THE JOURNAL OF CHEMICAL PHYSICS 122, 094507 共2005兲
A simple model of entropy relaxation for explaining effective activation
energy behavior below the glass transition temperature
Juan Bisquerta兲
Departament de Ciències Experimentals, Universitat Jaume I, 12080 Castelló, Spain
François Henn and Jean-Charles Giuntini
Laboratoire de Physicochimie de la Matière Condensée, UMR 5617 CNRS, Université Montpellier II,
34095 Montpellier cedex 5, France
共Received 14 September 2004; accepted 21 December 2004; published online 1 March 2005兲
Strong changes in relaxation rates observed at the glass transition region are frequently explained in
terms of a physical singularity of the molecular motions. We show that the unexpected trends and
values for activation energy and preexponential factor of the relaxation time ␶, obtained at the glass
transition from the analysis of the thermally stimulated current signal, result from the use of the
Arrhenius law for treating the experimental data obtained in nonstationary experimental conditions.
We then demonstrate that a simple model of structural relaxation based on a time dependent
configurational entropy and Adam–Gibbs relaxation time is sufficient to explain the experimental
behavior, without invoking a kinetic singularity at the glass transition region. The pronounced
variation of the effective activation energy appears as a dynamic signature of entropy relaxation that
governs the change of relaxation time in nonstationary conditions. A connection is demonstrated
between the peak of apparent activation energy measured in nonequilibrium dielectric techniques,
with the overshoot of the dynamic specific heat that is obtained in calorimetry techniques. © 2005
American Institute of Physics. 关DOI: 10.1063/1.1858862兴
I. INTRODUCTION
The glass transition is a common feature to quite different and important classes of materials: polymers, oxides,
chalcogenides, metals, molecular solids, etc. A useful way to
think about the glass transition is based on the energy landscape of the liquid.1 Indeed, when such materials are cooled
down at a finite rate from the liquid state, around the temperature of the glass transition, Tg, atomic reorganization becomes too slow and the material is trapped in local energy
minimum characterized by an excess of energy and entropy
with respect to the equilibrium state. So there are two main
aspects, i.e., thermodynamic and kinetic, to the glass transition in relation with the potential energy hypersurface topography of the considered system. The former is related to the
number of energy minima and to the corresponding multiplicity of configurations while the latter is coupled to the
transition rates between energy minima. Whether and how
the thermodynamic and kinetic behaviors are connected, is a
central question to the phenomenology of the glass transition. It is therefore particularly important to properly access
parameters as activation energy and configurational entropy
which characterize the relaxation time ␶ of the system around
T g.
Dielectric relaxation spectroscopy is the most sensitive
method to probe the relaxation times, but it is usually carried
out in stationary conditions at a constant temperature. So, it
does not provide a measure of the structural relaxation itself
at a finite rate of temperature variation. This last kind of
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2005/122共9兲/094507/9/$22.50
relaxation should therefore be accessed using nonstationary
conditions. A widely used procedure to determine relaxation
parameters in the transformation range across the glass transition is based on analysis of differential scanning calorimetry 共DSC兲 curves.2–4 This is a nonequilibrium technique in
which the system evolves through the glass transition at a
constant heating rate; however, calorimetry techniques do
not resolve separately the activation energies at different
temperatures but provide global parameters for the whole
transformation range.2–4 Recently, progress was made in relating the DSC and dielectric 共equilibrium兲 relaxation
parameters.5–7
The combined use of nonstationary conditions and of
dielectric measurements has been widely used to characterize
the glass transition in polymers, both by thermal sampling
共TS兲 of thermally stimulated depolarization current 共TSDC兲
共Refs. 8–15兲 and differential sampling 共DS兲 of dielectric
permittivity.16,17 However, the unexpected trends and values
for activation energy and preexponential factor of the relaxation time ␶, obtained at the glass transition,8,16,18–21 raise
questions about the physical significance of the results, and
the models, i.e., Arrhenius, used for treating the experimental
data.
