Chemistry Reference
In-Depth Information
4.3.3 Leibler Theory for Strongly Incompatible Systems and Its Modification
Diblock Copolymer Additives
Leibler [
75
] considered a flat interface with surface area
A
between phase-separated
A and B homopolymers. The thickness of the interfacial region
a
I
¼
1/2
and
b
(6
w
)
k
B
Tb
2
(
w
/6)
1/2
are independent of the number of
segments
P
A
and
P
B
of the two homopolymers [
27
] for a highly incompatible
situation of
wP
i
the interfacial tension
g
0
¼
1 (where
k
B
T
is the thermal energy). It was assumed that
both types of links have the same segmental volume
b
3
. Suppose that
Q
u ¼
copolymer chains with number of segments
N
¼
N
A
+
N
B
and composition
f
i
¼
N
i
/
N
are adsorbed at the A B interface (for most practical situations,
wN
i
1). It was expected that the copolymer joints will be localized in a thin interfacial
layer of thickness [
103
]
d
0
¼ðp=
a
I
(independent of
N
i
and
P
i
);
d
0
is equal to
the semiempirical parameter
d
of Noolandi et al. [
71
,
281
]in(
98
), as discussed
by Semenov [
103
]. The blocks A and B extend towards the respective bulk
layers and form two “adsorbed layers” of thicknesses
L
A
and
L
B
, respectively.
Since
d
0
2
Þ
L
i
, each side of the interfacial film resembles a layer of polymers
anchored by one end onto a wall. The free energy of the interfacial film can, thus,
be approximated as [
75
]:
F
interf
:
film
¼ g
0
A
þ
Qg
A
þ
ð
g
B
Þ
(107)
where
g
0
is the A B interfacial tension in the absence of the additive,
A
is the
interfacial area, and
g
A
,
g
B
represent the free energies per A B chain of the A and B
layers, respectively. The number of copolymer chains per unit interfacial area is
given by
s
=
Q
/
A
.
In most of the experimental studies, the copolymer chains are not so long relative
to the homopolymers. Thus, mixing of the copolymer and homopolymer chains
should be taken into account due to the penetration of homopolymers into the layer
of chains anchored at the interface, whereas the copolymer chains can be either
stretched (wet brush regime) or not (wet mushroom). Neglecting the composition
gradients in the brush (Flory approximation),
g
i
is given by [
40
,
75
,
287
]:
þ
L
i
N
i
b
2
g
i
k
B
T
¼
1
sb
3
1
P
i
3
2
ln
N
i
b
2
s
L
i
ð
1
i
Þ
ln 1
ð
i
Þþ
(108a)
i
¼ sN
i
b
3
where
=
L
i
is the average volume fraction of monomers of the A block in
the layer and (1
A
) is that of the P
A
monomers. The first two terms in (
108a
)
approximate the entropy of mixing between copolymer and homopolymer chains,
which tend to swell the copolymer blocks; the first term is associated with the
translational freedom of the copolymers in the two-dimensional film, whereas the
second term originates from the translational entropy of the homopolymer chains
and has a standard excluded volume form [
287
]. The last term represents the elastic
