Nonlinear Dielectric Behavior of a Secondary Relaxation - American

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Nonlinear Dielectric Behavior of a Secondary Relaxation: Glassy D‑Sorbitol Subarna Samanta and Ranko Richert* Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287, United States ABSTRACT: We have studied the nonlinear dielectric behavior of a glass-forming liquid, D-sorbitol, with particular attention to its exceptionally intense Johari−Goldstein (JG) type secondary relaxation. It is found that this β-relaxation displays significant nonlinear dielectric effects, but these differ qualitatively from their α-process counterparts. High fields increase the amplitudes of the secondary modes (rather than reducing their time constants), consistent with a field induced increase of fictive temperatures. This result implies that the amplitudes of the secondary modes fluctuate in the glassy state, consistent with MD simulations reported for a liquid displaying a JG relaxation. The nonlinear features of this secondary process are reminiscent of those found for the excess wing regime, suggesting that these two contributions to dynamics have common origins.

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on the absence of nonlinear effects within the excess wing,9 time-resolved studies on the same liquids have found considerable NDEs for frequencies as high as 107νmax.15 The apparent discrepancy originates from the unexpectedly long time it takes these fast modes to establish their steady state behavior, observed as a change of the loss spectrum, ε″(ω), as a function of the time that the high field is applied.15 For several typical glass-formers, propylene carbonate (PC), glycerol (GLY), and 2-methyltetrahydrofuran (MTHF), it was observed that the field induced change of ε″(ω) consists of a quasiinstantaneous contribution, followed by an approach to the steady state level with a dependence on time which follows the average structural relaxation, characterized by a time constant τα and stretching exponent β.15 While this slow structural recovery suggests that the majority of this change in fictive temperature, Tf, is slaved to macroscopic softening, the origin of the fast change has remained unclear. Here, the concept of a fictive (or configurational) temperature is used to characterize the state of a nonequilibrium system or subsystem via the value of Tf that would create the same state if the system were in equilibrium at a temperature T = Tf.16 Numerous other sources of nonlinear dielectric effects have been identified in studies of liquids; among them are dielectric saturation17,18 and chemical effects.19−21 Fortunately, the field induced modifications due to the absorption of energy from an electric field of sufficient amplitude display a rather characteristic frequency dependence,7−9 and can therefore be discriminated from other sources of NDEs. Moreover, these consequences of energy absorption can modify permittivity

INTRODUCTION The microscopic picture of structure and dynamics of glassforming materials near their glass transition temperature, Tg, is still incomplete and thus continues to attract attention from both experimental and theoretical research.1 While linear response broadband dielectric relaxation spectroscopy provides detailed information on the dynamics of the macroscopic dipole moment,2 the inferences regarding more microscopic properties are rather limited. The situation is different for dielectric experiments that drive the response beyond the linear regime, where the resulting polarization P is no longer proportional to the amplitude E of the electric field.3 A well documented example for such a nonlinear dielectric effect (NDE) is the approach known as (nonresonant) dielectric hole burning,4 which revealed the heterogeneous nature of orientational dielectric polarization in supercooled liquids near Tg.5,6 The advantageous feature of this nonlinear dielectric method is based upon the spectral selectivity with which energy is transferred from the high amplitude sinusoidal electric field, E(t) = E0 sin(ωt), to modes whose time constants τ match the frequency of the external field (τ ≈ 1/ω). Similar NDEs have been observed by quasi-steady-state and time-resolved impedance experiments performed at high field amplitudes, using electric fields of several hundred kV/cm.7−10 The field induced effects have been interpreted either as spectrally selective modifications of fictive temperatures (Tf)11−13 or, in analogy to spin glass systems, as the consequence of nontrivial dynamical correlations of the liquid’s constituents.14 In both cases, more microscopic details of the behavior are inferred from nonlinear dielectric experiments, and such conclusions could not have been derived from the linear response counterparts. More recently, high-field dielectric experiments have extended into the regime of higher frequencies (ν ≫ νmax) relative to the loss peak at νmax, i.e., into the so-called excess wing of the loss spectrum.9,15 While Bauer et al. have concluded © XXXX American Chemical Society

Special Issue: Branka M. Ladanyi Festschrift Received: July 9, 2014 Revised: July 24, 2014

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dx.doi.org/10.1021/jp506854k | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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values by up to 20% at moderately high fields, levels that are uncommon for other sources of nonlinear behavior. The present study is devoted to exploring the nonlinear dielectric response of the secondary relaxations of the Johari− Goldstein (JG) type,22 i.e., those β-processes that are considered an intrinsic feature of the dynamics of glass-forming liquids and glasses. We employ D-sorbitol for this study, as it possesses a well characterized secondary process with exceptional dielectric amplitude on an absolute scale (Δε ≈ 4) as well as relative to the primary relaxation amplitude (Δεβ/Δεα ≈ 0.1).23−26 Nonlinear features of both the primary and secondary processes are characterized using electric fields up to 230 kV/cm. It is found that high electric fields modify the dielectric loss of the β-process, even for temperatures well below Tg. In contrast to the NDEs associated with the primary relaxation, high external fields increase the amplitude of the secondary process modes, rather than reduce their time constants. In the glassy state, T < Tg, the magnitude of the nonlinear effect of the β-peak does not change in the course of the observation time window of approximately 2−500 s. Comparing the present results with observations of NDEs in the excess wing regime supports the notion that the excess wing and the JG type secondary relaxation share characteristics of the nonlinear behavior and may thus have the same origin.

measured loss values of the primary relaxation to the literature23,24 results on D-sorbitol indicated that the real sample thickness is 15 μm, i.e., 5 μm thicker than the nominal value determined by the spacer material. All permittivity values reported below are corrected for the actual geometry. This is the result of the high viscosity of the sample at the preparation temperature, which prohibits displacing all excess liquid from the capacitor.

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RESULTS For the sample temperatures relevant to this study, the linear regime dielectric loss spectra of D-sorbitol are depicted in Figure 1. The curves validate the notion that D-sorbitol is a very

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EXPERIMENT The compound D-sorbitol (98%) has been purchased from Aldrich and is used as received. Sorbitol is heated to about 400 K, i.e., sufficiently above its melting point Tm = 368 K, for 10 min prior to cooling to Tg at a rate of ∼5 K/min. It is important that the process generates an optically clear liquid while avoiding caramelization. Samples are prepared in a capacitance cell that uses a 10 μm thick Teflon ring spacer (with 14 mm inner diameter), which defines the distance between the polished stainless steel electrodes of 16 mm (top) and 20 mm (bottom) diameter. In order to avoid excessive stress on the Teflon ring and sample due to temperature changes, the electrode pair is spring loaded in order to achieve a constant force on the spacer ring and sample. For controlling the sample temperature, a closed cycle He refrigerator (Leybold RDK 6-320/Coolpak 6200) along with a temperature controller (Lakeshore Model 340) equipped with calibrated diode sensors are used. High voltages are achieved using a Trek PZD-700 high voltage amplifier boosting the generator signal of the Solarton SI-1260 gain/phase analyzer. Impedances are measured for frequencies from 1 Hz to 25 kHz at a density of 8 points per decade. For each frequency, a sequence of 1 ≤ i ≤ 100 measurements is programmed, where the field amplitude is high for 10 ≤ i ≤ 50 and low otherwise. To ensure that the system is held at isothermal conditions, sufficient waiting times were provided between two measurements. The software added a time stamp to each data record, thus providing a time scale with 1 s resolution. From these results, the time dependence of εhi″ − εlo″ was derived, where “hi” refers to the loss obtained at a high field amplitude (E0,hi) and “lo” is for the low field result (E0,lo = E0,hi/10). Sample temperatures were selected as follows. For experiments related to the α-peak, three temperatures close to Tg = 268 K of D-sorbitol were chosen, T = 267, 270, and 273 K, for which a sufficient spectral separation between the α- and βprocess facilitates discerning their contributions. For investigating the β-process, the temperatures were kept at 195, 220, and 245 K, respectively, i.e., well below Tg. Comparing the

Figure 1. Low field (linear response) dielectric spectra of D-sorbitol at the temperatures relevant to the present study. Higher temperatures (T = 267, 270, and 273 K) refer to the measurements on the α-peak, while the three lower temperatures (T = 195, 220, and 245 K) were employed to focus on the β-relaxation effects.

good candidate for the study of nonlinear effects of β-relaxation dynamics, as there is a clear separation between the α-process and the pronounced β-peak. Given the present experimental frequency window of the high field measurements, the three upper temperatures are chosen for focusing on the α-peak, while the three lower temperatures facilitate measuring the βpeak effects without strong interference from the primary modes. For temperatures such as the T = 267 or 270 K cases, the upper end of the frequency window used for the high field measurements reaches the onset of the β-peak. However, it is important to keep in mind that there still are significant contributions from the primary structural relaxation peak at those frequencies. The graphical matrix of Figure 2 compiles representative examples of the time dependence of the field induced dielectric effect in terms of the relative change of the dielectric loss, (εhi″ − εlo″ )/εlo″ , for the three temperatures located near or above Tg and for different frequencies. This array has a common sample temperature T for each row and a common measurement frequency ν for each column. The lines in Figure 2 are fits according to εhi″ − εlo″ = ϕX (t ) = ϕ∞ + (ϕs − ϕ∞)(1 − exp[− (t /τα)β ]) εlo″ (1) B



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Figure 2. Time-dependent field induced relative change of the dielectric loss of D-sorbitol, (ε″hi − ε″lo)/ε″lo, in response to the field amplitude increasing from E0,lo = E0,hi/10 to E0,hi = 190 kV/cm at t = 0 and then decreasing again to E0,lo at a later time. From top to bottom, the rows are for temperatures T = 267, 270, and 273 K, corresponding to structural relaxation times of τα = 200, 20, and 2 s, respectively. For each of the three temperatures, results for three measurement frequencies are shown, ν = 1, 10, and 100 Hz, with every column representing a common frequency for all temperatures.