In this paper we will be concerned with the explanation
of the properties of the effective activation energy obtained
by TS and DS methods, and with the interpretation of the
results in terms of more fundamental models such as the
Adams–Gibbs theory which is based on the configurational
entropy. We emphasize that the results obtained by TS and
DS methods are determined in nonequilibrium conditions, so
that the application of the Arrhenius law does not have im-
122, 094507-1
© 2005 American Institute of Physics
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094507-2
J. Chem. Phys. 122, 094507 共2005兲
Bisquert, Henn, and Giuntini
mediate physical interpretation. We discuss the possibility of
a singular behavior around Tg, due to the complexity of molecular rearrangement in this temperature region, which is a
hypothesis often considered in the literature for explaining
large activation energies and activation entropies.22 From the
model proposed in this paper, we aim to show that this behavior can be understood as a physically continuous and
monotonic evolution, determined by the influence of the
rapid variation of entropy in the relaxation time.
Calorimetry peaks are a common experimental feature of
the glass transition. Upon heating, the DSC curve exhibits an
overshoot in the dynamic heat capacitance before reaching
the liquid specific heat. However, this overshoot is recognized as a consequence of the nonequilibrium conditions of
the measurement, while the relaxation time is a smooth function of both the temperature and the fictive temperature.2–4
We believe that a similar situation exists in the activation
energy results of TS and DS, and will develop this idea in
detail below.
The experimental methods and models that are used to
investigate the kinetic properties of glass transition phenomena are briefly reviewed. Then, we present a different approach for explaining the dominant trends of experimental
observations, based on the computing of the evolution of
entropy across the glass transition and then on the analysis of
the implications for the effective parameters of the relaxation
time.
II. RELAXATION PARAMETERS IN THE GLASS
TRANSITION
A. Relaxation parameters near equilibrium
Let us first recall that the kinetic factor, namely, the relaxation time ␶, of a process that is thermally activated at a
given temperature T and in stationary conditions can be derived from the Boltzmann equation:
冋 册
Eact
,
k BT
␶共T兲 = ␶0exp
共1兲
where Eact共T兲 is the difference between the bottom of the
potential well and the saddle point of the potential surface.
Further, if Eact and ␶0 are constant in a given temperature
range then the relaxation follows an Arrhenius form. In addition, if the relaxation is interpreted with the Eyring transition state theory,23 then the effective activation entropy ⌬Sact
can be determined as
␶0 =
冋 册
⌬Sact
h
exp
,
k BT
kB
共2兲
where h is the Planck’s constant.
Usually, in the glass materials the Arrhenius plot of ln ␶
versus reciprocal temperature is not a straight line, but shows
a significant curvature.24 The results are well described by
the empirical Vogel–Tamman–Fulcher 共VTF兲 form
冋 册
␶共T兲 = A exp
B
,
T − T0
共3兲
where A and B are temperature independent parameters featuring the studied system. In general, from an experimental
procedure that provides ␶ as a function of temperature, the
effective activation energy can be obtained as the local slope
Eact共T兲 =
⳵ ln ␶
.
⳵ 共1/kBT兲
共4兲
An explanation of the temperature dependent activation
energy found in Eq. 共3兲 is made possible by the Adams and
Gibbs model 共AG兲.25 The central idea of this theory25–27 is
that for elementary conformal units to realize a relaxation
arrangement, their neighbors must also move in cooperation.
The effective domain size changes with the overall density,
so that the number of particles that cooperatively rearrange
increases with decreasing temperature. At high temperature,
the conformers can relax independently from their neighbors.
If equilibrium can be achieved at a low temperature limit TK,
every conformer becomes meshed with all others, and all the
units in the polymer belong to a unique domain. This is the
temperature at which the configurational component of the
total entropy goes to zero 共the Kauzmann temperature兲. In a
domain that contains a number z of conformal units, the observed activation energy for the conformers of the domain to
relax simultaneously is determined by their combined transition probability, and the apparent activation energy must be z
times the intramolecular activation energy for one conformer
to relax, ⌬␮,
⌬h * = z⌬␮ .
