Relation between configurational entropy and relaxation dynamics of glass-forming systems under volume and temperature reduction
ABSTRACT SUMMARY The structural relaxation dynamics of two molecular glass-forming systems have been analyzed by
means of dielectric spectroscopy, under cooling and compression conditions. The relation of the dynamic
slowing down with the reduction of the configurational entropy, SC, as predicted by Adam and Gibbs (AG),
was also investigated. As SC is not directly accessible by experiments, it was estimated, following a common
procedure in literature, from the excess entropy Sexc of the supercooled liquid with respect to the
crystal, determined from calorimetric and expansivity measurements over the same T–P range of dynamics
investigation. The AG relation, predicting linear dependence between the logarithmic of structural
relaxation time and the reciprocal of the product of temperature with configurational entropy, was successfully
tested. Actually, a bilinear relation between Sexc and SC was found, with different proportionality
factors in isothermal and isobaric conditions. Using such results, we derived an equation for predicting
the pressure dependence of the glass transition temperature, in good accordance with the experimental
values in literature.
Relation between Configurational Entropy and Relaxation Dynamics of
Glass-Forming Systems under Volume and Temperature Reduction
Simone Capaccioli1,2, Daniele Prevosto1,2, Mauro Lucchesi1,2, Masoud
Amirkhani1, Pierangelo Rolla1,2
1Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo 3, I-56127, Pisa, Italy;
2INFM-CNR, polyLab, Largo B. Pontecorvo 3, I-56127, Pisa, Italy;
Simone Capaccioli; Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo 3, I-
56127, Pisa, Italy; E-mail: email@example.com; Tel. +390502214322; Fax. +390502214333.
The structural relaxation dynamics of two molecular glass forming systems have been
analyzed by means of dielectric spectroscopy, under cooling and compression conditions. The
relation of the dynamic slowing down with the reduction of the configurational entropy, SC, as
predicted by Adam and Gibbs (AG), was also investigated. As SC is not directly accessible by
experiments, it was estimated, following a common procedure in literature, from the excess
entropy Sexc of the supercooled liquid with respect to the crystal, determined from
calorimetric and expansivity measurements over the same T-P range of dynamics
investigation. The AG relation, predicting linear dependence between the logarithmic of
structural relaxation time and the product of temperature with configurational entropy, was
successfully tested. Actually, a bilinear relation between Sexc and SC was found, with different
proportionality factors in isothermal and isobaric conditions. Using such results, we derived
an equation for predicting the pressure dependence of the glass transition temperature, in good
accordance with the experimental values in literature.
PACS codes: 64.70.Pf, 77.22.Gm, 05.70.Ce
Keywords: Glass transition, structural relaxation time, configurational entropy, glass
transition temperature, pressure.
According to the Adam and Gibbs (AG) model the slowing down of α-relaxation time τα in
glass-forming systems is linked to the the increasing size NCRR of the cooperatively
rearranging regions (CRR) . AG model postulates that the free energy activation barrier for
a CRR relaxation is linearly increasing with NCRR, that is reciprocal of configurational entropy
SC of the system. Thus, on approaching the glass transition by cooling or compressing, the
reduction of configurational entropy implies a slowing down of structural dynamics,
according to the equation:
0 τ τ
where τ0 is the value of τ in the limit of infinite (TSc), and CAG=(sc*∆µ/R), with R the gas
constant, sc*=kBln2 and ∆µ is the free energy activation barrier for an elementary transition,
i.e. at high temperatures where cooperativity does not take place and NCRR =1. AG model is
derived for an ensemble in isobaric-isothermal conditions. CAG was assumed as a constant,
since the increase of pressure and the decrease of temperature are believed to change only
NCRR∝1/Sc. Of course, for high compression (usually for pressures comparable than the
isothermal bulk modulus, around some GPa), changes in the structure of the material could
take to not negligible variations of ∆µ, but this is not the case of most of the experiments
reported in the literature. For instance, for pressure up to 1 GPa, it has been reported that the
shape of relaxation peak of a given glass-former is not depending on pressure or temperature,
but only on the relaxation time , indication that the mechanisms of relaxation are not
altered by the compression.
