Electrorheological Source of Nonlinear Dielectric Effects in Molecular

Article pubs.acs.org/JPCB

Electrorheological Source of Nonlinear Dielectric Effects in Molecular Glass-Forming Liquids Subarna Samanta and Ranko Richert* School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, United States ABSTRACT: We have measured the dielectric relaxation spectra of eight glass-forming liquids in the presence of electric direct current (dc)-bias fields ranging from 100 to 500 kV/cm. For every sample, we observe two distinct field-induced effects: a reduction in the relaxation amplitude and an increase in the primary structural relaxation time that is associated with viscous flow. Whereas amplitude change is typical of the wellknown dielectric saturation, the field-induced increase in viscosity is a source of nonlinear behavior that has been recognized only recently. We find that this electrorheological behavior occurs in all polar liquids of this study, and its magnitude is correlated with the field-induced change in thermodynamic entropy. It constitutes a significant source of nonlinear dielectric behavior, which occurs for both dc and alternating current fields.

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INTRODUCTION Dielectric relaxation spectroscopy is a powerful tool for characterizing the dynamics in simple liquids and complex systems.1 In the vast majority of such measurements, the external electric fields, E, applied to the sample do not exceed 1 kV/cm, and as a result, the energies involved remain negligible compared to the thermal energy, that is, μE ≪ kBT, where μ is the molecular dipole moment.2 In this linear regime, permittivity ε, which links polarization P to field E via P = εε0E, is field invariant. At significantly higher fields, say E ≥ 100 kV/cm, the quantity ε begins to depend on E to a measurable extent, and the resulting nonlinear dielectric effects (NDEs) have gained considerable interest in recent years.3−8 In the case of highly sensitive detection capabilities, it has been demonstrated that fields as low as 2 kV/cm can be sufficient to observe dielectric nonlinearity.9−11 Various sources of NDEs are known, and dielectric saturation is the first of these that has been studied in detail.12−14 Subsequently, a so-called “negative saturation” effect has been observed, which is now more correctly termed chemical effect,15,16 and it includes various mechanisms that enhance the macroscopic dipole moment as a result of a strong electric field.6,17 Considerable changes in the dielectric constant occur in mixtures,6,18 near critical points,19 and in the vicinity of phase transitions as well.20,21 However, proximity to the glass transition alone will not produce analogous effects. The common feature of the above two NDEs is that they affect predominantly relaxation amplitude Δε = εs − ε∞, where εs and ε∞ are the dielectric constants in the limit of low and high frequencies, respectively. There are also mechanisms leading to nonlinear behavior, where the relaxation time constants, τ, are modified more that the relaxation amplitude, Δε. One example is the increase in effective dipole temperatures, originating from © 2016 American Chemical Society

a polar liquid absorbing energy from a strong time-dependent electric field,22−24 leading to accelerated dynamics for reasons somewhat analogous to those on microwave heating.25,26 Because dipole reorientation in a liquid can lead to absorption of energy from only a time-varying field, this “heating” effect is absent in the case of static direct current (dc)-bias fields. Another case of a strong electric field modifying the relaxation times is not limited to time-dependent fields, and it results in larger time constants, equivalent to more frustrated dynamics or an increased glass-transition temperature, Tg. Such an effect was predicted by Moynihan and Lesikar27 on the basis of thermoelectrical stability criteria, similar to the approach leading to the Prigogine−Defay ratio but with electrical work replacing volume work. Assuming a constant relaxational volume, this model predicted a glass-transition shift of ΔETg = 2 mK for glycerol when applying 100 kV/cm. Using the Adam−Gibbs28 (AG) relation to link dynamics to entropy, Johari29 concluded more recently that the entropy change, ΔES, due to a strong static field should lead to measurable changes in relaxation time or viscosity. For the case of glycerol, nonlinear effects consistent with this idea have been observed.30,31 In this work, seven molecular glass-forming liquids and one polymeric material are studied with regard to the steady-state effect of a high-dc-bias field on the structural relaxation time, τ, which is measured by an additional small-amplitude harmonic field of varying frequency. In all cases, a reduction in the relaxation amplitude and an increase in time constant τ is found for fields EB between 100 and 500 kV/cm, relative to the zerofield case, EB = 0. These nonlinear effects are observed to scale Received: May 14, 2016 Revised: June 19, 2016 Published: July 12, 2016 7737

