Handbook of Solvents - George Wypych - ChemTech - Ventech!

310 E. Ya. Denisyuk, V. V. Tereshatov
Dimensionless diffusion coefficient is defined by the formula k(u,l) = D(u,l)/D 0.
The longitudinal stresses in the layer (15) are expressed in terms of dimensionless
stresses q(x,t) by
( )
−1
σ = σ = RTV q x, t
[6.1.28]
2 3 2
where according to Eqs. [6.1.15] and [6.1.27]
{ }
−13
/
2 6
( , ) = ε () −[ ( − ε) ( , ) + ε]
/ ()
qxt lt 1 1 uxt l t
[6.1.29]
Consider two functions
2
() = ( ) () = ( )
g t u x, t , g t u x, t
[6.1.30]
1 2
which are integral characteristics of swelling kinetics for a plane layer and can be determined
from experiments. The first function characterizes a relative amount of liquid absorbed
by a polymer in time t and the second function according to Eq. [6.1.27] is related to
longitudinal layer deformation. For high-swelling elastomers
6 () ()
g t ≈ l t
[6.1.31]
2
The numerical results obtained by solving model problem of Eqs. [6.1.23] - [6.1.25]
for a constant diffusion coefficient are plotted in Figure 6.1.1. 6 The obtained curves show
the evolution of penetrating liquid concentration and longitudinal stresses. It is seen that the
boundary liquid concentration during swelling monotonically increases.
6.1.3 DIFFUSION KINETICS OF PLANE LAYER SWELLING
Consider two stages of swelling process in a plane layer - the initial and final. In the initial
stage, the influence of the opposite layer boundary on the swelling process is inessential and
therefore diffusion in a layer of finite thickness at sufficiently small values of time can be
considered as the diffusion in half-space.
Figure 6.1.1. Distribution of penetrating liquid (a) and longitudinal stresses (b) during swelling of a plane layer
with constant diffusion coefficient k(u,l)=1atε=0.1andd=2/9:1-t=0.05; 2-t=0.2; 3-t=0.4;4-t=0.6; 5 - t
=1;6-t=1.8. [Adapted, by permission, from E. Ya. Denisyuk, V. V. Tereshatov, Vysokomol. soed., A42, 74
(2000)].