Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 197
which is also called the combinatorial contribution to solvent activity or chemical potential,
arising from the different configurations assumed by polymer and solvent molecules in solution,
ignoring energetic interactions between molecules and excess volume effects. The
Flory-Huggins derivation of the athermal combinatorial contribution contains the implicit
assumption that the r-mer chains placed on a lattice are perfectly flexible and that the flexibility
of the chain is independent of the concentration and of the nature of the solvent. Generalized
combinatorial arguments for molecules containing different kinds of energetic
contact points and shapes were developed by Barker 223 and Tompa, 224 respectively.
Lichtenthaler et al. 225 have used the generalized combinatorial arguments of Tompa to analyze
VLE of polymer solutions. Other modifications have been presented by
Guggenheim 226,227 or Staverman 228 (see below). The various combinatorial models are compared
in a review by Sayegh and Vera. 229 Recently, Freed and coworkers 230-232 developed a
lattice-field theory, that, in principle, provides an exact mathematical solution of the combinatorial
Flory-Huggins problem. Although the simple Flory-Huggins expression does not
always give the (presumably) correct, quantitative combinatorial entropy of mixing, it qualitatively
describes many features of athermal polymer solutions. Therefore, for simplicity,
it is used most in the further presentation of models for polymer solutions as reference state.
The total solvent activity/activity coefficient/chemical potential is simply the sum of
the athermal part as given above, plus a residual contribution:
or
athermal residual
lna = lna + lna
[4.4.69a]
1 1 1
0
μ = μ + μ + μ
1 1
athermal residual
1 1
[4.4.69b]
The residual part has to be explained by an additional model and a number of suitable
models is now listed in the following text.
The Flory-Huggins interaction function of the solvent is the residual function used
residual
2
first and is given by Equations [4.4.12 and 4.4.13] with μ1 / RT = χϕ2.
It was originally
based on van Laar’s concept of solutions where excess entropy and excess volume of mixing
could be neglected and χ is represented only in terms of an interchange energy Δε/kT. In
this case, the interchange energy refers not to the exchange of solvent and solute molecules
but rather to the exchange of solvent molecules and polymer segments. For athermal solutions,
χ is zero, and for mixtures of components that are chemically similar, χ is small compared
to unity. However, χ is not only a function of temperature (and pressure) as was
evident from this foundation, but it is also a function of composition and polymer molecular
mass, see e.g., Refs. 5,7,8 If we neglect these dependencies, then the Scatchard-Hildebrand
theory, 233,234 i.e., their solubility parameter concept, could be applied:
with
( )( )
residual 2
μ / RT = V / RT δ − δ ϕ
[4.4.70]
1 1 1 2
0 ( / 1)
δ 1 1
= Δ vap U V
12 /
2
2
[4.4.71]