共5兲
It is assumed that only the minimum value of z determines the relaxation time, because the processes which
larger z produce exponentially longer relaxation times. This z
is related to the molar configurational entropy Sc 共defined as
the total molar entropy subtracted of the contribution arising
from intramolecular and intermolecular vibrations兲 as
follows:25
共6兲
z = s * /Sc ,
where s* is the entropy of the minimum number of particles
that are able to rearrange. The transition state theory expression for the relaxation time then gives the result
冋 册
␶共T,Sc兲 = A exp
C
,
TSc共T兲
共7兲
where C = NAs * ⌬␮ / k is temperature independent 共NA is
Avogadro’s number兲. It is noteworthy that the AG model has
been recently supported by simulations28–30 and empirical
observations.5 The configurational entropy is determined
from
Sc共T兲 =
冕
T
TK
⌬C p共T⬘兲
dT⬘ ,
T⬘
共8兲
where ⌬C p共T兲 is the molar heat capacity increment due to
gaining access to the configurational states. ⌬C p is usually
identified with the experimentally observed difference between liquid 共or rubber兲 and glass molar heat capacities.27 It
is convenient to use the hyperbolic form
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094507-3
J. Chem. Phys. 122, 094507 共2005兲
Effective activation energy below the glass transition
FIG. 1. 共a兲 Schematics of the variation of configurational entropy by cooling
a glass-forming liquid. The Kauzmann temperature, glass transition temperature, and equilibrium entropy are indicated. 共b兲 Schematics of effective
activation energies obtained by thermal sampling in TSDC technique, as a
function of polarization temperature. Note that each activation energy is
obtained in a different experiment at constant heating rate. The equilibrium
line indicates the result of the Adam–Gibbs relaxation time.
⌬C p共T兲 = S⬁
TK
T
共9兲
which is found suitable for low weight glass-forming
materials.5 S⬁ is the temperature independent entropy of the
liquid well above Tg. Other forms of ⌬C p共T兲 reported in the
literature provide qualitatively similar results,31 so we will
use Eq. 共9兲 in our calculations. The stationary configurational
entropy at equilibrium is thus given by 关see Fig. 1共a兲兴
冉 冊
Seq
c 共T兲 = S⬁ 1 −
TK
.
T
共10兲
Then, the AG relaxation time, Eq. 共7兲, takes the VTF
form with B = C / S⬁ and T0 = TK. In the equilibrium case, Eqs.
共7兲 and 共10兲 in Eq. 共4兲 yield the effective activation energy to
be temperature dependent such as4
eq
Eact
共T兲 =
k BB
.
共1 − TK/T兲2
共11兲
Note in Eq. 共11兲 and Fig. 1共b兲 the monotonic decrease of
eq
with increasing temperature, which is a consequence of
Eact
the facilitated transitions by the decreasing size of the cooperative domains.
B. Experimental observations of the relaxation time
parameters
The coupling of dielectric relaxation to structural kinetics is easily achieved by measuring, upon heating, the depolarization current of a sample with initial polarization P0 at
FIG. 2. 共Top兲 Apparent activation energies versus temperature for PVC
obtained from TSC-TS and Steeman and van Turnhout’s 共Ref. 16兲 derivative
analysis method of ac dielectric data showing a prominent maximum at Tg.
Some literature TSC-TS and TSC-creep values were taken from Ref. 19, and
for all the thermally stimulated data, the temperature axis is the polarization
temperature. 共Bottom兲 Values of apparent activation entropies versus temperature for PVC showing the low temperature broadening of the glass
transition. 共Adapted from Ref. 18, © John Wiley & Sons, Inc., reprinted
with permission.兲
low temperature Ta. Assuming a first-order relaxation, the
rate of depolarization at constant heating rate, T = qht + Ta, is
given by
dP
P0
=−
.
dT
q h␶
共12兲
From Eq. 共11兲, the TSDC as a function of temperature is
I=
冋
1
P0
exp −
␶共T兲
qh
冕
T
Ta
册
1
dT⬘ .
␶共T⬘兲
共13兲
TSDC has high sensitivity and the ability to selectively
probe the relaxation kinetics during heating in a very restricted range of temperatures, by the TS method,32 in which
the polarization is applied in a narrow window ⌬T ⬇ 5 K.