AG model has been criticized from the theoretical point of view, especially concerning its two
the main assumptions: (a) the barrier height grows like the number of molecules forming a
CRR and (b) the critical configurational entropy sc* associated to a CRR is fixed and is equal
to kBln2, independently of the size of the CRR. The first hypothesis implies that even if the
number of molecules in a CRR is large, the number of configurations they can build is
constant. So, a finite number of possible rearrangements in a CRR is enough to make the
system relax, disregarding the volume of the CRR. When the volume is large, however, this
becomes a heavy requirement that is hard to justify a priori and not obvious at all. Another
objection to this hypothesis comes from ref., where it is shown that in a number of glass
formers if sc* is fixed to be kBln2 the extrapolated size of the cooperatively rearranging region
NCRR is less than one, and if sc* is left as a free parameter, though constant for the whole
experiment, NCRR remains small. Implying that, in this view the structural cooperative
relaxation is due to the simultaneous rearrangment of few molecules, contrary to what
evidenced from experiments . The mosaic model or "random first order transition theory"
 allows for an alternative derivation of a generalized AG law, where a proportionality
between log(τ) and a power law of (TSc)-1 holds, without resorting to the above mentioned
assumptions. Along this line, recent publications [4, 5, 6] provide careful and physically
reasonable revisions of AG model and predictions. In spite of this shortcomings, the Adam-
Gibbs relation (Eq.1) succeeded quite well in accounting for experimental data and gained a
considerable popularity. In fact, Eq.(1) has been successfully applied to reproduce the
dynamics of supercooled liquids above the glass transition in many cases, both in numerical
simulations [7, 8, 9] and experiments [10, 11, 12, 13, 14, 15 and references therein]. For
molecular dynamics simulations, the determination of configurational entropy SC is directly
obtained from the difference of the total entropy and the vibrational entropy of typical basins
sampled, Svib or by the distribution of energy minima in energy landscape. Anyway, the good
agreement found between Eq.(1) and data of molecular dynamics simulations is puzzling, as
the relaxation is there investigated on a very short timescale (ns-ps) where the cooperativity
should not be so important. Other doubts can be raised about the correct decomposition
between vibrational and configurational modes in a liquid at this short time-scale, rather than
in a solid, as a vibration inside a definite cage cannot be precisely defined for a liquid.
Concerning experiments, since Svib is not directly accessible, a common approximation for SC
in literature is to calculate the excess entropy Sexc of the supercooled liquid with respect to the
crystal. This approximation gave rise to several criticisms . The main one is that the
vibrational properties of a crystal are different from that of the supercooled liquid at the same
T-P conditions and so a vibrational excess term is considerably contributing to Sexc . As in
supercooled liquids a relation log[τ(T,P)]∝[TSexc(T,P)]-1 is usually fulfilled [12, 13, 14, 15], a
hypotesis about the proportionality Sexc∝SC was proposed, yielding again the relation of Eq.
(1) [7, 17, 18]. Notwithstanding the above mentioned results appear as convincing tests of the
AG model, the debate is still open about several issues: (a) the right way of calculating the
configurational entropy in an experiment [16, 19, 20, 21]; (b) the physical meaning of the size
of a CRR [21, 22, 23, 24]; (c) the amount of configurational part of Sexc [7, 17, 16, 25, 21, 26,
27, 28, 18]; (d) the independence from pressure and temperature of the coefficient CAG in Eq.
(1) [11, 13, 14, 29, 8, 9]. Our previous experimental studies confirmed that AG relation is
fullfilled also at high pressure [11, 13, 14]: so a proportionality can be postulated between Sexc
and SC or, better, a bilinear relation, with different proportionality factors gT(P) and gP(T) in
isobaric and isothermal conditions respectively  (see Eq. (2) below). As required by
general principle of thermodynamic, configurational entropy must be a function of state and
therefore gT(P) and gP(T) are mutally dependent. In particular, when SC(T,P) is calculated
along an isobaric path at atmospheric pressure, Patm, from the initial temperature TK till the
final temperature T, followed by an isothermal path at T till the final pressure P, the
configurational entropy is expressed by:
where TK is defined as the temperature where Sc(TK,P=0.1 MPa)=0, (∆Cp/T)dT=[(Cpmelt-
Cpcryst)/T]dT and ∆(∂V/∂T)pdP=-[(∂V/∂T)Pmelt-(∂V/∂T)Pcryst]dP, are the excess heat capacity
and the temperature derivative of the excess volume of the melt with respect to the crystal.
