Mechanical and dielectric relaxation spectra in seven highly viscous glass formers
U. Buchenau∗
Institut für Festkörperforschung, Forschungszentrum Jülich
Postfach 1913, D–52425 Jülich, Federal Republic of Germany
(Dated: April 2, 2007)
Published dielectric and shear data of six molecular glass formers and one polymer are evaluated
in terms of a spectrum of thermally activated processes, with the same barrier density for the
retardation spectrum of shear and dielectrics. The viscosity, an independent parameter of the fit,
seems to be related to the high-barrier cutoff time of the dielectric signal, in accordance with the
idea of a renewal of the relaxing entities after this critical time. In the five cases where one can fit
accurately, the temperature dependence of the high-barrier cutoff follows the shoving model. The
Johari-Goldstein peaks, seen in four of our seven cases, are describable in terms of gaussians in
the barrier density, superimposed on the high-frequency tail of the α-process. Dielectric and shear
measurements of the same substance find the same peak positions and widths of these gaussians,
but in general a different weight.
PACS numbers: 64.70.Pf, 77.22.Gm
I. INTRODUCTION
The publications of Kia Ngai deal with more sub-
stances and more measurement techniques than the work
of any other scientist in the field of undercooled liquids.
Within the past three decades, whenever a new develop-
ment appeared, he was the quickest to appreciate it, ana-
lyze it and bring it to the general attention, thus speeding
up the progress substantially.
Many scientists in the field share his conviction that
the flow process in highly viscous liquids can only be
understood by combining all possible techniques for its
study1,2,3,4,5,6. The present paper evaluates recently
published7,8 broadband shear and dielectric relaxation
data on seven glass formers. The mechanical shear data
were obtained with a new technique9 which allows to
cover a large dynamical range. Samples for dielectric and
shear measurements were taken from the same charge,
and the temperature sensors of both measurements were
calibrated to each other.
The data show a striking similarity of G′′(ω) and ǫ′′(ω)
on the right hand side of the α-peak, a similarity which is
sometimes perturbed by the secondary Johari-Goldstein
peak10,11. The similarity suggests a common origin of the
α-peak in dielectrics and shear. The fact that the shear
peak appears at a higher frequency than the dielectric
peak is explainable in terms of the viscosity, which in a
compliance treatment12 is a free parameter.
In order to identify the elementary processes with ther-
mally activated jumps over an energy barrier V , one can
use a recent translation13 of the textbook12 retardation
spectrum L(ln τ) into a barrier density function l(V ). We
will argue that the compliance barrier density of the shear
equals the electric dipole moment barrier density lǫ(V )
of the dielectric data.
The next section (section II) explains and motivates
this approach in more detail. The results of the data
treatment are described in section III. They are discussed
and compared to other approaches in section IV. Section
V summarizes and concludes the paper.
II. THE BARRIER DENSITY FUNCTIONS FOR
SHEAR AND DIELECTRICS
The choice of a retardation spectrum for the shear is
motivated by a surprising coincidence, which is more or
less visible in the data of all seven substances. We show
the two examples where it is most clearly seen in Fig. 1
and Fig. 2.
Fig. 1 compares G′′(ω) and −ǫ′′(ω) at the same tem-
perature for the type-A glass former PPE. PPE, 1,3-
Bis(3-phenoxyphenoxy)benzene, is a diffusion pump oil
with the commercial name Santovac-5P, a molecule con-
sisting of five phenyl rings connected by four oxygens to
form a short chain. In terms of the classification proposed
by the Bayreuth group14, it is a type-A glass former, a
-1 0 1 2 3 4 5 6 7
    G''
    - " scaled
Santovac-5P (PPE)
        254 K
log( /s-1)
FIG. 1: Comparison of G′′(ω) to a properly scaled −ǫ′′(ω) in
a double-log scale for PPE at 254 K.
http://arxiv.org/abs/0704.1356v1
glass former which shows no or at least no pronounced
secondary Johari-Goldstein peak. The (negative) ǫ′′(ω)-
data have been scaled to coincide with the G′′(ω)-data
on the right hand side of the peak. One finds good agree-
ment betweenG′′(ω) and−ǫ′′(ω) as soon as the frequency
is two decades higher than the one of the peak in G′′(ω).
In addition, Fig. 1 shows a change of slope of the α-
tail, from an ω−1/2- to an ω−1/3-behavior. The tendency
is seen most clearly in the dielectric data, but it is also in
the shear data; their fit improves markedly if one allows
for a ω−1/3-component. This might be the influence of a
hidden Johari-Goldstein peak, an explanation which has
been favored for glycerol on the basis of pressure, aging
and chemical series measurements (for a review, see Ngai
and Paluch15). But there is also impressive experimen-
tal evidence for a limiting ω−1/3-behavior in shear com-
pliance data of type-A molecular glass formers16, which
show the so-called Andrade17 creep, J(t) ∼ t1/3, in the
short-time limit. Therefore we will fit our data in terms
of a sum of an ω−β-term (with β as fit parameter) and
an ω−1/3-term, dominating at high frequency.
Toward lower frequency, the common slope terminates
for the shear data already at a higher frequency (a shorter
time) than for the dielectric data. The natural explana-
tion for this is that the parallel between dielectrics and
shear is in fact between shear compliance and dielectric
susceptibility. In a comparison of these two quantities,
the shear compliance starts to deviate from the dielectric
susceptibility as soon as the viscous flow sets in. This
suggests a treatment of the shear data in terms of a re-
tardation spectrum, with the viscosity as an independent
parameter12.
The good agreement at the right-hand side of the α-
peak is found to be a general feature of all seven mea-
sured substances, as long as there is no disturbing in-
fluence from the secondary Johari-Goldstein peak10,11.
This is seen in Fig. 2, which shows that the good
agreement between G′′ and −ǫ′′ disappears as the peaks
merge. The substance, tripropylene glycol (TPG),
-2 -1 0 1 2 3 4 5 6 7
1     G'' 188 K
    G'' 192 K
    - " 188 K scaled
    - " 192 K scaled
tripropylene glycol (TPG)
log( /s-1)
FIG. 2: Comparison of G′′(ω) to a properly scaled −ǫ′′(ω) in
a double-log scale for TPG at 188 and 192 K.
C6H20O4, is still a molecule and not yet a polymer (the
whole series from the small molecule propylene glycol to
long-chain polypropylene glycol is well-investigated by
dielectrics18,19,20). TPG itself has been studied under
aging21 and under pressure22. It consists of three con-
nected propylene groups, with a large dielectric moment
and a pronounced Johari-Goldstein peak (which is absent
or at least much less pronounced in propylene glycol18).
Fig. 2 shows another tendency which will be evaluated
quantitatively in this paper, namely a much stronger in-
crease of the imaginary quantities in the α-peak region
with temperature than in the Johari-Goldstein peak re-
gion.
If one wants to decompose a measured relaxation into
a spectrum of exponential decays in time, one can choose
between two equivalent possibilities12, the relaxation
spectrum in which the elementary exponential relaxators
add to decrease the modulus or the retardation spectrum
in which they add to increase a susceptibility. In princi-
ple, the choice is not crucial, because the two spectra can
be calculated from each other. Here, we choose the retar-
dation spectrum, with the viscosity η as an independent
variable.
For this choice, one has the textbook expressions12 for
the real and imaginary parts of the complex frequency-
dependent shear compliance
J ′(ω) = Jg +
L(ln τ)
1 + ω2τ2
d(ln τ) (1)
J ′′(ω) =
L(ln τ)
1 + ω2τ2
d(ln τ) +
, (2)
where τ is the relaxation time and L(ln τ) is the weight of
this relaxation time in the retardation spectrum. Jg, the
glass compliance, is the inverse of the infinite frequency
modulus G∞.
In an energy landscape picture23, one reckons with
thermally activated jumps over the energy barrier be-
tween two neighboring minima. In fact, one very of-
ten finds a broad secondary relaxation peak (the Johari-
Goldstein peak10,11) below the α-peak of the flow pro-
cess. This peak follows the Arrhenius relation in the
glass phase, indicating that it stems from local thermally
activated jumps. For a jump over an energy barrier of
height V , the Arrhenius relation for the relaxation time
τV reads
τV = τ0e
V/kBT , (3)
where τ0 = 10
−13 s and T is the temperature.
For a spectrum of thermally activated jumps, one
defines13 the barrier density function ls(V )
ls(V ) =
L(V/kBT + ln τ0). (4)
The index s stands for the shear. With this definition, the
complex shear compliance equations (1) and (2) trans-
form into
J ′(ω) = Jg + Jg
ls(V )
1 + ω2τ2V
dV (5)
J ′′(ω) = Jg
ls(V )
1 + ω2τ2V
. (6)
The dielectric susceptibility can also be described24,25
in terms of a dielectric barrier density function lǫ(V )
ǫ′(ω)− ǫ∞
ǫ(0)− ǫ∞
lǫ(V )
1 + ω2τ2V
dV (7)
ǫ′′(ω)
ǫ(0)− ǫ∞
lǫ(V )
1 + ω2τ2V
dV. (8)
Here ǫ(0) is the static dielectric susceptibility, ǫ∞ is the
real part of ǫ(ω) in the GHz range (larger than n2, the
square of the refractive index, because of vibrational
contributions6).
The above definitions of eqs. (5-8) imply a normaliza-
tion of both ls(V ) and lǫ(V ) with
ls(V )dV =
J0e − Jg
, (9)
where J0e is the recoverable compliance of the steady-
state flow12, and
lǫ(V )dV = 1. (10)
The dielectric α-peak occurs always at a lower fre-
quency than the shear one and seems to coincide
with the heat capacity and the structural relaxation
peaks1,2,3,4,5,6. Below, we will adopt the view that the
left side of the dielectric peak marks the disappearance
and renewal of the relaxing entities.
In order to describe this decay, one needs to multiply
the barrier density of the energy landscape with an ap-
propriate cutoff function at a cutoff barrier Vc. Here, we
will assume that the relaxing entities decay exponentially
in time with the critical relaxation time τc. With the Ar-
rhenius relation τc = τ0 exp(Vc/kBT ), this translates into
a double-exponential cutoff
c(V ) = exp(− exp((V − Vc)/kBT )). (11)
Equations (6) and (8) show that a Johari-Goldstein
peak in G′′(ω) or ǫ′′(ω) at the peak frequency ω1 cor-
responds to a peak in l(V ) at a peak barrier V1 =
kBT ln(1/ω1τ0). We will see that the Johari-Goldstein
peaks are reasonably well described by gaussians in l(V ).
To describe both the α-peak and the Johari-Goldstein-
peak in terms of a barrier density, ls(V ) and lǫ(V ) will
be fitted by the form
l(V ) = (aβe
βV/kBT + a1/3e
V/3kBT + a1e
−γ1(V−V1)
)c(V ).
The first two terms describe the high-frequency tail of
the α-process, the third term the Johari-Goldstein peak
(if there is one; in three of our seven examples, it is not
needed). The dimensionless parameter β determines the
slope ω−β at the beginning of the α-tail in the double-log
plot of Fig. 1.
Instead of using the three prefactors aβ , a1/3 and a1
as fit parameters, it is better to use the correspond-
ing weights wβ , w1/3 and w1 in the integral over the
barriers, equs. (9) and (10). A type-A glass former
without Johari-Goldstein peak with w1 = 0 is char-
acterized by the two dimensionless parameters β and
b2 = w1/3/(wβ + w1/3), at least as far as the form of
its spectrum is concerned. β and b2 have the advantage
to be reasonably temperature-independent.
With this prescription, one can fit the ǫ′′(ω) of a type-A
glass former with two temperature-independent parame-
ters, β and b2, and two temperature-dependent parame-
ters, ∆ǫ = ǫ(0)− ǫ∞ and Vc. Their temperature depen-
dence is a decrease with increasing temperature, which
-2 -1 0 1 2 3 4 5
log( /s-1)
 254 K
 262 K
 274 K
FIG. 3: (a) Data and fit of ǫ′′(ω) in a double-log scale for
TPE between 254 and 274 K (b) the same for G(ω).
is well fitted by an appropriate power law
∆ǫ(T ) = ∆ǫ(Tg)
, (13)
where Tg is the glass temperature. Similarly, one de-
scribes the decrease of Vc with the exponent γV and the
one of G∞ with γG.
The strategy of our evaluation is to fit l(V ) to the
dielectric data, and then use the same spectral form
to describe the shear. The fit of the shear data re-
quires three additional temperature-dependent parame-
ters, Jg, J
e and η. Again, it is worthwhile to look for
combinations which might turn out to be temperature-
independent. One of them is the ratio
J0e − Jg
, (14)
which appears in the normalization of the shear spec-
trum, eq. (9). A second interesting possibility is not to
fit the viscosity η, but the ratio
fjc =
f0Jgη
, (15)
where τc is the Arrhenius relaxation time of the terminal
barrier Vc. As we will show in the discussion, one can
argue that the ratio fjc should be 2 for a renewal of the
relaxing entities within the critical time τc.
In the case of a type-B glass former, Fig. 2 shows that
one needs another dimensionless parameter, because the
weight of the Johari-Goldstein peak is different in the
two quantities. In Fig. 2, the Johari-Goldstein peak is
more prominent in the shear signal, but this varies from
substance to substance.
205 210 215 220 225 230 235 240
 fit data
 from V
DC704
temperature (K)
FIG. 4: Fit values of G∞ in DC704 as a function of tem-
perature. The continuous line is the temperature dependence
Vc/∆v (with ∆v = 0.057 nm
3) expected from the shoving
model26.
III. DATA EVALUATION
A. The three type-A glass formers
Three of our seven substances, TPE, DC704 and PPE,
happen to have no or at least only a rather weak Johari-
Goldstein peak. Let us begin with TPE. TPE stands
for triphenylethylene, C20H16, a rather flexible molecule
with three phenyl rings attached to a central C = C
double bond. Fig. 3 (a) shows data and fit for ǫ′′(ω)
in a double-log plot, Fig. 3 (b) the ones for G(ω). The
dielectric data in Fig. 3 (a) are well fitted with only the
first two terms of eq. (12), without any Johari-Goldstein
peak. β and b2 turn out to be temperature-independent
within experimental accuracy.
One gets a good fit for the shear data in Fig. 3 (b), tak-
ing over β, b2 and the cutoff barrier Vc from the fit of the
dielectric data at the given temperature and fitting G∞,
f0 and fjc. G∞ is temperature-dependent, but f0 and
fjc are again temperature-independent within the exper-
imental accuracy, thus justifying our choice of variables.
The parameters and their temperature dependence are
listed in Table I.
The temperature exponents γV and γG of the critical
barrier Vc and the infinite frequency shear modulus G∞
are the same within their error bars (about 5 % for γV
and about 10 % for γG). This shows the validity of the
shoving model26, according to which the energy barrier
of the α-process should be proportional to the infinite
frequency shear modulus G∞. The shoving model postu-
lates that the α-process happens when the local energy
concentration exceeds the product G∞∆v, where ∆v is a
volume expansion.
