Handbook of Solvents - George Wypych - ChemTech - Ventech!

200 Christian Wohlfarth
* * *
P = P ψ + P ψ −ψ
θ Χ [4.4.75a]
1 1 2 2 1 2 12
[ ( 1 1 1) ( 2 2 2)
]
* * * * * *
T = P / P ψ / T + P ψ / T
[4.4.75b]
where the segment fractions ψi and the surface fractions θi have to be calculated according
to:
ψ i
* *
= nV i i / ∑nkVk
*
*
= mV i spez, i / ∑mkVspez,
k = xiri / ∑xkrk[4.4.76a]
θi = ψisi / ∑ψksk
[4.4.76b]
where:
mi mass of component i
xi mole fraction of component i
* *
ri number of segments of component i, here with ri /rk = Vi / Vk
and r1 =1
si number of contact sites per segment (proportional to the surface area per segment)
Now it becomes clear from Equation [4.4.74] that the classical Flory-Huggins χ-func-
2 residual
tion (χψ 2 = ln a 1 ) varies with composition, as found experimentally. However, to calculate
solvent activities by applying this model, a number of parameters have to be
considered. The characteristic parameters of the pure substances have to be obtained by fitting
to experimental PVT-data as explained above. The number of contact sites per segment
can be calculated from Bondi’s surface-to-volume parameter tables 264 but can also be used
as fitting parameter. The X12-interaction parameter has to be fitted to experimental data of
the mixture. Fitting to solvent activities, e.g. Refs., 265,266 does not always give satisfactorily
results. Fitting to data for the enthalpies of mixing gives comparable results. 266 Fitting to excess
volumes alone does not give acceptable results. 142 Therefore, a modification of Equation
[4.4.74] was made by Eichinger and Flory 142 *
2
by appending the term -(V1 / R)Q12θ
2
where the parameter Q12 represents the entropy of interaction between unlike segments and
is an entropic contribution to the residual chemical potential of the solvent. By adjusting the
parameter Q12, a better representation of solvent activities can be obtained.
There are many papers in the literature that applied the Prigogine-Flory-Patterson theory
to polymer solutions as well as to low-molecular mixtures. Various modifications and
improvements were suggested by many authors. Sugamiya 267 introduced polar terms by
adding dipole-dipole interactions. Brandani 268 discussed effects of non-random mixing on
the calculation of solvent activities. Kammer et al. 269 added a parameter reflecting differences
in segment size. Shiomi et al. 270,271 assumed non-additivity of the number of external
degrees of freedom with respect to segment fraction for mixtures and assumed the sizes of
hard-core segments in pure liquids and in solution to be different. Also Panayiotou 272 accounted
for differences in segment size by an additional parameter. Cheng and Bonner 273
modified the concept to obtain an equation of state which provides the correct zero pressure
limit of the ideal gas. An attractive feature of the theory is its straightforward extension to
multi-component mixtures, 274 requiring only parameters of pure components and binary
ones as explained above. A general limitation is its relatively poor description of the compressibility
behavior of polymer melts, as well as its deficiencies regarding the description
of the pressure dependence of thermodynamic data of mixtures.
Dee and Walsh 249 developed a modified version of Prigogine’s cell model that provides
an excellent description of the PVT-behavior of polymer melts: