Chemistry Reference
In-Depth Information
A
w
¼
2
þ
C
(13)
ð
1
B
'
Þ
where these differences are contained in the parameter
B
.
A
and
C
are considered to
be further constants for a given system and fixed variables of state.
The thermodynamic relations discussed so far were, above all, formulated for
the description of moderately to highly concentrated polymer solutions. The
information acquired in the context of the determination of molar masses, on the
other hand, refers to dilute solution and is usually expressed in terms of second
osmotic virial coefficients
A
2
and higher members of a series expansion of the
chemical potential of the solvent with respect to the polymer concentration
c
(mass/volume). For the determination of osmotic pressures, p
osm
,the
corresponding relation reads:
G
1
RT V
1
¼
D
p
osm
RT
¼
c
M
n
þ
A
2
c
2
A
3
c
3
þ
þ ...
(14)
Performing a similar series expansion for the logarithm in (
5
), inserting w from (
12
)
into this relation, and comparing the result with (
14
) yields [
21
]:
1
2
r
P
V
1
A
2
w
o
¼
(15)
and:
1
3
r
P
V
1
A
3
w
1
¼
(16)
where w
o
represents the Flory Huggins
int
eraction parameter in the limit of pair
interactions between polymer molecules.
V
1
is the molar volume of the solvent and
r
P
is the density of the polymer.
The need for a different view on the thermodynamics of polymer solutions
became, in the first place, obvious from experimental information on dilute sys-
tems. According to the original Flory Huggins theory, the second osmotic virial
coefficient should without exception decrease with rising molar mass of the poly-
mer. It is, however, well documented (even in an early work by Flory himself [
22
])
that the opposite dependence does also occur. Based on this finding and on the fact
that the Flory Huggins theory only accounts for chain connectivity in the course of
calculating the combinatorial entropy of mixing and for concentrated solutions, we
attacked the problem by starting from the highly dilute side.
The central idea of this approach is the treatment of a swollen isolated polymer
coil surrounded by a sea of pure solvent as a sort of microphase and applying the
usual equilibrium condition to such a system. In a thought experiment, one can
insert a single totally collapsed polymer molecule into pure solvent and let it swell
