Chemistry Reference
In-Depth Information
The expression for spherical micelles is the same as equation A-8 of Shull et al.
[
38
] and is consistent with equation 21 of Lyatskaya et al. [
85
] and equation 36 of
Semenov [
103
]. That for cylindrical micelles is the same as equation A-12 of Shull
et al. [
38
] and as equation 36 of Semenov [
103
]. Finally, the equation for the
chemical potential of lamellar micelles is the same as equation A-12 of Shull et al.
[
38
] and somehow different from equation 36 of Semenov [
103
], as is also
acknowledged by him [
103
].
When micelles are not present, the equilibrium is established between copoly-
mer chains homogeneously distributed within the homopolymer phase and copoly-
mer at the interface. The surface density
s
is, then, determined by:
m
int
ðs
;
N
Þ¼m
bulk
ðf
;
N
Þ
(119a)
where, in this case, it is assumed that
f ¼ f
f
add
. When micelles are present,
then at thermodynamic equilibrium
s
is determined by the equation:
m
int
ðs
;
N
Þ¼m
mic
ð
N
Þ¼m
bulk
ðf
;
N
Þ
(119b)
which also determines the volume fraction
f
of copolymers remaining homo-
geneously distributed in the bulk A or B phases.
For calculation of the interfacial tension reduction, one evaluates first the che-
mical potentials
m
mic
and
m
bulk
for
< m
mic
, then the
equilibrium is established between copolymers at the interface and copolymers
homogeneously mixed within the B-rich phase. The interfacial excess
s
is, then,
determined by (
119a
) together with (
115a
) and (
117a
), and the interfacial tension
reduction
f ¼ f
¼ f
add
.If
m
bulk
(
f
add
)
> m
mic
, equilibrium is established among the
three different states of the copolymer and
s
and
Dg
by (
113a
). If
m
bulk
(
f
add
)
f
are determined by (
119b
)
together with (
115a
), (
117a
), and (
118a
);
Dg
is evaluated by (
113a
).
The semiquantitative model was compared with the data on the effects of the
molecular weight of symmetric diblock copolymers on the polymer polymer
interfacial tension; the data showed a nonmonotonous dependence of the interfacial
tension increment on the additive molecular weight in the plateau region. Although
the assumptions involved in the model do not allow a quantitative comparison, the
behavior of
Dg
when the copolymer molecular weight increases at constant additive
concentration resembles the response seen experimentally. Figure
30
shows the
estimated surface density of copolymers at the A B interface,
s
, together with
the interfacial tension reduction,
Dg ¼ g
0
g
, as a function of the number of
segments of the copolymeric additive for
f
add
¼
0.02. The parameters used were
P
A
¼
0.04. Moreover, for the present range of
values of
P
i
and
N
i
, the wet-mushroom configuration for the adsorbed copolymer
chains was assumed, which was then verified by the extracted
s
values.
It was found that the magnitude of
P
PI
¼
81,
P
B
¼
P
PS
¼
112, and
w ¼
Dg
increases with copolymer molecular
weight, as long as the copolymer chains at the interface are at equilibrium with
only homogeneously mixed chains and micelles do not exist (regime I). At higher
molecular weights, when micelles are also present,
Dg
decreases with further
