (Fig. 3). The sample was cooled down to the measurement temperature with a cooling rate of 0.05 K/min and then isothermally annealed for 30 min before the measurements. After this treatment, the spectral density of the sample did not change appreciably on the time scale of tens of minutes. Strikingly, the extra noise contribution at low frequency was increasing on decreasing temperature, in opposite direction of the change of the predicted Johnson–Nyquist voltage noise. The low frequency noise appeared to be bigger and bigger, higher was the degree of out-of-equilibrium of the system. The spectral density had an inverse power law behavior at lower frequencies. Inset of Fig. 3 shows the behavior of spectral density at 278 K during the isothermal evolution after a quenching at 2 K/min. The intensity and the steepness of the extra noise contribution were much bigger in this case than for the case of the slowly cooled sample and both intensity and steepness decreased
4924
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
with waiting time. On the other hand, for all the temperatures, the measured spectral density for frequencies higher than 10 Hz well agreed with the expected signal. These results parallel with what was recently found on polymeric systems below Tg [26], with Teff reaching huge values if calculated from low frequency data and substantially Teff = T for frequencies higher than 10 Hz. Summarizing, a good agreement was found between the measured voltage spectral density and the Johnson–Nyquist predictions for the system at equilibrium. However, below Tg and in the low frequency range, an intense polarization noise was detected, much greater than those predicted by fluctuation–dissipation theorem. In this region, the power spectral density of the polarization fluctuations showed a power law behavior. 4. Discussion The evidence that fluctuation–dissipation theorem holds for the equilibrium liquid, whereas a substantial FDT violation occurs in the glassy state, should be better validated by calculating directly the ratio between Johnson–Nyquist spectral density SZ(x) and the real part of the impedance (see Eq. (3)). The noise spectral density of the amplifier Sa was subtracted from total spectral density SV in order to obtain SZ: in this way, Teff(x) can be calculated using the relation of Eq. (3). When the quantity SV Sa was lower than the standard deviation of SV, the value for SZ was not considered since not reliable. Results are shown in Fig. 4, where the ratio was estimated for three different frequencies: (a) at the highest frequency (20 Hz) the FDT relation holds with Teff = T above and below Tg (see inset); (b) at the lowest frequency (0.2 Hz) a strong deviation from the FDT prediction occurs just few degrees above Tg and Teff reaches huge values (>105 K) at low temperatures; (c) at 2 Hz the deviation from FDT occurs below Tg and the values of Teff are intermediate compared to the cases (a)
Fig. 4. Main figure: effective temperature Teff = (SV Sa)/[4kBRe(Z)] calculated at three different frequencies over the whole temperature range. Lines are guide for the eyes. Inset shows an enlarged view for the range above Tg, straight line represents Teff = T. Error bars were omitted when smaller than the symbol size.
and (b). It is evident that, for the out-of-equilibrium system, the total voltage fluctuation spectral density is of some orders of magnitude much more intense than both the amplifier noise spectral density Sa and the impedance spectral density SZ(x) predicted by Johnson–Nyquist’s formula. Teff(x) = (SV Sa)/[4kBRe(Z)] shows, at low frequencies, an inverse power law behavior xc, with 1.2 < c < 1.8, in agreement with some results in literature [24,26]. The intensity of such extra-contribution of noise is decreasing with aging time, but its persistence can last for several hours at the lower temperature (278 K), much more than the endurance of our cryogenic apparatus. Such phenomena should be related to the aging occurring in the system below Tg and to the structural evolution during aging, when the slow modes of the system, unable to follow the quench below Tg, are relaxing towards equilibrium. On the other hand, the strong violation of FDT occurring at low frequency, particularly strong during the first instants of aging, gave values of Teff almost infinite and changing with the probe frequency x. This behavior is not compatible with the predictions of most of the theories of aging (see Ref.[4] for an overview), that predict mainly a more or less sharp crossover between two time-scales, one fast for which Teff = T and one slow for which Teff is higher than T but in any case lower than the temperature of the glass transition or, anyway, than the temperature of the starting point. In other terms, the uprising of Teff at low temperatures shown in Fig. 4 should not occur: on the contrary Teff should attain a constant value. Indeed, a similar experiment to ours, but performed only at a single frequency of 7 Hz [25], reported a small violation of FDT, with Teff higher than T but lower than Tg according to the predictions of aging theories. But it is noteworthy that also in our case only a very small violation of FDT could be detected at 7 Hz, whereas even at 2 Hz the violation is strong. In a recent aging experiment of a spin-glass system [19] a value of Teff ffi 5Tg was measured. Almost infinite values for Teff were reported for aging polymers [26] and colloidal glasses [21,30]. A large Teff at low frequency is not peculiar in our experiment but it has been observed and predicted for domain growth models [3,31] and activated dynamics models [17,32]. An interpretation on the connection between such predictions and our evidences can be found in Ref. [6], where the evolution of the system close to Tg in deeper and deeper valleys of the energy landscape was associated to rare jumps between traps, giving rise to intermittent dynamics. The intermittency could cause the strong violation of FDT at low frequency and could be accountable for the huge values of Teff. A clear signal of intermittency can be found by inspecting the time series of voltage fluctuations V(t), where very large and rare peaks or ‘bursts’ can be observed over the mean square noise (see examples in [26]). In our case such events, although present, are quite rare and not so intense. Our voltage time-series are rather characterized by the occurrence of random slow modulations (see examples in Fig. 6(b)). Anyway, a clear indication of non-Gaussian behavior can come from a plot of
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
the voltage probability density function versus the voltage amplitude. In the case of Ref. [26] a strong non-Gaussian probability function (with asymmetric profile) was reported for out-of-equilibrium state and a Gaussian shape was recovered during aging. Fig. 5 shows the voltage noise probability P(V) versus voltage amplitude measured on TPMTGE over voltage time series of 120 s acquired at 278 K for different cooling rate and waiting time. For the glass obtained after a strong quench, a non-Gaussian behavior was evident, especially far from the maximum
Fig. 5. Voltage noise probability P(V) versus voltage fluctuation amplitude measured over voltage time series at 278 K for different cooling rate and waiting time. Lines represent Gaussian fitting curves.
4925
(Gaussian fit has v2m ¼ 6:1), with P(V) that was narrowing during the aging process and eventually reached a Gaussian shape. On the other hand, if slowly cooled, the sample showed a Gaussian behavior (v2m ¼ 0:63). Moreover, P(V) plot for T > Tg gave always Gaussian distribution centered on zero. So the presence of an extra-noise contribution in low frequency spectral density for out-of-equilibrium states seems to be related to the non-Gaussian behavior of the voltage fluctuations. However, a non-Gaussian statistics could look quite odd for the macroscopic size of our sample (100 mm3). The same concern could be applied to the other cases (of macroscopic samples) of non-Gaussian statistics reported in literature [21,26]. In order to verify if our result is truthful, it is important to avoid in our experiments artifacts or extrinsic effects that could give rise to a non-Gaussian statistics, like the possible occurrence of steps or drifts in the voltage time-series. In our case no step and no appreciable drift were revealed in the voltage time series of 120 s, from which the probabilities P(V) of Fig. 5 were extracted. Moreover, the non-Gaussian character of the fluctuations can be observed even for a short time series, without occurrence of peaks or ‘bursts’: a Jarque– Bera test performed on the fluctuations of Fig. 6(b), recorded over a segment of 4 s in out-of-equilibrium state, rejected the hypothesis of a normal distribution, as resulted from the values of skewness (0.09) and kurtosis (2.60). Besides, the deviation from the Gaussian behavior is evident at the edges of a normal probability plot (Fig. 6(a)), constructed by plotting the sorted values of voltage of Fig. 6(b) versus the associated theoretical values from the
Fig. 6. (a) Normal probability plot and (b) voltage vs. time series at 30 min. of time elapsed after a quenching at 278 K. (c) and (d) show the same plots at equilibrium after a slow cooling at 278 K. Crosses are the experimental values. Straight lines in (a) and (c) show the expected values for a normal distribution. Voltage values are amplified by a factor of 1800, gain of the amplifier.
4926
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
standard normal distribution. On the other hand, voltage time series (Fig. 6(d)) recorded at equilibrium state yielded normal distribution statistics (Fig. 6(c)). All indicates that the non-Gaussian statistics has a real origin and it is not a result of artifacts. Recent numerical simulation works have determined a non-Gaussian behavior of the voltage fluctuations coupled to FDT violation [33]. Such non-Gaussian behavior together with the values of effective temperature exceeding by four orders of magnitude the temperature of the thermal bath were interpreted also in the framework of coarsening dynamics (where Teff tends to infinite) [34]. Anyway, a conclusive explanation of the origin of these high values for Teff is still unknown. In fact, contrary to what would be expected, no significant detection of FDT violations resulted from noise measurements in a rheological experiment for colloidal glasses, whereas in the same systems dielectric measurement reported FDT violation and intermittent noise [26,27]. However, the presence of this strong, intermittent, bursting noise, although it is not yet a wellunderstood phenomenon, deserves to be further accurately studied. Additionally, more experiments clearly demonstrating the existence of effective temperatures related to FDT violations, also in absence of intermittent behavior, are needed. Our experimental setup could be useful to extend such investigations to many glass-forming systems.
copy. The polarization noise followed the predictions of the fluctuation–dissipation theorem which connects noise and susceptibility when the material is at the thermodynamic equilibrium or weakly out. At temperatures well below the glass transition an intense polarization noise was detected, much more than those predicted by permittivity using the fluctuation–dissipation theorem. In this region, the power spectral density of the polarization fluctuation showed an inverse power law form and unusual aging effects were observed. Such extra-noise contribution decreased with aging. Moreover a non-Gaussian distribution of the probability density function of the polarization fluctuations was observed immediately after a rapid quenching of the sample below the glass transition temperature. The width of the distribution reduced during the aging process and the its shape tended towards a Gaussian distribution as the equilibrium state was approached. These evidences were compared to recent theories of aging. Acknowledgements This research was supported at Pisa by MIUR (PRIN 2005). The authors thank Mr Alfio Pistoresi for his help in the development of the home-made electronic setup. References
5. Conclusions A recent intense interest has risen in glass transition community about the nature of fluctuations of global quantities, such as power dissipation, in systems held far from equilibrium. It is of particular importance how fluctuating excitations on different spatial and temporal scales give rise to variations in globally-measured quantities with particular regard to the probability density function of the fluctuations of these global quantities. Therefore, it is important to develop experimental setups able to measure simultaneously fluctuations and response to external applied field of a selected observable linked to structural motions. Polarization has been proved to be an excellent choice for the observable. So we developed an apparatus able to acquire simultaneously the complex permittivity by dielectric spectroscopy and the polarization fluctuations observed via voltage noise, produced by a capacitor cell filled with the glass-former under test. A low noise current–voltage converter, home made, in series with an ultra low noise pre-amplifier gave to the apparatus high sensitivity and accuracy in the low frequency region (0.1–100 Hz). As a test, measurements of thermal dielectric noise were performed on an epoxy organic glass former both above and below the glass transition temperature. The sample was driven to the vitreous state by cooling with different rapidity and then it was thermally annealed, while the power spectral density of noise was measured at different times and compared with related spectra of the imaginary part of permittivity determined by usual dielectric spectros-
[1] L.C.E. Struik, Physical Aging in Amorphous Polymers and Other Materials, Elsevier, Amsterdam, 1978. [2] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer Ser. Solid-State Sci., Berlin, vol. 31, 1992. [3] L.F. Cugliandolo, J. Kurchan, L. Peliti, Phys. Rev. E 55 (1997) 3898. [4] L. Cipelletti, L. Ramos, J. Phys.: Condens. Matter 17 (2005) R253. [5] L.F. Cugliandolo, J. Kurchan, Phys. Rev. Lett. 71 (1993) 173. [6] G. Parisi, Phys. Rev. Lett. 79 (1997) 3660. [7] M. Sellitto, Eur. Phys. J. B 4 (1998) 135. [8] E. Marinari, G. Parisi, F. Ricci-Tersenghi, J.J. Ruiz-Lorenzo, J. Phys. A: Math. Gen. 31 (1998) 2611. [9] F. Sciortino, P. Tartaglia, Phys. Rev. Lett. 86 (2001) 107. [10] J.L. Barrat, W. Kob, Europhys. Lett. 46 (1999) 637. [11] W. Kob, J.L. Barrat, Eur. Phys. J. B 13 (2000) 319. [12] J.L. Barrat, L. Berthier, Phys. Rev. E 63 (2001) 0125031. [13] S.M. Fielding, P. Sollich, Phys. Rev. E 67 (2003) 011101. [14] L. Berthier, J.L. Barrat, Phys. Rev. Lett. 89 (2002) 095702. [15] L. Berthier, J.L. Barrat, J. Chem. Phys. 116 (2002) 6228. [16] L. Berthier, J.L. Barrat, J. Kurchan, Phys. Rev. E 61 (2000) 5464. [17] S.M. Fielding, P. Sollich, Phys. Rev. Lett. 88 (2002) 050603. [18] C.S. O’Hern, A.J. Liu, S.R. Nagel, Phys. Rev. Lett. 93 (2004) 165702. [19] D. Herisson, M. Ocio, Phys. Rev. Lett. 88 (2002) 257702. [20] B. Abou, F. Gallet, Phys. Rev. Lett. 93 (2004) 160603. [21] L. Buisson, S. Ciliberto, A. Garcimartin, Europhys. Lett. 63 (2003) 603. [22] N.E. Israeloff, Phys. Rev. B 53 (1996) R11913. [23] N.E. Israeloff, X. Wang, Rev. Sci. Instrum. 68 (1997) 1543. [24] N.E. Israeloff, T.S. Grigera, Europhys. Lett. 43 (1999) 308. [25] T.S. Grigera, N.E. Israeloff, Phys. Rev. Lett. 81 (1999) 5038. [26] L. Buisson, L. Bellon, S. Ciliberto, J. Phys.: Condens. Matter 15 (2003) S1163. [27] L. Bellon, S. Ciliberto, Physica D 168 (2002) 325–335. [28] D. Pisignano, S. Capaccioli, R. Casalini, M. Lucchesi, P.A. Rolla, A. Justl, E. Ro¨ssler, J. Phys.: Condens. Matter 13 (2001) 4405.
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927 [29] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113. [30] L. Bellon, S. Ciliberto, C. Laroche, Europhys. Lett. 53 (2001) 511. [31] A. Barrat, Phys. Rev. E 57 (1998) 3629.
4927
[32] A. Perez-Madrid, D. Reguera, J.M. Rubi, 2002, cond-mat/0210089. [33] A. Crisanti, F. Ritort, Europhys. Lett. 66 (2004) 253. [34] A. Crisanti, F. Ritort, J. Phys. A: Math. Gen. 36 (2003) R181.
Polarization fluctuations near the glass transition M. Lucchesi a
a,c,* ,
A. Dominjon a, S. Capaccioli
a,b
, D. Prevosto
a,c
, P.A. Rolla
a,c
Dipartimento di Fisica and INFM, Universita` di Pisa, Largo Bruno Pontecorvo, 3, I-56127 Pisa, Italy b CNR-INFM, CRS SOFT, Piazzale Aldo Moro 2, I-00185 Roma, Italy c CNR-INFM, Polylab, Largo Bruno Pontecorvo, 3, I-56127 Pisa, Italy Available online 7 September 2006
Abstract Measurements of polarization fluctuations were performed for an epoxy glass former. The voltage noise produced by polarization fluctuations of the sample filling a capacitor was acquired via a home-made very high input impedance current–voltage converter, in series with an ultra low noise pre-amplifier, achieving high sensitivity and accuracy in the range 0.1–1000 Hz. The temperature and frequency dependence of polarization noise was investigated above and below the glass transition temperature Tg. The sample was driven to the glassy state with different cooling rate and then isothermally aged, while the noise spectral density was measured at different times and compared with the Johnson–Nyquist noise determined by the sample impedance measured by conventional dielectric spectroscopy. At thermodynamic equilibrium the polarization noise agreed with the predictions of the fluctuation–dissipation theorem linking noise spectral density to susceptibility. On the contrary, a strong violation of the theorem was observed after a fast cooling below Tg: an intense polarization noise was detected, with a power spectral density following an inverse power law frequency behavior, whose intensity was decreasing with aging time. At the same time, the amplitude of polarization fluctuations showed a non-Gaussian distribution, whose width reduced during the aging process, up to recover the Gaussian statistics on approaching the equilibrium. Ó 2006 Elsevier B.V. All rights reserved. PACS: 64.70.Pf; 77.22.Gm; 05.40.a Keywords: Dielectric properties; Relaxation, electric modulus; Glass transition; Fluctuation-dissipation theorem
1. Introduction Systems characterized by slow dynamics can be driven out of thermodynamic equilibrium by means of a rapid change of appropriate external variables, for instance a temperature jump taking a liquid down to a glassy state. After such treatment, the system spontaneously tends to recover the equilibrium state, and its properties (dynamic, thermodynamic, mechanical) depend on the waiting time tw, elapsed after the achievement of the out-of-equilibrium state [1]. The evolution towards equilibrium with time is commonly called ‘aging’. In an out-of-equilibrium system * Corresponding author. Address: Dipartimento di Fisica and INFM, Universita` di Pisa, Largo Bruno Pontecorvo, 3, I-56127 Pisa, Italy. Tel.: +39 0502214514; fax: +39 0502214333. E-mail address: [email protected] (M. Lucchesi).