DOI: 10.1021/acs.jpcb.6b04903 J. Phys. Chem. B 2016, 120, 7737−7744

Article

The Journal of Physical Chemistry B with EB2, and the magnitude of the electrorheological effect varies by a factor of 12 as a matter of the compound if compared for the same field. We find that the field-induced relative change in relaxation time, ΔE ln τ, correlates strongly with the extent of reduction in thermodynamic entropy that originates from applying the field at a constant temperature.

average of two measurements, one with a positive-bias and one with a negative-bias field, to eliminate the direct polarization response to the bias-field steps. The resulting averaged traces, V(t) and I(t), are subject to period-by-period Fourier analysis at fundamental frequency ω. This yields amplitudes (A) and phases (φ) of the voltage (AV, φV) and current (AI, φI). The effect of the high-bias field is quantified by means of the out-ofphase “permittivity” evaluated for each period of the measurement, using

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EXPERIMENT The materials investigated in this study are N-methyl-εcaprolactam (NMEC, 99%), propylene glycol (PG, 99.5%), propylene carbonate (PC, 99.7%), 2-methyltetrahydrofuran (MTHF, 99+%, anhydrous), phenyl salicylate (SAL, 99%), and poly(vinylacetate) (PVAc, Mw =12 800). All compounds were purchased from Sigma-Aldrich and used as received. Low-field (≤1 kVrms/cm) impedance experiments are performed using a system based on a Solartron SI-1260 gain/ phase analyzer and Mestec DM-1360 transimpedance amplifier. For these experiments, an Invar-steel/sapphire cell described earlier32 is used within a Novocontrol Quatro nitrogen gas cryostat. Where literature data are not available, these low-field results are used to obtain accurate Δε(T) values needed to calibrate the sample thickness, d, in the high-field cell and to calculate the slope, ∂εs/∂T. Although the nominal electrode separation of the high-field cell is given by the 10 μm Teflon spacer, deviations of up to 30% occur, and these lead to considerable uncertainties in the relaxation amplitudes and applied electric field, unless the actual value of d is determined. The high-field capacitor cell consists of a pair of springloaded polished stainless steel disks (16 and 20 mm ⌀) separated by a Teflon ring of d = 10 μm thickness and 14 mm inner diameter, which defines the sample area. This cell is mounted onto the cold stage of a closed-cycle He-refrigerator cryostat, Leybold RDK 6-320, with a Leybold Coolpak-6200 compressor. Temperature stability within several mK is achieved using a Lakeshore Model 340 temperature controller equipped with DT-470-CU sensor diodes. Steady-state high-dcbias-field spectra are obtained with the SI-1260 gain/phase analyzer and a high-voltage amplifier, Trek PZD-350 or PZD700, with the sample current measured as voltage drop across a shunt, connected to the analyzer via a protective voltage follower and dc-blocking capacitor. At each frequency, the system performs one measurement with the high dc field, followed by a measurement at the zero dc field. The timing is designed to obtain quasi-steady-state results while maintaining a low duty cycle for the high electric field. All components involved are calibrated using low-loss styroflex capacitors. For these measurements, the root mean square (rms) amplitude of the alternating field is set to 20% of the dc-bias field. For recording the time dependence of nonlinear effects, the voltage applied to the sample is generated by a programmable function generator (Stanford Research Systems DS-345) and amplified using a Trek PZD-350 or PZD-700 high-voltage amplifier. The programmed voltage waveforms consist of an integer number of periods of a low-field sinusoidal signal, V(t) = V0 sin(ωt), with a dc-bias voltage of amplitude VB superposed for some of the periods, using a voltage ratio of VB/V0 = 4. To avoid excessive current spikes, changes in the bias level are realized through linear voltage ramps that use 10% of the duration of a period. The waveform is repeated once every 5 s, ensuring that the duty cycle of the high-bias field remains small. Voltage and current signals are recorded with a digitizing oscilloscope (Nicolet Sigma 100) as averages over 3000 waveform repetitions. All signal traces shown refer to the

