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Discussion
Comment on ‘‘Decrease in the configurational entropy during a
meltÕs polymerization’’ [Chem. Phys. 305 (2004) 231]
S. Corezzi a,b, *, D. Fioretto b,c , P.A. Rolla d
a Dipartimento di Fisica, Università di Roma ‘‘La Sapienza’’, P.le A. Moro 2, I-00185 Roma, Italy
b INFM-CRS Soft, c/o Università di Roma ‘‘La Sapienza’’, P.le A. Moro 2, I-00185 Roma, Italy
c Dipartimento di Fisica and INFM, Università di Perugia, Via A. Pascoli, I-06123 Perugia, Italy
d INFM and Dipartimento di Fisica, Università di Pisa, Via F. Buonarroti 2, I-56127 Pisa, Italy
In a recent paper, Johari [1] examines the manner in
which, in a paper published by us [2], the reduction in configurational
entropy during a meltÕs polymerization was
connected to the number of chemical bonds, and a relationship
that links the relaxation time to the advancement of
reaction was derived. The author claims to have demonstrated
that the connection we made is untenable on experimental
grounds. He also suggests that the apparent success
in fitting the relaxation time data may just be an artifact of
an improper use of the Adam–Gibbs entropy equation [3].
In this Comment, we discuss the main points of the JohariÕs
argument, concerning the validity of our premise and its
use in combination with the Adam–Gibbs equation.
(1) To investigate the existence of a correlation between
the dynamical slowdown during a meltÕs polymerization
and the loss of configurations caused by the formation of
chemical bonds, we have considered a convenient class of
reactions and chosen convenient experimental conditions,
in which such a correlation could be more clearly exposed.
It is important for the following discussion to recall them
here. We have considered step polymerizations where ring
formation is negligible, and focused on slow reactions (i.e.,
low T) to stay in thermodynamic quasi-equilibrium condition.
A pronounced slowdown of the dynamics usually corresponds
to a number-average degree of polymerization xn
between two and few units (

with s 0 approximately independent of both T and a, and
B =[C/TSc(0)] only dependent on temperature, on the
assumption that the variation of C with a is negligible –
i.e., the relaxation time data vs a should lie on a Vogel–Fulcher-like
curve that extrapolates to infinity at a = a0. Contrary
to what is done by Johari [1], the consequences of Eq.
(1) should be analyzed within the physical conditions that
allow this relation to hold valid. In particular, a tendency
of S c to zero (i.e., of s to diverge) at a finite a 0, as extrapolated
from data at a < a0, does not imply a true vanishing
of Sc (i.e., a divergence of s) ata0. The reaction, in fact, is
not prevented from departing from Eq. (1) and from proceeding
to an extent greater than a0, if the conditions for
Eq. (1) are progressively removed. As a matter of fact
any polymerization can be brought to its full extent by progressively
increasing the reaction temperature, and this is
normally used to estimate the total heat of reaction via calorimetry.
It is also possible when the reaction temperature
is high that the system remains fluid at even full reaction, as
recognized in [1].
While data at a close to or higher than a0 are of no use
for testing a correlation between Sc and s through a test of
Eq. (2), a large amount of relaxation time data in the expected
range of validity of Eq. (1) have been analyzed,
from which an excellent agreement with Eq. (2) has turned
out with the parameter a0 predicted on a molecular basis
[2,6,7]. The observation that this relationship applies to a
wide range of systems investigated, in different conditions,
using different experimental methods, by numerous authors
is hard to be seen as coincidental. Instead, it strongly suggests
that the essence of the relaxational slowdown on polymerization
can be traced back to a bond-controlled
reduction in the number of configurations available to
the system.
Further experiments by our group – in which the characteristics
of the systems investigated and the experimental
conditions have been chosen to meet the requirements for
Eq. (2), mentioned above – have strengthened the basis
for Eq. (1) in combination with the Adam–Gibbs relation,
particularly persuasive being the results for a number of
epoxy–amine mixtures composed of two reagents (DGE-
BA–DETA) in different molar ratios [8]. Because a0 may
be controlled by varying the molar ratio of reagents in
the initial mixture (solid line in Fig. 1), according to Eq.
(2) the extent of reaction at which the relaxation time
behavior appears to diverge would also be varied in the
same way. We have found, for each reaction carried out
isothermally, that Eq. (2) represents the relaxation time
data well, in a wide relaxation range (at least five decades
for s >10 4 s), where xn(a) is few units [8]. The derived values
for the fitting parameter a0 are reported in Fig. 1. Indeed,
Fig. 1 reveals that a0 not only changes with the
mixture composition but (within the experimental error)
actually follows the expected variation with the molar ratio
(N a/N e) of the reacting molecules, in a wide N a/N e range.
Moreover, the same a 0 is obtained by reacting a given mixture
at different temperatures [9], despite the time-depen-
S. Corezzi et al. / Chemical Physics 323 (2006) 622–624 623
α 0
1.0
0.9
0.8
0.7
0.6
10:3
5:2
5:2.8
10:9
0.2 0.3 0.4 0.5 0.6
Na /Ne 0.7 0.8 0.9 1.0
Fig. 1. Variation of the parameter a 0 with the molar ratio N a/N e between
the amino and the epoxy monomer constituents of the system DGEBA–
DETA Ne:Na. The experimental points for different compositions are
labelled by their Ne:Na values. Open symbols are used to distinguish a
different reaction temperature. The solid line is a 0 =[1+(N a/N e)]/2,
calculated from the functionality and the molar ratio of the reacting
molecules.
dent evolution of the reaction is greatly affected. This is
shown in Fig. 1 by closed and open symbols. In the same
figure, isoabscissa closed-points refer to reactions repeated
at the same temperature to check reproducibility.
(2) A number of other observations reported by Johari
deserve our comment. It is asserted in [1] that a connection
between the relaxation time and a by using the Adam–
Gibbs equation can be made only by allowing Dl, and
therefore, the quantity B / Dl to change with a, being
Dl a free-energy barrier per cooperative unit. We notice
that this is the opposite of what the same author maintained
in previous studies of polymerization [10–12]. Apart
from this contradiction, we observe that as long as multimers
behave effectively like cooperative units a variation
of Dl with a seems not to be required by the theory [3].
On the other hand, if one would be forcing Eq. (2) with
wrong but constant values of B, to fit the experimental data
for polymerizations with Dl strongly varying with a, the
values a0 which describe the data could scarcely correspond
to the values of a physical quantity, i.e., the molecular
parameter a0 calculated from the functionality of the reagents.
This suggests that a sensible variation of Dl with
a is not required either by the experimental data. A minor
variation can, of course, account for details.
Moreover, it is asserted in [1] that s 0 in Eq. (2) should
necessarily correspond to the time scale of phonons, i.e.,
10 13 –10 14 s. There is no question that in simple glassforming
liquids analyses using the Adam–Gibbs approach
usually do give pre-exponents not too far from the phonon
time scale, though the values obtained notoriously span
several decades (e.g., from 10 10.5 to 10 17.3 sin[14]). We
remark that for it to be a general expectation the pre-exponent
s 0 should represent the true value of s in the limit
T !1, and different spectroscopic techniques (such as
dielectric and photon-correlation spectroscopy) should
measure the same s. Neither of these is the case. It is
4:3

624 S. Corezzi et al. / Chemical Physics 323 (2006) 622–624
well-known that relaxation times measured by different
techniques all sensitive to structural dynamics may even
differ by a few decades, which obviously reflects in different
values of s0. It is also well-known that at some intermediate
temperature on heating above Tg, the relaxation mechanism
becomes different from cooperative transitions governed
by the configurational entropy, and a marked
change in the dynamics is observed [13]. Because of this
change the relaxation time of a liquid at high temperature,
which actually corresponds to the phonon time scale, may
differ from s0 in Eq. (2), which is the value extrapolated
from the cooperative regime. The difference between the
two could be less pronounced if the process under analysis
is a simple physical process. However, if it is a chemical
process like polymerization then chemical step activation
entropies could become involved and then all bets should
be off about the value of the pre-exponent s 0.
(3) In [1], the author reports as a feature of Eq. (1) that
‘‘... CP, c of a polymerizing melt at a fixed T would not
change as a increases from zero to one’’, and similarly,
‘‘... the product bcVc, [where bc and Vc are the configurational
contribution to the thermal expansion coefficient
and the volume of the melt] would not change’’. On the contrary,
we note that Eq. (1) yields a factor in C P, c and b c V c
explicitly dependent on a [e.g., C P, c(a) =T(oS c(0)/oT) P
(1 a/a0) =CP, c(0)(1 a/a0)], which factor has been erroneously
omitted in [1]. Therefore, our Sc(a) relation does
not conflict with the observation that changes in CP, b,
and V during a meltÕs polymerization are also to be ascribed
to changes in the configurational contribution to
these quantities.
(4) In our opinion, the calculations of S c performed by
Wang and Johari [15] in a lattice–hole model are interesting
but cannot prevail over the indications coming from experiments,
as we have described. In addition, we observe that
Sc, calculated with plausible values of the adjustable
parameters, in the Wang and JohariÕs model would be
expected to initially increase and then decrease with a in
such a way that the value at the full extent of polymerization
is nearly the same as (or even higher than) in the
monomeric liquid (see Fig. 2 in [15]). This result can hardly
be reconciled with an entropy-based picture of the process
of polymerization; which picture, instead, Johari upholds.
On the other hand, different lattice models exist and provide
different results for the statistics of polymer chains.
Interestingly, in a Flory–Huggins model of equilibrium
polymerization [16] the average length of polymer chains
scales inversely with the excess entropy of the polymer
solution, similarly to our premise Sc / 1/xn.
In conclusion, we believe there are sufficient experimental
indications that Eq. (2) captures the principal physics
behind the dynamical slowdown that occurs in a wide class
of polymerizations, and therefore, goes beyond a mere phenomenological
description of the process. We stress once
more that it is appropriate in a limited range of a and T
but this range usually covers the most part of relaxation
data from experiments.
Acknowledgement
S.C. acknowledges financial support from MIUR-
FIRB.
References
[1] G.P. Johari, Chem. Phys. 305 (2004) 231.
[2] S. Corezzi, D. Fioretto, P. Rolla, Nature 420 (2002) 653.
[3] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139.
[4] xn represents the average number of monomers per molecule in the
system. a, also called the chemical conversion, provides a measure of
the extent of reaction.
[5] High temperature also tends: (i) to favor cyclization, which changes
the variation of xn with a (e.g., xn would grow less rapidly and tend to
diverge at a > a 0), and (ii) to favor departure from the quasiequilibrium
condition.
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[9] For all the polymerizations studied, the temperature was near the
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rate of 10 °C/min.
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