Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 139

in Eq. 3.90 needs to be replaced with the value that increases with increasing ʱ .
A similar approach is used to adequately describe the temperature dependence of
viscosity in a system undergoing reactive polymerization [ 174 ]. A rather simple
modification of Eq. 3.90:

2 T 1

=−

(3.91)

E εα A −

( )

3

( )

2

TTTT − −

gel gel

has proven [ 169 ] to be sufficient to account for the effect. Equation 3.91 includes
three parameters. The parameter A represents a contribution of the nucleation
process to the overall temperature dependence. This contribution manifests itself
through a decreasing dependence of E on T . The increasing portion of the depen-
dence corresponds to the diffusion contribution that is represented by the param-
eters ʵ and n. By its meaning, ʵ is the activation energy of diffusion at complete
conversion, i.e., at ʱ = 1. The parameter n characterizes the strength of diffusion
contribution: the smaller the n, the stronger the contribution, i.e., becomes operative
earlier and contributes more.
Fitting Eq. 3.91 to the experimental dependencies (Fig. 3.66 ) requires knowledge
of the T gel values. They have been reported by Takahashi et al. [ 175 ] and are about
40, 38, and 36 ﾰC for 2, 4, and 8 % solutions, respectively. The resulting fits are sta-
tistically significant. The values of the adjustable parameters are given in Table 3.3 .
The diffusion parameters reflect an increasing contribution of diffusion to the overall
temperature dependence with increasing concentration. When comparing the 2 and
8 % solutions, an increase in the concentration results in larger ʵ (i.e., larger activation
energy of diffusion at ʱ = 1) as well as in smaller n (i.e., larger strength of the con-
tribution). The behavior of the 4 % solution is a bit more complex. Compared to the
2 % solution, it shows 10 % smaller ʵ that would be consistent with a slightly smaller
diffusional contribution. However, it also demonstrates 2.5 smaller n that suggests the
overall diffusional contribution to be stronger and larger compared to the 2 % solution.
On the other hand, the nucleation parameter A can be used to estimate of the free
energy barrier to nucleation at any temperature as follows (see Eq. 3.44):

A
TT

*

∆ G = . (3.92)

2

( − )

gel

For comparison purposes, the Δ G * values can be estimated at T = 60 ﾰC, which
is the threshold temperature above which gelation becomes detectable by DSC

Table 3.3 The results of fitting Eq . 3.91 to experimental E ʱ versus T dependencies
C (%) A (K 2 kJ mol −1 ) ʵ (kJ mol −1 ) n Δ G * (60 ﾰC; kJ mol −1 )

2 3315.4 95.0 5 8.3

4 4879.8 85.2 2 10.1

8 6915.0 151.9 3 12.0