Chemistry Reference
In-Depth Information
entropy term, which limits the swelling. For low values of
A
1, (
108a
) can be
written as [
40
,
75
]:
þ
g
i
k
B
T
¼
N
i
i
L
i
N
i
b
2
1
2
3
2
ln
N
i
b
2
s
P
i
þ
(109a)
For stretched chains, the brush thickness
L
i
and the block monomer concentra-
tion
i
are obtained from (
109a
) by minimization with respect to
L
i
. In that case,
1/3
N
i
b
(
sb
2
)
1/3
P
i
1/3
,
6
1/3
(
sb
2
)
2/3
P
i
1/3
and:
L
i
¼
6
i
¼
þ
3
4
=
3
2
5
=
3
2
=
3
N
i
P
2
=
3
i
g
i
k
B
T
¼
ln
N
i
b
2
sb
2
s
ð
wet brush
Þ
(110a)
3/2
< sb
2
1/2
. For nonstretched chains,
L
i
N
i
1/2
b
and
which is valid for
P
i
N
i
<
P
i
the last term of (
109a
) can be neglected; this applies for
sb
2
3/2
[
287
]. Then,
<
P
i
N
i
i
¼ sb
2
N
i
1/2
and:
N
3
=
2
i
þ
sb
2
P
i
g
i
k
B
T
¼
1
2
ln
N
i
b
2
s
ð
wet mushroom
Þ
(111a)
The interfacial tension in the presence of the copolymer is calculated as
1
:
Q
¼ g
0
s
g ¼
@
F
interf
:
film
@
@
@s
þ
@
g
A
g
B
@s
2
(112)
A
Therefore, the interfacial tension reduction,
Dg ¼ g
0
g
, is given by:
8
<
h
i
2
=
3
N
A
P
2
=
3
N
B
P
2
=
3
3
1
=
3
2
2
=
3
s
2
þ
ðÞ
sb
2
þ
ð
wet brush
Þ
k
B
T
¼
g
0
g
Dg
A
B
k
B
T
¼
h
i
:
N
3
=
2
A
N
3
=
2
B
1
2
sb
2
P
A
þ
P
1
B
s
2
þ
ð
wet mushroom
Þ:
(113a)
At equilibrium,
s
is determined by equating the chemical potential of the
copolymer chains at the interface with that of the copolymer chains either
homogeneously mixed with the homopolymers or at micelles formed within the
1
It is noted that Noolandi [
288
] objects to the use of (
107
) and (
112
) because he claims that the
main contribution to the interfacial tension reduction is of enthalpic and not entropic origin (as
(
112
) suggests), i.e., that it is due to the favorable energetics of the orientation of the copolymer
blocks into their respective homopolymers and that entropic effects are second order. He suggests
that (
107
) should be corrected by adding the contributions of the orientational entropy of the
blocks and their entropy of localization. The latter was introduced by Shull and Kramer [
77
]by
replacing
g
0
by
g
0
0
g
0
þ sk
B
T
ln
L
A
þ
½
ð
L
B
Þ=
d
0
. In the present analysis, the expression of Leibler
[
75
,
76
,
40
] is utilized.