e″ =

AI cos(φI − φV ) ωAV C0

(1)

where C0 = ε0A/d is the geometrical capacitance. The notation e″ instead of ε″ is used because e″ changes with time in this nonequilibrium situation, in which permittivity ε is not defined. Details of all techniques used here have been provided elsewhere.31,33

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RESULTS Prior to presenting the experimental data, schematic Figure 1 is employed to demonstrate the manner in which the nonlinear

Figure 1. (a) Calculated dielectric storage (ε′lo) and loss (ε″lo) components for an HN-type system, with ε∞ = 2, Δε = 10, τHN = 1 s, αHN = 1, and γHN = 0.5 in eq 2, representing the low-field limit (‘lo’). The high-field counterparts (ε′hi, ε″hi) are based on the same parameters but with Δε reduced by 10% to mimic saturation and τHN increased by 50% to mimic the electrorheological effect. All curves are shown on a reduced frequency scale, ν/νmax, where νmax is the loss of peak frequency in the low-field limit. (b) The solid line represents the “field-induced” change in dielectric loss, (ε″hi − ε″lo)/ε″lo, relative to the low-field limit, ε″lo. The “field” effects, quantified by ΔE ln τHN = +50% and ΔE ln Δε = −10%, are strongly exaggerated. The saturation level (ΔE ln Δε = −0.1) is indicated by the horizontal dashed line; the dash-dotted line reflects the case without saturation, i.e., ΔE ln τHN = +50% and ΔE ln Δε = 0.

dielectric results are reported. These curves are based on Havriliak−Negami (HN)-type dielectric permittivity functions,34 which are also employed for the data analysis outlined below. The output of the impedance instrumentation includes storage and loss components of the zero-bias-field permittivity, ε′lo(ω) and ε″lo(ω), as well as their high-dc-field counterparts, ε′hi(ω) and ε″hi(ω), analogous to the four curves outlined in 7738


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The Journal of Physical Chemistry B Figure 1a. As the aim of the data analysis is to derive values for the field-induced changes in relaxation time (τ) and relaxation amplitude (Δε), see Figure 1a , we focus on loss components ε″(ω) because these do not involve uncertainties with regard to the value of ε∞. Therefore, the results are shown as the relative change in dielectric loss, (ε″hi − ε″lo)/ε″lo, as in Figure 1b, which for small effects is equal to the field-induced difference in ln ε″(ω), designated ΔE ln ε″(ω). Throughout the remainder of this article, ΔEX represents the difference between the value of X obtained at dc field EB and its zero-field counterpart. The following five figures display the relative change in dielectric loss, (ε″hi − ε″lo)/ε″lo, versus reduced frequency, ν/νmax, where νmax is the loss of peak frequency in the low-field limit. The field effect for NMEC is depicted in Figure 2 for various temperatures, as indicated, whereas all curves are taken at a

Figure 3. Steady-state values of the field-induced relative changes in the dielectric loss component, (ε″hi − ε″lo)/ε″lo, for PC at T = 168 K vs reduced frequency, ν/νmax. The subscripts ‘hi’ and ‘lo’ refer to bias electric fields EB = 360 and 0 kV/cm, respectively. Different symbol sets represent different field polarities, as indicated. The solid line is based on an HN fit, eq 2, to the ε″lo data and the same HN curve for the ε″hi case but subject to ΔE ln τHN = +8.1%, ΔE ln γHN = −1.7%, and ΔE ln Δε = −6.0%, the latter saturation level being indicated by the dashed line.

