Chemical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 183

introduces the diffusion term as a multiplicand ˆ ( η ), where η is viscosity. The latter
obviously involves dependencies on both temperature and molecular weight [ 37 ].
A more practically convenient approach is to introduce a multiplicand term that
depends on temperature and conversion [ 45 ]:

d α

(4.29)

= kT f ( ) ( αϕα ) ( , T ).

d t

Variants of this approach are discussed by van Assche et al. [ 45 ] and by Schawe
[ 40 ].
An alternative approach is to make use of the addition of kinetic resistances [ 46 ,
47 ] (Eq. 1.20) and introduce a diffusion term, k D ( T , ʱ ), as an addend:

1 1 1

(4.30)

=

() (,) .

+

kT kT α

k

D

ef

The term can be introduced in the form of a diffusion rate constant with separable
contributions of temperature and conversion [ 35 ]:

E
RT

D

kT D (,) α = exp + B α ,

(4.31)

D 0

where D 0 is the preexponential factor, E D the activation energy of diffusion, and B is
a constant that accounts for a change in the conditions of diffusion due to the prog-
ress of cross-linking. A similar type of equation is used to describe the temperature
dependence of viscosity in a system undergoing reactive polymerization [ 48 ].
Essentially, the Bʱ term in Eq. 4.31 introduces the effective variable activation
energy of diffusion:

(4.32)

E

=+α .

E BRT

D,ef

There are a number of factors that may cause the activation energy of diffusion to
vary during the process of curing. For example, it is known that the activation en-
ergy of diffusion of gases in rubber increases with the degree of cross-linking [ 49 ].
On the other hand, it is also known that for the same small molecule diffusant the
activation energy is smaller when diffusion occurs in the glassy state than when it
takes place in the liquid (rubbery) state of a polymer [ 17 , 49 , 50 ].
From Eq. 4.30, the effective rate constant for the cross-linking kinetics accompa-
nied by diffusion is as follows:

(4.33)

kTkT
kT kT

() (,)
() (,) .

α
α

D

k =

ef

+

D

Equation 4.33 can further be used to derive the isoconversional activation energy
in the usual manner: