Chemistry Reference
In-Depth Information
ferent values of
ʱ
T
at different temperatures. It can be converted to the relative value
by dividing the absolute
ʱ
over
ʱ
T
. The relative
ʱ
would then run from 0 to 1.
It has been demonstrated [
56
] that the correct approach is to the use the rela-
tive values of conversion. It is explained easily by assuming that cure obeys, for
example, a reaction-order model,
f
(
ʱ
) = (1
− ʱ
)
n
. To adjust this model to conditions
of incomplete cure, the model needs to be modified as:
n
f
() (
ααα
=−
) .
(4.37)
T
This adjustment is necessary to secure the fact that when cure reaches its ultimate
conversion
ʱ
T
at a given temperature,
f
(
ʱ
) and, thus, the reaction rate turns to zero.
As discussed in Chap. 1 (Eqs. 1.12 and 1.13), the derivative of the reaction rate at a
constant conversion is equal to the activation energy:
∂
ln(d
α
/ d )
t
∂
ln
kT
(
)
∂
ln
f
(
α
)
ER
=−
=−
R
−
R
,
(4.38)
α
−
1
−
1
−
1
∂
T
∂
T
∂
T
α
α
α
because the second term in the right-hand side of Eq. 4.38 is zero. However, if one is
to use the absolute conversion for estimating the activation energy, then the reaction
model in Eq. 4.38 would have to take the form that contains
ʱ
T
,
such as in Eq. 4.37.
Then, the second term in Eq. 4.39
∂
ln
kT
(
)
∂
ln(
αα
−
)
T
ER
=−
−
R
(4.39)
α
−
1
−
1
∂
T
∂
T
α
α
is no longer zero because
ʱ
T
depends on temperature.
That is, the use of the absolute values of
ʱ
would result in estimating the
E
ʱ
val-
ues distorted by the temperature dependence of
ʱ
T
in the second term of Eq. 4.39.
Needless to say, exactly the same issue would arise when using any other
f
(
ʱ
) model
as long as it is adjusted to the condition of incomplete cure. On the other hand,
transforming the absolute conversions for incomplete cure to the relative ones
would change
ʱ
T
in Eq. 4.37 to 1 that would make the second term in Eq. 4.39
zero and, thus, would produce undistorted values of
E
ʱ
. The systematic error in
E
ʱ
caused by using the absolute conversions increases as
ʱ
approaches
ʱ
T
and can be
quite significant as illustrated in Fig.
4.18
for simulated data [
56
]. Significant dif-
ferences in
E
ʱ
estimated by using absolute and relative conversions are also found
for experimental data [
56
-
58
].
In the beginning of Sect. 4.2.3, we discuss briefly that in a kinetic regime the
initial portions of the
E
ʱ
dependence are likely to be descending that follows from
the model Eqs. 4.22 and 4.24 as long as
E
1
for the uncatalyzed reaction is larger than
E
2
for the catalyzed reaction. However, this is not a general rule. The
E
ʱ
values at
the smallest conversions (i.e., at
ʱ
ₒ 0) are determined by the activation energy of
initiation that can be larger or smaller depending on the mechanism. For example,
the activation energy of initiation of epoxy ring opening can depend strongly on
how this process is catalyzed. An instructive example is provided in Fig.
4.19
that
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