Handbook of Solvents - George Wypych - ChemTech - Ventech!

340 George Wypych
where:
τM the molecular relaxation time
τD the characteristic diffusion time
If the diffusion Deborah number is small (small molecular relaxation time or large diffusion
time) molecular relaxation is much faster than diffusive transport (in fact, it is almost
instantaneous). 2 In this case the diffusion process is similar to simple liquids. For example,
diluted solutions and polymer solutions above glass transition temperature fall in this
category.
If the Deborah number is large (large molecular relaxation time or small diffusion
time), the diffusion process is described by Fickian kinetics and is denoted by an elastic diffusion
process. 1 The polymeric structure in this process is essentially unaffected and coefficients
of mutual and self-diffusion become identical. Elastic diffusion is observed at low
solvent concentrations below the glass transition temperature. 2
The relationships below give the energy required for the diffusion process and compare
the sizes of holes required for the solvent and polymer jumping unit to move within the
system. The free-volume coefficient of self-diffusion is given by the equation: 2
where:
( V�* 1 1+ 2 V�*
2)
E
D1= Do
−
RT
VFH
⎡
⎡ γ ω ω ξ
⎤
exp
⎣
⎢
⎦
⎥
× exp⎢−
⎢ �
⎣
⎤
⎥
⎥
⎦
[7.1.2]
D1 self-diffusion coefficient
Do pre-exponential factor
E energy per molecule required by the molecule to overcome attractive forces
R gas constant
T temperature
γ overlap factor introduced to address the fact that the same free volume is available
for more than one molecule
ω mass fraction (index 1 for solvent, index 2 for polymer)
V ^
* specific free hole volume (indices the same as above)
ξ the ratio of the critical molar volume of the solvent jumping unit to the critical molar
volume of the polymer jumping unit (see equation [7.1.3])
average hole free volume per gram of mixture.
�V FH
ξ= � / = � � / �
* * * *
V V V M V M
1 2 1 1 2 2 [7.1.3]
where:
M molecular weight (1 - solvent, 2 - polymer jumping unit)
The first exponent in equation [7.1.2] is the energy term and the second exponent is the
free-volume term. Figure 7.1.1 shows three regions of temperature dependence of
free-volume: I - above glass transition temperature, II - close to transition temperature, and
III - below the transition temperature. In the region I, the second term of the equation [7.1.2]
is negligible and thus diffusion is energy-driven. In the region II both terms are significant.
In the region III the diffusion is free volume-driven. 3