of εhi″ over εlo″ that is faster than about 2 s. Steady state nonlinear effects such as those represented by the solid symbols in Figure 3, ϕs(ν), have been interpreted as a field induced modification of time constants, rather than a change in relaxation amplitudes.4,7,8 Therefore, it is useful to assess the so-called “horizontal” difference curves that gauge the separation between the high field spectrum, ε″hi(ω), and its low field counterpart, ε″lo(ω), measured along the logarithmic frequency scale, ln ν. Because points satisfying εhi″ (ωhi) = εlo″ (ωlo) are generally not available from the experiment performed at a fixed set of frequencies, the “horizontal” difference, ln νhi − ln νlo, is calculated according to

with ϕ∞ and ϕs representing the “instantaneous” (not timeresolved in the present experiment) and the steady state level of the relative change of the loss, respectively. The values for ϕ∞ and ϕs are adjustable fit parameters, while τα and β are the Kohlrausch−Williams−Watts27,28 parameters that describe the α-relaxation in the time domain at the respective temperature. The frequency dependencies of ϕ∞ and ϕs are depicted for the three liquid state temperatures in Figure 3. According to the fits, the instantaneous values, ϕ∞, for the temperatures T = 270 and 273 K were zero, and thus, only one curve is shown for ϕ∞(ν). Note that the contribution to ϕ∞ contains all nonlinear effects that are not time-resolved in this experiment, i.e., the rise

ln νhi − ln νlo =

(εhi″ − εlo″)/εlo″ −d log εlo″/d log ν

(2)

The results for the field induced frequency shifts, ln(νhi/νlo), that are derived from the three ϕs(ν) curves of Figure 3 are shown in Figure 4. Because the slope, d ln ε″/d ln ν, attains values of zero at the loss minimum between primary and secondary peak, the ln(νhi/νlo) curves for T = 267 and 270 K display poles at 100 and 1000 Hz, respectively. Time-resolved measurements analogous to those shown in Figure 2 have been performed for the three lower temperatures, T = 195, 220, and 245 K, i.e., in the glassy state where the secondary relaxation peak is within the frequency window of the high field experiments and with little interference from the primary structural relaxation signal. A selection of these results is compiled in matrix form in Figure 5, which is analogous to Figure 2 but with the highest frequency column showing 1 or 10 kHz results instead of a common value of 100 Hz. In contrast to the results obtained for T > Tg, these curves observed for T < Tg show practically no time dependence and an overall smaller effect in the 1−2% range. A spectrum of the field induced relative change, ϕs(ν), of the dielectric loss at T = 220 K has been derived from the long time plateau values of the (εhi″ − εlo″ )/εlo″ data of Figure 5. The frequency dependence of this nonlinear dielectric effect of the

Figure 3. Quasi-instantaneous (open symbols, ϕ∞) and steady state (solid symbols, ϕs) values of the “vertical” relative difference, (ε″hi − εlo″ )/εlo″ , between loss spectra recorded at E0,hi = 190 kV/cm and E0,lo = E0,hi/10. Data represent D-sorbitol at T = 267, 270, and 273 K. Values are obtained by fitting eq 1 to curves such as those shown in Figure 2. The values of ϕ∞ for T = 270 and 273 K are zero and thus not shown. Lines serve as guides only. C



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Figure 6. Dielectric relaxation results for the β-process of D-sorbitol at a temperature of T = 220 K. (a) Experimental data for the low field (linear response) dielectric loss spectrum (symbols), along with an HN fit (line) based on the parameters listed in the legend. (b) The “vertical” relative difference, (ε″hi − ε″lo)/ε″lo, between loss spectra recorded at E0,hi = 230 kV/cm and E0,lo = E0,hi/10, as derived from the virtually time invariant curves of Figure 5. The line serves as a guide only.

Figure 4. Spectra of field induced “horizontal” differences, ln νhi − ln νlo, for D-sorbitol for three temperatures, T = 267, 270, and 273 K, where the indices “hi” and “lo” refer to fields of E0,hi = 190 kV/cm and E0,lo = E0,hi/10, respectively. The “horizontal” differences are obtained by dividing the “vertical” effect, (εhi″ − εlo″ )/εlo″ ≈ Δ ln ε″, by the slope, −d ln ε″/d ln ν. For T = 267 and 270 K, this slope is zero at the respective frequencies indicated by arrows, leading to the poles for ln νhi − ln νlo. Lines serve as guides only.

experiments, these measurements are often performed with zero bias field with a high amplitude sinusoidal field, E(t) = E0 sin(ωt). In the case that the detection of nonlinear effects focuses on the polarization P at the fundamental frequency ω, the subscript “1” is used to identify the order of the Fourier component of interest. Contributions to P1 will be dominated by the linear susceptibility, χ1, and at moderate departures from the linear regime, a further contribution arises from a term that is proportional to the third power of the electric field, with susceptibility χ(3) 1 . If higher order terms in the field (n = 5, 7, ...)

secondary relaxation is depicted in Figure 6, which also shows the low field (linear response) loss spectrum for comparison.

