Chemistry Reference
In-Depth Information
replacement of the isolated l by 1/2, as formulated below for the differential
interaction parameter w:
a
w
2
zl 1
ð
þ
2
'
Þ
(32)
ð
1
n
'
Þ
The analogous relation for the integral interaction parameter reads:
a
g
Þ
zl 2
ð
þ '
Þ
(33)
ð
Þ
ð
'
1
n
1
n
By this means, the number of adjustable parameter reduces to three. As will be
shown in the section dealing with experimental data (Sect.
4
), further simplifica-
tions are possible, for instance because of a theoretically expected interrelation
between the parameters a (first step of mixing) and zl (second step of mixing) for a
given class of polymer solutions. In its general form this equation reads:
zl
¼
E
2 a
ð
1
Þ
(34)
where
E
is a constant, typically assuming values between 0.6 and 0.95. Equation
(
34
) is in accord with the typical case of theta conditions where z
0.5.
As long as such an interrelation exists, the number of parameters required for the
quantitative description of the isothermal behavior of polymer solutions reduces to
two. Like with the expression for w
o
(high dilution), the contributions of chain
connectivity and conformational relaxation are in (
32
) (arbitrary polymer concen-
tration) exclusively contained in the second term. Another aspect also deserves
mentioning, namely the fact that (
32
) is not confined to the modeling of polymer-
containing systems but can also be successfully applied to mixtures of low molecu-
lar weight liquids, as will be shown in Sect.
4
.
According to expectation, and in agreement with measurements, all system-
specific parameters p (namely a, n, z, and l) vary more or less with temperature
(and pressure). The following relation is very versatile to model p(
T
):
!
0 and a
!
p
1
T
þ
p
¼
p
o
þ
p
2
T
(35)
where either p
1
or p
2
can be set to zero in most cases.
Up to now, it was the chemical potential of the solvent that constituted the object
of prime interest. The last part of this section is dedicated to the modeling of liquid/
liquid phase separation by means of the integral Gibbs energy of mixing in the case
of polymer solutions. The equations presented in this context can, however, be
easily generalized to polymer blends and to multinary systems. Such calculations
are made possible by using the minimum Gibbs energy a system can achieve via
phase separation as the criterion for equilibria, instead of equality of the chemical
potentials of the components in the coexisting phases. The method of a direct