The same results, maybe even a bit clearer because of
the stronger dielectric signals, are obtained for the two
other type-A glass formers PPE and DC704. Again, the
fit parameters are listed in Table I. In particular, the
ω−1/3-contribution is much better seen, as illustrated
in Fig. 1 for PPE. In DC704, again a diffusion pump
oil (1,3,3,5-tetramethyl-1,1,5,5-tetraphenyl-trisiloxane, a
rather large molecule) we have the additional advan-
tage of a large temperature range of the measurement,
from 209 to 239 K. As in TPE, we find temperature-
independent parameters β, b2, f0 and fjc. Again, we
find the shoving model26 confirmed in both glass form-
ers. In DC704, one even sees the curvature of both curves
(see Fig. 4), which justifies our temperature exponent
Ansatz, eq. (13).
Table I comprises the fit parameters for these three
type-A glass formers. Note that our formalism allows to
describe both shear and dielectric data over the whole
temperature range with eleven temperature-independent
parameters.
glass former TPE DC704 PPE
Tg (K) 249 211 244
∆ǫ 0.0491 0.257 2.011
γǫ 1.85 2.26 1.90
β 0.77 0.85 1.04
b2 0.18 0.27 0.215
Vc(Tg) (eV) 0.767 0.639 0.755
γV 4.3 4.6 4.6
G∞(Tg) (GPa) 1.38 1.80 1.27
γG 4.7 4.2 4.7
f0 1.65 2.38 2.22
fjc 2.5 2.45 2.05
TABLE I: Parameters of the three type-A glass formers. Up-
per part ǫ(ω), lower part G(ω).
B. The four type-B glass formers
In the type-B glass formers DHIQ, PB20, Squalane and
TPG, one needs to fit a Johari-Goldstein peak on top of
the high-frequency tail of the α-process. This is illus-
trated in Fig. 5 for our first type-B example, squalane.
Squalane, C30H62, is a short chain molecule with 24
carbon atoms in the backbone and 6 attached CH3-
groups, rather polymerlike. It has a strong and well-
separated Johari-Goldstein peak (see Fig. 5), much bet-
ter visible in the shear data than in the dielectric data.
The dielectric dipole moment is very weak. Nevertheless,
it is possible to fit both sets of data with the same retar-
dation spectrum, attaching a substantially higher weight
-3 -2 -1 0 1 2 3 4 5 6
 168 K
 174 K
 180 K
    fits  
squalane
log( /s-1)
FIG. 5: Data and fits of (a) ǫ′′(ω) (b) G(ω) in squalane.
to the Johari-Goldstein peak in the shear (see Table II).
In this substance, it is not possible to fit the shear data
with a temperature-independent parameter f0; one has
to postulate a rather strong increase of f0 with increasing
temperature
f0(T ) = f0(Tg) + f
0(T − Tg), (16)
but one can keep the parameter fjc constant (see Table
PB20 is a true polymer, relatively short (5000 g/mol),
composed of 80 % 1,4-polybutadiene monomers and 20
% 1,2-polybutadiene monomers. The results look very
similar to those of squalane, and the resulting fit param-
eters in Table II are in fact close to those of squalane.
Even more than squalane, it has the polymer feature of
a relatively slow decrease of ǫ′′(ω) at low frequency, ex-
plainable in terms of chain modes with long relaxation
times12. This is illustrated in Fig. 6, which shows the
deviation between fit and data at low frequency. As a
consequence, the resulting parameters have a larger er-
ror bar in squalane and polybutadiene than in the two
molecular substances TPG and DHIQ. In particular, the
deviations between γV and γG do not demonstrate a fail-
ure of the shoving model.
TPG is a much more favorable case, with a very large
dipole moment and no problems at the cutoff barrier. As
Fig. 7 (a) shows, our spectrum of eq. (12) provides beau-
tiful fits over a large temperature range. One needs to
take the temperature dependence of the Johari-Goldstein
peak position V1 into account. Our fit found
V1 = 0.295
, (17)
a bit smaller shift than the one found in aging
experiments21.
Fig. 7 (b) shows that the shear data are well described
in terms of the dielectric retardation spectrum. There is
a small temperature dependence of f0, but fjc is again a
temperature-independent constant. The shoving model
is found to be well fulfilled (see Fig. 8).
-2 -1 0 1 2 3 4 5 6 7
  fit
polybutadiene 180 K
log( /s-1)
FIG. 6: Data and fit of ǫ′′(ω) in polybutadiene at 180 K.
Finally, DHIQ, decahydroisoquinoline, C9H17N , is
best described as two cyclohexanol rings sharing one
C−C-bond, one of the two rings having anNH replacing
a CH2-group. In this case, the Johari-Goldstein peak is
very prominent in the dielectric data27,28, comparable to
the one in G(ω). The dipole moment is large; both Vc
and G∞ can be determined with high accuracy. Again,
their temperature exponents γV and γG agree within the
error bars (see Table II), in agreement with the shov-
ing model26. Since both are exceptionally large (DHIQ
is very fragile, m=158 in Angell’s scheme29), their good
agreement provides a strong argument for the validity of
the model.
In Table II, the Johari-Goldstein peak is characterized
by the weight of the peak
w(T ) = a1
(π/γ1) = a1FWHM
(π/4 ln 2) (18)
which shows a Boltzmann factor behavior
w(T ) = w(Tg) exp(−Ea(1/kBT − 1/kBTg)), (19)
with a formation energy Ea which is on the average 2/3
of the peak position V1.
IV. DISCUSSION
The preceding section presented a quantitative descrip-
tion of the α- and the β-process in dielectrics and shear
-3 -2 -1 0 1 2 3 4 5 6 7
 182 K
 188 K
 194 K
 200 K
 206 K
 212 K
 218 K
 224 K
  fits TPG
log( /s-1)
FIG. 7: Data and fits of (a) ǫ′′(ω) (b) G(ω) in TPG.
glass former Squalane PB20 TPG DHIQ
Tg (K) 167 176 184 175
∆ǫ(Tg) 0.0155 0.132 23.3 1.707
γǫ 2.1 0.0 1.51 0.0
β 0.6 0.44 0.85 0.4
b2 0.2 0.2 0.22 0.2
Vc(Tg) (eV) 0.517 0.54 0.63 0.635
γV 3.2 3.8 3.0 6.4
G∞(Tg) (GPa) 1.33 1.63 2.69 3.1
γG 2.5 2.7 2.8 6.3
f0(Tg) 2.4 3.25 6.7 1.56
f ′0 (1/K) 0.4 0.41 -0.04 0.25
fjc 2.7 2.4 2.5 2.0
V1 (eV) 0.27 0.28 0.32* 0.32
FWHM(eV) 0.135 0.170 0.154 0.16
ws(Tg) 0.56 0.55 0.15 0.50
wǫ(Tg) 0.03 0.23 0.02 0.48
Ea (eV) 0.25 0.12 0.19 0.25
*average value, see eq. (17)
TABLE II: Parameters of the four type-B glass formers. Up-
per part G(ω), middle part ǫ(ω), lower part Johari-Goldstein
peak parameters for both.
for seven different glass formers, a description which is
based on the concept of isolated and independent ther-
mally activated jumps in the energy landscape. The de-
scription allows for a reasonable fit of the temperature
dependence in terms of temperature-independent param-
eters. The number of parameters is not small; one needs
eleven or twelve parameters for a type-A glass former
(depending on whether f0 is temperature-independent or
not, see Table I and II) and five additional parameters
for the description of the β- or Johari-Goldstein peak (see
Table II).
