Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 199
The first successful theoretical approach of an equation of state model for polymer solutions
was the Prigogine-Flory-Patterson theory. 236-242 It became popular in the version by
Flory, Orwoll and Vrij 236 and is a van-der-Waals-like theory based on the corresponding-states
principle. Details of its derivation can be found in numerous papers and books
and need not be repeated here. The equation of state is usually expressed in reduced form
and reads:
~ ~
~ / 13
PV V
~ ~
T V VT
/ ~ ~
1
= −
13
−1
where the reduced PVT-variables are defined by
[4.4.72]
~ ~ ~
P = P / P*, T = T / T*, V = V / T*, P * V* = rcRT *
[4.4.73]
and where a parameter c is used (per segment) such that 3rc is the number of effective external
degrees of freedom per (macro)molecule. This effective number follows from
Prigogine’s approximation that external rotational and vibrational degrees of freedom can
be considered as equivalent translational degrees of freedom. The equation of state is valid
for liquid-like densities and qualitatively incorrect at low densities because it does not fulfill
the ideal gas limit. To use the equation of state, one must know the reducing or characteristic
parameters P*, V*, T*. These have to be fitted to experimental PVT-data. Parameter tables
can be found in the literature - here we refer to the book by Prausnitz et al., 49 a review by
Rodgers, 262 and the contribution by Cho and Sanchez 263 to the new edition of the Polymer
Handbook.
To extend the Flory-Orwoll-Vrij model to mixtures, one has to use two assumptions:
(i) the hard-core volumes υ* of the segments of all components are additive and (ii) the
intermolecular energy depends in a simple way on the surface areas of contact between solvent
molecules and/or polymer segments. Without any derivation, the final result for the residual
solvent activity in a binary polymer solution reads:
~
* *
~
ln ln
/
~ / ~
13 ⎡
residual PV 1 1 V 1 −1⎛
1 1⎞⎤
a1
= ⎢3T1
+ ⎜ − ⎟⎥
+
13
~
RT ⎢
⎜ ⎟⎥
V −1⎝V1
⎣⎢
V⎠⎦⎥
*
V ⎛
+ ⎜
Χ
RT ⎜ ~
⎝ V
1 12
⎞
*
⎟
PV ~ ~
2 1
θ 2 + ( V−V1) [4.4.74]
⎟
⎠
RT
where:
X12 interaction parameter
θ2 surface fraction of the polymer
The last term in Equation [4.4.74] is negligible at normal pressures. The reduced volume
of the solvent 1 and the reduced volume of the mixture are to be calculated from the
same equation of state, Equation [4.4.72], but for the mixture the following mixing rules
have to be used (if random mixing is assumed):