The derivation of Eq. (2) is valid for systems in thermodynamic equilibrium (i.e. above the
glass transition) and cannot be used for systems in the glassy state. Further details about such
derivation and the compatibility with thermodynamic requirements for a function of state can
be found in ref. . Previous applications of Eq.(2) to represent set of isobaric data in
molecular and polymeric glass-formers can be found in ref. [30, 32, 33]. The present
experimental study comprises an extensive study of dynamics under isothermal compression
of two glass formers: diphenyl-vinylene carbonate and polyphenylglycidylether, already
partially studied [34, 35, 31, 36]. Moreover, using the above mentioned procedure, Sexc is
calculated from calorimetric and expansivity measurements over a wide T-P range, allowing a
direct comparison with the dielectric relaxation times τα and a succesful test of the AG model.
In particular we show that log[τ(T,P)] is proportional to [TSc(T,P)]-1 and the proportional
coefficient is constant for low-pressure values. Moreover, from our procedure, we derive a
general expression for predicting from configurational entropy the values for the pressure
dependence of the glass transition temperature in the limit of low pressure.
2. Experimental procedures
Diphenyl-vinylene carbonate (DPVC) and polyphenylglycidylether (PPGE) were obtained
from Aldrich. Their molecular weight is 348.15 g mol-1 (DPVC) and 345 g mol-1 (PPGE). The
glass transtion temperature Tg, obtained by DSC (onset), is 251 K for DPVC and 258 K for
PPGE. Dielectric measurements, both at atmospheric and at high pressure, were carried out
using a Novocontrol Alpha-Analyzer (ν=10-3-107 Hz). For atmospheric pressure
measurements, a parallel plate capacitor separated by a quartz spacer (geometric capacitance
~ 90 pF) and filled by the sample was placed in the nitrogen flow Quatro cryostat. For high
pressure measurements, a sample-holder multi-layer capacitor (geometric capacitance ~ 30pF)
was isolated from the pressurizing fluid (silicon oil) by a Teflon membrane. The dielectric
cell was then placed in a Cu-Be alloy high pressure chamber, provided by UNIPRESS,
connected to a manually operated pump with a pressure intensifier able to reach 700 MPa.
The high pressure chamber was surrounded by a metallic jacket, whose temperature was
varied in the interval 200–360 K and controlled within 0.1 K by means of a thermally
conditioned liquid flow. Dielectric measurements were performed after equilibrating
temperature and pressure for enough time to have data in thermodynamic equilibrium (at least
30 minutes or more, up to 4 times the relaxation time τ, on approaching the glass transition).
PVT data were obtained by GNOMIX high pressure dilatometer, while absolute density
measurements, at ambient temperature, were made using a helium pycnometer. Calorimetric
data were obtained using a Mettler 30 Differential Scanning Calorimeter.
The relaxation map of structural relaxation time of DPVC in isobaric and isothermal
conditions is shown in Fig.1(a)-(b), respectively. The Vogel-Fulcher-Tamman (VFT) function
fitting the data at 0.1 MPa has the parameters: log10(τ0/s)=-18.6±0.4, B=1200±50 K and the
Vogel temperature T0=193±5 K. The dielectric glass transition, defined as the temperature at
which log10(τ/s)=2, is 251 K and the steepness index m=d[log10(τ)]/d[Tg/T]⎜Tg=89. The
isothermal scans show the linear relation log10(τ)∝(P∆V#)/RT, where ∆V# is the so-called
activation volume, that is decreasing with temperature: ∆V#= 240±15 cm3mol-1 at T=271 K
and ∆V#= 170±10 cm3mol-1 at T=306 K. From the extrapolation of the linear behavior it is
possible to estimate the value Tg(P). It is increasing with pressure as a second order
polynomial, but at low pressure it has a slope dTg(P)/dP= 190 K GPa-1.
Similar considerations are valid also for data of PPGE shown in Fig.2-(a)-(b). The VFT
function fitting parameters can be found in refs.[13, 35, 36]. Fit of data at 0.1 MPa have the
parameters: log(τ0/s)=-15.1±0.2, B=708±10 K and the Vogel temperature T0=217±2 K. At
times shorter than 0.1 µs times deviates from the VFT behavior, according to a crossover
behavior often reported in literature . The dielectric glass transition is 256 K and the
steepness index m=107. The isothermal scans show a non-linear relation between logarithm
of times and pressure, of the kind log(τ)∝B/(P0-P). A deviation from the linear behavior is
expected for a wide pressure range, but the pressure value at which the non-linearity takes
place is material dependent. Usually the non linearity shows up at lower pressure for systems
with higher m index. From the extrapolation of the non-linear behavior of log10(τ/s) to 2 it is
possible to estimate the value Tg(P). It is increasing with pressure as a second order
polynomial, but at low pressure it has a slope dTg(P)/dP= 154 K GPa-1.
PVT data and thermograms of DPVC and PPGE are not reported for lack of space and a
detailed description about this procedure will be published elsewhere . Anyway,
expansivity and calorimetry data of PPGE are reported partially in refs. [34, 35, 36].
For PPGE the specific heat was measured in both liquid and crystalline state. On the other
hand, the temperature derivative of the volume of the crystal of PPGE was not available, and
it was obtained from that of the glassy state. In fact, the data of the few systems reported in
literature  suggest that the temperature derivative of specific volume in the glassy state is
not more than 15% higher than in the crystalline one. Since in the supercooled liquid
is nearly 3 times higher than that in the crystal phase at the same temperature, the
error in estimating the excess expansion (and then the term ()isoth
) using the glassy value
instead of the crystalline one is negligible (less than 5%). For DPVC all the thermodynamic
data used in the present paper are obtained directly from the comparison of supercooled and
crystalline state in the same range of the dielectric measurements. In this case, no fitting
procedure was applied: isothermal and isobaric reduction of excess entropy (the two terms in
Eq.(2)) were obtained by interpolating and integrating the data. Excess entropy of DPVC is
well represented by the function Sexc=S∞(1-TK/T), with S∞=130±1 J K-1 mol-1 and
TK=198.7±0.8 K. Specific volume of DPVC is well represented by Tait equation V=V0(1-
0.094ln(1+P/B(T))), where V0=1.0023+6.16⋅10-4(T(°C))+4.22⋅10-7(T(°C))2 cm3/g and
B=229⋅exp(-3.95⋅10-3 (T(°C)) MPa. The temperature derivative of the volume of the crystal
was estimated to be (∂V/∂T)Pcryst=1.79⋅10-4 cm3g-1 K-1, only slightly changing with P
(∂ ln[(∂V/∂T)Pcryst]/∂P∼0.6 GPa-1). Using these data the isothermal reduction of entropy
can be calculated according to Eq.(2). If Eq.(1) is valid at any
pressure, and CAG does not change with pressure as postulated in section 1, it is possible to
obtain an expression for the “dynamic” configurational entropy, SCDYN, i.e. the value of
configurational entropy fulfilling AG relation:
where τ0 and CAG are fixed to the values obtained at P=0.1 MPa. The test on isothermal data is
indeed a direct check of Eq.(2). This test is shown in Fig.3-(a)-(b) for DPVC and PPGE
respectively. Dynamic data are corresponding to those in Fig.1-(b) and Fig.2-(b).
A proportionality between SCDYN and ∆(Sexc)isoth is found, as expected when the bilinear
relation of Eq.(2) is valid. The slope is equal to the ratio gP(T)/gT(Patm). From these data, it
results that gP(T) changes negligibly with T. Moreover the ratio R=gP(T)/gT(Patm) is always
bigger than 1. R is 1.44±0.04 for DPVC and 1.35±0.15 for PPGE. The two functions gT(P)
and gP(T) are expected to be different because isobaric changes include configurational,
harmonic, and anharmonic terms of the potential energy, whereas isothermal ones, depending
on the expansivity, have no contribution from the harmonic term of the potential energy [37,
32]. If R was 1, the expression in Eq. (2) would reduce to that originally proposed by AG or
to that proposed by Angell and Borick  for gT=gP=const different from unity. In our case
we have that gT(Patm)/gP(T)=0.70±0.02 for DPVC and 0.74±0.08 for PPGE, that seems to
suggest that isothermal reduction of excess entropy is much more effective in term of
reduction of configurational entropy. Similar values of gT(Patm)/gP(T)≈0.7 have been found for
the glass-forming van der Waals molecular liquids OTP and TPCM . It is to be pointed
out that our procedure estimates only the ratio of these two quantities and it is not possible to
claim what is the fraction of the excess entropy that is configurational, differently from what
stated in ref. .
Following our approach, Eq.(2) can be represented in this case as a bilinear equation, where
the parameter gP(T)/gT(Patm) is constant, determined by the plot of isothermal data in Fig.3. In
this way, taking into account the relative reduction of configurational entropy under pressure,
it is possible to determine a “corrected” excess entropy, SexcC, i.e. a quantity that is always
proportional to the true configurational entropy:
Eq.(1) can be expressed by using the new quantities as:
in the case gP(T)/gT(Patm) ≈1 configurational entropy can be replaced by excess entropy.