Figure 2. Steady-state values of the field-induced relative changes in the dielectric loss component, (ε″hi − ε″lo)/ε″lo, for NMEC vs reduced frequency, ν/νmax. The subscripts ‘hi’ and ‘lo’ refer to bias electric fields EB = 128 and 0 kV/cm, respectively. Different symbol sets represent different temperatures, as indicated. The solid line is based on an HN fit, eq 2, to the ε″lo data and the same HN curve for the ε″hi case but subject to ΔE ln τHN = +2.7% and ΔE ln Δε = −0.67%, the latter saturation level being indicated by the dashed line.

common field, EB = 128 kV/cm. The obvious qualitative feature seen similarly for the other samples is that relative to the saturation level the loss is enhanced for frequencies below the loss peak position and suppressed for frequencies ν > νmax. The modifications are not perfectly symmetric about νmax, that is, zero NDE does not occur at ν = νmax. The case of PC is shown in Figure 3 for T = 168 K and field EB = 360 kV/cm, for two distinct field polarities, which expectedly generate the same result. This plot has features observed in Figure 2, but all ΔE ln ε″(ω) values are negative and the high-frequency side has a stronger tendency to approach zero with increasing frequency. Figure 4 represents an NDE spectrum for MTHF, recorded at T = 97.5 K and EB = 210 kV/cm. The final two examples are glass-forming systems with lower dielectric constants, and the spectra are recorded for different electric fields. The curves compiled for SAL at T = 234 K in Figure 5 are measured at different fields, EB, ranging from 330 to 480 kV/cm. These traces superpose due to normalization to reflect a common field, EB → 450 kV/cm, achieved by multiplying all data by (450 kV/cm−1/EB)2. Figure 6 depicts the results for polymer PVAc at T = 323 K, again measured for various fields but normalized to EB → 327 kV/cm.

Figure 4. Steady-state values of the field-induced relative changes in the dielectric loss component, (ε″hi − ε″lo)/ε″lo, for MTHF at T = 97.5 K vs reduced frequency, ν/νmax. The subscripts ‘hi’ and ‘lo’ refer to bias electric fields EB = 210 and 0 kV/cm, respectively. The solid line is based on an HN fit, eq 2, to the ε″lo data and the same HN curve for the ε″hi case but subject to ΔE ln τHN = +2.7% and ΔE ln Δε = −0.66%, the latter saturation level being indicated by the dashed line. The star represents the steady-state plateau of a time-resolved result measurement for T = 99.5 K.

An example for a time-resolved measurement is presented in Figure 7 for NMEC at T = 187.5 K, using the bias-field pattern depicted in Figure 7a. The highest field level reaches 184 kV/ cm, whereas the intermediate field plateaus are at 130 kV/cm, a factor of 1/√2 below the 184 kV/cm maximum. The symbols in Figure 7b reflect the relative change in the loss at a fixed frequency, ν = 4.0 kHz, according to (e″hi − ε″lo)/ε″lo. Here, the time-dependent quantity is determined on the basis of eq 1 and labeled e″hi instead of ε″hi because dielectric loss is not defined outside of a steady-state situation. The feature that the NDE levels differ by a factor of 2 for 184 and 184/√2 kV/cm 7739


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The Journal of Physical Chemistry B

Figure 5. Steady-state values of the field-induced relative changes in the dielectric loss component, (ε″hi − ε″lo)/ε″lo, for salol at T = 234 K vs reduced frequency, ν/νmax. The subscripts ‘hi’ and ‘lo’ refer to bias electric fields EB = 450 and 0 kV/cm, respectively. Different symbol sets represent different fields, as indicated, all rescaled to 450 kV/cm. The solid line is based on an HN fit, eq 2, to the ε″lo data and the same HN curve for the ε″hi case but subject to ΔE ln τHN = +3.2% and ΔE ln Δε = −0.57%, the latter saturation level being indicated by the dashed line. The star represents the steady-state plateau of a timeresolved result measurement.