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DISCUSSION High field impedance spectroscopy is an emerging novel tool to study the structure and dynamics of glass-forming liquids, exploiting the notion that more microscopic insight can be obtained from nonlinear dielectric effects, relative to the standard linear response experiments. As in the present

Figure 5. Time-dependent field induced relative change of the dielectric loss of D-sorbitol, (εhi″ − εlo″ )/εlo″ , in response to the field amplitude increasing from E0,lo = E0,hi/10 to E0,hi = 230 kV/cm at t = 0 and then decreasing again to E0,lo at a later time. From top to bottom, the rows are for temperatures T = 195, 220, and 245 K. For each of the three temperatures, results for three measurement frequencies are shown, ν = 1 Hz, 10 Hz, and 1 or 10 kHz. The nonlinear effects amount to 1−2% and are largely time invariant, with the exception of the T = 195 K, ν = 10 kHz case, which is dominated by electrode heating. D



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with KWW parameters τα and β that closely resemble the average α-relaxation in the time domain at the given temperature. The fit examples shown in Figure 2 demonstrate that eq 1 captures the time dependence of the field induced relative increase of the dielectric loss with τα values derived from the Havriliak−Negami32 fits to the α-peak and with βvalues that vary only slightly around the low value typical for this highly fragile (m = 127) liquid. This universally slow equilibration or structural recovery time for modes that are associated with fast structural relaxation time scales within the high frequency wing is equivalent to the behavior observed with the same type of experiment for PC, GLY, and MTHF. The magnitudes, ϕs, of the relative enhancements of the loss are compiled in Figure 3, which spans a range of 6.5 to 1.5 × 106 regarding the ratio ν/νmax. At the field of E0,hi = 190 kV/cm, levels of ϕs between 4 and 12% are found, with the tendency to decrease monotonically with frequency in the range of ν > 102νmax. Again, these observations are reminiscent of the high frequency nonlinear dielectric effects seen with other molecular glass formers. Expressing the field effects of Figure 3 in terms of the “horizontal” difference, ln νhi − ln νlo, as defined in eq 2, a picture emerges in Figure 4 that differs qualitatively from what has been observed previously for molecular glass formers. Here, the values of ln νhi − ln νlo increase along the frequency axis, prior to displaying a pole followed by negative values at even higher frequencies; see Figure 4. The pole originates from the division by the slope d ln ε″/d ln ν in eq 2, which becomes zero at the loss minimum between the α- and β-peaks. As the materials studied previously in a similar fashion (PC, GLY, MTHF) do not exhibit a distinct β-peak, such poles and changes in sign as in Figure 4 have not been observed before. Consequently, it is expected that a closer scrutiny of the nonlinear dielectric behavior of the β-peak will shed light on these features; see the next section “Secondary Relaxation of Sorbitol”. The comparatively slow approach to equilibrium observed for D-sorbitol even for frequencies that are 106 × νmax or higher is analogous to the behavior seen for other molecular glass formers, PC, GLY, and MTHF. This slow process is the equivalent of structural recovery,16 i.e., the approach of time constants toward their equilibrium value. In the case of physical aging or calorimetry, structural recovery occurs in response to a change in temperature.16 By contrast, the present high field experiments displace the equilibrium state from its low field counterpart through an influx of energy, without a change in the temperature.33 For the fastest modes of the relaxation time distribution, the present results indicate that structural recovery is orders of magnitude slower than their structural relaxation time. Confinement of these fast modes to a rigid matrix has been offered as an explanation.34,35 Secondary Relaxation of Sorbitol. The distinct secondary relaxation of D-sorbitol can be observed clearly above and below Tg in Figure 1. Its origin is generally understood as being of the JG type, i.e., a dynamic feature intrinsic in the behavior of viscous liquids, rather than reflecting intramolecular degrees of freedom. According to experimental evidence collected for Dsorbitol, this type of “slow” β-relaxation is associated with dynamic heterogeneity,36 and a correlation between primary and secondary relaxation times has been found.37 Several examples of time-resolved experiments across the βpeak range are shown in Figure 5, with all cases being associated with temperatures well below Tg and thus with primary relaxation times that are strongly separated from the β

are disregarded, the relation between P1 and E0 can be written as P1(̂ ω) 3 = χ1̂ (ω) + E0 2χ1̂(3) (ω) ε0E0 4

(3)

The present analysis follows common practice in evaluating the data using the standard relations for obtaining permittivity ε in the linear regime but with different results for the low (linear response) and high field case, identified by the respective indices “lo” and “hi”, i.e., εlô (ω) = 1 + χ1̂ (ω)

and

εhî (ω) = 1 + χ1̂ (ω) +

3 2 (3) E0 χ1̂ (ω) 4

(4)

The energy that a sample absorbs from an external timedependent field of sufficiently large amplitude has been identified as a source of considerable nonlinear dielectric behavior.7−9 In the extreme case of microwave heating and related experiments, the field induced modification is governed by a change in sample temperature.29,30 At lower frequencies and accordingly reduced power levels, changes in permittivity can be detected, even under isothermal conditions. Using the concept of dynamic heterogeneity, these field induced modifications can be rationalized on the basis of spectrally selective increases in configurational temperatures (ΔTcfg), which result in a reduction of relaxation time constants, implying that modes shift along the ln ν scale with a negligible change of their amplitudes. For steady state conditions and for time constants not too far away from the most probable value, τmax, the difference between ln τ values at high (τhi) and low (τ = τlo, linear response value) fields, E0 sin(ωt), can be approximated by