Nevertheless, the exercise is not completely meaning-
less. One does indeed get meaningful quantitative in-
formation, which is impossible to obtain otherwise. The
180 185 190 195 200 205 210 215 220 225 230
  fit data
  from V
temperature (K)
FIG. 8: Fit values ofG∞ in TPG as a function of temperature.
The continuous line is the temperature dependence Vc/∆v
(with ∆v = 0.037 nm3) expected from the shoving model26.
first and rather important one is the probable equality of
the retardation spectra of shear and dielectrics (but with
a different weight of the Johari-Goldstein peak), an infor-
mation which one can guess from the raw data (see Figs.
1 and 2), but which requires a full fit for its quantitative
check.
The second and equally important quantitative infor-
mation concerns the dimensionless ratio fjc between the
terminal dielectric relaxation time τc and the product
of the viscosity with the total retardation compliance,
eq. (15). The seven fitted values lie between 2 and 2.7
(average value 2.37). This indicates a general relation
between the dielectric terminal time and the viscosity.
Since the dielectric terminal time seems to coincide with
the structural lifetime5, it is probably also the lifetime of
the double-well potentials which are responsible for the
retardation spectrum.
Question: What do we expect for the ratio fjc if this
is indeed the case? To answer this question, consider
a constant applied shear stress. After the time τc, all
the double-well potentials of the spectrum would have
reached thermal equilibrium, giving their full contribu-
tion to the compliance. From this consideration, if we
renew them at the time τc, we would naively expect them
to be able to give their contribution again after this time,
yielding fjc = 1.
But this answer is not correct. To get the correct an-
swer, one must consider the difference between energy
and free energy in these double-well potentials. To keep
the argument simple, let us restrict ourselves to the spe-
cial case of a symmetric double-well; it applies as well to
the asymmetric case.
If the double-well is initially symmetric and if it couples
to the shear stress σ with a coupling constant v (the
coupling constant has the dimension of a volume), then
the asymmetry ∆ under the stress is σv. One well has
the energy −σv/2, the other has the energy +σv/2.
In thermal equilibrium, the population of the two wells
is given by their Boltzmann factors. It is easy to calculate
the energy U of the equilibrated system in the limit of a
small stress
U = −
. (20)
This is the energy transported to the heat bath in the
equilibration of the relaxing entity after switching on the
stress.
The free energy F is
F = −
, (21)
only half of the energy itself. If one thinks about it, the
reason is clear: spending the energy, one has spanned an
entropic spring by the population difference in the two
wells. If one removes the stress slowly, one gets half the
energy back. But if the double-well potential decays, one
gets nothing back.
The contribution of the relaxing entity to the compli-
ance is given by the second derivative of the free energy
with respect to the stress. But if we now deal with the
effect of a renewal of the double-well potential on the vis-
cosity, we have to count the energy. This means we spend
twice as much energy under a constant stress as the one
calculated above in our first oversimplified picture. And
this means the viscosity must be a factor of 2 smaller,
which implies fjc = 2. This is reasonably close to the
fitted values in Table I and II.
A third quantitative conclusion of the present study
is a surprising agreement with the conclusions of Plazek
et al16 from their recoverable shear compliance experi-
ments. If one takes the parameters of Table I to calculate
the recoverable compliance, one gets curves which closely
resemble those reported by them. Obviously, it is exper-
imentally much easier to detect the Andrade creep17 in
creep experiments than in dynamical ones. If one calcu-
lates f0 from their data, one finds values between 1.5 and
2.3, similar to those in Table I.
Here, however, a word of caution is in place. Our data,
taken as they are, do not imply a limited recoverable
shear compliance. In fact, they are well fitted by the BEL
model30, which has a divergent recoverable compliance.
The values in the two tables stem from the assumption
that the two retardation spectra of dielectrics and shear
(at least as far as the α-peak is concerned) are the same.
The same is true for the fourth conclusion, the valid-
ity of the shoving model26. The fitted G∞-values were
obtained under the same assumption.
Finally, the Johari-Goldstein peak increases its height
with increasing temperature. The increase follows a
Boltzmann factor, with a formation energy of about two
thirds of the barrier height at the center of the peak.
V. SUMMARY AND CONCLUSIONS
Dielectric and shear relaxation data in seven highly
viscous liquids, most of them molecular liquids, were
evaluated in terms of a barrier density of independent
thermally activated relaxation centers. Three of the sub-
stances are type-A glass formers without or with only a
very small Johari-Goldstein peak, four of them show a
pronounced Johari-Goldstein peak.
The most important conclusion is the probable equal-
ity of the dielectric and shear retardation spectra,
guessed from the raw data and confirmed by a quanti-
tative fit. The difference in the peak positions is due to
the influence of the viscosity. The Johari-Goldstein peak
has different weight in dielectrics and shear.
The second important conclusion concerns the viscos-
ity. It seems probable that the viscosity results from
the constant renewal of the double-well potentials in the
sample within the terminal dielectric relaxation time.
Our data support earlier recovery compliance results
by Plazek et al16, according to which one has an
Andrade17 creep J ∼ t1/3 at short times in type-A glass
formers (glass formers without Johari-Goldstein peak).
They further support the shoving model26, which pos-
tulates a proportionality between the infinite frequency
shear modulus and the Arrhenius barrier of the terminal
relaxation time.
Acknowledgement: The author is deeply thankful to
Kristine Niss and Bo Jakobsen for communicating their
beautiful data to him, to Niels Boye Olsen and Tage
Christensen for enlightening discussions and to Jeppe
Dyre for constant encouragement and a lot of helpful
advice.
∗ Electronic address: buchenau-juelich@t-online.de
1 N. O. Birge and S. R. Nagel, Phys. Rev. Lett. 54, 2674
(1985); N. O. Birge, Phys. Rev. B 34, 1631 (1986)
2 K. L. Ngai and R. W. Rendell, Phys. Rev. B 41, 754 (1990)
3 I. Chang and H. Sillescu, J. Chem. Phys. 101, 8794 (1997)
and further references therein
4 K. Schröter and E. Donth, J. Non-Cryst. Solids 307-310,
270 (2002)
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124, 094505 (2006)
6 U. Buchenau, R. Zorn, M. Ohl and A. Wischnewski,
cond-mat/0607056 and Phil. Mag. 2006 (in press)
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234510 (2005)
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234511 (2005)
9 T. Christensen and N. B. Olsen, Rev. Sci. Instrum. 66,
5019 (1995)
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(1970)
11 G. P. Johari and M. Goldstein, J. Chem. Phys. 55, 4245
(1971)
12 D. J. Ferry, ”Viscoelastic properties of polymers”, 3rd ed.,
John Wiley, New York 1980
13 U. Buchenau, Phys. Rev. B 63, 104203 (2001)
14 A. Kudlik, Ch. Tschirwitz, S. Benkhof, T. Blochowicz and
E. Rössler, Europhys. Lett. 40, 649 (1997)
15 K. L. Ngai and M. Paluch, J. Chem. Phys. 120, 857 (2004)
16 D. J. Plazek, C. A. Bero and I.-C. Chay, J. Non-Cryst.
Solids 172-174, 181 (1994)
17 E. N. da C. Andrade, Proc. Roy. Soc. A 84, 1 (1910)
18 C. Leon, K. L. Ngai and C. M. Roland, J. Chem. Phys.
110, 11585 (1999)
19 J. Mattsson, R. Bergman, P. Jacobsson and L. Börjesson,
Phys. Rev. Lett. 90, 075702 (2003)
20 J. Mattsson, R. Bergman, P. Jacobsson and L. Börjesson,
Phys. Rev. Lett. 94, 165701 (2005)
21 J. C. Dyre and N. B. Olsen, Phys. Rev. Lett. 91, 155703
(2003)
22 S. Pawlus, S. Hensel-Bielowska, K. Grzybowska, J. Ziolo
and M. Paluch, Phys. Rev. B 71, 174107 (2005)
23 M. Goldstein, J. Chem. Phys. 51, 3728 (1968)
24 M. Pollak and G. E. Pike, Phys. Rev. Lett. 28, 1449 (1972)
25 K. S. Gilroy and W. A. Phillips, Phil. Mag. B 43, 735
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53, 2171 (1996)
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D. Prevosto, S. Capaccioli, P. A. Rolla and K. L. Ngai, J.
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mailto:buchenau-juelich@t-online.de
http://arxiv.org/abs/cond-mat/0607056

  Published dielectric and shear data of six molecular glass formers and one
polymer are evaluated in terms of a spectrum of thermally activated processes,
with the same barrier density for the retardation spectrum of shear and
dielectrics. The viscosity, an independent parameter of the fit, seems to be
related to the high-barrier cutoff time of the dielectric signal, in accordance
with the idea of a renewal of the relaxing entities after this critical time.
In the five cases where one can fit accurately, the temperature dependence of
the high-barrier cutoff follows the shoving model. The Johari-Goldstein peaks,
seen in four of our seven cases, are describable in terms of gaussians in the
barrier density, superimposed on the high-frequency tail of the
$\alpha$-process. Dielectric and shear measurements of the same substance find
the same peak positions and widths of these gaussians, but in general a
different weight.

Mechanical and dielectric relaxation spectra in seven highly viscous glass formers
U. Buchenau∗
Institut für Festkörperforschung, Forschungszentrum Jülich
Postfach 1913, D–52425 Jülich, Federal Republic of Germany
(Dated: April 2, 2007)
Published dielectric and shear data of six molecular glass formers and one polymer are evaluated
in terms of a spectrum of thermally activated processes, with the same barrier density for the
retardation spectrum of shear and dielectrics. The viscosity, an independent parameter of the fit,
seems to be related to the high-barrier cutoff time of the dielectric signal, in accordance with the
idea of a renewal of the relaxing entities after this critical time. In the five cases where one can fit
accurately, the temperature dependence of the high-barrier cutoff follows the shoving model. The
Johari-Goldstein peaks, seen in four of our seven cases, are describable in terms of gaussians in
the barrier density, superimposed on the high-frequency tail of the α-process. Dielectric and shear
measurements of the same substance find the same peak positions and widths of these gaussians,
but in general a different weight.
PACS numbers: 64.70.Pf, 77.22.Gm
I. INTRODUCTION
The publications of Kia Ngai deal with more sub-
stances and more measurement techniques than the work
of any other scientist in the field of undercooled liquids.
Within the past three decades, whenever a new develop-
ment appeared, he was the quickest to appreciate it, ana-
lyze it and bring it to the general attention, thus speeding
up the progress substantially.
Many scientists in the field share his conviction that
the flow process in highly viscous liquids can only be
understood by combining all possible techniques for its
study1,2,3,4,5,6. The present paper evaluates recently
published7,8 broadband shear and dielectric relaxation
data on seven glass formers. The mechanical shear data
were obtained with a new technique9 which allows to
cover a large dynamical range. Samples for dielectric and
shear measurements were taken from the same charge,
and the temperature sensors of both measurements were
calibrated to each other.
The data show a striking similarity of G′′(ω) and ǫ′′(ω)
on the right hand side of the α-peak, a similarity which is
sometimes perturbed by the secondary Johari-Goldstein
peak10,11. The similarity suggests a common origin of the
α-peak in dielectrics and shear. The fact that the shear
peak appears at a higher frequency than the dielectric
peak is explainable in terms of the viscosity, which in a
compliance treatment12 is a free parameter.
In order to identify the elementary processes with ther-
mally activated jumps over an energy barrier V , one can
use a recent translation13 of the textbook12 retardation
spectrum L(ln τ) into a barrier density function l(V ). We
will argue that the compliance barrier density of the shear
equals the electric dipole moment barrier density lǫ(V )
of the dielectric data.
The next section (section II) explains and motivates
this approach in more detail. The results of the data
treatment are described in section III. They are discussed
and compared to other approaches in section IV. Section
V summarizes and concludes the paper.
II. THE BARRIER DENSITY FUNCTIONS FOR
SHEAR AND DIELECTRICS
The choice of a retardation spectrum for the shear is
motivated by a surprising coincidence, which is more or
less visible in the data of all seven substances. We show
the two examples where it is most clearly seen in Fig. 1
and Fig. 2.
Fig. 1 compares G′′(ω) and −ǫ′′(ω) at the same tem-
perature for the type-A glass former PPE. PPE, 1,3-
Bis(3-phenoxyphenoxy)benzene, is a diffusion pump oil
with the commercial name Santovac-5P, a molecule con-
sisting of five phenyl rings connected by four oxygens to
form a short chain. In terms of the classification proposed
by the Bayreuth group14, it is a type-A glass former, a
-1 0 1 2 3 4 5 6 7
    G''
    - " scaled
Santovac-5P (PPE)
        254 K
log( /s-1)
FIG. 1: Comparison of G′′(ω) to a properly scaled −ǫ′′(ω) in
a double-log scale for PPE at 254 K.
http://arxiv.org/abs/0704.1356v1
glass former which shows no or at least no pronounced
secondary Johari-Goldstein peak. The (negative) ǫ′′(ω)-
data have been scaled to coincide with the G′′(ω)-data
on the right hand side of the peak. One finds good agree-
ment betweenG′′(ω) and−ǫ′′(ω) as soon as the frequency
is two decades higher than the one of the peak in G′′(ω).
In addition, Fig. 1 shows a change of slope of the α-
tail, from an ω−1/2- to an ω−1/3-behavior. The tendency
is seen most clearly in the dielectric data, but it is also in
the shear data; their fit improves markedly if one allows
for a ω−1/3-component. This might be the influence of a
hidden Johari-Goldstein peak, an explanation which has
been favored for glycerol on the basis of pressure, aging
and chemical series measurements (for a review, see Ngai
and Paluch15). But there is also impressive experimen-
tal evidence for a limiting ω−1/3-behavior in shear com-
pliance data of type-A molecular glass formers16, which
show the so-called Andrade17 creep, J(t) ∼ t1/3, in the
short-time limit. Therefore we will fit our data in terms
of a sum of an ω−β-term (with β as fit parameter) and
an ω−1/3-term, dominating at high frequency.
Toward lower frequency, the common slope terminates
for the shear data already at a higher frequency (a shorter
time) than for the dielectric data. The natural explana-
tion for this is that the parallel between dielectrics and
shear is in fact between shear compliance and dielectric
susceptibility. In a comparison of these two quantities,
the shear compliance starts to deviate from the dielectric
susceptibility as soon as the viscous flow sets in. This
suggests a treatment of the shear data in terms of a re-
tardation spectrum, with the viscosity as an independent
parameter12.
The good agreement at the right-hand side of the α-
peak is found to be a general feature of all seven mea-
sured substances, as long as there is no disturbing in-
fluence from the secondary Johari-Goldstein peak10,11.
This is seen in Fig. 2, which shows that the good
agreement between G′′ and −ǫ′′ disappears as the peaks
merge. The substance, tripropylene glycol (TPG),
-2 -1 0 1 2 3 4 5 6 7
1     G'' 188 K
    G'' 192 K
    - " 188 K scaled
    - " 192 K scaled
tripropylene glycol (TPG)
log( /s-1)
FIG. 2: Comparison of G′′(ω) to a properly scaled −ǫ′′(ω) in
a double-log scale for TPG at 188 and 192 K.
C6H20O4, is still a molecule and not yet a polymer (the
whole series from the small molecule propylene glycol to
long-chain polypropylene glycol is well-investigated by
dielectrics18,19,20). TPG itself has been studied under
aging21 and under pressure22. It consists of three con-
nected propylene groups, with a large dielectric moment
and a pronounced Johari-Goldstein peak (which is absent
or at least much less pronounced in propylene glycol18).
Fig. 2 shows another tendency which will be evaluated
quantitatively in this paper, namely a much stronger in-
crease of the imaginary quantities in the α-peak region
with temperature than in the Johari-Goldstein peak re-
gion.
If one wants to decompose a measured relaxation into
a spectrum of exponential decays in time, one can choose
between two equivalent possibilities12, the relaxation
spectrum in which the elementary exponential relaxators
add to decrease the modulus or the retardation spectrum
in which they add to increase a susceptibility. In princi-
ple, the choice is not crucial, because the two spectra can
be calculated from each other. Here, we choose the retar-
dation spectrum, with the viscosity η as an independent
variable.
For this choice, one has the textbook expressions12 for
the real and imaginary parts of the complex frequency-
dependent shear compliance
J ′(ω) = Jg +
L(ln τ)
1 + ω2τ2
d(ln τ) (1)
J ′′(ω) =
L(ln τ)
1 + ω2τ2
d(ln τ) +
, (2)
where τ is the relaxation time and L(ln τ) is the weight of
this relaxation time in the retardation spectrum. Jg, the
glass compliance, is the inverse of the infinite frequency
modulus G∞.
In an energy landscape picture23, one reckons with
thermally activated jumps over the energy barrier be-
tween two neighboring minima. In fact, one very of-
ten finds a broad secondary relaxation peak (the Johari-
Goldstein peak10,11) below the α-peak of the flow pro-
cess. This peak follows the Arrhenius relation in the
glass phase, indicating that it stems from local thermally
activated jumps. For a jump over an energy barrier of
height V , the Arrhenius relation for the relaxation time
τV reads
τV = τ0e
V/kBT , (3)
where τ0 = 10
−13 s and T is the temperature.
For a spectrum of thermally activated jumps, one
defines13 the barrier density function ls(V )
ls(V ) =
L(V/kBT + ln τ0). (4)
The index s stands for the shear. With this definition, the
complex shear compliance equations (1) and (2) trans-
form into
J ′(ω) = Jg + Jg
ls(V )
1 + ω2τ2V
dV (5)
J ′′(ω) = Jg
ls(V )
1 + ω2τ2V
. (6)
The dielectric susceptibility can also be described24,25
in terms of a dielectric barrier density function lǫ(V )
ǫ′(ω)− ǫ∞
ǫ(0)− ǫ∞
lǫ(V )
1 + ω2τ2V
dV (7)
ǫ′′(ω)
ǫ(0)− ǫ∞
lǫ(V )
1 + ω2τ2V
dV. (8)
Here ǫ(0) is the static dielectric susceptibility, ǫ∞ is the
real part of ǫ(ω) in the GHz range (larger than n2, the
square of the refractive index, because of vibrational
contributions6).
The above definitions of eqs. (5-8) imply a normaliza-
tion of both ls(V ) and lǫ(V ) with
ls(V )dV =
J0e − Jg
, (9)
where J0e is the recoverable compliance of the steady-
state flow12, and
lǫ(V )dV = 1. (10)
The dielectric α-peak occurs always at a lower fre-
quency than the shear one and seems to coincide
with the heat capacity and the structural relaxation
peaks1,2,3,4,5,6. Below, we will adopt the view that the
left side of the dielectric peak marks the disappearance
and renewal of the relaxing entities.
In order to describe this decay, one needs to multiply
the barrier density of the energy landscape with an ap-
propriate cutoff function at a cutoff barrier Vc. Here, we
will assume that the relaxing entities decay exponentially
in time with the critical relaxation time τc. With the Ar-
rhenius relation τc = τ0 exp(Vc/kBT ), this translates into
a double-exponential cutoff
c(V ) = exp(− exp((V − Vc)/kBT )). (11)
Equations (6) and (8) show that a Johari-Goldstein
peak in G′′(ω) or ǫ′′(ω) at the peak frequency ω1 cor-
responds to a peak in l(V ) at a peak barrier V1 =
kBT ln(1/ω1τ0). We will see that the Johari-Goldstein
peaks are reasonably well described by gaussians in l(V ).
To describe both the α-peak and the Johari-Goldstein-
peak in terms of a barrier density, ls(V ) and lǫ(V ) will
be fitted by the form
l(V ) = (aβe
βV/kBT + a1/3e
V/3kBT + a1e
−γ1(V−V1)
)c(V ).
The first two terms describe the high-frequency tail of
the α-process, the third term the Johari-Goldstein peak
(if there is one; in three of our seven examples, it is not
needed). The dimensionless parameter β determines the
slope ω−β at the beginning of the α-tail in the double-log
plot of Fig. 1.
Instead of using the three prefactors aβ , a1/3 and a1
as fit parameters, it is better to use the correspond-
ing weights wβ , w1/3 and w1 in the integral over the
barriers, equs. (9) and (10). A type-A glass former
without Johari-Goldstein peak with w1 = 0 is char-
acterized by the two dimensionless parameters β and
b2 = w1/3/(wβ + w1/3), at least as far as the form of
its spectrum is concerned. β and b2 have the advantage
to be reasonably temperature-independent.
With this prescription, one can fit the ǫ′′(ω) of a type-A
glass former with two temperature-independent parame-
ters, β and b2, and two temperature-dependent parame-
ters, ∆ǫ = ǫ(0)− ǫ∞ and Vc. Their temperature depen-
dence is a decrease with increasing temperature, which
-2 -1 0 1 2 3 4 5
log( /s-1)
 254 K
 262 K
 274 K
FIG. 3: (a) Data and fit of ǫ′′(ω) in a double-log scale for
TPE between 254 and 274 K (b) the same for G(ω).
is well fitted by an appropriate power law
∆ǫ(T ) = ∆ǫ(Tg)
, (13)
where Tg is the glass temperature. Similarly, one de-
scribes the decrease of Vc with the exponent γV and the
one of G∞ with γG.
The strategy of our evaluation is to fit l(V ) to the
dielectric data, and then use the same spectral form
to describe the shear. The fit of the shear data re-
quires three additional temperature-dependent parame-
ters, Jg, J
e and η. Again, it is worthwhile to look for
combinations which might turn out to be temperature-
independent. One of them is the ratio
J0e − Jg
, (14)
which appears in the normalization of the shear spec-
trum, eq. (9). A second interesting possibility is not to
fit the viscosity η, but the ratio
fjc =
f0Jgη
, (15)
where τc is the Arrhenius relaxation time of the terminal
barrier Vc. As we will show in the discussion, one can
argue that the ratio fjc should be 2 for a renewal of the
relaxing entities within the critical time τc.
In the case of a type-B glass former, Fig. 2 shows that
one needs another dimensionless parameter, because the
weight of the Johari-Goldstein peak is different in the
two quantities. In Fig. 2, the Johari-Goldstein peak is
more prominent in the shear signal, but this varies from
substance to substance.
205 210 215 220 225 230 235 240
 fit data
 from V
DC704
temperature (K)
FIG. 4: Fit values of G∞ in DC704 as a function of tem-
perature. The continuous line is the temperature dependence
Vc/∆v (with ∆v = 0.057 nm
3) expected from the shoving
model26.
III. DATA EVALUATION
A. The three type-A glass formers
Three of our seven substances, TPE, DC704 and PPE,
happen to have no or at least only a rather weak Johari-
Goldstein peak. Let us begin with TPE. TPE stands
for triphenylethylene, C20H16, a rather flexible molecule
with three phenyl rings attached to a central C = C
double bond. Fig. 3 (a) shows data and fit for ǫ′′(ω)
in a double-log plot, Fig. 3 (b) the ones for G(ω). The
dielectric data in Fig. 3 (a) are well fitted with only the
first two terms of eq. (12), without any Johari-Goldstein
peak. β and b2 turn out to be temperature-independent
within experimental accuracy.
One gets a good fit for the shear data in Fig. 3 (b), tak-
ing over β, b2 and the cutoff barrier Vc from the fit of the
dielectric data at the given temperature and fitting G∞,
f0 and fjc. G∞ is temperature-dependent, but f0 and
fjc are again temperature-independent within the exper-
imental accuracy, thus justifying our choice of variables.
The parameters and their temperature dependence are
listed in Table I.
The temperature exponents γV and γG of the critical
barrier Vc and the infinite frequency shear modulus G∞
are the same within their error bars (about 5 % for γV
and about 10 % for γG). This shows the validity of the
shoving model26, according to which the energy barrier
of the α-process should be proportional to the infinite
frequency shear modulus G∞. The shoving model postu-
lates that the α-process happens when the local energy
concentration exceeds the product G∞∆v, where ∆v is a
volume expansion.
The same results, maybe even a bit clearer because of
the stronger dielectric signals, are obtained for the two
other type-A glass formers PPE and DC704. Again, the
fit parameters are listed in Table I. In particular, the
ω−1/3-contribution is much better seen, as illustrated
in Fig. 1 for PPE. In DC704, again a diffusion pump
oil (1,3,3,5-tetramethyl-1,1,5,5-tetraphenyl-trisiloxane, a
rather large molecule) we have the additional advan-
tage of a large temperature range of the measurement,
from 209 to 239 K. As in TPE, we find temperature-
independent parameters β, b2, f0 and fjc. Again, we
find the shoving model26 confirmed in both glass form-
ers. In DC704, one even sees the curvature of both curves
(see Fig. 4), which justifies our temperature exponent
Ansatz, eq. (13).
Table I comprises the fit parameters for these three
type-A glass formers. Note that our formalism allows to
describe both shear and dielectric data over the whole
temperature range with eleven temperature-independent
parameters.
glass former TPE DC704 PPE
Tg (K) 249 211 244
∆ǫ 0.0491 0.257 2.011
γǫ 1.85 2.26 1.90
β 0.77 0.85 1.04
b2 0.18 0.27 0.215
Vc(Tg) (eV) 0.767 0.639 0.755
γV 4.3 4.6 4.6
G∞(Tg) (GPa) 1.38 1.80 1.27
γG 4.7 4.2 4.7
f0 1.65 2.38 2.22
fjc 2.5 2.45 2.05
TABLE I: Parameters of the three type-A glass formers. Up-
per part ǫ(ω), lower part G(ω).
B. The four type-B glass formers
In the type-B glass formers DHIQ, PB20, Squalane and
TPG, one needs to fit a Johari-Goldstein peak on top of
the high-frequency tail of the α-process. This is illus-
trated in Fig. 5 for our first type-B example, squalane.
Squalane, C30H62, is a short chain molecule with 24
carbon atoms in the backbone and 6 attached CH3-
groups, rather polymerlike. It has a strong and well-
separated Johari-Goldstein peak (see Fig. 5), much bet-
ter visible in the shear data than in the dielectric data.
The dielectric dipole moment is very weak. Nevertheless,
it is possible to fit both sets of data with the same retar-
dation spectrum, attaching a substantially higher weight
-3 -2 -1 0 1 2 3 4 5 6
 168 K
 174 K
 180 K
    fits  
squalane
log( /s-1)
FIG. 5: Data and fits of (a) ǫ′′(ω) (b) G(ω) in squalane.
to the Johari-Goldstein peak in the shear (see Table II).
In this substance, it is not possible to fit the shear data
with a temperature-independent parameter f0; one has
to postulate a rather strong increase of f0 with increasing
temperature
f0(T ) = f0(Tg) + f
0(T − Tg), (16)
but one can keep the parameter fjc constant (see Table
PB20 is a true polymer, relatively short (5000 g/mol),
composed of 80 % 1,4-polybutadiene monomers and 20
% 1,2-polybutadiene monomers. The results look very
similar to those of squalane, and the resulting fit param-
eters in Table II are in fact close to those of squalane.
Even more than squalane, it has the polymer feature of
a relatively slow decrease of ǫ′′(ω) at low frequency, ex-
plainable in terms of chain modes with long relaxation
times12. This is illustrated in Fig. 6, which shows the
deviation between fit and data at low frequency. As a
consequence, the resulting parameters have a larger er-
ror bar in squalane and polybutadiene than in the two
molecular substances TPG and DHIQ. In particular, the
deviations between γV and γG do not demonstrate a fail-
ure of the shoving model.
TPG is a much more favorable case, with a very large
dipole moment and no problems at the cutoff barrier. As
Fig. 7 (a) shows, our spectrum of eq. (12) provides beau-
tiful fits over a large temperature range. One needs to
take the temperature dependence of the Johari-Goldstein
peak position V1 into account. Our fit found
V1 = 0.295
, (17)
a bit smaller shift than the one found in aging
experiments21.
Fig. 7 (b) shows that the shear data are well described
in terms of the dielectric retardation spectrum. There is
a small temperature dependence of f0, but fjc is again a
temperature-independent constant. The shoving model
is found to be well fulfilled (see Fig. 8).
-2 -1 0 1 2 3 4 5 6 7
  fit
polybutadiene 180 K
log( /s-1)
FIG. 6: Data and fit of ǫ′′(ω) in polybutadiene at 180 K.
Finally, DHIQ, decahydroisoquinoline, C9H17N , is
best described as two cyclohexanol rings sharing one
C−C-bond, one of the two rings having anNH replacing
a CH2-group. In this case, the Johari-Goldstein peak is
very prominent in the dielectric data27,28, comparable to
the one in G(ω). The dipole moment is large; both Vc
and G∞ can be determined with high accuracy. Again,
their temperature exponents γV and γG agree within the
error bars (see Table II), in agreement with the shov-
ing model26. Since both are exceptionally large (DHIQ
is very fragile, m=158 in Angell’s scheme29), their good
agreement provides a strong argument for the validity of
the model.
In Table II, the Johari-Goldstein peak is characterized
by the weight of the peak
w(T ) = a1
(π/γ1) = a1FWHM
(π/4 ln 2) (18)
which shows a Boltzmann factor behavior
w(T ) = w(Tg) exp(−Ea(1/kBT − 1/kBTg)), (19)
with a formation energy Ea which is on the average 2/3
of the peak position V1.