Figure 7. (a) Bias-field pattern, EB(t), used to measure the data in this plot. (b) Field-induced relative change in the “dielectric loss” component, e″(t), probed at ν = 4.0 kHz for NMEC at temperature T = 187.5 K. Symbols represent the results from the average of the two signals with positive and negative EB(t) patterns, where the direct response to the bias step is canceled. The results are corrected for heating effects that originate from the field transitions according to ref 33. The lines are fits to the rise and decay of the NDE, with the common parameters indicated.

result of the time-resolved experiment is that the true steadystate values for (ε″hi − ε″lo)/ε″lo can be derived from the plateau levels of measurements, such as the one in Figure 7. For the cases of MTHF, SAL, and PVAc, these steady-state values are included as stars in the spectra in Figures 4−6. Their coincidence with the frequency-dependent measurements demonstrates that the spectral data represent steady-state situations, which are reached in the same time that it takes for the structural relaxation to attain equilibrium.

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DISCUSSION To quantify the effect of a dc field of magnitude EB on the dielectric permittivity, the spectra are subject to data reduction by HN-type permittivity functions,34 using

Figure 6. Steady-state values of the field-induced relative changes in the dielectric loss component, (ε″hi − ε″lo)/ε″lo, for PVAc at T = 323 K vs reduced frequency, ν/νmax. The subscripts ‘hi’ and ‘lo’ refer to bias electric fields EB = 327 and 0 kV/cm, respectively. Different symbol sets represent different fields, as indicated, all rescaled to 327 kV/cm. The solid line is based on an HN fit, eq 2, to the ε″lo data and the same HN curve for the ε″hi case but subject to ΔE ln τHN = +1.5% and ΔE ln Δε = −0.45%, the latter saturation level being indicated by the dashed line. The star represents the steady-state plateau of a timeresolved result measurement.

ε(̂ ω) = ε′(ω) − iε″(ω) = ε∞ + Δε × [1 + (iωτ )α ]−γ (2)

Fits based on eq 2 provide values for dielectric amplitude Δε, the characteristic time constant τ; and shape parameters α and γ. These latter two variables define the limiting power-law behavior of the loss profile, d log ε″/d log ω = α for ν ≪ νmax and d log ε″/d log ω = −αγ for ν ≫ νmax. The first step is to find the HN parameters for the low-field loss spectrum, ε″lo(ω). Then, the high-field case, ε″hi(ω), is represented by the same HN curve but with modified values for Δε and τ. The relative field-induced changes in these two quantities are designated ΔE ln Δε and ΔE ln τ, respectively. The HN-based model curve (ε″hi − ε″lo)/ε″lo obtained in this manner is then tuned via ΔE ln Δε and ΔE ln τ for best agreement with the experimental result. The only exception to this approach is the case of PC,

confirms the EB2 dependence of the change. It can be seen that the material requires several milliseconds to equilibrate to a new dc-field level; that is, this approach to steady state is similar to the primary structural relaxation, which is characterized by the stretched exponential, exp[−(t/τα)β], with τα = 0.5 ms and β = 0.65. This observation for NMEC is analogous to the time dependence observed previously for glycerol (GLY), and a detailed discussion of the rise and decay behaviors of this NDE can be found elsewhere.33 In the present context, the main 7740


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The Journal of Physical Chemistry B Table 1. Nonlinear Dielectric Results and Entropy Changes for the Materials in this Studya material

T (K)

Δε

100 × ΔE ln Δε [obs.]

100 × ΔE ln τ [obs.]

υ (cm3/mol)

∂εs/∂T (K−1)

ΔES (mJ/(K mol))

NMEC PG PC GLY CPDE MTHF SAL PVAc

187.5 190 168 216.8 335 97.5 234 323

52.7 55.6 94 62 15 13.2 5.3 5.8

−0.41 −0.34 −0.46 −0.089 −0.15 −0.15 −0.028 −0.042

1.65 0.55 0.63 0.43 0.16 0.61 0.16 0.14

128.0 80.0 75.7 72.0 293.8 100.9 171.3 72.4

−0.49 −0.54 −0.50 −0.41 −0.073 −0.18 −0.044 −0.083

−27.8 −19.1 −16.8 −13.2 −9.5 −7.8 −3.3 −2.6

Measurement temperature T, relaxation amplitude Δε at T, observed field-induced change in relaxation amplitude ΔE ln Δε, observed field-induced change in relaxation time ΔE ln τ, molar volume υ, temperature derivative of dielectric constant ∂εs/∂T, and field-induced change in entropy ΔES. All field-induced changes labeled ΔEX are normalized to a field of 100 kV/cm. a

where even γ has been adjusted by ΔE ln γ. Because such HN fits describe only the results in the vicinity of the primary loss peak with sufficient accuracy, only experimental data within a decade of the loss peak frequency, νmax, have been considered for determining ΔE ln Δε and ΔE ln τ. The resulting HN fits to the NDE results in terms of (ε″hi − ε″lo)/ε″lo are included as red solid lines in Figures 2−6, and the numerical values of the parameter changes are listed within each figure. A better comparison of the field effects across different materials is obtained after normalizing the values to a common field of 100 kV/cm, and these quantities are compiled in Table 1. All field-induced changes in the relaxation amplitudes are negative, and these effects are understood to reflect dielectric saturation. Most earlier saturation experiments have been performed near room temperature,35,36 which prevents direct comparison with the literature data. Results from alternating current (ac) high-field measurements of PG near Tg have shown agreement with theoretical approaches.37,38 As seen in Table 1, these ΔE ln Δε values are correlated with the relaxation amplitude, Δε, of the material at the measurement temperature. In the HN fits, no frequency dependence of ΔE ln Δε is considered because the experiments are designed to capture the steady-state effect of high dc (opposed to ac) fields. In each figure showing NDE spectra, the level of the saturation effect is included as a dashed horizontal line. Because the loss profile is horizontal at ν = νmax, a small change in τ has no effect on ε″ at that frequency position. Accordingly, a change in ε″ at ν = νmax is solely determined by the saturation, leading to the (ε″hi − ε″lo)/ε″lo data crossing the ΔE ln Δε level (dashed line) at ν = νmax, see Figures 1−6. An exception is the case of PC, Figure 3, where field-induced modifications are excessive and the crossing frequency is thus shifted away from νmax. Because of the static nature of the high (dc) electric fields involved in the present study, heating effects remain negligible because they require the sample absorbing energy from a timedependent field. An apparent nonlinear effect that should be addressed at the high field used here is electrostriction, the reduction in the electrode separation by Coulombic attraction. The resulting relative increase in permittivity is given by Δε/ε = εsε0AEB2/(4aY), where A is the electrode area, a is the surface area of the support separating the electrodes, and Y is the Young modulus of the supporting material. For PG in the present cell at EB = 100 kV/cm, an estimate reveals that Δε/ε is safely below 10−4;37 thus, it is small compared to the saturation levels observed here. The focus of this article is on the field-induced changes in relaxation times, quantified by ΔE ln τ. For all cases, a positive ΔE ln τ is observed, consistent with the bias field increasing the

dielectric relaxation time. For the present molecular materials, ΔE ln τ can be considered a reliable indicator of changes in the structural relaxation time and viscosity and thus in the electrorheological effects. According to Table 1, the more polar liquids tend to display more pronounced electrorheological effects, but the correlation between ΔE ln τ and Δε is not very high. Therefore, it is of interest to find out what determines the extent of ΔE ln τ at a given field and for a certain material. As early as 1981, Moynihan and Lesikar27 predicted a change, ΔETg, of 2 mK for glycerol for field EB = 100 kV/cm and for the conditions of constant temperature and volume, which approximate the present experimental situations well. The value of ΔE ln τ = 0.0043 observed for glycerol at the same field (see Table 1) translates into a glass-transition shift, ΔETg, of about 12 mK, that is, considerably higher than Moynihan’s prediction. A more recent approach to electrorheological effects in molecular liquids has been advanced by Johari,29 who realized that the field-induced change in entropy, ΔES, should affect the relaxation times via the AG28 relation that links dynamics to configurational entropy. According to Fröhlich,39 applying a field of magnitude EB to a dielectric leads to a change in the thermodynamic (intensive) entropy, which is given by ΔES =