Note that in TS the Bucci method33 is applied for the calculation of the relaxation time, so that each peak is considered,
to a first approximation, as correspondent to a Debye process, characterized by a single value of the relaxation time
parameters. The TS method in polymers therefore consists on
fitting the depolarization peaks to Arrhenius- or Eyring-type
equation and provides a value of Eact associated to the chosen polarization temperature T p. It is important to remark
that Eq. 共1兲 is used while the system is studied under nonstationary condition and varies with a relaxation rate comparable to the experimental time scale.
In Fig. 2 we show the representative experimental results
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094507-4
J. Chem. Phys. 122, 094507 共2005兲
Bisquert, Henn, and Giuntini
for the apparent activation energy that is measured in poly共vinyl chloride兲 共PVC兲 when approaching Tg from below,
using both the TS-TSDC and DS of dielectric
permittivity.16,18,19 It is noted that the different experimental
techniques record a very high activation energy 共enthalpy兲 at
temperatures slightly below the glass transition 共␣ transition
in polymers兲. This behavior has been found in a wide variety
of polymers.16–21
In principle this common experimental result cannot be
explained directly with Eq. 共6兲 of AG theory, because the
size of the minimum cooperative domain decreases at increasing temperature, so that ⌬h* decreases monotonically.
A conventional explanation for relaxation parameters along
the glass transition assumes that a unique process with a
distribution of activation energies 共or activation entropies兲
occurs over a range of temperatures. While the distribution
of activation energies is well supported by the existence of
broad relaxation peaks in dielectric spectroscopy, this approach could not explain the very high activation energy recorded at temperatures below the ␣ relaxation.
It has been often remarked that relaxation parameters
obtained from TS peaks obey the compensation law,11,13,14
which means a linear relationship of the parameters derived
from Eq. 共1兲:
ln ␶0 = ln ␶c +
Eact
,
k BT c
共14兲
where Tc and ␶c are the compensation temperature and time,
respectively. If the relaxation time is rationalized with the
Eyring-type expression, then the compensation law implies a
strong correlation between the activation energy and activation entropy 关Eq. 共2兲兴 of the transitions in the temperature
range of the glass transition8 共see Fig. 2兲. This view of the
compensation phenomenon frequently observed in polymers
suggests a dramatic change of both the activation energy and
activation entropy due to the complexity of the molecular
motions involved during the glass transition.13,14,22 It should
be remarked, however, that the compensation law is a purely
empirical relationship which lacks further detailed interpretation.
In addition, TSDC shows a number of peculiarities that
indicate the relevance of nonequilibrium effects. For example, two of us showed that, in some polymers, the width
of the TSDC peak decreases with increasing heating rate34
and that this cannot be explained by any of the existing models based, on the one hand, on a static distribution of activation energies or of preexponential factor ␶0 or, on the other
hand, on a stretched exponential relaxation, exp关−共t / ␶兲␤兴, in
which ␶ and ␤ are constants.
III. ENTROPY RELAXATION MODEL AND ACTIVATION
ENERGY IN NONSTATIONARY CONDITIONS
A. Relaxation parameters in nonequilibrium
conditions
Here we propose an explanation for the experimental
observations on the sharp variation of effective activation
energy as one approaches Tg from below. Our model starts
with the observation that the AG relaxation time describes
structural rearrangement below Tg and is still valid when the
glass departs strongly from equilibrium with the liquid, i.e.,
when Sc Ⰷ Seq
c 共Fig. 1兲. The effective activation energy takes
the form
Eact共T兲 =
冉
冊
k BC
T dSc
1+
.
Sc
Sc dT
共15兲
We remark that AG model has been used for calorimetric
determination of relaxation parameters,27,35,36 and Eq. 共15兲
was derived in this context.35,36
We consider in more detail the main trends of the observed activation energy and their interpretation. Figure 1共b兲
shows schematically Eact as a function of T p 共the experimental data showing this behavior are given in Fig. 2兲, in connection with the variation of entropy indicated in Fig. 1共a兲.
eq
Figure 1共b兲 shows also the expected equilibrium value Eact
according to Eq. 共11兲.
In the region A of structural arrest, Eact shows a nearly
constant and physically reasonable value typically around 1
eV. We note that this is in agreement with the AG expressions, Eqs. 共7兲 and 共15兲, because the configurational entropy
is constant. In the quasiequilibrium region C, TSDC is difficult to measure, however, the data are expected to simply
eq
.
follow the liquidlike equilibrium Eact
As already mentioned, in the transition region B below
Tg, the effective activation energy increases to values
⬇5 eV. This increase is too large to be interpreted as an
activated process. Besides, the preexponential factor ␶0 takes
unphysical values in the order of 10−40 s.
When the system undergoes a transformation from glass
to liquid, the entropy will depart from the equilibrium value.
Let us denote by ⌬Sc = Sc − Seq
c the extent of departure. Then
using dSc = dSeq
c + d⌬Sc, we obtain from Eq. 共15兲
eq
Eact共T兲 ⬇ Eact
共T兲 + kBC
T d⌬Sc
,
S2c dT
if Sc ⬇ Seq
c .
共16兲
Near the glass transition, ⌬Sc / Sc need not be too large
共see later on Fig. 4兲. However, even if Sc ⬇ Seq
c the derivative
d⌬Sc / dT may be large, and will cause the observation of an
anomalously high apparent activation energy. Note that apparent Eact in Eq. 共16兲 is defined 共and experimentally determined兲 by adapting the relaxation time to the Arrhenius
form, through Eq. 共4兲. In contrast to this, kinetic transitions
for structural rearrangement are described by the constant
C = NAs * ⌬␮ / k of AG theory. In conclusion, the variation of
excess entropy in nonequilibrium conditions, which occurs
below Tg, introduces a component in the effective activation
energy that is not related to the kinetic transitions in the
material, so that Eact will differ significantly from its equilibrium value in Eq. 共11兲. It is interesting to note that some
authors16 indicated the nonequilibrium conditions of measurement below the glass transition as the possible origin of
the very high activation enthalpies observed. However, no
detailed model was proposed to explain such effect. In the
following we develop a model that determines the variation
of dSc / dT under a heating scan, in order to explain the results usually obtained for the apparent Eact in nonequilibrium
conditions, in terms of the real relaxation time of AG theory.
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094507-5
J. Chem. Phys. 122, 094507 共2005兲
Effective activation energy below the glass transition
B. Phenomenological model for entropy relaxation
As mentioned in the Introduction, the main method used
to investigate the relaxation time parameters of structural relaxation is based on DSC technique. An array of phenomenological methods have been developed to treat DSC data and
derive the relaxation parameters.2,3,37 The main idea behind
these methods is that the glass is characterized by frozen
degrees of freedom that relax to equilibrium. Far below the
glass transition the structural rearrangement is kinetically arrested because the relaxation time is orders of magnitude
larger than observation times, and measured changes in properties such as enthalpy and entropy in response to changes in
temperature do not contain contributions from configurational degrees of freedom. When the system is heated, at
some point the temperature-dependent relaxation time becomes smaller than accessible times, hence, the system begins to relax and continues the relaxation until it achieves the
equilibrium with the liquid state. The relaxation process is
described by kinetic models2,3,37 that usually adopt the original method for handling nonlinearity due to Tool,38 who expressed the relaxation time as a function of the departure
from equilibrium. This is accomplished by assuming that the
relaxation time is a function of the fictive temperature39 T f ,
which is defined as the temperature at which some property
of the nonequilibrium glass would be at equilibrium.
The form most commonly adopted is the Tool–
Narayanaswamy–Moynihan2,3,37 共TNM兲 equation
冋
␶共T,T f 兲 = A exp
册
x⌬h* 共1 − x兲⌬h*
+
,
RT
RT f
共17兲
where the extent of nonlinearity is determined by the parameter 0 ⬍ x 艋 1.
The normal phenomenology for DSC is based on the
relaxation on enthalpy,2,3,37 which is the quantity measured in
this method. However we are more interested in the relaxation of configurational entropy, which provides the apparent
activation energy in TS and DS techniques following Eq.
共15兲. Since our aim is to explain the experimental features of
these quantities, we will keep the phenomenological model
for structural relaxation as simple as possible. Therefore, we
take the entropy as the primary relaxing quantity in the phenomenological model. Configurational entropy will be assumed to obey a first-order relaxation governed by AG relaxation time:
1
dSc
=−
⌬S ,
dt
␶共T,Sc兲 c
共18兲
where ␶共T , Sc兲 is given in Eq. 共7兲. Equation 共18兲 is equivalent
to the relaxation equation that can be obtained from a simple
Erhenfest’s two urns model.40 We remark that the literature
contains many discussions of entropy relaxation applied to
calorimetry techniques. Scherer41 and Hodge4,27 applied the
AG theory to structural relaxation, and the fictive
temperature39 was defined in terms of configurational
entropy.4 Hodge35 has remarked that in a good approximation enthalpic and entropic fictive temperatures are the same.
A complete formalism for relaxation of Sc has been
developed42–44 which shows excellent accord with calorim-
etry data for polymer glasses. In addition, recent analysis of
nonequilibrium glassy state have emphasized the role of
Sc.31,45
Equation 共18兲 is assumed to hold under any temperature
history. In particular, for a constant cooling or heating rate q
we obtain
1
dSc
=−
⌬Sc .
dT
q␶共T,Sc兲
共19兲
We can now discuss the connection of our model with conventional DSC phenomenology.2,3,37 From the definition of
the fictive temperature39 T f it comes out Sc共T兲 = Seq
c 共T f 兲, implying T f = TK / 共1 − Sc / S⬁兲, so Eq. 共19兲 becomes
1
dT f
Tf
=−
关T f − T兴,
dT
q␶共T,T f 兲 T
共20兲
where ␶, using the AG formalism, takes the form31 关see
Eq. 共7兲兴
冋
␶共T,T f 兲 = A exp
册
B
.
T共1 − TK/T f 兲
共21兲
Equation 共20兲 is similar to Tool’s relaxation equation38
and Eq. 共21兲 is experimentally indistinguishable35 from
Eq. 共17兲.
Let us now confirm that our model reproduces the usual
phenomenological features of the glass transition.2–4 Simulations were performed using Eq. 共18兲 or Eq. 共19兲 with the
parameters reported for 2-methyltetra-hydrofuran in Ref. 5.
Figure 3共a兲 shows the results of isothermal recoveries of
temperature steps of the same magnitude but opposite sign to
a common final temperature. The asymmetry of the isotherms relates to nonlinear behavior as described by
Kovacs.24,46 Another important aspect of the phenomenology
is the determination of the effective activation energy from
the values of fictive temperature at different cooling rates
qc.3 Following the standard phenomenology, it can be
shown3,47 that the final value of the fictive temperature in the
cooling curves, T⬘f = T f 共T ⬇ 0兲 ⬇ Tg 共corresponding to the excess entropy as defined above兲, varies with the cooling rate
qc as predicted by
Eact共Tg兲 = −
⳵ ln qc
.
⳵ 共1/kBTg兲
共22兲
This last equation can be derived from general arguments on
the invariance of the cooling curves.47,48 Because the T⬘f
value is obtained from cooling curves as those of Fig. 3共b兲,
the activation energy of Eq. 共22兲 is usually taken as the equilibrium one, i.e., Eq. 共11兲 evaluated at the glass transition
temperature. Then, equating Eqs. 共11兲 and 共22兲 and integrating we obtain the relationship
ln
冉
冊
1
1
qc
=−B
−
,
qc,ref
T⬘f − TK T⬘f,ref − TK
共23兲
where qc,ref and T⬘f,ref are reference values. The cooling
curves obtained from Eq. 共19兲 are reported in Fig. 3共b兲. The
data for T⬘f 共qc兲 derived from the integration of entropy variation 关Eq. 共19兲兴 at different cooling rates are shown in Fig.
3共c兲. By fitting the data to Eq. 共23兲 the parameter B of AG
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094507-6
Bisquert, Henn, and Giuntini
J. Chem. Phys. 122, 094507 共2005兲
FIG. 4. Simulation of configurational entropy evolution during heating at
different rates, as indicated, the glass at initial fictive temperature T f
= 90 K. Same parameters as in Fig. 3.
for the activation energy behavior. Indeed we show that this
behavior is well described by AG theory and a minimal relaxation theory, while the distribution of relaxation times is
unessential to account for such behavior. As a consequence,
the peaks of activation energy and heat capacity that will be
obtained, will be less broad than the experimental ones. We
remark that additional use of a distribution of relaxation
times following standard procedures2,3,37 共see in particular
the work of Gómez–Ribelles and co-workers42–44兲 will improve the accord of the model with measurements. This extension of the model will not be considered here.