0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.02.153
the response to a small external perturbation, especially at long time, is no longer related, via the temperature of the thermal bath, to the spontaneous fluctuations of the observable under test, as predicted by the fluctuation–dissipation theorem (FDT) [2], depending themselves on the thermal history of the system. In such a situation the concept of temperature of the system is not well defined any more, and the response of the system to an external perturbation is unable to give new insight into the molecular dynamics. Therefore, from an experimental point of view, it is of importance not only to study the response of the system to external fields but also its thermal fluctuations. Another significant reason of interest in the simultaneous study of response and fluctuations of a system out-of-equilibrium involves the definition of temperature. Indeed a description of weakly out-of-equilibrium dynamics has been recently proposed, extending the concept of temperature by
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
a generalization of the fluctuation–dissipation theorem (FDT) [3]. According to FDT the response R(t) to an external perturbation h is related to the normalized time autocorrelation function C(t) of the observable A to which the perturbation field h is conjugated: for systems at equilibrium, the time derivative of the correlation function is proportional to the response: dCðtÞ ¼ k B TRðtÞ dt
ð1Þ
where kB is the Boltzmann constant and T is the temperature of the thermal bath contacting the system. Eq. (1) provides an alternative definition for the concept of ‘temperature of the system’ [2]. As FDT is derived by assuming time-translation invariance [2], its violation is expected for out-of-equilibrium systems: in this case, a generalization of FDT has been proposed [3,5] by introducing an effective temperature, Teff, that should replace T in Eq. (1) as a proportionality constant between the time derivative of the correlation function and the response function dCðt; tw Þ T eff ðt; tw Þ ¼ ð2Þ k B Rðt; tw Þ dt At equilibrium, of course Teff = T. It is noteworthy that Teff is expected to depend on the time-scale t (or on the angular frequency x) of observation, on the waiting time tw during aging and to be higher than the temperature of the thermal bath contacting the system [5]. Indeed in a ‘slow dynamics’ system driven out-of-equilibrium it is possible to find fast processes, that equilibrate with thermal bath and attain Teff = T, and slower processes, that slowly relax towards the equilibrium and keep memory of higher temperature configurations for a longer time, and that are characterized by Teff higher than the temperature of thermal bath. A study of evolution of Teff(x, tw) and its dependence on the observation time-scale and aging time could verify the conflicting predictions on FDT violation of the different models of aging and out-of-equilibrium dynamics [3,4,6–8]. A number of recent simulation works deal with the study of FDT violation in out-of-equilibrium state for aging systems [9–11] and also for stationary systems undergoing strong external perturbation (e.g. shear deformations) [12,13]: numerical simulations, in agreement with theoretical models, predict that, in an out-of-equilibrium system, fluctuations of dynamic variables on very long time scales (comparable to those of structural relaxation) feel a fictive temperature Teff (defined by the generalized FDT) higher than T (thermal bath temperature), whereas on short time scales (comparable to those of vibrational or cage rattling motions) Teff = T [3,9–18]. The value of Teff depends on tw (for aging systems) or, in the case of stationary out-of-equilibrium states, on the frequency of the external driving [3,10–18]. On the contrary, few experimental works verifying the violation or the generalization of FDT in out-of-equilibrium systems have been performed (for an extensive review see Ref.[4]). Some examples concern aging spin glasses [19]
4921
and colloidal solutions during the liquid-gel transition [20,21], but studies on structural glasses are almost lacking. Concerning this issue, the measurements of polarization at low frequencies (or long times), by means of dielectric spectroscopy to characterize the response function and by means of electric noise measurements to acquire the fluctuations, proved to be well suitable, so that up till now the only experimental checks of FDT violation or generalization in structural glasses deal with polarization as observable under test. Usually the voltage spontaneous fluctuations of a capacitor filled by the material under test (with complex permittivity e = e 0 je00 and empty capacitance C0) are acquired and compared with the Johnson– Nyquist noise determined by electric impedance Z(x) of the capacitor assessed by dielectric spectroscopy [22–27]. In this case FDT has the form of Nyquist’s theorem that for the voltage fluctuation spectral density SV(x) = hjV(x)j2i predicts S V ðx; tw Þ ¼ 4k B T eff ðx; tw ÞRe½Zðx; tw Þ ¼ 4k B T eff ðx; tw Þ
e00 ðx; tw Þ 1 e0 ðx; tw Þ e0 ðx; tw ÞC 0 x
1 00 2 e ðx;tw Þ 1 þ e0 ðx;tw Þ
ð3Þ
Investigations performed above the temperature of glass transition Tg on molecular and polymeric glass-formers verified Eq. (3) with Teff = T [22,23]. On the contrary, for glycerol, studies done few degrees below Tg during the instants following a quenching revealed a weak FDT violation, characterized by Teff slowly decreasing with time until the value of the temperature T was reached [25]. In a broadband study on polycarbonate, a stronger violation of FDT was observed after fast cooling from liquid to glassy state [26,27]. The amplitude and the persistence time of this violation resulted to be decreasing functions of frequency and to depend on the cooling rate. The origin of such a violation has been ascribed to the presence of a highly intermittent dynamics characterized by large fluctuations with amplitude distributed according to a strongly non-Gaussian statistics [21,26,27]. The aim of our paper is to present a new experimental setup able to acquire voltage fluctuations and impedance and to assess the ratio Teff over a wide frequency range. Measurements have been performed on an epoxy glass-former in both equilibrium liquid state and out-of-equilibrium aging glass. 2. Experimental procedures Triphenylolmethane triglycidyl ether (TPMTGE) was commercially available1 and used as received. Its molecular weight is 460.5 g mol1 and Tg is 287 K. Further details on 1
Sigma–Aldrich.