where EA = d ln τmax/d(1/kBT) is the overall apparent activation energy, Δε = εs − ε∞ is the (linear response) dielectric relaxation amplitude, ΔCcfg is the configurational heat capacity step, and ρ is the density. This relation has been found to provide reasonable predictions for the magnitude of the nonlinear effects for numerous materials, including D-sorbitol.31 Toward higher frequencies relative to the peak loss frequency νmax, i.e., in the excess wing regime with ν > 102νmax, the magnitude of the nonlinear dielectric effect is reduced relative to the prediction of eq 5 by a factor of up to about 2 at ν = 107νmax. Moreover, it was found for several glass formers that the approach to the steady state value of εhi″ occurs as slow as the average structural relaxation, even for modes as fast as τ = 10−7τmax.15 Primary Relaxation of Sorbitol. The nonlinear dielectric behavior of D-sorbitol is studied at three temperatures, T = 267, 270, and 273 K, with the respective average α-process dynamics characterized by the relaxation times τα = 200, 20, and 2 s. In the same order, the peak frequencies of the linear response dielectric loss profiles are positioned at νmax = 0.002, 0.02, and 0.2 Hz; see Figure 1. Regardless of temperature T and measurement frequency ν, it is found that the field induced change as gauged by the quantity (ε″hi − ε″lo)/ε″lo approaches its steady state level via eq 1, i.e., with an “instantaneous” step of magnitude ϕ∞, followed by a stretched exponential rise to ϕs E



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that study, local values of Δεβ and τβ did not display a significant correlation. This field induced increase of the β-peak amplitudes by about 1% can be expressed in terms of a higher fictive temperature for the E0 = 230 kV/cm case relative to the linear response limit. According to a previous study of the β-relaxation of D-sorbitol, its amplitude Δεβ varies approximately as d ln Δεβ/dT = 0.03 K−1 in the glassy state.24 Therefore, an increase in Tcfg of ∼0.35 K would explain the 1% effect. The steady state expression ΔTcfg = ε0E02Δε/(2ρΔCp) is derived from the balance of the influx of energy (∝ ε″E2) and relaxation to the thermal bath, and it has been demonstrated to compare favorably with experimental data for the α-process, including the case of Dsorbitol.31 Applying this approach to the secondary peak case of Figure 6 requires using Δεβ = 3.7 for the relaxation strength and E0 = 230 kV/cm for the field amplitude. The glass-to-liquid heat capacity step associated with the α-process of D-sorbtiol is ρΔCp,α = 1.65 J K−1 cm−3.40 If this value were used to determine ΔTcfg, the predicted field effect would be negligible compared to what is actually observed. However, because ρΔCp enters in the denominator, a much smaller β-relaxation heat capacity step of order ΔCp,β = 1.5% × ΔCp,α would be able to explain the field induced amplitude effect on the basis of an increased configurational (or fictive) temperature. While calorimetric data for a quantitative test of this notion is not available, heat capacity steps of the secondary process are known to be much smaller than those associated with the αrelaxation.41 Excess Wing Behavior. The excess wing of a susceptibility refers to the high frequency contribution to ε″(ω), with ω ≫ ωmax, that is in excess of the extrapolated power law that describes the high frequency side of the loss profile closer to the peak. In practice, the excess wing loss is considered to be the additional high frequency signal that is not captured by the typical fit functions with limiting power law behavior, such as the HN function and its special cases.32 It is being discussed frequently whether the excess wing is an unresolved JG βrelaxation or some independent feature of liquid dynamics.42,43 Supercooled liquids with a lower fragility index, say m < 70, will tend to have a small JG type secondary process that has little spectral separation from the α-peak, which then might appear as an excess wing rather than a distinct peak.38,43 The three liquids, PC, GLY, and MTHF, that have been analyzed in terms of their ϕX(t) behavior, see eq 1, all possess excess wings in their liquid states (T > Tg) but no obvious βprocess loss maxima. The connection to the present case of Dsorbitol is founded in the observation that PC, GLY, and MTHF all display instantaneous contributions (ϕ∞) to (ε″hi − εlo″ )/εlo″ for frequencies exceeding 102−103 times νmax, whereas ϕ∞ is practically zero for frequencies closer to the loss peak position. A clear example for the onset of ϕ∞ coinciding with the onset of the excess wing at 200 × νmax for PC at T = 156.5 K is provided in Figure 7, with the level of ϕ∞ also reaching values around 2% at E0,hi = 200 kV/cm. In the case of Dsorbitol, no considerable instantaneous field induced changes are seen across the high frequency wing of the α-process. An apparent exception is the 0.5% effect for the T = 267 K curve in Figure 3, but the loss for frequencies ν > 30 Hz contains significant contributions from the β-process. Thus, in the present experiments, it is only the secondary process that displays these quasi-“instantaneous” rises of the loss at high fields. The coincidence of a quickly developed nonlinear effect with loss contributions from either excess wing or JG type

