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Limitations of heterogeneous models of liquid dynamics: Very slow rate exchange in
the excess wing
Subarna Samanta, and Ranko Richert
Citation: The Journal of Chemical Physics 140, 054503 (2014);
View online: https://doi.org/10.1063/1.4863347
View Table of Contents: http://aip.scitation.org/toc/jcp/140/5
Published by the American Institute of Physics
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THE JOURNAL OF CHEMICAL PHYSICS 140, 054503 (2014)
Limitations of heterogeneous models of liquid dynamics: Very slow rate
exchange in the excess wing
Subarna Samanta and Ranko Richert
Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287-1604, USA
(Received 19 November 2013; accepted 13 January 2014; published online 4 February 2014)
For several molecular glass formers, the nonlinear dielectric effects (NDE’s) are investigated for the
so-called excess wing regime, i.e., for the relatively high frequencies between 102 and 107 times the
peak loss frequency. It is found that significant nonlinear behavior persists across the entire frequency
window of this study, and that its magnitude traces the temperature dependence of the activation
energy. A time resolved measurement of the dielectric loss at fields up to 480 kV/cm across tens
of thousands of periods reveals that it takes an unexpectedly long time for the steady state NDE
to develop. For various materials and at different temperatures and frequencies, it is found that the
average structural relaxation with time scale τ α governs the equilibration of these fast modes that
are associated with time constants τ which are up to 107 times shorter than τ α . It is argued that true
indicators of structural relaxation (such as rate exchange and aging) of these fast modes are slaved to
macroscopic softening on the time scale of τ α , and thus many orders of magnitude slower than the
time constant of the mode itself. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863347]
I. INTRODUCTION
The glass transition temperature Tg indicates a kinetic
transition from a supercooled liquid to a glassy solid state.
The primary (or α-) structural relaxations involved in this
regime are characterized by non-exponential correlation functions and super-Arrhenius behavior.1 For a simple glassforming liquid, one could argue that the chemical identity of all molecules should result in spatially uniform
dynamics and serial relaxation patterns,2 referred to as homogeneous dynamics. However, numerous experimental results are qualitatively inconsistent with such a picture, e.g.,
those derived from nuclear magnetic resonance,3 dielectric
hole-burning,4, 5 deep-photobleaching,6 solvation dynamics,7
probe kinetics,8 and single molecule studies.9, 10 These experiments go beyond measuring two-point correlation functions
and were therefore able to lead to the recognition of the heterogeneous nature of the dynamics, characterized by the feature that fast and slow contributions to the dispersion of relaxation times are mutually independent and even spatially
segregated.11, 12 In fact, the prevailing picture implies that local relaxation processes are purely exponential for times that
do not exceed the average relaxation time, τ α .13, 14 Eventually, ergodicity is restored by rate exchange, i.e., by temporal fluctuations of the local time constants on a time scale
τ x ≈ 3τ α .15–17
A simple approach to heterogeneous dynamics justifies
writing the overall dispersive relaxation process, φ(t), as superposition of independent “modes” (e−t/τ ) characterized by
their time constants τ and relative amplitude or volume fraction gi = g(τ )dτ ,
∞
φ(t) = φ0
g(τ )e−t/τ dτ ≈ φ0
0
0021-9606/2014/140(5)/054503/7/$30.00
N
i=1
gi e−t/τi .