4922
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
the thermodynamic and dielectric properties of this epoxy compound can be found in Ref. [28]. This sample is particularly suitable for at least three reasons: (a) Tg is almost at room temperature, making easier the environment for the experimental setup; (b) it is very fragile according to Angell classification [29], i.e. very sensitive to temperature, allowing to get a strong out-of-equilibrium state not too much below Tg even for moderate quenching; (c) it has appreciable dielectric losses e00 /e 0 in the investigated frequency–temperature range, so enabling noise power spectral density (see Eq. (3)) to be measured. In the present experiment, polarization fluctuations were observed via voltage thermal noise produced by a capacitor cell filled with the material. The cell consists of two metallic electrodes of 3 cm diameter separated by quartz spacers (100 lm thick). The empty capacitance, C0 of this cell is 62 pF. In order to achieve an enough good signal-to-noise ratio, voltage fluctuations have to be amplified, taking care of reducing the additional noise coming from the amplifier. As a first step, the commercial low noise amplifier Stanford Research 550 was directly used at the end of the capacitor to amplify the voltage fluctuations. In Fig. 1 current (ei) and voltage (ev) spectral densities coming from the electric noise of the SR550 amplifier are compared to the Johnson– Nyquist noise spectral density expected according to Eq. (3) from the impedance of our cell filled with TPMTGE at T = 293 K: for frequencies below 1 kHz, the signal is dominated by the current generator noise of the SR550 amplifier. Then, in order to extend the available spectral band to lower frequencies, we set up a home-made low-noise preamplifier based on an Analog Devices 549 L ultra-low current noise junction field-effect transistor op-amp. The scheme of the electronic setup is shown in Fig. 1, as well as the current (ei) and voltage (ev) spectral densities of the home-made amplifier: the available spectral band for measurements now is extended down to less 1 Hz, as the current
noise level of the AD549 has been reduced. On the other hand, the voltage noise level is increased, limiting the high frequency band at 1 kHz. However, by combining the two setups, it is possible to measure the sample polarization fluctuations on a frequency range from less than 1 Hz up to more than 10 kHz. As in the present experiment our interest was restricted to low frequency, the scheme shown in Fig. 1 is related to the home-made low-noise setup (based on AD549), whose signal was further amplified by the commercial low-noise amplifier. The feedback impedance (Z2) on the AD549 amplifier consists of a 100 GX resistance in parallel of a capacitance of 1 pF. The input capacitance Z1 is the capacitor filled by the sample under test. Before each measure of the output fluctuation, the transfer function of the system was recorded by applying a weak sinusoidal signal to the input capacitance Z1 (switch of the circuit in the position 1 (Fig. 1)). The transfer function usually displayed an almost constant gain over the all bandwidth (1 Hz – 10 kHz). Then, switching the circuit to position 2, the amplified output total voltage fluctuations were measured with a signal analyzer.2 The test cell was connected to the amplification circuit by minimizing the length of the connections and the stray capacitance. All the system, capacitor and electronic circuit, was placed in a shielded cryogenic chamber with double walls (vacuum thermally insulated). The electronic circuit was kept at 298 K in a separate box, whereas the capacitor was placed in a nitrogen flow controlled by a PID controller.3 In order to avoid excess noise introduced by vibrations induced by the gas, the flow was reduced: a flow generated by a gradient of 10–30 mbar had a negligible effect on the noise spectral density measurement but it was enough to control the temperature in the range 270–320 K with accuracy better than 0.1 K, high stability (several hours) and reproducibility, and able to cool with rate between 0.05 and 2 K/min (enough to induce strong out-of-equilibrium states in our system). In order to avoid extra electric noise coming from thermocouples or platinum thermometers, the temperature of the capacitor was measured using a fluoroptic thermometer, consisting of an optical fiber (inserted inside a capacitor plate) measuring a T-dependent fluorescence of a phosphor sensor. This technology achieves an accuracy better than ±0.1 K and a measurement range of 170–570 K. 3. Results
Fig. 1. Square root of the power spectral density for current (dotted lines) and voltage (continuous lines) noise of the used amplifiers. The spectral density of the voltage fluctuation of the capacitor filled with TPMTGE at 293 K is shown as open triangles. In the inset the scheme of the homemade low-noise amplifier is shown.
Measurements of noise power spectral density were performed in the range 0.1 Hz–1 kHz at different temperatures above and below the glass transition temperature of TPMTGE. Fig. 2 shows some selected spectra acquired above Tg. The total spectral density (displayed as symbols) SV = hjVoutj2i measured at different temperatures is shown together with the expected spectra (continuous lines)
2 3
Stanford Research SR785. Lake Shore Model 330.
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
4923
Fig. 3. Noise power spectral density of the signal versus frequency acquired for temperatures: 283 K (black squares), 280 K (open circles), 278 K (black triangles). Continuous lines represent the total expected spectral density. Dashed lines represent the noise spectral density predicted by Johnson–Nyquist’s formula. Arrows indicate the direction of decreasing temperature. Inset shows the evolution of spectral density with aging after a quenching to 278 K for different waiting times tw.