time scales. An important feature of these results obtained with 230 kV/cm is that a nonlinear effect of order 1.5% in terms of (εhi″ − εlo″ )/εlo″ is observed across the β-process spectrum. This level corresponds to about 1.0% if a field of 190 kV/cm had been used (as is the case for the α-process experiments), as it is observed (and expected) that the magnitude of the relative change of the loss scales with E02. Moreover, within the present experimental ranges, this field induced change is almost independent of temperature, frequency, and time. This invariance regarding T, ν, and t is strikingly different from the corresponding results observed for the α-process; see Figure 2. The lack of any considerable time dependence in Figure 5 of the ϕX(t) signals as defined in eq 1 can be explained as follows. At the relevant temperatures, T = 195, 220, and 245 K, the time scale of the primary structural relaxation, τα, is much longer than the time (≤500 s) over which the high field loss has been monitored. Consequently, a possible increase of ϕX(t) that traces the average structural relaxation would have remained inaccessible by this experiment. In other words, instantaneous and steady state amplitudes of the field effect are virtually the same in the glassy state with T ≪ Tg. If the ∼1.5% change had been established on the time scale of the β-process, that rise would remain unresolved given the present time resolution of about 2 s. The slight rise seen for the T = 245 K and ν = 1 Hz case may be due to residual overlap with the α-peak. The linear increase for the T = 195 K and ν = 10 kHz situation can be ascribed to sample heating as a result of excessive power.33 We now ask whether the field effect on the permittivity of the β-process can also be rationalized in terms of a shift of relaxation time constants, a picture that explains much of the nonlinear features associated with the primary loss peak.4,7,8 This question is answered by the result shown in Figure 6, which shows a uniform effect of around +1.25% field induced gain of the loss, regardless of the slope of the β-peak spectrum. By contrast, a shift of secondary modes toward higher frequencies (at constant amplitudes) would result in a reduction of ε″ for ν < νβ,max, an enhancement of ε″ for ν > νβ,max, and a field invariant loss for ν ≈ νβ,max, where νβ,max is the loss peak position of the secondary process. We conclude that the field induced change of the β-process is dominated by modifications of the amplitudes rather than time constants, whereas these roles are reversed for the nonlinear features observed for the α-relaxation. This immediately explains the negative “horizontal” effects in Figure 4, because the loss increases with field also in the range in which the slope d ln ε″/ d ln ν is positive. A field induced reduction of time constants τβ of the secondary relaxation process is not excluded but appears to be a minor contribution. In the common picture of the JG process reflecting librational motion with a longer-lived cage that confines reorientation to a certain cone angle,38 the present field effect would correspond to a widening of that cone angle (thus increasing Δεβ) with less effect on the time constant associated with that motion. The amplitude change in the βregime is a response to fields where the energy involved is much smaller than the thermal energy (μE ≪ kB T). Qualitatively, this implies thermally induced fluctuations of the β-amplitude in the absence of an external field (analogous to the fluctuation−dissipation theorem for linear response). Similarly, a distribution of Δεβ values derived via MD simulations for individual molecules has been discussed by Fragiadakis and Roland for a glass consisting of asymmetric diatomic molecules displaying a JG type secondary process.39 In F



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process, whereas both τβ and Δεβ change in comparable ways with T for the secondary relaxation. Therefore, the high-field effects for both relaxation modes can be described by field induced increases in the fictive temperature. The two distinct effects also differ in their time scale for reaching steady state conditions. Modifying the time constants of the α-process requires the average primary structural relaxation time, even for contributions to frequencies that exceed the peak value, νmax, by more than a factor of 106. Increases of the amplitudes of the secondary process or excess wing modes are quasi-instantaneous compared with the α-relaxation time. The similarity of the nonlinear dielectric behavior of the JG secondary process with that of the excess wing modes supports the notion that both dynamic features share a common microscopic origin, differing only in the spectral separation from the primary peak. This is emphasized by the qualitative difference of these field effects from the nonlinear response of the entire loss spectrum associated with the α-process.

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Figure 7. Dielectric relaxation results for the α-process of propylene carbonate at a temperature of T = 156.5 K. (a) Experimental data for the low field (linear response) dielectric loss spectrum (symbols), along with an HN fit (line) that is extrapolated to show the loss peak. (b) The “vertical” relative difference, (εhi″ − εlo″ )/εlo″ , between loss spectra recorded at E0,hi = 200 kV/cm and E0,lo = E0,hi/10, as derived from time-resolved curves from ref 15, analogous to those in Figure 2. Open symbols represent quasi-instantaneous levels, ϕ∞, while solid symbols are for steady state values, ϕs.

AUTHOR INFORMATION

Corresponding Author

*Phone: (480) 727-7052. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. CHE 1026124.

secondary relaxations suggests that both dynamics share a common origin. In this context, we note that the α-process of D-sorbitol displays time−temperature superposition (TTS) for a large range of structural relaxation times: from 102 to 10−6 s.44 Obeying TTS or thermorheological simplicity requires that temperature predominantly affects the average time constant of the relaxation process, without modifying the profile of the frequency-dependent susceptibility. TTS extending over a number of decades in terms of the peak loss frequency is more commonly observed for the more fragile cases, where the JG β-process is well separated from the α-peak.44,45 Liquids exhibiting excess wings are notorious for violating TTS when τα changes several orders of magnitude, and the explanation based on the identity of excess wing and JG process is that primary and secondary peaks are subject to different activation behaviors, which prohibits the occurrence of TTS if the secondary peak is submerged below the primary signal. A graph of βKWW versus log νmax for 14 different liquids as provided by Wang et al.44 supports this idea that TTS is a consequence of the excess wing/JG β-process not interfering with the primary loss profile.

