DHm¼RT^2 lN 1 ’ 2 (8.46)DSm¼R½X 0 þðwTlÞN 1 ’ 2 (8.47)whereX 0 ¼N 1 ln’ 1 þN 2 ln’ 2 þðN 1 þN 2 ÞlnVV^0 þN 2 lnee 0 , and l¼ð@@wTÞf 2 ;P.
Maron’s theory has been proved to be effective across the whole concentration
range of the solutions for natural rubber/benzene and polystyrene/benzene, toluene,
cyclohexane, methyl ethyl ketone, ethyl acetate, chloroform, acetone, ethylbenzene
and chlorobenzene solvents (Maron and Nakajima 1959 ).
8.3.4 Concentration Dependence of Interaction Parameters
The Flory-Huggins interaction parameterw¼(q2)B/(kT), andwseems to be
independent off 2. However, from experiments (Flory 1953 ),wis found to vary
withf 2. For the sake of convenience, we use
weff¼qeffB
kT(8.48)
In 1971, Koningsveld and Kleijens made the first-order correction (Koningsveld
and Kleijens 1971 ), as
wKK¼
ðq 2 þ 2 ’ 2 ÞB
kT(8.49)
But this expression appears not enough precise in comparison to experimental
results.
In 1988, Bawendi and Freed considered the free-volume contribution, and made
the second-order correction (Bawendi and Freed 1988 ), as given by
wBF¼ðq 2 þ 2 ’ 2 ÞB
kTq’ 2 ð 1 ’ 2 ÞðB
kTÞ^2 (8.50)
In principle, the expansion can be two dimensional in addition with respect to
1/q, and both go to higher order corrections.
8.3.5 Lattice-Cluster Theory Considering Molecular Geometry
In polymer blends, the chain-unit volumes of different species are often not
identical, for instance, propylene containing one more methyl than ethylene, and
162 8 Statistical Thermodynamics of Polymer Solutions