Chemistry Reference
In-Depth Information
For the calculation of the chemical potential of the AB
2
graft copolymer chains in
a micelle, one has to distinguish two different cases: (1) when the micelle is formed
within the B homopolymer, i.e., when the “tooth” A block forms the core of the
micelle and the two B blocks form the corona; and (2) when the micelle is formed
within the A homopolymer phase, i.e., when the two B blocks form the core and the
A block forms the corona of the micelle. The two cases were considered by Retsos
et al. [
56
] along the lines of Leibler [
75
] and Semenov [
103
] for the three different
cases of formation of spherical, cylindrical, or lamellar micelles. The chemical
potential of an AB
2
chain in a micelle formed within the B phase is, thus, [
56
]:
1
=
3
spherical
mic
m
4
=
3
f
4
=
9
A
1
=
3
1
=
3
k
B
T
¼ð
3
=
2
Þ
4
:
74
f
4
ðÞ
wN
A
cylindrical
mic
m
1
=
3
0
1
=
3
k
B
T
¼
1
:
89
wf
A
N
ð
Þ
ð
:
41
ln
f
A
Þ
lamellar
mic
k
B
T
m
1
=
3
1
1
=
3
¼
1
:
75
wN
ðÞ
ð
:
26
f
A
Þ
(118b)
whereas the chemical potential of an AB
2
chain in a micelle formed within the
A phase is given by [
56
]:
h
i
1
=
3
spherical
mic
m
4
=
3
4
=
9
1
=
3
1
=
3
k
B
T
¼ð
3
=
2
Þ
ð
1
f
A
Þ
3
:
96 1
ð
f
A
Þ
1
ðÞ
wN
cylindrical
mic
m
1
=
3
6
1
=
3
k
B
T
¼
1
:
19
wf
A
N
ð
Þ
½
:
57
ln 1
ð
f
A
Þ
lamellar
mic
k
B
T
m
1
=
3
1
1
=
3
¼
1
:
57
wN
ðÞ
ð
:
44
f
A
Þ
(118c)
When micelles are not present, the equilibrium between copolymer chains
homogeneously mixed with the respective homopolymer and chains at the interface
is established (
119a
) whereas, when micelles are present, (
119b
) determines the
thermodynamic equilibrium. The equations result in the surface density of copoly-
mer chains at the interface,
s
. Again, it is assumed that
f
f
add
, with
f
being
the volume fraction of copolymers remaining homogeneously mixed in the bulk
A or B phases.
Therefore, for the calculation of the interfacial tension reduction, one again
evaluates the chemical potentials
m
mic
,
m
bulk
for
< m
mic
,
then the equilibrium is established between copolymers at the interface and
copolymers homogeneously mixed within the B-rich (or A-rich) phase. The inter-
facial excess
s
is, then, determined by (
119a
) together with (
115b
) and (
117a
)
(or
117b
, respectively), and the interfacial tension reduction
f¼f
¼f
add
.If
m
bulk
(
f
add
)
Dg
by (
113b
). If
m
bulk
(
f
add
)
> m
mic
, equilibrium is established among the three different states of
