Chemistry Reference
In-Depth Information
Fig. 4.15
Fitting the experi-
mental
E
ʱ
dependence for
epoxy-amine curing data
(
circles
) by Eq. 4.34.
Solid
lines
are plotted for
E
= 63
and
E
D
= 20 kJ mol
− 1
and dif-
ferent values of
B
: −2 (1), −4
(2), and −6 (3). (Reproduced
from Vyazovkin and Sbirraz-
zuoli [
35
] with permission
of ACS)
α
ER
k
T
∂
∂
ln
kTEkTE
kT kT
() (,)
() (,)
+
+
α
α
ef
DD
D
=−
=
,
(4.34)
α
−
1
α
where
E
D
and
E,
respectively, are the activation energies of diffusion and chemi-
cal reaction, e.g., catalyzed epoxy-amine reaction. Depending on the sign of
B
in
Eq. 4.31, the
E
ʱ
dependence can either decrease (
B
< 0) or increase (
B
> 0). As dis-
cussed earlier (Sect. 4.2.1), one can expect the
E
ʱ
dependence to decrease when
the rate becomes limited by the diffusion of small molecules such as a monomer
or a short segment of a polymer chain. The increasing dependence should be ex-
pected when the process becomes determined by the diffusion of large molecules
such as polymer chains or their long segments. Figure
4.15
provides an example of
an epoxy-amine reaction that demonstrates a transition from a kinetic to diffusion
regime accompanied by a decrease in
E
ʱ
. It is seen that the aforementioned model
(Eqs. 4.30, 4.31, and 4.34) is capable of adequately reproducing the actual variation
in the effective activation energy.
It should be mentioned that vitrification during cross-linking is not the sole rea-
son for diffusion control. The latter generally becomes operating when the charac-
teristic time of relaxation,
˄,
of the reaction medium exceeds markedly the charac-
teristic time of the reaction itself. The diffusion rate constant in a viscous medium
can be expressed as [
41
]:
E
RT
1
τ
η
k
D
==
C
exp n
T
−
,
(4.35)
where
C
is the preexponential factor and
E
η
is the activation energy of viscous flow.
Unlike Eq. 4.31 that describes a complex variation of
k
D
with
T
and
ʱ,
Eq. 4.35
describes a simple increase of the diffusion rate constant with temperature. This
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