a) b - École Polytechnique de Montréal

Some other studies(Everaert, Aerts, & Groeninckx, 1999; Ho, Wu, & Su, 1990) developed
Equation 2-18 and added excess factors for given blend systems to correlate their results with a
semi-empirical model. A model was derived by Metelkin et al.(Metelkin & Blekht, 1984) based
on theory of filament instability described by Tomotika (Tomotika, 1935a) in terms of viscosity
ratio:
Equation 2-20.
where λ is viscosity ratio and f(λ) is defined as :
1
φ = I 2
1+
λf
( λ)
2
Equation 2-21. f ( λ ) = 1.
25 log( λ ) + 1.
81(log(
λ ))
with λ being the viscosity ratio of the blend components at the blending shear rate. Again all of
these equations can predict the phase inversion point with a very good accuracy when both
components have approximately equal viscosity. Results of samples with high viscosity ratio
demonstrate that these models are unable to predict the phase inversion point and significant
errors appear. For such cases, some authors used the percolation threshold definition to develop
an equation based on maximum packing volume fraction. Utracki(Utracki, 1991) proposed a
model based on theory for mono-dispersed hard spheres(Krieger & Dougherty, 1959).
Equation 2-22.
where
1
[ η ] φ
⎛η
⎞ 1
φ + ( 1−
φ ) ⎜ ⎟
m
m
⎝η
2
φ =
⎠
I 2
1
[ η ] φm
⎛η
⎞ 1 ⎜ ⎟ + 1
⎝η
2 ⎠
φ is the phase inversion point, φ
I 2
m is the maximum packing volume fraction, [η] is
intrinsic viscosity, and indices refer to components. For polymer blends, φ is defined as
m
φ −
m = 1 φc
, where c
φ stands for the percolation threshold and amounts to 0.156 for dispersions of
spheres. The definition of the percolation threshold is explained in detail in the next sections.
This equation explains that the addition of the first component to second one leads to an increase
m
17