Fig. 2. (a) Noise power spectral density of the signal acquired for different temperatures: square symbols (313, 311, 309, 307, 305, 303 K). Arrow indicates the direction of decreasing temperature. Continuous lines are the total expected signal composed of the instrumental background noise and of the noise spectral density predicted by Johnson–Nyquist’s formula. (b) Noise power spectral density at 303 K together with the total expected signal (continuous line), the Johnson–Nyquist contribution (dashed line) and the instrumental background noise (dotted line). (c) Noise power spectral density of the signal acquired for different temperatures: square symbols (301, 299, 297, 293, 289 K). Arrow indicates the direction of decreasing temperature. Continuous lines are the total expected signal composed of the instrumental background noise and of the noise spectral density predicted by Johnson–Nyquist’s formula. (d) Noise power spectral density at 293 K together with the total expected signal (continuous line), the Johnson–Nyquist contribution (dashed line) and the instrumental background noise (dotted line).
consisting of the sum of the noise spectral density Sa of the amplifier (dotted lines in Fig. 2(b)–(d)) and of the Johnson– Nyquist voltage noise (dashed lines) of the impedance SZ(x) = 4kBTRe[Z(x)], where T is the temperature of the sample and Z(x) is the complex impedance of the capacitor filled by the sample, measured by common dielectric spectroscopy [28]. The instrumental noise background Sa was measured by assessing the voltage and current contribution to the amplifier noise spectral density, using the procedure of replacing the sample capacitor with an empty capacitor with the same value of C at that temperature (see Ref. [24] for details). Spectral density data of Fig. 2 demonstrate
that FDT predictions are quite well satisfied, as expected when the material is at the thermodynamic equilibrium. A good agreement is obtained over most of the frequency–temperature range. It is noteworthy that the Johnson–Nyquist voltage noise of the sample impedance is well detectable over the instrumental noise background for a range of at least two decades, that shifts to lower frequency on decreasing the temperature (see Fig. 2(b)–(d)). An interesting behavior is the increase of the total spectral density with decreasing temperature (up to 303 K), followed by a decrease related to the passing, on cooling, of the maximum of dielectric loss peak through the experimental window. At the lower temperature (289 K), quite close to Tg, the low frequency data show an extra-noise contribution, exceeding the expected signal well above the background noise. Such disagreement with the FDT predictions is even more evident for measurements performed below Tg (Fig. 3). The sample was cooled down to the measurement temperature with a cooling rate of 0.05 K/min and then isothermally annealed for 30 min before the measurements. After this treatment, the spectral density of the sample did not change appreciably on the time scale of tens of minutes. Strikingly, the extra noise contribution at low frequency was increasing on decreasing temperature, in opposite direction of the change of the predicted Johnson–Nyquist voltage noise. The low frequency noise appeared to be bigger and bigger, higher was the degree of out-of-equilibrium of the system. The spectral density had an inverse power law behavior at lower frequencies. Inset of Fig. 3 shows the behavior of spectral density at 278 K during the isothermal evolution after a quenching at 2 K/min. The intensity and the steepness of the extra noise contribution were much bigger in this case than for the case of the slowly cooled sample and both intensity and steepness decreased
4924
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
with waiting time. On the other hand, for all the temperatures, the measured spectral density for frequencies higher than 10 Hz well agreed with the expected signal. These results parallel with what was recently found on polymeric systems below Tg [26], with Teff reaching huge values if calculated from low frequency data and substantially Teff = T for frequencies higher than 10 Hz. Summarizing, a good agreement was found between the measured voltage spectral density and the Johnson–Nyquist predictions for the system at equilibrium. However, below Tg and in the low frequency range, an intense polarization noise was detected, much greater than those predicted by fluctuation–dissipation theorem. In this region, the power spectral density of the polarization fluctuations showed a power law behavior. 4. Discussion The evidence that fluctuation–dissipation theorem holds for the equilibrium liquid, whereas a substantial FDT violation occurs in the glassy state, should be better validated by calculating directly the ratio between Johnson–Nyquist spectral density SZ(x) and the real part of the impedance (see Eq. (3)). The noise spectral density of the amplifier Sa was subtracted from total spectral density SV in order to obtain SZ: in this way, Teff(x) can be calculated using the relation of Eq. (3). When the quantity SV Sa was lower than the standard deviation of SV, the value for SZ was not considered since not reliable. Results are shown in Fig. 4, where the ratio was estimated for three different frequencies: (a) at the highest frequency (20 Hz) the FDT relation holds with Teff = T above and below Tg (see inset); (b) at the lowest frequency (0.2 Hz) a strong deviation from the FDT prediction occurs just few degrees above Tg and Teff reaches huge values (>105 K) at low temperatures; (c) at 2 Hz the deviation from FDT occurs below Tg and the values of Teff are intermediate compared to the cases (a)
Fig. 4. Main figure: effective temperature Teff = (SV Sa)/[4kBRe(Z)] calculated at three different frequencies over the whole temperature range. Lines are guide for the eyes. Inset shows an enlarged view for the range above Tg, straight line represents Teff = T. Error bars were omitted when smaller than the symbol size.
