Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 203
with measured solvent activities. The model and its modifications have been successfully
used to represent thermodynamic excess properties, VLE and LLE for a variety of mixtures,
as summarized by Sanchez and Panayiotou. 275 In addition to mixtures of polymers with normal
solvents, the model can also be applied to polymer-gas systems, Sanchez and
Rodgers. 280
In lattice-hole theories, vacant cells or holes are introduced into the lattice, which describe
the major part of thermal expansion, but changes in cell volume are also allowed
which influence excess thermodynamic properties as well. The hole-theory for polymeric
liquids as developed by Simha and Somcynsky 245 provides a very accurate equation of state
that works much better than the Prigogine-Flory-Patterson equation of state or the
Sanchez-Lacombe lattice-fluid model with respect to the precision how experimental
PVT-data can be reproduced. However, the Dee-Walsh equation of state, Equation [4.4.77],
with its more simple structure, works equally well. The Simha-Somcynsky equation of state
must be solved simultaneously with an expression that minimizes the partition function
with respect to the fraction of occupied sites and the final resulting equations for the chemical
potential are more complicated. Details of the model will not be provided here. Characteristic
parameters for many polymers have recently been given by Rodgers 262 or Cho and
Sanchez. 263 The model is applicable to solutions of small molecules as well as to polymer
solutions. Binary parameters have to be fitted to experimental data as with the models explained
above. Again, one can assume random mixing of contact sites rather than random
mixing of segments as well as non-random mixing, as was discussed, for example, by Nies
and Stroeks 281 or Xie et al. 282,283
Whereas the models given above can be used to correlate solvent activities in polymer
solutions, attempts also have been made in the literature to develop concepts to predict solvent
activities. Based on the success of the UNIFAC concept for low-molecular liquid mixtures,
284 Oishi and Prausnitz 285 developed an analogous concept by combining the
UNIFAC-model with the free-volume model of Flory, Orwoll and Vrij. 236 The mass fraction
based activity coefficient of a solvent in a polymer solution is given by:
( a w )
comb res fv
lnΩ ln / = lnΩ + lnΩ + lnΩ
[4.4.83]
1 1 1 1 1 1
where:
Ω1 mass fraction based activity coefficient of solvent 1 at temperature T
comb
Ω1 combinatorial contribution to the activity coefficient
res
Ω1 residual contribution to the activity coefficient
fv
Ω1 free-volume contribution to the activity coefficient
Instead of the Flory-Huggins combinatorial contribution, Equation [4.4.68], the
Staverman relation 228 is used.
ψ1z θ1ψ1M
ψ M
1
lnΩ 1 = ln + q1ln + l1−
∑
w 2 ψ w w
comb j j
1
1
1
j
j
[4.4.84]
where the segment fractions ψ i and the surface area fractions θ i have to be calculated according
to
( wr M) ( w r M )
ψi = i i / i / k k / k
Σ [4.4.85a]