Chemistry Reference
In-Depth Information
T
curve. The transition temperature reduced by
T
c
increases with increasing molec-
ular weight.
Joanny and Leibler [
246
] predicted the same critical exponents for the tempera-
ture dependence; however, they found that the interfacial tension decreases with
increasing chain length
r
as
r
1/2
, while the dependence of the interfacial width on
chain length is the same as that of Nose [
249
]. For a symmetric system, their final
expressions were:
2
3
k
B
Tb
2
r
1
=
2
3
=
2
g ¼
e
(92)
1
3
br
1
=
2
1
=
2
a
I
¼
e
(93)
where
b
is the Kuhn statistical segment length of the polymers.
Sanchez [
181
] used a Taylor expansion of the Flory-Huggins equation for the
free energy density, and the Cahn Hilliard theory with a constant coefficient for the
gradient terms. He found the same classical mean field exponents for the tempera-
ture dependence of interfacial tension and thickness, but he predicted that, for the
symmetric case, both the interfacial tension and the thickness are independent of
chain length. Sanchez explained this result to be due to the fact that, in his approach,
chain connectivity was only implicitly taken into consideration through the entropy
of mixing. The theories of Nose [
249
] and Joanny and Leibler [
246
] take explicitly
into account chain connectivity in various approximations.
Ronca and Russell [
232
] calculated the interfacial tension near the critical point.
They used the Cahn Hilliard expansion of the free energy with the Flory-Huggins
approximation in modeling the spinodal decomposition in polymer mixtures. For
a symmetric system, the interfacial tension was found to follow the classical
dependence:
1
=
2
3
=
2
h
g /
Tr
e
ð
r
Þ
(94)
where the function
h
(
r
) depends on the chain length [
232
].
de Gennes [
250
] has argued that a polymer blend should behave nearly classi-
cally; thus, the predicted classical behavior of
g / e
1/2
may be very
close to being correct. With respect to the molecular weight dependence, the
situation is not clear. The results of Joanny and Leibler [
246
] and Ronca and
Russell [
232
] would be similar to those of Nose if the temperature,
T
, appearing
in (
92
) and (
94
), respectively, were equated to the critical temperature,
T
c
,as
suggested by Sanchez [
181
]. Our opinion is that Sanchez's suggestion is correct.
In that case, the theories would predict that, near the critical point, the interfacial
tension increases with molecular weight to the 1/2 power, as:
3/2
and
a
I
/ e
r
1
=
2
3
=
2
g /
e
(95)
except for a correction introduced in the Ronca and Russell derivation [
232
].