C. Behavior of the entropy and relaxation time
parameters in a constant rate heating
FIG. 3. Simulation of entropy relaxation for a glassformer with paramaters
TK = 69.3 K , A = 10−17.3 s , B = 935 K , S⬁ = 97.5 J K−1. 共a兲 Isothermal entropy
relaxation following temperature jumps ⌬T = ± 2 K towards final equilibrium at T = 90 K. 共b兲 Configurational entropy evolution during cooling at
different heating rate, as indicated, from the liquid state. 共c兲 Results of final
fictive temperature at different cooling rates. The linear regression gives a
slope B = 950 K.
relaxation time is recovered within an error of 2%, confirming that the entropy relaxation model obeys Eq. 共22兲.
We have shown that the model presented is nonlinear as
required in glass transition phenomenology 共the connection
between TNM parameter x and AG parameters is discussed
by Hodge and O’Reilly31兲. Another aspect of standard
phenomenology2,3,37 is the nonexponential relaxation, usually represented by the parameter ␤ of the stretched exponential. The memory effect is very important for the good
accord of theory with experimental DSC curves, and its description requires a distribution of relaxation times in the
internal relaxing units.49 Here, these additional complications
have been omitted to prevent obfuscation of the basic reason
Having presented the kinetic model for the variation of
entropy, we can now discuss the properties of the effective
relaxation time through the glass transition. The evolution of
entropy and of the corresponding relaxation time are shown
in Figs. 4 and 5共a兲, respectively. ␶ is characterized by a near
Arrhenius temperature dependence below Tg, corresponding
to the region of nearly constant entropy, and then a sudden
drop to the equilibrium VTF curved line. This behavior of ␶
is reported experimentally.50,51 Figure 5共b兲 shows the change
of activation energy 关determined with Eq. 共15兲兴 in the heating excursion. The result is quite similar to the characteristic
experimental observations summarized in Fig. 1共b兲 and indicated in Fig. 2. At low temperatures the activation energy is
nearly constant, corresponding to the Arrhenius region of the
relaxation time. In the final stages, the activation energy
drops to and follows the equilibrium curve. The interpretation of the peak is discussed in the following.
We have mentioned in the Introduction the similarity of
the peak of apparent activation energy obtained in TSDC-TS,
displayed in Fig. 2, with the overshoot of the dynamic specific heat that is measured in DSC, which is generally found
and is used to define the glass transition temperature.52 We
remark that the latter is a nonequilibrium effect that is obtained when the sample undergoes the transition from the
specific heat of the glass, Cglass, to that of the liquid, Cliquid,
which is larger than the former due to the release of the
frozen configurational changes of the liquid.52 Now we can
demonstrate the relationship between the two features.
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094507-7
J. Chem. Phys. 122, 094507 共2005兲
Effective activation energy below the glass transition
FIG. 6. 共a兲 Simulation of temperature derivative of configurational entropy
during heating at constant rate, with the glass at initial fictive temperature
T f = 90 K as in Fig. 4. The dotted line is the temperature derivative of equilibrium configurational entropy, dSceq / dT = ⌬C p / T = S⬁TK / T2 关see Eq. 共9兲兴.
共b兲 Simulation of temperature derivative of enthalpy during heating at constant rate, with the glass at initial fictive temperature T f = 90 K as in Fig. 4.
The glass specific heat Cglass is taken as the baseline. The dotted line is the
heat capacity increment Cliquid − Cglass = ⌬C p = S⬁TK / T 关see Eq. 共9兲兴.
FIG. 5. Simulation of the evolution of relaxation time during heating at
different rates, as indicated, for the glass at initial fictive temperature T f
= 90 K. Same parameters as in Fig. 3. Dotted line indicates the equilibrium
value at each temperature. 共a兲 Representation of the relaxation time with
respect to reciprocal temperature. 共b兲 Activation energy. 共c兲 Activation
entropy.
First we show in Fig. 6共a兲 the derivative of the configurational entropy during a heating at constant rate. Let us
analyze the relationship of Fig. 5共b兲 to Fig. 6共a兲. We note that
eq
共T兲 in
according to Eq. 共16兲, the point at which Eact共T兲 = Eact
Fig. 5共b兲 corresponds to that at which dSc / dT
= dSeq
c / dT共d⌬Sc / dT = 0兲 in Fig. 6. The temperature region of
sharp increase of both magnitudes Eact共T兲 and dSc / dT corresponds to the region where Sc overcomes the equilibrium line
in Fig. 4. Indeed, in Fig. 4 configurational entropy Sc intersects the equilibrium line at zero slope, i.e., the local minimum of Sc共dSc / dT = 0兲 occurs at the line Seq
c 共T兲. 关We remark
incidentally that in the real glasses the local minimum occurs
before the Seq
c 共T兲 line, so that the intersect shows a positive
slope, due to the memory effect,37,49 however, as already
mentioned this memory effect is not considered here.兴 In
conclusion, the strong increase of the activation energy in
Fig. 5共b兲 is caused by the rapid increase of entropy when
entropy overcomes the equilibrium line, see Figs. 4 and 6共a兲.
The peak in apparent activation energy displayed in Fig. 5共b兲
is a nonequilibrium effect that has its origin in the derivative
dSc / dT, as previously discussed in relation to Eqs. 共15兲 and
共16兲.
Figure 6共b兲 shows the dynamic specific heat that is obtained from the same data, taking into account that
Cp =
dS
dSc
dH
,
= T = Cglass + T
dT
dT
dT
共24兲
where H is the enthalpy and Cglass contains the temperature
derivative of the entropy component due to vibrational degrees of freedom. The dynamic heat capacitance displays the
overshoot that is found in measurements of DSC across the
glass transition. Comparing Figs. 6共a兲 and 6共b兲, it is seen that
both peaks have the same origin.
Next we determine the prefactor ␶0 from Eq. 共1兲 and
then the activation entropy ⌬Sact from Eq. 共2兲 关Fig. 5共c兲兴
which follows a similar trend as the activation energy. Thus
both magnitudes are correlated linearly in the compensation
plot 共Fig. 7兲. Note also the artificial anomaly in the activation
entropy which becomes negative for the lowest heating rate
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094507-8
J. Chem. Phys. 122, 094507 共2005兲
Bisquert, Henn, and Giuntini
model explains the peculiar behavior of peak narrowing at
increasing heating rate that was also experimentally
observed.34
IV. FINAL REMARKS AND CONCLUSIONS
FIG. 7. Compensation plot of the simulation data of Fig. 5.
共dashed line in Fig. 4兲. This is because at this slow heating
rate the entropy relaxes already at the starting temperature.
D. Thermally stimulated currents
Finally, Fig. 8共a兲 shows the characteristic shape of global
peaks of TSDC 关Eq. 共13兲兴. Interestingly, the curves show a
shoulder in the region below Tg as it was reported
experimentally9 and interpreted as a secondary relaxation
preceding the glass transition. Figure 8共b兲 shows the normalized TSDC peaks. It is clear that the entropy relaxation
The characteristic behavior of the effective activation energy in nonequilibrium measurements can be explained by
coupling a simple model on entropy relaxation with the
Adam–Gibbs relaxation time. The model indicates that the
strong variation of activation energy is not related to a kinetic singularity around Tg, but to the rapid variation of entropy with temperature in this region. It is therefore demonstrated that the resulting change in the relaxation time, if
analyzed using a Arrhenius treatment, leads to the compensation law and to inappropriate singularity for Eact and ␶0.
Furthermore, the model explains qualitatively many experimental observations on thermally stimulated current reported
in the literature.
A referee of this paper brought to our attention a paper
by Goitiandia and Alegria53 that was published recently with
a similar goal as ours, namely, to test the apparent activation
energy out of equilibrium in glasses, using TSDC, and deriving the nonequilibrium relaxation time. Our results are in
general agreement with theirs.
ACKNOWLEDGMENT
J.B. is deeply grateful to the Université Montpellier II
共France兲 for the visiting Professor fellowship and its financial support.
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