and (b). It is evident that, for the out-of-equilibrium system, the total voltage fluctuation spectral density is of some orders of magnitude much more intense than both the amplifier noise spectral density Sa and the impedance spectral density SZ(x) predicted by Johnson–Nyquist’s formula. Teff(x) = (SV Sa)/[4kBRe(Z)] shows, at low frequencies, an inverse power law behavior xc, with 1.2 < c < 1.8, in agreement with some results in literature [24,26]. The intensity of such extra-contribution of noise is decreasing with aging time, but its persistence can last for several hours at the lower temperature (278 K), much more than the endurance of our cryogenic apparatus. Such phenomena should be related to the aging occurring in the system below Tg and to the structural evolution during aging, when the slow modes of the system, unable to follow the quench below Tg, are relaxing towards equilibrium. On the other hand, the strong violation of FDT occurring at low frequency, particularly strong during the first instants of aging, gave values of Teff almost infinite and changing with the probe frequency x. This behavior is not compatible with the predictions of most of the theories of aging (see Ref.[4] for an overview), that predict mainly a more or less sharp crossover between two time-scales, one fast for which Teff = T and one slow for which Teff is higher than T but in any case lower than the temperature of the glass transition or, anyway, than the temperature of the starting point. In other terms, the uprising of Teff at low temperatures shown in Fig. 4 should not occur: on the contrary Teff should attain a constant value. Indeed, a similar experiment to ours, but performed only at a single frequency of 7 Hz [25], reported a small violation of FDT, with Teff higher than T but lower than Tg according to the predictions of aging theories. But it is noteworthy that also in our case only a very small violation of FDT could be detected at 7 Hz, whereas even at 2 Hz the violation is strong. In a recent aging experiment of a spin-glass system [19] a value of Teff ffi 5Tg was measured. Almost infinite values for Teff were reported for aging polymers [26] and colloidal glasses [21,30]. A large Teff at low frequency is not peculiar in our experiment but it has been observed and predicted for domain growth models [3,31] and activated dynamics models [17,32]. An interpretation on the connection between such predictions and our evidences can be found in Ref. [6], where the evolution of the system close to Tg in deeper and deeper valleys of the energy landscape was associated to rare jumps between traps, giving rise to intermittent dynamics. The intermittency could cause the strong violation of FDT at low frequency and could be accountable for the huge values of Teff. A clear signal of intermittency can be found by inspecting the time series of voltage fluctuations V(t), where very large and rare peaks or ‘bursts’ can be observed over the mean square noise (see examples in [26]). In our case such events, although present, are quite rare and not so intense. Our voltage time-series are rather characterized by the occurrence of random slow modulations (see examples in Fig. 6(b)). Anyway, a clear indication of non-Gaussian behavior can come from a plot of
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
the voltage probability density function versus the voltage amplitude. In the case of Ref. [26] a strong non-Gaussian probability function (with asymmetric profile) was reported for out-of-equilibrium state and a Gaussian shape was recovered during aging. Fig. 5 shows the voltage noise probability P(V) versus voltage amplitude measured on TPMTGE over voltage time series of 120 s acquired at 278 K for different cooling rate and waiting time. For the glass obtained after a strong quench, a non-Gaussian behavior was evident, especially far from the maximum
Fig. 5. Voltage noise probability P(V) versus voltage fluctuation amplitude measured over voltage time series at 278 K for different cooling rate and waiting time. Lines represent Gaussian fitting curves.
4925
(Gaussian fit has v2m ¼ 6:1), with P(V) that was narrowing during the aging process and eventually reached a Gaussian shape. On the other hand, if slowly cooled, the sample showed a Gaussian behavior (v2m ¼ 0:63). Moreover, P(V) plot for T > Tg gave always Gaussian distribution centered on zero. So the presence of an extra-noise contribution in low frequency spectral density for out-of-equilibrium states seems to be related to the non-Gaussian behavior of the voltage fluctuations. However, a non-Gaussian statistics could look quite odd for the macroscopic size of our sample (100 mm3). The same concern could be applied to the other cases (of macroscopic samples) of non-Gaussian statistics reported in literature [21,26]. In order to verify if our result is truthful, it is important to avoid in our experiments artifacts or extrinsic effects that could give rise to a non-Gaussian statistics, like the possible occurrence of steps or drifts in the voltage time-series. In our case no step and no appreciable drift were revealed in the voltage time series of 120 s, from which the probabilities P(V) of Fig. 5 were extracted. Moreover, the non-Gaussian character of the fluctuations can be observed even for a short time series, without occurrence of peaks or ‘bursts’: a Jarque– Bera test performed on the fluctuations of Fig. 6(b), recorded over a segment of 4 s in out-of-equilibrium state, rejected the hypothesis of a normal distribution, as resulted from the values of skewness (0.09) and kurtosis (2.60). Besides, the deviation from the Gaussian behavior is evident at the edges of a normal probability plot (Fig. 6(a)), constructed by plotting the sorted values of voltage of Fig. 6(b) versus the associated theoretical values from the
Fig. 6. (a) Normal probability plot and (b) voltage vs. time series at 30 min. of time elapsed after a quenching at 278 K. (c) and (d) show the same plots at equilibrium after a slow cooling at 278 K. Crosses are the experimental values. Straight lines in (a) and (c) show the expected values for a normal distribution. Voltage values are amplified by a factor of 1800, gain of the amplifier.
4926
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927
standard normal distribution. On the other hand, voltage time series (Fig. 6(d)) recorded at equilibrium state yielded normal distribution statistics (Fig. 6(c)). All indicates that the non-Gaussian statistics has a real origin and it is not a result of artifacts. Recent numerical simulation works have determined a non-Gaussian behavior of the voltage fluctuations coupled to FDT violation [33]. Such non-Gaussian behavior together with the values of effective temperature exceeding by four orders of magnitude the temperature of the thermal bath were interpreted also in the framework of coarsening dynamics (where Teff tends to infinite) [34]. Anyway, a conclusive explanation of the origin of these high values for Teff is still unknown. In fact, contrary to what would be expected, no significant detection of FDT violations resulted from noise measurements in a rheological experiment for colloidal glasses, whereas in the same systems dielectric measurement reported FDT violation and intermittent noise [26,27]. However, the presence of this strong, intermittent, bursting noise, although it is not yet a wellunderstood phenomenon, deserves to be further accurately studied. Additionally, more experiments clearly demonstrating the existence of effective temperatures related to FDT violations, also in absence of intermittent behavior, are needed. Our experimental setup could be useful to extend such investigations to many glass-forming systems.