IV. DISCUSSION
The preceding section presented a quantitative descrip-
tion of the α- and the β-process in dielectrics and shear
-3 -2 -1 0 1 2 3 4 5 6 7
 182 K
 188 K
 194 K
 200 K
 206 K
 212 K
 218 K
 224 K
  fits TPG
log( /s-1)
FIG. 7: Data and fits of (a) ǫ′′(ω) (b) G(ω) in TPG.
glass former Squalane PB20 TPG DHIQ
Tg (K) 167 176 184 175
∆ǫ(Tg) 0.0155 0.132 23.3 1.707
γǫ 2.1 0.0 1.51 0.0
β 0.6 0.44 0.85 0.4
b2 0.2 0.2 0.22 0.2
Vc(Tg) (eV) 0.517 0.54 0.63 0.635
γV 3.2 3.8 3.0 6.4
G∞(Tg) (GPa) 1.33 1.63 2.69 3.1
γG 2.5 2.7 2.8 6.3
f0(Tg) 2.4 3.25 6.7 1.56
f ′0 (1/K) 0.4 0.41 -0.04 0.25
fjc 2.7 2.4 2.5 2.0
V1 (eV) 0.27 0.28 0.32* 0.32
FWHM(eV) 0.135 0.170 0.154 0.16
ws(Tg) 0.56 0.55 0.15 0.50
wǫ(Tg) 0.03 0.23 0.02 0.48
Ea (eV) 0.25 0.12 0.19 0.25
*average value, see eq. (17)
TABLE II: Parameters of the four type-B glass formers. Up-
per part G(ω), middle part ǫ(ω), lower part Johari-Goldstein
peak parameters for both.
for seven different glass formers, a description which is
based on the concept of isolated and independent ther-
mally activated jumps in the energy landscape. The de-
scription allows for a reasonable fit of the temperature
dependence in terms of temperature-independent param-
eters. The number of parameters is not small; one needs
eleven or twelve parameters for a type-A glass former
(depending on whether f0 is temperature-independent or
not, see Table I and II) and five additional parameters
for the description of the β- or Johari-Goldstein peak (see
Table II).
Nevertheless, the exercise is not completely meaning-
less. One does indeed get meaningful quantitative in-
formation, which is impossible to obtain otherwise. The
180 185 190 195 200 205 210 215 220 225 230
  fit data
  from V
temperature (K)
FIG. 8: Fit values ofG∞ in TPG as a function of temperature.
The continuous line is the temperature dependence Vc/∆v
(with ∆v = 0.037 nm3) expected from the shoving model26.
first and rather important one is the probable equality of
the retardation spectra of shear and dielectrics (but with
a different weight of the Johari-Goldstein peak), an infor-
mation which one can guess from the raw data (see Figs.
1 and 2), but which requires a full fit for its quantitative
check.
The second and equally important quantitative infor-
mation concerns the dimensionless ratio fjc between the
terminal dielectric relaxation time τc and the product
of the viscosity with the total retardation compliance,
eq. (15). The seven fitted values lie between 2 and 2.7
(average value 2.37). This indicates a general relation
between the dielectric terminal time and the viscosity.
Since the dielectric terminal time seems to coincide with
the structural lifetime5, it is probably also the lifetime of
the double-well potentials which are responsible for the
retardation spectrum.
Question: What do we expect for the ratio fjc if this
is indeed the case? To answer this question, consider
a constant applied shear stress. After the time τc, all
the double-well potentials of the spectrum would have
reached thermal equilibrium, giving their full contribu-
tion to the compliance. From this consideration, if we
renew them at the time τc, we would naively expect them
to be able to give their contribution again after this time,
yielding fjc = 1.
But this answer is not correct. To get the correct an-
swer, one must consider the difference between energy
and free energy in these double-well potentials. To keep
the argument simple, let us restrict ourselves to the spe-
cial case of a symmetric double-well; it applies as well to
the asymmetric case.
If the double-well is initially symmetric and if it couples
to the shear stress σ with a coupling constant v (the
coupling constant has the dimension of a volume), then
the asymmetry ∆ under the stress is σv. One well has
the energy −σv/2, the other has the energy +σv/2.
In thermal equilibrium, the population of the two wells
is given by their Boltzmann factors. It is easy to calculate
the energy U of the equilibrated system in the limit of a
small stress
U = −
. (20)
This is the energy transported to the heat bath in the
equilibration of the relaxing entity after switching on the
stress.
The free energy F is
F = −
, (21)
only half of the energy itself. If one thinks about it, the
reason is clear: spending the energy, one has spanned an
entropic spring by the population difference in the two
wells. If one removes the stress slowly, one gets half the
energy back. But if the double-well potential decays, one
gets nothing back.
The contribution of the relaxing entity to the compli-
ance is given by the second derivative of the free energy
with respect to the stress. But if we now deal with the
effect of a renewal of the double-well potential on the vis-
cosity, we have to count the energy. This means we spend
twice as much energy under a constant stress as the one
calculated above in our first oversimplified picture. And
this means the viscosity must be a factor of 2 smaller,
which implies fjc = 2. This is reasonably close to the
fitted values in Table I and II.
A third quantitative conclusion of the present study
is a surprising agreement with the conclusions of Plazek
et al16 from their recoverable shear compliance experi-
ments. If one takes the parameters of Table I to calculate
the recoverable compliance, one gets curves which closely
resemble those reported by them. Obviously, it is exper-
imentally much easier to detect the Andrade creep17 in
creep experiments than in dynamical ones. If one calcu-
lates f0 from their data, one finds values between 1.5 and
2.3, similar to those in Table I.
Here, however, a word of caution is in place. Our data,
taken as they are, do not imply a limited recoverable
shear compliance. In fact, they are well fitted by the BEL
model30, which has a divergent recoverable compliance.
The values in the two tables stem from the assumption
that the two retardation spectra of dielectrics and shear
(at least as far as the α-peak is concerned) are the same.
The same is true for the fourth conclusion, the valid-
ity of the shoving model26. The fitted G∞-values were
obtained under the same assumption.
Finally, the Johari-Goldstein peak increases its height
with increasing temperature. The increase follows a
Boltzmann factor, with a formation energy of about two
thirds of the barrier height at the center of the peak.
V. SUMMARY AND CONCLUSIONS
Dielectric and shear relaxation data in seven highly
viscous liquids, most of them molecular liquids, were
evaluated in terms of a barrier density of independent
thermally activated relaxation centers. Three of the sub-
stances are type-A glass formers without or with only a
very small Johari-Goldstein peak, four of them show a
pronounced Johari-Goldstein peak.
The most important conclusion is the probable equal-
ity of the dielectric and shear retardation spectra,
guessed from the raw data and confirmed by a quanti-
tative fit. The difference in the peak positions is due to
the influence of the viscosity. The Johari-Goldstein peak
has different weight in dielectrics and shear.
The second important conclusion concerns the viscos-
ity. It seems probable that the viscosity results from
the constant renewal of the double-well potentials in the
sample within the terminal dielectric relaxation time.
Our data support earlier recovery compliance results
by Plazek et al16, according to which one has an
Andrade17 creep J ∼ t1/3 at short times in type-A glass
formers (glass formers without Johari-Goldstein peak).
They further support the shoving model26, which pos-
tulates a proportionality between the infinite frequency
shear modulus and the Arrhenius barrier of the terminal
relaxation time.
Acknowledgement: The author is deeply thankful to
Kristine Niss and Bo Jakobsen for communicating their
beautiful data to him, to Niels Boye Olsen and Tage
Christensen for enlightening discussions and to Jeppe
Dyre for constant encouragement and a lot of helpful
advice.
∗ Electronic address: buchenau-juelich@t-online.de
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mailto:buchenau-juelich@t-online.de
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