(1)
In the absence of rate exchange, the quantities τ i are time invariant. In a system with rate exchange, the values of τ i in
Eq. (1) represent the eigenvalues of more complicated
kinetics.18
Nonlinear dielectric techniques19, 20 aimed at identifying
heterogeneous dynamics are based on the following idea:4
the power density p absorbed by a certain relaxation mode
with time constant τ from an external sinusoidal electric field,
E(t) = E0 sin(ωt), will depend on the magnitude of the dielectric loss (ε ) of that mode at the frequency ω via Joule’s law,
p = ε0 E02 ε (ω)ω/2. For the case of independent modes and
at sufficiently high field amplitudes E0 , energy absorption will
displace the system from equilibrium, a change that can be
modeled by an increase of the configurational (fictive) temperature, Tcfg , of particularly those modes with τ ≈ 1/ω. A
change in Tcfg will initiate an equilibration process that involves adjustments of the relaxation time constants, τ , thereby
facilitating experimental access to the time scale τ x of rate
exchange at a constant temperature. Unlike standard heating,
this process modifies the dynamics of a subset of the relaxation time dispersion, and this spectral selectivity has been
verified experimentally.4, 5, 13, 16
The phenomenological model of the nonlinear dielectric
effects (NDE’s) that originate from the absorption of energy
from the field rests on the approximate identity of the spectral shapes for the dielectric permittivity, ε (ω), and dynamical heat capacity, cp (ω), so that their respective probability
densities, g(τ ), cancel.4, 5 It is further assumed that the configurational temperature relaxes to the true temperature with the
time constant of the mode in question, τ i . This is equivalent
to claiming that τ x = τ i for each individual mode, which reveals the heterogeneous character of this model with respect
to both relaxation dynamics and rate exchange. The steady
state prediction for the field induced modification of a single
140, 054503-1
© 2014 AIP Publishing LLC
054503-2
S. Samanta and R. Richert
J. Chem. Phys. 140, 054503 (2014)
mode with time constant τ is
ln τhi − ln τlo = −
ε0 E02 ε
EA
ω2 τ 2
×
×
,
2
kB T
2ρcp
1 + ω2 τ 2
(2)
Tcfg
where ε is the dielectric relaxation amplitude, cp is the
(glass to liquid) heat capacity step, ρ is the density, ε0 is the
permittivity of vacuum, and kB is Boltzmann’s constant. Indices “hi” and “lo” refer to the field amplitudes E0 and ≈0,
respectively. The activation parameter EA quantifies the sensitivity of τ to changes in the configurational temperature,
EA = dlnτ /d(1/Tcfg ), and the value is taken from its analog,
dlnτ α /d(1/T), for the overall linear response dynamics. For
high frequencies, ω τ , the ω dependent term in Eq. (2)
remains unity, and this feature leads to the expectation of a
frequency invariant magnitude of this NDE across the high
frequency wing of the α-process.19, 20
For many molecular glass-formers, the relation implied
in Eq. (2) has been verified in detail for frequencies up
ω ≈ 500ωmax , where ωmax is the peak frequency of the loss
profile.19, 20 Recently, Bauer et al. have extended such experiments to ω ≈ 107 ωmax , and concluded that the loss in the
excess wing region does not show any nonlinear behavior,21
which could imply the absence of rate exchange in that frequency regime. The excess wing is understood as the high
frequency regime characterized by loss in excess of the power
law ε (ω) ∝ ω−γ .22 Whether this part of the structural relaxation is qualitatively different from the lower frequency
counterpart, particularly regarding dynamical heterogeneity,
has been the subject of intense scrutiny.23–25
The present study investigates NDE’s in the excess wing
(i.e., for high frequencies with ω ωmax ) of typical molecular glass-formers, with emphasis on clarifying the effects
associated with rate exchange and the implications on models based upon heterogeneous dynamics. To this end, timeresolved high-field dielectric relaxation measurements are
performed at frequencies up to 107 times the peak frequency
ωmax of the loss profile. It is found that it requires an unexpectedly long time for these otherwise fast modes to equilibrate regarding their time constants. The connections of these
findings to structural relaxation, physical aging, calorimetry,
and rate exchange are discussed.
ming a sequence of measurements (1 ≤ i ≤ 100) at the same
frequency, with the field amplitude set to a high value of E0
for the entire time interval for which 10 ≤ i ≤ 50 and with a
low (linear response) field otherwise. In this mode, frequencies between 1 Hz and 10 kHz are used and a permittivity
value is stored once every few seconds. The behavior shortly
after increasing the field has been recorded with a time resolution of one period (1/ν) with an oscilloscope based technique described in detail previously.20, 27 In this case, voltage
and current traces are recorded with a resolution of about
1000 points per period for hundreds of periods which include field amplitude transitions, and the dielectric properties
are derived from a period-by-period Fourier analysis of these
curves.