copy. The polarization noise followed the predictions of the fluctuation–dissipation theorem which connects noise and susceptibility when the material is at the thermodynamic equilibrium or weakly out. At temperatures well below the glass transition an intense polarization noise was detected, much more than those predicted by permittivity using the fluctuation–dissipation theorem. In this region, the power spectral density of the polarization fluctuation showed an inverse power law form and unusual aging effects were observed. Such extra-noise contribution decreased with aging. Moreover a non-Gaussian distribution of the probability density function of the polarization fluctuations was observed immediately after a rapid quenching of the sample below the glass transition temperature. The width of the distribution reduced during the aging process and the its shape tended towards a Gaussian distribution as the equilibrium state was approached. These evidences were compared to recent theories of aging. Acknowledgements This research was supported at Pisa by MIUR (PRIN 2005). The authors thank Mr Alfio Pistoresi for his help in the development of the home-made electronic setup. References
5. Conclusions A recent intense interest has risen in glass transition community about the nature of fluctuations of global quantities, such as power dissipation, in systems held far from equilibrium. It is of particular importance how fluctuating excitations on different spatial and temporal scales give rise to variations in globally-measured quantities with particular regard to the probability density function of the fluctuations of these global quantities. Therefore, it is important to develop experimental setups able to measure simultaneously fluctuations and response to external applied field of a selected observable linked to structural motions. Polarization has been proved to be an excellent choice for the observable. So we developed an apparatus able to acquire simultaneously the complex permittivity by dielectric spectroscopy and the polarization fluctuations observed via voltage noise, produced by a capacitor cell filled with the glass-former under test. A low noise current–voltage converter, home made, in series with an ultra low noise pre-amplifier gave to the apparatus high sensitivity and accuracy in the low frequency region (0.1–100 Hz). As a test, measurements of thermal dielectric noise were performed on an epoxy organic glass former both above and below the glass transition temperature. The sample was driven to the vitreous state by cooling with different rapidity and then it was thermally annealed, while the power spectral density of noise was measured at different times and compared with related spectra of the imaginary part of permittivity determined by usual dielectric spectros-
[1] L.C.E. Struik, Physical Aging in Amorphous Polymers and Other Materials, Elsevier, Amsterdam, 1978. [2] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer Ser. Solid-State Sci., Berlin, vol. 31, 1992. [3] L.F. Cugliandolo, J. Kurchan, L. Peliti, Phys. Rev. E 55 (1997) 3898. [4] L. Cipelletti, L. Ramos, J. Phys.: Condens. Matter 17 (2005) R253. [5] L.F. Cugliandolo, J. Kurchan, Phys. Rev. Lett. 71 (1993) 173. [6] G. Parisi, Phys. Rev. Lett. 79 (1997) 3660. [7] M. Sellitto, Eur. Phys. J. B 4 (1998) 135. [8] E. Marinari, G. Parisi, F. Ricci-Tersenghi, J.J. Ruiz-Lorenzo, J. Phys. A: Math. Gen. 31 (1998) 2611. [9] F. Sciortino, P. Tartaglia, Phys. Rev. Lett. 86 (2001) 107. [10] J.L. Barrat, W. Kob, Europhys. Lett. 46 (1999) 637. [11] W. Kob, J.L. Barrat, Eur. Phys. J. B 13 (2000) 319. [12] J.L. Barrat, L. Berthier, Phys. Rev. E 63 (2001) 0125031. [13] S.M. Fielding, P. Sollich, Phys. Rev. E 67 (2003) 011101. [14] L. Berthier, J.L. Barrat, Phys. Rev. Lett. 89 (2002) 095702. [15] L. Berthier, J.L. Barrat, J. Chem. Phys. 116 (2002) 6228. [16] L. Berthier, J.L. Barrat, J. Kurchan, Phys. Rev. E 61 (2000) 5464. [17] S.M. Fielding, P. Sollich, Phys. Rev. Lett. 88 (2002) 050603. [18] C.S. O’Hern, A.J. Liu, S.R. Nagel, Phys. Rev. Lett. 93 (2004) 165702. [19] D. Herisson, M. Ocio, Phys. Rev. Lett. 88 (2002) 257702. [20] B. Abou, F. Gallet, Phys. Rev. Lett. 93 (2004) 160603. [21] L. Buisson, S. Ciliberto, A. Garcimartin, Europhys. Lett. 63 (2003) 603. [22] N.E. Israeloff, Phys. Rev. B 53 (1996) R11913. [23] N.E. Israeloff, X. Wang, Rev. Sci. Instrum. 68 (1997) 1543. [24] N.E. Israeloff, T.S. Grigera, Europhys. Lett. 43 (1999) 308. [25] T.S. Grigera, N.E. Israeloff, Phys. Rev. Lett. 81 (1999) 5038. [26] L. Buisson, L. Bellon, S. Ciliberto, J. Phys.: Condens. Matter 15 (2003) S1163. [27] L. Bellon, S. Ciliberto, Physica D 168 (2002) 325–335. [28] D. Pisignano, S. Capaccioli, R. Casalini, M. Lucchesi, P.A. Rolla, A. Justl, E. Ro¨ssler, J. Phys.: Condens. Matter 13 (2001) 4405.
M. Lucchesi et al. / Journal of Non-Crystalline Solids 352 (2006) 4920–4927 [29] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113. [30] L. Bellon, S. Ciliberto, C. Laroche, Europhys. Lett. 53 (2001) 511. [31] A. Barrat, Phys. Rev. E 57 (1998) 3629.
4927
[32] A. Perez-Madrid, D. Reguera, J.M. Rubi, 2002, cond-mat/0210089. [33] A. Crisanti, F. Ritort, Europhys. Lett. 66 (2004) 253. [34] A. Crisanti, F. Ritort, J. Phys. A: Math. Gen. 36 (2003) R181.