Chemistry Reference
In-Depth Information
w
c
o
¼
w
after
w
before
(19)
Choosing l, the interaction between polymer segments and solvent molecules in the
isolated state, as a clear cut reference point for the contribution of the rearrange-
ment in the second step of dilution, and assuming that the effect will be proportional
to l, we can write:
w
c
o
¼
zl
(20)
where the negative sign in the above expression has been chosen to obtain positive
values for this parameter in the great majorit of cases. Denoting:
w
f
o
¼
a
(21)
(
18
) and (
20
) yield the following simple expression for the Flory Huggins interac-
tion parameter in the limit of high dilution:
w
o
¼
a
zl
(22)
For sufficiently dilute polymer solutions, the only difference between the new
approach and the original Flory Huggins theory is in the second term. According
to theoretical considerations and in accord with experimental findings, z becomes
zero under theta conditions (where the coils assume their unperturbed dimensions)
and the conformational relaxation no longer contributes to w
o
.
In order to generalize (
22
) to arbitrary polymer concentrations, we assume that
the composition dependence of its first term can be formulated by analogy to (
13
).
The necessity of a composition dependence for the second term results from the fact
that the insertion of a solvent molecule between contacting polymer segments
(belonging to different polymer chains) opens only one binary contact within the
composition range of pair interactions, whereas there are inevitably more segments
affected at higher polymer concentrations. For the second term, we suppose a linear
dependence of the integral interaction parameter
g
on
. Comparing the coefficients
of this ansatz (as they appear in the expression for differential interaction parame-
ter) with (
22
) for w
o
results in (
23
):
'
a
w
¼
2
zl
ð
þ
21
ð
l
Þ'
Þ
(23)
ð
1
n
'
Þ
The symbol n instead of
B
(
13
) in the above relation indicates that this parameter is
related to g [
21
], the geometrical differences of solvent molecules and polymer
segments as formulated in the next equation, but not identical with g;
ð
polymer
s
s
=
v
g
1
(24)
ð
solvent
=
v