III. RESULTS
All NDE features shown below are qualitatively equal for
PC, MTHF, and GLY, but the effects are best resolved for
PC. All magnitudes of field induced changes scale with E0 2
and are thus proportional to the energy associated with that
field amplitude. Linear response dielectric loss spectra are depicted for PC for several temperatures on a reduced frequency
scale in Fig. 1. In this temperature range, time-temperature superposition (TTS) is valid, and the dielectric constant εs and
the relaxation strength ε are virtually temperature invariant. The graph also includes a loss curve for one temperature
(T = 165 K) obtained at a high field of E0 = 283 kV/cm, and
the deviation from its low field counterpart is clearly visible at
frequencies exceeding the loss peak position (ν > ν max ). The
figure also clarifies the two distinct methods of reporting the
field induced changes in what follows: in terms of the “vertical” difference, ε hi − ε lo , or as the “horizontal” difference,
lg(ν hi ) − lg(ν lo ) or equivalently lg(τ hi ) − lg(τ lo ), calculated
propylene carbonate
10
lg( hi)
''hi
E0, lo = 14 kV/cm:
II. EXPERIMENT
The glass-forming liquids of this study are propylene
carbonate (PC, 99.7%, anhydrous), 2-methyltetrahydrofuran
(MTHF, 99+%, anhydrous), and glycerol (GLY, 99.5+%,
spectrophotometric grade), all purchased from Sigma-Aldrich
and used as received. High field impedance experiments are
performed for frequencies from 0.1 Hz to 100 kHz using a
system based upon a Solartron Si-1260 gain/phase analyzer
and voltage booster Trek PZD-700 that has been describer
earlier.26 The capacitor consists of two polished stainless steel
disks (16 and 20 mm ø), separated by a 10 μm thick Teflon
ring that leaves an inner area of 14 mm ø for the sample. For
those cases where a time resolution of a few seconds is sufficient, time resolved experiments are performed by program-
lg( lo)
E0, hi = 283 kV/cm:
T = 165 K
T = 156 K
1
T = 159 K
T = 162 K
T = 165 K
10
-1
10
0
''lo
-5
= 9.3 10 Hz
max
-3
max
= 7.9 10 Hz
-1
= 2.8 10 Hz
max
= 5.0 Hz
max
10
/
1
10
2
10
3
max
FIG. 1. Symbols represent linear response (E0, lo = 14 kV/cm) dielectric
loss spectra, εlo (ν/ν max ), for propylene carbonate at the temperatures indicated. The superposition is obtained by normalizing the frequency axis to
the respective peak values, ν max , listed in the legend, while the amplitudes
are not subject to normalization. The dashed line shows the high field (E0, hi
= 283 kV/cm) loss spectrum, ε hi (ν/ν max ), for T = 165 K, using the same
ν max = 5.0 Hz value as for its low field counterpart. Arrows at the high frequency wing indicate the two distinct approaches to reporting the field effect:
the “vertical” difference, ε hi − ε lo , and the “horizontal” difference, lg(ν hi )
− lg(ν lo ), calculated according to Eq. (3).
S. Samanta and R. Richert
J. Chem. Phys. 140, 054503 (2014)
propylene carbonate
ln (
lo
T = 154 K
T = 155 K
T = 156 K
E0 = 283 kV/cm
-1.0
0.2
lo
)/
propylene carbonate
T = 156 K
T = 159 K
T = 162 K
T = 165 K
E0 = 283 kV/cm
)
0.3
lo
054503-3
-0.8
= 1250 s
= 200 s
=
40 s
max
max
-0.4
max
time
ln (
0.1
(
hi
hi
)
-0.6
-0.2
0.0
10
-1
10
0
10
1
10
2
10
3
10
0.0
4
10
3
10
4
10
/ Hz
/
10
7
max
-0.8 (a) 2-methyltetrahydrofuran
)
E0 = 283 kV/cm
T = 88.4 K
T = 89.0 K
T = 89.6 K
-0.4
lo
Spectra of field induced changes in the loss ε for
propylene carbonate at different temperatures are depicted in
Fig. 2, where the NDE’s are shown as relative change of the
dielectric loss, (ε hi − ε lo )/ε lo . This relative vertical difference is equal to lnε for small values, where refers to the
difference between high and low field result. Consistent with
similar experiments reported recently,21 the values of lnε
approach zero for frequencies exceeding about 105 ωmax . Note
that these data are recorded in a standard impedance mode,
i.e., with only moderately long measurement times that would
guarantee steady state results only if the polarization had been
in the regime of linear response.