REFERENCES

(1) Angell, C. A.; Ngai, K. L.; McKenna, G. B.; McMillan, P. F.; Martin, S. W. Relaxation in Glassforming Liquids and Amorphous Solids. J. Appl. Phys. 2000, 88, 3113−3157. (2) Broadband Dielectric Spectroscopy; Kremer, F., Schönhals, A., Eds.; Springer: Berlin, 2002. (3) Małecki, J. The Relaxation of the Nonlinear Dielectric Effect. J. Mol. Struct. 1997, 436−437, 595−604. (4) Schiener, B.; Bö hmer, R.; Loidl, A.; Chamberlin, R. V. Nonresonant Spectral Hole Burning in the Slow Dielectric Response of Supercooled Liquids. Science 1996, 274, 752−754. (5) Ediger, M. D. Spatially Heterogeneous Dynamics in Supercooled Liquids. Annu. Rev. Phys. Chem. 2000, 51, 99−128. (6) Richert, R. Heterogeneous Dynamics in Liquids: Fluctuations in Space and Time. J. Phys.: Condens. Matter 2002, 14, R703−R738. (7) Richert, R.; Weinstein, S. Nonlinear Dielectric Response and Thermodynamic Heterogeneity in Liquids. Phys. Rev. Lett. 2006, 97, 095703. (8) Huang, W.; Richert, R. Dynamics of Glass-Forming Liquids. XIII. Microwave Heating in Slow Motion. J. Chem. Phys. 2009, 130, 194509. (9) Bauer, Th.; Lunkenheimer, P.; Kastner, S.; Loidl, A. Nonlinear Dielectric Response at the Excess Wing of Glass-forming Liquids. Phys. Rev. Lett. 2013, 110, 107603. (10) Brun, C.; Ladieu, F.; L’Hôte, D.; Tarzia, M.; Biroli, G.; Bouchaud, J.-P. Nonlinear Dielectric Susceptibilities: Accurate Determination of the Growing Correlation Volume in a Supercooled Liquid. Phys. Rev. B 2011, 84, 104204. (11) Chamberlin, R. V.; Schiener, B.; Böhmer, R. Slow Dielectric Relaxation of Supercooled Liquids Investigated by Nonresonant Spectral Hole Burning. Mater. Res. Soc. Symp. Proc. 1997, 455, 117− 125. (12) Schiener, B.; Chamberlin, R. V.; Diezemann, G.; Böhmer, R. Nonresonant Dielectric Hole Burning Spectroscopy of Supercooled Liquids. J. Chem. Phys. 1997, 107, 7746−7761. (13) Jeffrey, K. R.; Richert, R.; Duvvuri, K. Dielectric Hole Burning: Signature of Dielectric and Thermal Relaxation Time Heterogeneity. J. Chem. Phys. 2003, 119, 6150−6156.

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CONCLUSIONS We have explored the nonlinear dielectric response of the JG type secondary process of D-sorbitol and find field induced increases of the dielectric loss in the β-peak regime of order 1− 2% at fields of around 200 kV/cm. These nonlinear dielectric effects are compared with those of the primary relaxation process and with those associated with the excess wing of other glass-forming liquids. The effects of high fields at frequencies near the α-peak can be described by modes being accelerated, i.e., subject to a field induced reduction of their time constants. By contrast, high fields enhance the amplitudes of both the JG type β-relaxation and the excess wing. Similarly, in equilibrium, temperature affects τα much more than Δεα for the primary G



