Handbook of Solvents - George Wypych - ChemTech - Ventech!

198 Christian Wohlfarth
where:
V1 molar volume of the pure liquid solvent 1 at temperature T
0
Δ vapU1 molar energy of vaporization of the pure liquid solvent 1 at temperature T
δ1 solubility parameter of the pure liquid solvent
δ2 solubility parameter of the polymer
Solubility parameters of polymers cannot be calculated from energy of vaporization
since polymers do not evaporate. Usually they have been measured according to Equation
[4.4.70], but details need not be explained here. Equation [4.4.70] is not useful for an accurate
quantitative description of polymer solutions but it provides a good guide for a qualitative
consideration of polymer solubility. For good solubility, the difference between both
solubility parameters should be small (the complete residual chemical potential term cannot
be negative, which is one of the disadvantages of the solubility parameter approach). Several
approximate generalizations have been suggested by different authors - a summary of
all these models and many data can be found in the books by Barton. 11,12 Calculations applying
additive group contributions to obtain solubility parameters, especially of polymers, are
also explained in the book by Van Krevelen. 235
Better-founded lattice models have been developed in the literature. The ideas of
Koningsveld and Kleintjens, e.g., Ref., 51 lead to useful and easy to handle expressions, as is
given above with Equations [4.4.15, 4.4.17 and 4.4.46] that have been widely used, but
mainly for liquid-liquid demixing and not so much for vapor-liquid equilibrium and solvent
activity data. Comprehensive examples can be found in the books by Fujita 41 or Kamide. 42
The simple Flory-Huggins approach and the solubility parameter concept are inadequate
when tested against experimental data for polymer solutions. Even for mixtures of
n-alkanes, the excess thermodynamic properties cannot be described satisfactorily - Flory et
al. 236-239 In particular, changes of volume upon mixing are excluded and observed excess
entropies of mixing often deviate from simple combinatorial contributions. To account for
these effects, the PVT-behavior has to be included in theoretical considerations by an equation
of state. Pure fluids have different free volumes, i.e., different degrees of thermal expansion
depending on temperature and pressure. When liquids with different free volumes
are mixed, that difference contributes to the excess functions. Differences in free volumes
must be taken into account, especially for mixtures of liquids whose molecules differ
greatly in size, i.e., the free volume dissimilarity is significant for polymer solutions and has
important effects on their properties, such as solvent activities, excess volume, excess
enthalpy and excess entropy. Additionally, the free volume effect is the main reason for liquid-liquid
demixing with LCST behavior at high temperatures. 240,241
Today, there are two principal ways to develop an equation of state for polymer solutions:
first, to start with an expression for the canonical partition function utilizing concepts
similar to those used by van der Waals (e.g., Prigogine, 242 Flory et al., 236-239 Patterson, 243,244
Simha and Somcynsky, 245 Sanchez and Lacombe, 246-248 Dee and Walsh, 249 Donohue and
Prausnitz, 250 Chien et al. 251 ), and second, which is more sophisticated, to use statistical thermodynamics
perturbation theory for freely-jointed tangent-sphere chain-like fluids (e.g.,
Hall and coworkers, 252-255 Chapman et al., 256-258 Song et al. 259,260 ). A comprehensive review
about equations of state for molten polymers and polymer solutions was given by Lambert
et al. 261 Here, only some resulting equations will be summarized under the aspect of calculating
solvent activities in polymer solutions.
The theories that are usually applied within activity coefficient models are given now,
the other theories are summarized in Subchapter 4.4.4.2.