Chemistry Reference
In-Depth Information
Sestak-Berggren model (Eq. 2.32) that provides enough flexibility to fit most of the
types of the
f
(
ʱ
) dependencies.
With regard to physically meaningful models, we should mention a work of San-
chez-Jimenez et al. [
90
], who have found a way of converting the random scission
model of polymer degradation [
91
] to the convenient
f
(
ʱ
) form. The original work
by Simha and Wall [
91
] describes the kinetics of bond breaking as:
d
d
x
t
(4.57)
=
kT x
()(
1
−
),
where
x
is the fraction of broken bonds. Since not every broken bond produces a
volatile fragment,
x
is not equal to the extent of conversion of polymer to volatile
fragments,
ʱ
. These two values are related to each other as [
91
]:
x
NLL
N
(
−
)(
−
1
)
L
−
1
(4.58)
1
−=−
α (
1
x
)
1
+
,
where
N
is the initial degree of polymerization and
L
is the number of monomer
units in the shortest chain fragment that does not evaporate before being degraded.
By assuming that normally
L
<<
N
, Sanchez-Jimenez et al. [
90
] simplify Eq. 4.58
to 4.59:
−
11 1
L
1
(4.59)
α=− −
(
x
)
[
+
x L
(
−
1
)],
and take its derivative with respect to time:
d
α
d
x
−
=−−
L
2
(4.60)
LL
(
1)(1
x
)
x
.
d
t
d
t
Replacing d
x
/d
t
with the right-hand side of Eq. 4.57 yields:
d
d
α
t
L
−
1
(4.61)
=
kTLL xx
()(
−
11
)(
−
)
.
Comparing Eq. 4.61 with the regular rate equation (Eq. 1.1) suggests that every-
thing but
k
(
T
) in the right-hand side is
f
(
ʱ
). Thus,
f
(
ʱ
) for random scission must
have the following form:
(4.62)
L
−
1
f
() (
α =−−
L L
11
)(
x
)
x
.
Equation 4.62 does not provide an analytical form of
f
(
ʱ
) dependence on
ʱ
as the
models considered earlier (Table 1.1) do. In general, this dependence can be found
in numerical form by substituting the same values of
x
and
L
in Eqs. 4.59 and 4.62.
An analytical expression for
f
(
ʱ
) was determined by fitting numerical dependencies
of
f
(
ʱ
) on
ʱ
to an approximating equation of the following form:
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