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a) b - École Polytechnique de Montréal

a) b - École Polytechnique de Montréal

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Some other studies(Everaert, Aerts, & Groeninckx, 1999; Ho, Wu, & Su, 1990) <strong>de</strong>veloped<br />

Equation 2-18 and ad<strong>de</strong>d excess factors for given blend systems to correlate their results with a<br />

semi-empirical mo<strong>de</strong>l. A mo<strong>de</strong>l was <strong>de</strong>rived by Metelkin et al.(Metelkin & Blekht, 1984) based<br />

on theory of filament instability <strong>de</strong>scribed by Tomotika (Tomotika, 1935a) in terms of viscosity<br />

ratio:<br />

Equation 2-20.<br />

where λ is viscosity ratio and f(λ) is <strong>de</strong>fined as :<br />

1<br />

φ = I 2<br />

1+<br />

λf<br />

( λ)<br />

2<br />

Equation 2-21. f ( λ ) = 1.<br />

25 log( λ ) + 1.<br />

81(log(<br />

λ ))<br />

with λ being the viscosity ratio of the blend components at the blending shear rate. Again all of<br />

these equations can predict the phase inversion point with a very good accuracy when both<br />

components have approximately equal viscosity. Results of samples with high viscosity ratio<br />

<strong>de</strong>monstrate that these mo<strong>de</strong>ls are unable to predict the phase inversion point and significant<br />

errors appear. For such cases, some authors used the percolation threshold <strong>de</strong>finition to <strong>de</strong>velop<br />

an equation based on maximum packing volume fraction. Utracki(Utracki, 1991) proposed a<br />

mo<strong>de</strong>l based on theory for mono-dispersed hard spheres(Krieger & Dougherty, 1959).<br />

Equation 2-22.<br />

where<br />

1<br />

[ η ] φ<br />

⎛η<br />

⎞ 1<br />

φ + ( 1−<br />

φ ) ⎜ ⎟<br />

m<br />

m<br />

⎝η<br />

2<br />

φ =<br />

⎠<br />

I 2<br />

1<br />

[ η ] φm<br />

⎛η<br />

⎞ 1 ⎜ ⎟ + 1<br />

⎝η<br />

2 ⎠<br />

φ is the phase inversion point, φ<br />

I 2<br />

m is the maximum packing volume fraction, [η] is<br />

intrinsic viscosity, and indices refer to components. For polymer blends, φ is <strong>de</strong>fined as<br />

m<br />

φ −<br />

m = 1 φc<br />

, where c<br />

φ stands for the percolation threshold and amounts to 0.156 for dispersions of<br />

spheres. The <strong>de</strong>finition of the percolation threshold is explained in <strong>de</strong>tail in the next sections.<br />

This equation explains that the addition of the first component to second one leads to an increase<br />

m<br />

17

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