SexcC was calculated for DPVC and PPGE, by using the respective values for gP(T)/gT(Patm)
obtained from isothermal data. Fig.4 shows the test of Eq.(5) for data in isobaric (for PPGE)
and isothermal (for DPVC) conditions. The data of the two systems collapse in two master-
curves, showing the linear behavior predicted by Eq.(1). We postulated the invariance of τ0
and CAG on pressure, so it is true that such result is partially expected but the master-plot of
Fig.4 is possible only because the bilinear relation between configurational entropy and
excess entropy of the melt with respect to the crystal expressed by Eq.(2) holds. Moreover, for
each system, the data measured in different isobaric conditions fall into one single master
curve: that is resulting from the fact that the ratio gP(T)/gT(Patm) assumes a single value
independent of temperature.
The independence on pressure of CAG and τ0 is consistent to data in Fig.4. In previous
analyses these two parameters were usually assumed as pressure-independent (for example
Refs. [12, 13, 14, 15, 30]), but never checked. It is to be reminded that in AG model the
energy barrier CAG should be not dependent on pressure, as long as the effective barrier for a
CRR is increasing as CAG/SC, while configurational entropy is reducing.
Starting from this speculation recently the assumption of CAG=const. was questioned and the
test of the AG were retained misleading . In the present work we clearly demonstrate that
at least in the low-pressure regime, i.e. for P<300 MPa in the case of DPVC and PPGE, this
dependence is negligible. In fact, for each system the slope of the curve log[1/τ(T,P)] vs
1/(TSexcC) for different isobaric condition is the same, meaning that CAG is the same.
Noteworthy, our finding agrees with previous results obtained by numerical simulations
reporting a negligible variation of CAG with pressure for density changes up to the order of
20% [8, 9] (the larger density variations in our measurements are of the order of 5-8%).
It is important to note that sometimes an incorrect evaluation of Sc(T,P) can led to an incorrect
evaluation of the behaviour of CAG, as well as of τ0. For example, if we consider
Sc(T,P)∝Sexc(T,P) (gT=gP=constant) and log[1/τ(T,P)] is plotted as a function (TSexc)-1, we can
find again a linear behaviour, but we do not find a master curve describing the data in
different isobaric conditions (an example for PPGE is in Fig.5). From a least squares linear fit
of the data of PPGE at different isobaric conditions from 0.1 to 240 MPa, we obtain log(τ0/s)≈
-15.7 and CAG/gT(Patm) changing from 45 to 56 kJ mol-1 (see inset Fig.5). This approach gives
fitting curves completely consistent to the data, but the energy barrier is found to change too
much (more than 25%) for density changes around 5%. Similar density effects should be
relevant already even for cooling at ambient pressure. Summarizing, fitting the relation
log[1/τ(T,P)] versus (TSexc)-1 with a CAG dependent on pressure is possible, but the results are
not in agreement with the physical meaning of the quantities involed in AG model. On the
other hand, assuming a negligible dependence of logτ0 and CAG on pressure, at least for very
low pressure, is quite natural. Moreover, by estimating the ratio between excess and
configurational entropy under isothermal reduction, a straightforward method to calculate
SexcC allows the construction of the master-curve predicted by Eq.(1).
However the present study shows how, from the experimental point of view, the AG equation
could represent very well the data with CAG(P) strongly changing with pressure or,
alternatively, with a bilinear relation between excess and configurational entropy. Choosing
one option rather than the other is just a matter of belief and of general considerations and it
cannnot come unequivocally from the present experimental data. We prefer the bilinear
approach since it gives a master-curve of the data (Fig.4). Moreover, there is a physical reason
to rationalise the result, since a different proportionality between the reduction of excess and
configurational entropy is expected if such reduction is performed isothermally or
isobarically. In fact, the vibrational contribution of the excess entropy should change less
under isothermal compression than under isobaric cooling. On the other hand, this does not
exclude the other possibility.
From Eq.(1) it is possible to calculate the pressure dependence of the glass transition
temperature, Tg, defined such as the isochronal locus where τ(P,Tg(P))=100 sec. According to
the AG model, T-P couples of values corresponding to the condition τ(P,Tg(P))=100 sec are
those satisfying the condition (TSc)=const. Solving the equation d(TSc)Tg=0 with Eq.(2) we
obtain that in the low P limit the pressure dependence of the glass transition temperature can
be described by:
where Vg is the specific volume of the glass, ∆α the excess expansivity, and ∆Cp excess heat
capacity calculated at the glass transition at ambient pressure.
