Chemistry Reference
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weights and the interactions and is calculated by the model. Good agreement was
observed between the predicted micelle core radii and experimental data [
284
] for
PS-b-PBD within PBD, obtained using small angle neutron scattering.
Shull and Kramer [
77
] developed and applied the Noolandi Hong theory for the
case of polymer A/polymer B/diblock copolymer AB, but without solvent. They
found that, at a given value of the chemical potential of the copolymer in the bulk
phases
m
c
, the ability of a copolymer to reduce
g
is highest for small
N
and small
w
.
However, at a given value of
f
c
, higher values of
N
result in much higher values
of
Dg
due to the exponential dependence of
m
c
on
wN
and because an increase in
m
c
results in an increase in the density of copolymer chains at the interface.
Theoretical determination of the limiting value of
m
c
associated with the formation
of micelles was made separately [
38
], since the possibility of micelle formation was
not explicitly introduced in the theory. A good agreement was found with
the experimental data [
38
] for the total amount of copolymer segregating to a
polymer polymer interface for concentrations below CMC using only
w
as an
adjustable parameter. Using the best-fit value of
w
, they estimated
Dg
for con-
centrations when micelles are not present. For concentrations higher than the
CMC, more micelles will be formed without, however, significantly increasing
the copolymer chemical potential; thus, the interfacial tension will not decrease
further. For the copolymer molecular weights used, a significant increase in the
total copolymer amount adsorbed at the interface was observed at higher copoly-
mer concentrations, which was attributed [
38
,
102
] to segregation of micelles to
the polymer polymer interface (as well to the polymer air surface [
38
,
102
,
285
]). The location of the upturn was used to estimate the copolymer chemical
potential at the CMC, which was in good agreement with a full self-consistent-
field theoretical estimate [
286
].
4.3.2 Leibler Theory for Nearly Compatible Systems
Leibler [
282
] developed a simple mean-field formalism to study the interfacial
properties of nearly compatible mixtures of two homopolymers, A and B, and a
copolymer AB. The free energy was expressed in terms of monomer concentration
correlation functions, which were calculated in a self-consistent way within the
random phase approximation introduced by de Gennes [
242
]. For the very broad
interface of nearly miscible systems, a gradient expansion was carried out giving a
generalization of the Cahn Hilliard theory [
216
]. As mentioned by Noolandi and
Hong [
71
], with the gradient expansion in the theory of Leibler, the diblock
copolymer was effectively treated as a small-molecule solvent compared to the
large width of the interfacial region, and the structure of the copolymer became
irrelevant. The system, thus, behaved as a mixture of two homopolymers driven to
the consolute point by the addition of an excess of solvent. As pointed out by
Leibler, for nearly compatible species (2
4
p
2), two mechanisms of the
interfacial activity of the copolymer chains had to be distinguished: (1) the species
A and B are more closely mixed as copolymer chains and are present in both phases,
< wN
<