Time resolution has been added to the above approach
in order to facilitate extracting the “instantaneous” contribution to the NDE as well as the long time limit at which ε hi
has reached a steady state value. The results are shown in
Fig. 3, and for reasons that are explained below, these effects are quantified in terms of the “horizontal” difference,
i.e., the separation of the curves ε hi (ω) and ε lo (ω) along the
lnω scale, which is derived from the “vertical” difference via
Eq. (3). Clearly, Fig. 3 demonstrates that the NDE does not
approach zero for ω ≤ 107 ωmax when its steady state value is
considered. Analogous results for MTHF and glycerol are depicted in Fig. 4, and the overall pattern is again that the steady
state magnitude of the NDE significantly exceeds its short
time value. As indicated by the arrow in Fig. 3, all NDE’s
shown in Figs. 3 and 4 are evolving in time from the level of
the open symbol to that of the closed symbol.
The time evolution of the NDE that is implied in the results of Figs. 3 and 4 have been measured for all open/closed
= 400 s
= 170 s
= 42 s
max
max
ln (
(3)
)
− εlo
)/εlo
(εhi
.
d lg εlo /d lg ω
6
symbol pairs of these two figures. Characteristic results of
such measurements are depicted as symbols in Fig. 5 for PC
and in Fig. 6 for MTHF and GLY. The curves are selected
to show the variation of the average relaxation time τ α (via
changing temperature) at a fixed test frequency ν, as well as
the variation of the test frequency ν at a fixed average relaxation time τ α . Values of the ratios ν/ν max range from about
160 to 70,000 for the six situations included in Figs. 5 and 6.
hi
according to
10
FIG. 3. Nonlinear dielectric effect based on a field of E0 = 283 kV/cm for
propylene carbonate at three different temperatures, T = 154, 155, and 156 K,
equivalent to the respective peak relaxation times τ max = 1250, 200, and 40 s.
The effects are shown as “horizontal” difference between high (“hi”) and
low (“lo”) field loss, ln(τ hi ) − ln(τ lo ), according to Eq. (3). Open symbols
represent the effect observed within the first few periods after switching from
low to high field amplitude (φ ∞ ); solid symbols are for the steady state values
(φ s ), see Eq. (4).
max
0.0
-0.2 (b) glycerol
E0 = 481 kV/cm
ln (
FIG. 2. Field induced relative change of the dielectric loss, (ε hi
− ε lo )/ε lo , for propylene carbonate at the temperatures indicated. The
subscripts “hi” and “lo” refer to electric fields of E0 = 283 kV/cm and
E0 = 14 kV/cm, respectively. Spikes recurring at three distinct frequency
positions for all temperatures have been removed for clarity. Open symbols
indicate situations in which the temperature is raised by excessive energy absorption. For the temperatures of 162 and 165 K, the peak loss frequency positions are indicated by arrows at ν max = 0.28 and 5.0 Hz, respectively. The
vertical height of the hexagons indicate the relative apparent activation parameter, dln(τ max )/d(1/T), while their abscissa positions match the frequency
at which the maximal field induced effect is observed at the corresponding
temperature.
ln τhi − ln τlo =
5
T = 186 K
T = 188 K
T = 190 K
-0.1
= 112 s
= 32 s
= 10 s
max
max
max
0.0 2
10
10
3
10
/
4
10
5
10
6
max
FIG. 4. Nonlinear dielectric effects as in Fig. 3, reported as “horizontal” difference between high (“hi”) and low (“lo”) field loss, ln(τ hi ) − ln(τ lo ), according to Eq. (3). Open symbols represent the effect observed within the first
few periods after switching from low to high field amplitude (φ ∞ ), solid symbols are for the steady state values (φ s ), see Eq. (4). (a) Results for a field of
E0 = 283 kV/cm for 2-methyltetrahydrofuran at the temperatures T = 88.4,
89.0, and 89.6 K, equivalent to the respective peak relaxation times τ max
= 400, 170, and 42 s. (b) Results for a field of E0 = 481 kV/cm for glycerol
at the temperatures T = 186, 188, and 190 K, equivalent to the respective
peak relaxation times τ max = 112, 32, and 10 s.
