Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 201
~ / 13
~ ~
PV V
~ ~
T V q T V V
/ ~ ⎛
⎞
2 . .
=
−
⎜12045
1011
−
⎟
13 ⎜ ~ 2 ~ 4 ⎟
−0.
8909 ⎝
⎠
[4.4.77]
where the reduced variables and characteristic parameters have the same definitions as in
the Flory model above. Equation [4.4.77] is formally identical with Prigogine’s result, except
for the additional constant parameter q, which can also be viewed as a correction to the
hard-core cell volume. The value of q = 1.07 corresponds approximately to a 25% increase
in the hard-core volume in comparison with the original Prigogine model. Characteristic parameters
for this model are given in Refs. 249,262 The final result for the residual solvent activity
in a binary polymer solution reads:
~
* * ~ .
ln ln
.
/
~ /
13
residual PV 1 1 V1 −08909q
⎛
12045 12045 05055 0505
a1
= 3T1
+
⎜ . .
⎞
−
⎟ . . 5
− −
13
2 2
4
RT
⎜ ~ ~ ⎟ ~
4
V −08909q ⎝ V1 V ⎠ V1
V ~
⎡
⎛
⎞⎤
⎢
⎜
⎟⎥
⎢
⎜
⎟⎥
⎣
⎝
⎠⎦
* 2
V
~ ⎛
⎞ *
1θ2 ⎛
⎞
+ ⎜ −
⎜12045
. 0. 5055
Χ12
TQ12 V ⎟ −
⎟ PV ~ ~
1⎛
⎞
RT ⎝
⎠⎜
~ 2 ~ 4 ⎟
+ ⎜V−V1⎟
RT ⎝ ⎠
⎝ V V ⎠
[4.4.78]
The last term in Equation [4.4.78] is again negligible at normal pressures, which is the
case for the calculation of solvent activities of common polymer solutions. The reduced volume
of the mixture is to be calculated from the equation of state where the same mixing
rules are valid, as given by Equations [4.4.75, 4.4.76] if random mixing is assumed. Equation
[4.4.78] is somewhat more flexible than Equation [4.4.74]. Again, entropic parameter
Q 12 and interaction parameter X 12 have to be fitted to experimental data of the mixture.
There is not much experience with the model regarding thermodynamic data of polymer solutions
because it was mainly applied to polymer blends, where it provides much better results
than the simple Flory model.
To improve on the cell model, two other classes of models were developed, namely,
lattice-fluid and lattice-hole theories. In these theories, vacant cells or holes are introduced
into the lattice to describe the extra entropy change in the system as a function of volume
and temperature. The lattice size, or cell volume, is fixed so that the changes in volume can
only occur by the appearance of new holes, or vacant sites, on the lattice. The most popular
theories of such kind were developed by Simha and Somcynsky 245 or Sanchez and
Lacombe. 246-248
The Sanchez-Lacombe lattice-fluid equation of state reads:
~ ~
PV
V
T
r
V 1 ~ −1 1
= −1− ln −
~
~ ~ ~
V VT
~
[4.4.79]
where the reduced parameters are given in Equation [4.4.73], but no c-parameter is included,
and the size parameter, r, and the characteristic parameters are related by
P * V* = ( r / M) RT *
[4.4.80]