PDF (119 KB) - Fisica

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