ε0υ ∂εs × E B2 2 ∂T

(3)

for isothermal and isochoric conditions (dT = dV = 0), a relation that is also the key to the electrocaloric effects.40 The sign of ΔES is governed by slope ∂εs/∂T, which is negative for most liquids.29 The molar volume (υ), the derivative of the dielectric constant with respect to temperature (∂εs/∂T), and the resulting entropy change (ΔES) for field EB = 100 kV/cm are included in Table 1. Unless estimated from standard data, molar volumes for GLY, MTHF, PG, cresolphthaleindimethylether (CPDE), and PC are determined from refs 41−45, respectively. Values for slopes ∂εs/∂T are determined in this study or are otherwise taken for GLY, MTHF, PG, and CPDE from refs 46−48 and 44, respectively. For the materials in this study, the values for ΔES range from 2.6 to 28 mJ/(K mol). All results are obtained under isothermal rather than adiabatic conditions because the samples are thin (d ≈ 10 μm), the electrodes have high heat capacities, and the structural relaxation times are long. A widely tested and accepted link between entropy and relaxation time is the model of Adam and Gibbs.28 The key feature of the model is the linear dependence of log τ on 1/ [TScfg(T)], equivalent to the apparent activation energy being 7741


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The Journal of Physical Chemistry B Table 2. AG Parameters and Nonlinear Dielectric Results for the Materials in this Studya material

T (K)

NMEC PG PC GLY CPDE MTHF SAL PVAc

187.5 190 168 216.8 335 97.5 234 323

CAG (kJ/mol) 35.47 44.9 67.4 174 39.0 112.8

Sexc(T) (J/(K mol))

EB (kV/cm)

ΔES (mJ/(K mol))

100 × ΔE ln τ [pred.]

100 × ΔE ln τ [obs.]

17.24 21.0 37.3 42.7 28.2 34.8

128 233 360 225 217 210 450 327

−45.5 −103.8 −217.2 −66.8 −44.7 −34.5 −67.6 −28.3

15.0 30.3 3.44 2.93 4.00 6.20

2.7 3.0 8.1 2.2 0.75 2.7 3.2 1.5

Measurement temperature T, AG constant CAG, excess entropy Sexc(T), electric field strength EB, field-induced change in entropy ΔES, AG prediction of field-induced change in relaxation time ΔE ln τ, and observed field-induced change in relaxation time ΔE ln τ. All field-induced changes labeled ΔEX are reported for the actual field, EB. a

fractions f S and f C is required55 before the AG relation can be tested in the context of the present field effects. In a search for the quantity underlying the material-specific magnitude of the observed ΔE ln τ for a common field, EB, correlation coefficients R of a linear relation of ΔE ln τobs with all other quantities listed in the two tables have been scrutinized. It turns out that the best correlation exists between ΔE ln τobs and ΔES (with R = 0.87), which is depicted in Figure 8 for a common field, EB = 100 kV/cm.

proportional to the reciprocal configurational entropy, 1/Scfg(T). Because Scfg cannot be measured directly, typical tests of the AG relation are based on the excess entropy, Sexc, determined as the difference between the entropy of the liquid/ glass state and that of the crystal.49 For practical purposes, the relation employed commonly thus reads log(τ /s) = AAG +

CAG TSexc(T )