054503-4
S. Samanta and R. Richert
0.20
0.15
J. Chem. Phys. 140, 054503 (2014)
IV. DISCUSSION
(a)
PC
E0 = 200 kV/cm
T = 156 K
= 1 Hz
= 25 s
= 0.61
0.10
0.05
( ''hi
''lo ) / ''lo
0.00
0
0.40 (b)
0.30
0.20
0.10
0.00
0
0.15
200
400
600
PC
E0 = 283 kV/cm
T = 155 K
= 1 Hz
= 105 s
= 0.61
200
800
400
600
800
(c)
PC
E0 = 283 kV/cm
T = 155 K
0.10
= 100 Hz
= 105 s
= 0.61
0.05
0.00
0
100
200
300
time / s
FIG. 5. Time resolved relative change of the dielectric loss, (ε hi
− ε lo )/ε lo , in propylene carbonate for an increase of the field amplitude
from E0 = 14 kV/cm (“lo”) to a significantly higher field (“hi”) at t = 0. For
the three panels, the high field amplitudes E0 , temperatures T, peak relaxation times τ max , and test frequencies ν are as follows: (a) E0 = 200 kV/cm,
T = 156 K, τ max = 25 s, ν = 1 Hz; (b) E0 = 283 kV/cm, T = 155 K,
τ max = 105 s, ν = 1 Hz; (c) E0 = 283 kV/cm, T = 155 K, τ max = 105 s,
ν = 100 Hz.
0.15
(a)
0.10
MTHF
E0 = 283 kV/cm
T = 89 K
= 1 Hz
= 170 s
= 0.52
0.05
0.00
0.08
0.06
0.04
0.02
0.00
200
400
(b)
200
600
800
GLY
E0 = 481 kV/cm
T = 186 K
= 1 Hz
= 112 s
= 0.63
0
(
hi
lo
)/
lo
0
400
600
800
0.03 (c)
0.02
GLY
E0 = 481 kV/cm
T = 186 K
= 100 Hz
= 112 s
= 0.63
0.01
0.00
0
50
100
150
200
250
time / s
FIG. 6. Time resolved relative change of the dielectric loss, (ε hi
− ε lo )/ε lo , in 2-methyltetrahydrofuran and glycerol for an increase of the
field amplitude from E0 = 14 kV/cm (“lo”) to a significantly higher field
(“hi”) at t = 0. For the three panels, the material, high field amplitudes E0 ,
temperatures T, peak relaxation times τ max , and test frequencies ν are as follows: (a) MTHF, E0 = 283 kV/cm, T = 89.0 K, τ max = 170 s, ν = 1 Hz;
(b) GLY, E0 = 481 kV/cm, T = 186 K, τ max = 112 s, ν = 1 Hz; (c) GLY,
E0 = 481 kV/cm, T = 186 K, τ max = 112 s, ν = 100 Hz.
A. Nonlinear effects in the excess wing
Numerous nonlinear effects have been observed in
dielectric studies of liquids; among them are dielectric
saturation,28, 29 chemical effects,30–32 and field induced modifications due to the absorption of energy from an electric field of sufficient amplitude.4, 5, 19–21 Even at moderately
high fields, this later NDE that is analogous to microwave
heating can change the dielectric loss by 20% or more.20
Most initial such experiments were limited to frequencies of
ω < 500ωmax , where ωmax is the peak frequency of the loss
profile. Recently, Bauer et al. have extended this range to ω
≈ 107 ωmax , and concluded on the “lack of nonlinearity in the
excess wing region” on the basis of lnε data approaching
zero at high frequencies for PC and GLY.21 The present results shown in Fig. 2 confirm this tendency of the field induced relative change in the dielectric loss, but those loss
values are obtained after the high field is applied only for a
moderate number of cycles and thus may not reflect steady
state situations.
The results of Figs. 3 and 4 demonstrate that the addition of time-resolution capabilities is required to determining the time after which the high field loss has reached its
steady state level, and the steady state values obtained in this
manner are considerably above the short time values for the
NDE’s. Instead of the NDE disappearing for high frequencies, reductions in the steady state magnitude of ln(τ hi /τ lo )
with increasing frequency range from 40% to 60%, and occur
predominantly for frequencies ω > 104 ωmax . Therefore, even
deep within the excess wing regime, the extent of nonlinearity
regarding dielectric polarization is practically as pronounced
as for frequencies that are much closer to the loss peak. The
novel feature observed at present is the exceedingly long time
or number of periods it takes for the NDE to establish its
steady state magnitude.
One might wonder why the solid symbols in Figs. 3 and
4 do not collapse onto a master curve, although the frequency
axis is normalized as ω/ωmax and the linear response spectra
shows TTS behavior (see Fig. 1). The reason is that the magnitude of the NDE scales with the apparent activation energy,
dln(τ max )/d(1/T), as shown by the hexagons in Fig. 2 for the
case of PC. According to Figs. 3 and 4, the resulting separation of the steady state curves increases in the order GLY,
MTHF, PC. This order correlates with the respective fragility
indices, m = 53, 65, 99,33 in support of the above idea that
the temperature dependence of the activation parameter is responsible for the lack of superposition.
Several examples of the present time resolved measurements are depicted in Fig. 5, with each panel being associated
with a different frequency relative to the loss peak: (a) 160, (b)
660, and (c) 66,000 in terms of ω/ωmax . Lowering the temperature from T = 156 K to 155 K and thus increasing τ α leads to
a slower rise of the quantity (ε hi − ε lo )/ε lo , even at a constant frequency of ν = 1 Hz, see Fig. 5(a) and 5(b). Increasing
the frequency from ν = 1 Hz to 100 Hz at a constant temperature of T = 155 K has no effect on the time dependence of
the NDE, see Fig. 5(b) and 5(c). One can observe in Fig. 5(c)
that the NDE has not yet reached a steady state plateau after
054503-5
S. Samanta and R. Richert
J. Chem. Phys. 140, 054503 (2014)
300 s for a frequency of ν = 100 Hz, i.e., after 30,000 periods. It turns out that all such curves for ω > 102 ωmax are well
represented by the following fit function designated φ x (t),
εhi
− εlo
= φx (t) = φ∞ + (φs − φ∞ )(1 − exp[−(t/τα )β ]),
εlo
(4)
with φ ∞ and φ s representing the “instantaneous” (not timeresolved) and the steady state level of the relative change
of the loss, respectively. The values for φ ∞ and φ s are adjustable fit parameters, while τ α and β are the KohlrauschWilliams-Watts34, 35 parameters that describe the α-relaxation
in the time domain, φ(t) = φ 0 exp −(t/τ α )β at that temperature. Solid lines in Fig. 5 demonstrate how well Eq. (4)
represents the time dependence of the NDE, and the values for φ ∞ and φ s are shown as open and solid symbols
in Fig. 3, respectively. Fig. 5 also verifies that all field induced changes are completely reversible, as the values for
ε hi − ε lo are seen to revert to zero after the electric field
amplitude is lowered again. While Fig. 5 focuses on PC, the
analogous behavior for MTHF and GLY is depicted in Fig. 6.
In conclusion, the average structural relaxation (characterized by φ(t) with parameters τ α and β) determines the
time evolution of field induced changes in the excess wing
regime.
B. Comparison with model predictions
High field dielectric studies that address the nonlinear
effect resulting from energy absorption began with dielectric hole burning experiments.4 These field induced and spectrally selective modifications has been seen to agree with
the predictions of the so-called ‘box’-model mentioned in
the Introduction.4, 36–38 Basic ideas of this phenomenological model have been used also to provide a quantitative description of high field impedance results such as those obtained in the present work.5, 19, 20, 27 One important aspect of
that model is that for the steady state “horizontal” difference,
a frequency invariant plateau at high frequencies is predicted
for a given temperature, ln τhi − ln τlo ∝ E02 ε/cp , see
Eq. (2). The simplicity of this relation rests on the assumption
that the contributions of a mode to cp and to ε are proportional quantities, which leads to permittivity and heat capacity entering only as the ratio ε/cp . Near the peak, this
feature is confirmed by comparing the profiles of ε (ω) and
cp (ω), which suggest proportionality of the two quantities
at every frequency.39 It means that the lower absorptivity at
higher frequencies is compensated by a reduction of the heat
capacity, and the net change in fictive temperature remains
frequency invariant at a given field amplitude E0 . The resulting high frequency plateau behavior of the NDE has been observed in many instances, but those findings were limited to
the regime of ω < 500ωmax , i.e., not too far above the loss
peak frequency.19, 20
The frequency dependence of (ε hi − ε lo )/ε lo reported
by Bauer et al.21 and seen in Fig. 3 appears to contrast the
frequency invariance implied in Eq. (2) for ω ωmax . The
change from the “vertical” to the “horizontal” difference us-
ing Eq. (3) will not solve the issue, because the high frequency
slope of the loss profile does not change by much more than
a factor of about 2. However, the steady state data (solid symbols in Figs. 3 and 4) display a comparatively mild frequency
variation, with the level of the magnitude of ln(τ hi /τ lo ) changing by 40% to 60% for frequencies ω > 104 ωmax . For this
range 104 ωmax to 107 ωmax , it is very possible that the ratio
ε (ω)/cp (ω) has changed by about 50% relative to the smaller
frequencies. Therefore, our steady state data for ln(τ hi /τ lo ) are
compatible with the model leading to the steady state relation
of Eq. (2), but with ε/cp replaced by ε (ω)/cp (ω). Accordingly, there is no reason to question the concept of nonlinear heterogeneous dynamics for frequencies ωmax to 107 ωmax
on the basis of the short time NDE becoming practically
zero.