In Fig.6 experimental values of dTg/dP at ambient pressure, dTg/dP|P=atm for DPVC and PPGE,
are compared with those predicted by the theory (Eq. (6)), using the values of gP(T)/gT(Patm)
found from the analysis of relaxation time data. A good agreement is obtained. Moreover, the
value of dTg/dP|P=atm is reported in literature for many systems, and the comparison between
the experimental value and that calculated through Eq.(6) can be another way to test AG
model. Actually such comparison was considered in literature to test the validity of different
theories of the glass transition [38, 39, 40, 41, 42]. Also the AG theory was tested with this
procedure, but considering that Sc(T,P)∝Sexc(T,P). As the ratio gP(T)/gT(Patm) is different than
1, this assumption led to an incorrect evaluation of dTg/dP|P=atm and consequently to
sometimes incorrect conclusion [38, 39]. In Fig.6 the experimental values of dTg/dP|P=atm of
several glass-formers are plotted versus the quantity X, obtained from thermodynamic data
from literature [42, 43, 44]. There is a fairly good agreement, with the two quantities roughly
proportional, and a proportionality constant in the range where usually gP(T)/gT(Patm) is
found. This further result confirm that AG model, despite many weak points, is able to
describe experimental data of glass formers. The introduction of Eqs. (4) for the calculation of
configurational entropy makes possible a fully consistent analysis of the relaxation behavior.
In this paper we proposed a straightforward test of the AG model for the glass
transition. The analysis was performed on two molecular glass-formers, DPVC and PPGE.
These systems are suitable for this type of analysis since we know their thermodynamic and
dynamic data in the same (T,P) interval, so that Sc(T,P) can be calculated in the same interval
of τ(T,P) without the necessity of any extrapolation procedure. First, we compared the
isothermal reduction of excess entropy (Fig.3) to what should be expected from AG model for
τ(T,P) . The quantity
should shift linearly with the isothermal
reduction of excess entropy only if AG model is valid and we found directly such
proportionality for a wide range of pressures and temperatures and for both systems. So AG
prediction, i.e. the inverse of logarithm of relaxation time proportional to 1/(TSexc), was found
valid. On the other hand, our test evidenced that the slope was not 1, as expected, but higher:
the proportionality constant is therefore different if the entropy is reduced isothermally rather
than isobarically. As a second approach, we followed the suggestion of ref. , that,
criticizing AG test to pressure data, postulated a variation of the proportionality constant CAG
with pressure. Surpisingly we found (Fig.6) a linear behaviour between log[1/τ(T,P)] and
(TSexc)-1, but with a slope CAG/gT(Patm) strongly pressure dependent. Such a variation, at least
for moderate pressure, would be in contrast with the assumption of original AG model, where
the activation energy was assumed to increase with the CRR size and the free energy barrier
∆µ was taken as constant. Of course, such assumption may be contradict by the present
results. Summarising, the AG model - in conjunction with the conventional assumption that
excess and configurational entropy are identical or strictly proportional- is disproved by the
present paper. On the other hand, a linear behavior can still be found between log[1/τ] and
(TSexc)-1 along isobaric or isothermal path variations, but with different proportionality
constants. The AG model can be so preserved only if a bilinear relation between excess and
configurational entropy is considered or if a strong pressure dependence of activation barrier
CAG is allowed even for small density variations. This last condition is at odds with the
assumptions of the original AG model  and with the results of recent numerical simulations
[8, 9]. Moreover, by using the bilinear approach and the calculation of Sc by Eqs. (4), from the
AG equation we can calculate values of dTg/dP|P=atm in agreement with the experimental ones.
Summarizing, a careful comparison of the pressure dependence of relaxation time with the
isothermal reduction of excess entropy allows a direct check of the AG model. The procedure
shown here can be extended to any other systems where thermodynamic and dynamic data are
obtained in the same range close to the glass transition at different pressures and
Andreas Best (Max Planck Institute for Polymer Research, Mainz) is acknowledged for the
assistance in getting expansivity and calorimetry data of DPVC. Financial support by MIUR-
FIRB 2003 D.D.2186 grant RBNE03R78E is kindly acknowledged.
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