(4)

where AAG and CAG are constants, and it compares very favorably with experimental data in the viscous regime near Tg.50−54 In light of the unknown factor involved, f S = Scfg/Sexc,55 the common explanation for the success of eq 4 is that f S is temperature invariant and thus part of adjustable parameter CAG. Here, the aim is to assess whether the AG relation has predictive power regarding ΔE ln τ on the basis of the fieldinduced changes in entropy, ΔES. For six materials in the present study, excess entropy data is available from the following literature: GLY,56,57 MTHF,58 SAL,59 PG,60 CPDE,44 and PC.61 The values required to test the AG approach are compiled in Table 2, in this case for the electric fields, EB, tabulated and used in the experiments. Predicted quantities for ΔE ln τ are obtained via the difference in relaxation times that emerges from eq 4 when replacing Sexc by Sexc + ΔES for the entropy term. Because the configurational entropy fractions, f S = Scfg/ Sexc, relevant when tuning temperature ( f S,T) or electric field (f S,E) may differ, the dependence of ΔE ln τ on ΔES, and thus on EB2 via eq 3, is expressed as ΔE ln τ = −

ln(10)CAG fS,E ΔES 2 T × Sexc (T ) fS,T

Figure 8. Compilation of the field-induced relative change in the relaxation time, ΔE ln τ, vs the field-induced entropy change, ΔES, for all eight compounds in this study. The values of ΔE ln τ are rescaled by the field squared to reflect the magnitude expected for field EB = 100 kV/cm.

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(5)

SUMMARY AND CONCLUSIONS In summary, we have subjected several glass formers to electric dc-bias fields between 100 and 500 kV/cm and determined the change in permittivity at the fundamental frequency. For all materials investigated here, we find a reduction in the relaxation amplitude (Δε), consistent with dielectric saturation as well as an electrorheological effect in terms of a field-induced increase in the structural relaxation time (τ). The eight viscous liquids displaying these NDEs vary in polarity (5.3 < Δε < 94) and fragility (52 < m < 104), suggesting that the dependence of relaxation times on the external electric field is a generic feature of polar liquids. The magnitude of this electrorheological effect, however, is material-specific, and for the present cases, the values varied from 0.14 to 1.65 in terms of ΔE ln τ at a field of 100 kV/cm.

The results of these predictions using eq 5 with f S,E = f S,T are compared to the observed values in Table 2. Their ratios, ΔE ln τobs/ΔE ln τpred, vary between 0.20 and 0.68, with an average of 0.43 and a standard deviation of 0.21. Expecting unity for these ratios, that is, ΔE ln τobs = ΔE ln τpred, would require that the configurational fractions, f S, be the same for field (E)- and temperature (T)-induced entropy changes, that is, f S,E = f S,T, which is not readily expected.53 Instead, the present data support that, on average, f S,E is 43% of f S,T, leading to lower ΔE ln τpred values if inserted into eq 5 and thus to better agreement with the observed values. The results compiled in Table 2 also fail to support the suggestion29 that ΔES is entirely configurational ( f S,E = 1), as this would imply the unphysical case of f S,T > 1. It appears that a deeper understanding of configurational 7742


Article

The Journal of Physical Chemistry B

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Among the various correlations considered here, the best predictor for the magnitude of the relative change in relaxation time constants, ΔE ln τ, turned out to be the field-induced change in thermodynamic entropy, ΔES, which is proportional to molar volume υ, slope ∂εs/∂T, and squared electric field EB2. For most simple liquids, the slopes of dielectric constant versus temperature, ∂εs/∂T, fall in the range −0.1 to −3, with only few hydrogen bonding cases displaying values around +0.01.29 It would be interesting to examine whether systems with positive ∂εs/∂T are associated with field-induced lowering of viscosity. As a change in τ impacts the dielectric permittivity where dielectric loss occurs, this electrorheological effect can be a significant source of nonlinear dielectric behavior in most liquids. This has been observed in dc-bias-field experiments for the eight systems in this study,30,31,33,44 as well as for the plastic crystal cyclooctanol.31 Analogous observations based on highamplitude ac fields (with zero bias) can also be found,26,62−65 partly reported before the relation to entropy was recognized.

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

The authors declare no competing financial interest.

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REFERENCES

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Electrorheological Source of Nonlinear Dielectric Effects in Molecular

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