As demonstrated for PC in Fig. 2, the extent of the nonlinear effect changes considerably with temperature, and this
change traces the apparent activation energy (indicated by
hexagons and dotted line). A similar scaling is also noted by
Bauer et al.,40 and their work relates the activation parameter
to the number, Ncorr , of correlated molecules, where Ncorr is
derived from a model that relates the nonlinear cubic susceptibility to a four-point correlation function for systems near
a critical point.41 In the framework of the phenomenological
model leading to Eq. (2), the scaling of the effect with the activation energy EA is implied in Eq. (2), as the other quantities
are virtually constant within the narrow temperature interval
covered in Fig. 2.
We now address the time scale with which the steady
state level of the NDE is reached. It has been recognized
previously that the time for the quantity (ε hi − ε lo )/ε lo
to saturate is frequency invariant and much longer at high
frequencies than predicted by the phenomenological model,
while the model is quite accurate regarding the frequency
dependent rate of change at lower frequencies.20, 42 The
threshold for this failure of the model on the frequency
scale coincides with the onset of the excess wing,20 i.e., at
about 102 ωmax , but the time dependence of the configurational temperature for ω > 102 ωmax had not been quantified
previously.
The above results demonstrate that all fast modes (τ
< 10−2 τ max , within the excess wing) are equally slaved to
the average structural relaxation regarding their rate (1/τ x ) at
which time constants are approaching their equilibrium values. The conclusion from these observations is that modes
within the excess wing change their relaxation time constant
(or, equivalently, configurational, or fictive temperature) on
the time scale of the overall primary structural relaxation, instead of on the time scale of the mode itself as suggested by
the “box”-model. Confinement by slower modes has been discussed as possible explanation for this slaving effect.43 This
appears to contrast the basic idea of heterogeneous dynamics
that fast and slow modes are entirely independent, and assuming the identity τ x = τ i for each mode is incompatible with the
present findings of τ x exceeding τ i by many orders of magnitude. Accordingly, modeling the impact of energy absoption
on the permittivity of the liquid requires a revision regarding
the time scales associated with τ x in the regime of the excess
wing.
054503-6
S. Samanta and R. Richert
C. Relation to rate exchange and physical aging
fluctuations:
independent
slaved
aging isotherm
-2
-1
0
1
2
3
4
5
2
3
4
5
lo g
FDT
rate exchange
The above discussion has shown that it is necessary to
discriminate two distinct time scales associated with each
mode “i” that contributes to the overall relaxation with time
constant τ i : On the one hand, the time constant τ i that defines a local correlation time of some experimental variable
(e.g., polarization in case of a dielectric experiment, enthalpy
content in case of calorimetry), and, on the other hand, the decay function φ x (t) associated with the time τ x required for τ i
to attain its equilibrium value after some perturbation (e.g., a
change in temperature as in physical aging or calorimetric experiments, or a change in structure or configurational temperature as in the present case of an influx of energy). Actually, it
is more the latter time correlation function φ x (t) that deserves
the label “structural relaxation.” Typical experiments aimed at
quantifying the relaxation behavior or identifying the heterogeneous nature of dynamics are performed at a constant temperature, i.e., under quasi-equilibrium conditions.16, 17 In the
context of such experiments, the term “dynamical heterogeneity” typically refers to the independence of the local processes
characterized by the τ i ’s, determined at a constant structure
(i.e., with temperature, fictive temperatures, and time constants not changing in the course of the relaxation process).
