Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 207
To obtain the complete activity coefficient, only the residual term from the UNIFAC
model, Equation [4.4.87], has to be added. An attempt to incorporate differences in shape
between solvent molecules and polymer segments was made by Kontogeorgis et al. 302 by
adding the Staverman correction term to obtain:
fv
fv
fv ϕiϕizq ⎛ ψ ψ ⎞
i i
i
lnγ i = ln + 1−
− ⎜ln
xix ⎜
+ 1 − ⎟
i ⎝ θi
θ ⎟
[4.4.93c]
2
i ⎠
where the segment fractions ψi and the surface area fractions θi have to be calculated according
to Equations [4.4.85a+b]. Using this correction, they get somewhat better results only
when Equation [4.4.93] leads to predictions lower than the experimental data.
Different approaches utilizing group contribution methods to predict solvent activities
in polymer solutions have been developed after the success of the UNIFAC-fv model.
Misovich et al. 304 have applied the Analytical Solution of Groups (ASOG) model to polymer
solutions. Recent improvements of polymer-ASOG have been reported by Tochigi et
al. 305-307 Various other group-contribution methods including an equation-of-state were developed
by Holten-Anderson et al., 308,309 Chen et al., 310 High and Danner, 311-313 Tochigi et
al., 314 Lee and Danner, 315 Bertucco and Mio, 316 or Wang et al., 317 respectively. Some of them
were presented again in Danner’s Handbook. 2 Detail are not provided here.
4.4.4.2 Fugacity coefficients from equations of state
Total equation-of-state approaches usually apply equations for the fugacity coefficients instead
of relations for chemical potentials to calculate thermodynamic equilibria and start
from Equations [4.4.2 to 6]. Since the final relations for the fugacity coefficients are usually
much more lengthy and depend, additionally, on the chosen mixing rules, only the equations
of state are listed below. Fugacity coefficients have to be derived by solving Equation
[4.4.6]. After obtaining the equilibrium fugacities of the liquid mixture at equilibrium temperature
and pressure, the solvent activity can be calculated from Equation [4.4.1]. The
standard state fugacity of the solvent can also be calculated from the same equation of state
by solving the same equations but for the pure liquid. Details of this procedure can be found
in textbooks, e.g., Refs. 318,319
Equations of state for polymer systems that will be applied within such an approach
have to be valid for the liquid as well as for the gaseous state like lattice-fluid models based
on Sanchez-Lacombe theory, but not the free-volume equations based on
Prigogine-Flory-Patterson theory, as stated above. However, most equations of state applied
within such an approach have not been developed specially for polymer systems, but,
first, for common non-electrolyte mixtures and gases. Today, one can distinguish between
cubic and non-cubic equations of state for phase equilibrium calculations in polymer systems.
Starting from the free-volume idea in polymeric systems, non-cubic equations of state
should be applied to polymers. Thus, the following text presents first some examples of this
class of equations of state. Cubic equations of state came later into consideration for polymer
systems, mainly due to increasing demands from engineers and engineering software
where three-volume-roots equations of state are easier to solve and more stable in computational
cycles.
About ten years after Flory’s development of an equation of state for polymer systems,
one began to apply methods of thermodynamic perturbation theory to calculate the thermo-