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(14) Bouchaud, J.-P.; Biroli, G. Nonlinear Susceptibility in Glassy Systems: A Probe for Cooperative Dynamical Length Scales. Phys. Rev. B 2005, 72, 064204. (15) Samanta, S.; Richert, R. Limitations of Heterogeneous Models of Liquid Dynamics: Very Slow Rate Exchange in the Excess Wing. J. Chem. Phys. 2014, 140, 054503. (16) Hodge, I. M. Enthalpy Relaxation and Recovery in Amorphous Materials. J. Non-Cryst. Solids 1994, 169, 211−266. (17) De Smet, K.; Hellemans, L.; Rouleau, J. F.; Corteau, R.; Bose, T. K. Rotational Relaxation of Rigid Dipolar Molecules in Nonlinear Dielectric Spectra. Phys. Rev. E 1998, 57, 1384−1387. (18) Jadżyn, J.; Kędziora, P.; Hellemans, L.; De Smet, K. Relaxation of the Langevin Saturation in Dilute Solution of Mesogenic 10-TPEB Molecules. Chem. Phys. Lett. 1999, 302, 337−340. (19) Piekara, A.; Chelkowski, A. New Experiments on Dielectric Saturation in Polar Liquids. J. Chem. Phys. 1956, 25, 794−795. (20) Małecki, J. Dielectric Saturation in Aliphatic Alcohols. J. Chem. Phys. 1962, 36, 2144−2145. (21) Singh, L. P.; Richert, R. Watching Hydrogen Bonded Structures in an Alcohol Convert from Rings to Chains. Phys. Rev. Lett. 2012, 109, 167802. (22) Johari, G. P.; Goldstein, M. Viscous Liquids and the Glass Transition. II. Secondary Relaxations in Glasses of Rigid Molecules. J. Chem. Phys. 1970, 53, 2372−2388. (23) Olsen, N. B. Scaling of β-relaxation in the equilibrium liquid state of sorbitol. J. Non-Cryst. Solids 1998, 235−237, 399−405. (24) Wagner, H.; Richert, R. Equilibrium and Non-Equilibrium Type β-Relaxations: D-sorbitol versus o-terphenyl. J. Phys. Chem. B 1999, 103, 4071−4077. (25) Faivre, A.; Niquet, G.; Maglione, M.; Fornazero, J.; Jal, J. F.; David, L. Dynamics of Sorbitol and Maltitol Over a Wide TimeTemperature Range. Eur. Phys. J. B 1999, 10, 277−286. (26) Power, G.; Johari, G. P.; Vij, J. K. Relaxation Strength of Localized Motions in D-sorbitol and Mimicry of Glass-softening Thermodynamics. J. Chem. Phys. 2003, 119, 435−442. (27) Kohlrausch, R. Theorie des elektrischen Rückstandes in der Leidener Flasche. Ann. Phys. 1854, 167, 179−214. (28) Williams, G.; Watts, D. C. Non-symmetrical Dielectric Relaxation Behaviour Arising from a Simple Empirical Decay Function. Trans. Faraday Soc. 1970, 66, 80−85. (29) Matsuo, T.; Suga, H.; Seki, S. Dielectric Loss Measurement by Differential Thermal Analysis (DTA). Bull. Chem. Soc. Jpn. 1966, 39, 1827. (30) Galema, S. A. Microwave Chemistry. Chem. Soc. Rev. 1997, 26, 233−238. (31) Wang, L.-M.; Richert, R. Measuring the Configurational Heat Capacity of Liquids. Phys. Rev. Lett. 2007, 99, 185701. (32) Havriliak, S.; Negami, S. A Complex Plane Representation of Dielectric and Mechanical Relaxation Processes in Some Polymers. Polymer 1967, 8, 161−210. (33) Khalife, A.; Pathak, U.; Richert, R. Heating Liquid Dielectrics by Time Dependent Fields. Eur. Phys. J. B 2011, 83, 429−435. (34) Richert, R. Physical Aging and Heterogeneous Dynamics. Phys. Rev. Lett. 2010, 104, 085702. (35) Richert, R. Confinement Effects in Bulk Supercooled Liquids. Eur. Phys. J.: Spec. Top. 2010, 189, 223−229. (36) Richert, R. Spectral Selectivity in the Slow β-Relaxation of a Molecular Glass. Europhys. Lett. 2001, 54, 767−773. (37) Böhmer, R.; Diezemann, G.; Geil, B.; Hinze, G.; Nowaczyk, A.; Winterlich, M. Correlation of Primary and Secondary Relaxations in a Supercooled Liquid. Phys. Rev. Lett. 2006, 97, 135701. (38) Döß, A.; Paluch, M.; Sillescu, H.; Hinze, G. From Strong to Fragile Glass Formers: Secondary Relaxation in Polyalcohols. Phys. Rev. Lett. 2002, 88, 095701. (39) Fragiadakis, D.; Roland, C. M. Molecular Dynamics Simulation of the Johari-Goldstein Relaxation in a Molecular Liquid. Phys. Rev. E 2012, 86, 020501.

(40) Wang, L.-M.; Angell, C. A.; Richert, R. Fragility and Thermodynamics in Non-Polymeric Glass-Forming Liquids. J. Chem. Phys. 2006, 125, 074505. (41) Fujimori, H.; Oguni, M. Calorimetric Study of D,L-propene Carbonate: Observation of the β- as well as α-Glass Transition in the Supercooled Liquid. J. Chem. Thermodyn. 1994, 26, 367−378. (42) Schneider, U.; Brand, R.; Lunkenheimer, P.; Loidl, A. Excess Wing in the Dielectric Loss of Glass Formers: A Johari-Goldstein β Relaxation? Phys. Rev. Lett. 2000, 84, 5560−5563. (43) Ngai, K. L.; Paluch, M. Classification of Secondary Relaxation in Glass-formers Based on Dynamic Properties. J. Chem. Phys. 2004, 120, 857−873. (44) Wang, L.-M.; Richert, R. Primary and Secondary Relaxation Time Dispersions in Fragile Supercooled Liquids. Phys. Rev. B 2007, 76, 064201. (45) Nielsen, A. I.; Christensen, T.; Jakobsen, B.; Niss, K.; Olsen, N. B.; Richert, R.; Dyre, J. C. Prevalence of Approximate √t Relaxation for the Dielectric α Process in Viscous Organic Liquids. J. Chem. Phys. 2009, 130, 154508.

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Nonlinear Dielectric Behavior of a Secondary Relaxation - American

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