Apart from the present high field dielectric technique
that involves an influx of energy to the sample, experiments
that involve the equilibration processes characterized by φ x (t)
are physical aging experiments,44, 45 and calorimetric techniques such as differential scanning calorimetry (DSC) and
modulated DSC.46 All these experiments have in common
that the structure is changing to an extent that time constants
(τ i ) are modified as time proceeds. Understanding these nonequilibrium processes quantitatively is complicated by the
fact that relaxation times (τ i ) depend not only on temperature
but also on structure, i.e., τ i = τ i (T,Tf ), where structure can
be mapped onto the fictive temperature, Tf .47, 48 A common
feature of such studies of enthalpy recovery is the assumption of a single fictive temperature which slaves the Tf ’s of
all modes to the overall structural relaxation.49, 50 The experimental signature of a single fictive temperature is the common observation of time-aging time superposition (TaTS).51
A single Tf implies that all mode specific τ i ’s move towards
their new equilibrium values in a concerted fashion, consistent
with the present findings regarding the excess wing regime.
Therefore, the slow equilibration of excess wing modes observed here is equivalent to the slow physical aging of these
fast modes, for instance in response to a step in temperature to
a lower value.51 Such a response is depicted schematically in
Fig. 7, where the upper part is meant to represent the shift
of the five distinct modes to lower frequencies (higher τ i ’s)
in response to a small temperature down-jump. By virtue of
the fluctuation dissipation theorem (FDT),52 there should be
corresponding fluctuations at a constant temperature, as illustrated in Fig. 7. The FDT does not provide a quantitative
connection in this case, because physical aging is not a linear response situation These fluctuations are known as rate
exchange,15, 17 and again a single global exchange rate τ x is
typically assumed for all modes. Therefore, aging isotherms
and the current high field experiments are considered direct
J. Chem. Phys. 140, 054503 (2014)
-2
-1
0
1
l olog
g
FIG. 7. Schematic representation of heterogeneity in terms of five distinct
Debye type modes whose superposition yields a dispersive dielectric loss
peak. Arrows in the top part indicate the equilibration of fictive temperatures or relaxation time constants, the process of an aging isotherm occurring
in response to a temperature down-jump. Arrows in the bottom part indicate
rate exchange at a constant temperature. The two scenarios are linked only
qualitatively by the fluctuation dissipation theorem.
measures of rate exchange effects, φ x (t), for which heterogeneity does not apply in the same way as for linear response
isothermal relaxations, φ(t).
V. SUMMARY AND CONCLUSIONS
High field dielectric relaxation experiments are performed in the high frequency wing of the dielectric loss of
three molecular glass-forming liquids, at field amplitude that
are sufficient to drive the systems beyond the regime of linear responses. The novel feature of these measurements is the
time resolution of nonlinear dielectric effects that extends to
about 30,000 periods of the applied field. For frequencies positioned between factors of 102 to 107 above the peak loss
frequency, ωmax , it is found that it takes an unexpectedly long
time for the system to reach steady state conditions. The level
of the steady state nonlinear effect is consistent with a phenomenological model of nonlinearity resulting from energy
absorption, whereas the time scale required to reach that level
is incompatible with the previous understanding of such high
field effects.
In conclusion, the present nonlinear dielectric experiments provide evidence for the relaxation of fictive temperatures and thus of structures not being heterogeneous for
modes that are much faster than the most probable time constants, a frequency range that appears to coincide with the excess wing regime of susceptibilities. The present results do
not indicate that the excess wing should be viewed as a process that is distinct from the main α-relaxation. True structural
relaxation (i.e., the time correlation function of fictive temperatures) is slaved to the macroscopically average α-relaxation
054503-7
S. Samanta and R. Richert
in this higher frequency range, 102 ωmax < ω < 107 ωmax . The
same technique has demonstrated heterogeneous relaxation or
mutual independence of modes with spectral positions closer
to the loss peak. The observations are consistent with the
heterogeneous nature of relaxation dynamics, φ(t), of typical
variables used to probe signatures of structural relaxation in
the regime of linear response (e.g., dielectric polarization, enthalpy, mechanical stresses) across the entire frequency spectrum of the α-process. In other words, such linear response
experiments do not involve systematic changes in the time
constants τ i and the constraint on how fast τ i can be altered
or fast modes remains irrelevant.
ACKNOWLEDGMENTS
Stimulating discussions with Th. Bauer, P. Lunkenheimer, and A. Loidl, as well as hospitality supported by
the Deutsche Forschungsgemeinschaft through the Research
Unit FOR 1394 are greatly appreciated. This material is based
on work supported by the National Science Foundation under Grant No. CHE 1026124 (International Collaboration in
Chemistry).
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